International Mathematics Research Notices

Tamburelli, Andrea

doi: 10.1093/imrn/rnw278pmid: N/A

Abstract We prove that given two metrics $$g_{+}$$ and $$g_{-}$$ with curvature $$\kappa <-1$$ on a closed, oriented surface $$S$$ of genus $$\tau\geq 2$$, there exists an $$AdS_{3}$$ manifold $$N$$ with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components of $$\partial N$$ are equal to $$g_{+}$$ and $$g_{-}$$. Using the duality between convex space-like surfaces in $$AdS_{3}$$, we obtain an equivalent result about the prescription of the third fundamental form. This answers partially Question 3.5 in [1]. 1 Introduction The three-dimensional Anti-de Sitter space $$AdS_{3}$$ is the Lorentzian analogue of hyperbolic space, that is, it is the local model of Lorentzian $$3$$-manifolds with constant sectional curvature $$-1$$. An $$AdS_{3}$$ spacetime is an oriented and time-oriented manifold locally modelled on $$AdS_{3}$$. A particular class of Anti-de Sitter $$3$$-manifold, called globally hyperbolic maximal compact (GHMC), has attracted much attention since the pioneering work of Mess [17], who pointed out many connections to Teichmüller theory and many similarities to hyperbolic quasi-Fuchsian manifolds. A GHMC $$AdS_{3}$$ spacetime $$M$$ is topologically a product $$S\times \mathbb{R}$$, where $$S$$ is a closed, oriented surface of genus $$\tau\ge 2$$, diffeomorphic to a Cauchy surface embedded in $$M$$. The holonomy representation of the fundamental group of $$S$$ into the isometry group of orientation and time-orientation preserving isometries of $$AdS_{3}$$, which can be identified with $$\mathbb{P}SL(2,\mathbb{R})\times \mathbb{P}SL(2,\mathbb{R})$$, provides a bijection between the space of GHMC $$AdS_{3}$$ structures on $$M$$ and the product of two copies of the Teichmüller space of $$S$$. Moreover, $$M$$ contains a convex core $$C(M)$$, whose boundary (when $$C(M)$$ is not a totally geodesic $$2$$-manifold) consists of the disjoint union of two hyperbolic surfaces pleated along a geodesic lamination. As a consequence, it is possible to formulate many classical questions of quasi-Fuchsian manifolds even in this Lorentzian setting. For example one can ask if it is possible to prescribe the induced metrics on the boundary of the convex core and the answer is very similar in both settings, where the existence of a quasi-Fuchsian manifold (as a consequence of results in [8] and [15]) and a GHMC $$AdS_{3}$$ manifold ([7]) with a prescribed metric on the boundary of the convex core has been proved, but uniqueness is still unknown. Another interesting question deals with the prescription of the metrics on the boundary of a larger compact, convex subset $$K$$ with two smooth, strictly convex, space-like boundary components in a GHMC $$AdS_{3}$$ manifold. By the Gauss formula, the boundaries have curvature $$\kappa <-1$$. We can ask if it is possible to realise every couple of metrics, satisfying the condition on the curvature, on a surface $$S$$ via this construction. The analogous question has a positive answer in a hyperbolic setting ([15]), where even a uniqueness result holds ([19]). In this article, we will follow a construction inspired by the work of Labourie ([15]), in order to obtain a positive answer in the Anti-de Sitter world. The main result of the article is thus the following: Corollary 4.3For every couple of metrics $$g_{+}$$ and $$g_{-}$$ on $$S$$ with curvature less than $$-1$$, there exists a globally hyperbolic convex compact $$AdS_{3}$$ manifold $$K\cong S\times [0,1]$$, whose induced metrics on the boundary are exactly $$g_{\pm}$$. Using the duality between space-like surfaces in Anti-de Sitter space, we obtain an analogous result about the prescription of the third fundamental form: Corollary 4.4For every couple of metrics $$g_{+}$$ and $$g_{-}$$ on $$S$$ with curvature less than $$-1$$, there exists a globally hyperbolic convex compact $$AdS_{3}$$ manifold $$K \cong S\times [0,1]$$, such that the third fundamental forms on the boundary components are $$g_{+}$$ and $$g_{-}$$. We outline here the main steps of the proof for the convenience of the reader. The first observation to be done is that Corollary 4.3 is equivalent to proving that there exists a GHMC $$AdS_{3}$$ manifold $$M$$ containing a future-convex space-like surface isometric to $$(S, g_{-})$$ and a past-convex space-like surface isometric to $$(S, g_{+})$$. Adapting the work of Labourie ([15]) to this Lorentzian setting, we prove that the space of isometric embeddings $$I(S,g_{\pm})^{\pm}$$ of $$(S, g_{\pm})$$ into a GHMC $$AdS_{3}$$ manifold as a future-convex (or past-convex) space-like surface is a manifold of dimension $$6\tau -6$$. On the other hand, by the work of Mess ([17]), the space of GHMC $$AdS_{3}$$ structures is parameterised by two copies of Teichmüller space, hence a manifold of dimension $$12\tau -12$$. This allows us to translate our original question into a question about the existence of an intersection between subsets in $$\mathrm{Teich}(S) \times \mathrm{Teich}(S)$$. More precisely, we will define in Section 4 two maps   ϕg±±:I(S,g±)±→Teich(S)×Teich(S) sending an isometric embedding of $$(S, g_{\pm})$$ to the holonomy of the GHMC $$AdS_{3}$$ manifold containing it. Corollary 4.3 is then equivalent to the following: Theorem 4.2For every couple of metrics $$g_{+}$$ and $$g_{-}$$ on $$S$$ with curvature less than $$-1$$, we have  ϕg++(I(S,g+)+)∩ϕg−−(I(S,g−)−)≠∅. In order to prove this theorem we will use tools from topological intersection theory, which we recall in Section 5. For instance, Theorem 4.2 is already known to hold under particular hypothesis on the curvatures [5], hence we only need to check that the intersection persists when deforming one of the two metrics on the boundary, as the space of smooth metrics with curvature less than $$-1$$ is connected (see e.g., Lemma 2.3 in [16]). More precisely, given any smooth paths of metrics $$g^{\pm}_{t}$$ with curvature less than $$-1$$, we will define the manifolds   W±=⋃t∈[0,1]I(S,gt±)± and the maps   Φ±:W±→Teich(S)×Teich(S) with the property that the restrictions of $$\Phi^{\pm}$$ to the two boundary components coincide with $$\phi_{g_{0}^{\pm}}^{\pm}$$ and $$\phi_{g_{1}^{\pm}}^{\pm}$$. We will then prove the following: Proposition 6.1The maps $$\Phi^{\pm}$$ are smooth. Hence, we will have the necessary regularity to apply tools from intersection theory. In particular, we can talk about transverse maps and under this condition we can define the intersection number (mod $$2$$) of the maps $$\phi_{g_{+}}^{+}$$ and $$\phi_{g_{-}}^{-}$$ as the cardinality (mod $$2$$), if finite, of $$(\phi_{g_{+}}^{+} \times \phi_{g_{-}}^{-})^{-1}(\Delta)$$, where   ϕg++×ϕg−−:I(S,g+)+×I(S,g−)−→(Teich(S))2×(Teich(S))2 and $$\Delta$$ is the diagonal in $$(\mathrm{Teich}(S))^{2} \times (\mathrm{Teich}(S))^{2}$$. We will compute explicitly this intersection number (see Section 8) under particular hypothesis on the curvatures of $$g_{+}$$ and $$g_{-}$$: the reason for this being that the transversality condition is in general difficult to check when the metrics do not have constant curvature. It turns out that in that case the intersection number is $$1$$. We then start to deform one of the two metrics and check that an intersection persists. Here, one has to be careful that, since the maps are defined on non-compact manifolds, the intersection does not escape to infinity. This is probably the main technical part of the article and requires results about the convergence of isometric embeddings (Corollary 6.5), estimates in Anti-de Sitter geometry (Lemma 6.12) and results in Teichmüller theory (Lemma 6.11). In particular, applying these tools, we prove Proposition 6.13For every metric $$g^{-}$$ and for every smooth path of metrics $$\{g_{t}^{+}\}_{t \in [0,1]}$$ on S with curvature less than $$-1$$, the set $$(\Phi^{+}\times \phi^{-})^{-1}(\Delta)$$ is compact This guarantees that when deforming one of the two metrics the variation of the intersection locus is always contained in a compact set. The proof of Theorem 4.2 then follows applying standard argument of topological intersection theory. In Section 7, we study the map   p1∘Φ+:W+→Teich(S), where $$p_{1}: \mathrm{Teich}(S) \times \mathrm{Teich}(S) \rightarrow \mathrm{Teich}(S)$$ is the projection on to the left factor. The main result we obtain is the following: Proposition 7.1Let $$g$$ be a metric on $$S$$ with curvature less than $$-1$$ and let $$h$$ be a hyperbolic metric on $$S$$. Then there exists a GHMC $$AdS_{3}$$ manifold $$M$$ with left metric isotopic to $$h$$ containing a past-convex space-like surface isometric to $$(S,g)$$. This is proved by showing that $$p_{1} \circ \phi^{+}_{g}$$ is proper of degree $$1$$ (mod $$2$$). Again, we are able to compute explicitly the degree of the map when $$g$$ has constant curvature and the general statement then follows since for any couple of metrics $$g$$ and $$g'$$ with curvature less than $$-1$$, the maps $$p_{1}\circ \phi_{g}$$ and $$p_{1} \circ \phi_{g'}$$ are connected by a proper cobordism. 1.1 Outline of the article In Section 2, we give a brief introduction to Anti-de Sitter space. We then describe a parameterisation of the space of GHMC $$AdS_{3}$$ structures in Section 4. In Section 3, we study the space of isometric embeddings. In Section 5, we recall the main tools of topological intersection theory. Section 6, contains the most technical proofs: in particular, we prove the smoothness and properness of the maps $$\Phi^{\pm}$$ and Proposition 6.13. Section 7, deals with Proposition 7.1. The main theorem (Theorem 4.2) is proved in Section 8. 2 Anti-de Sitter space The three-dimensional Anti-de Sitter space $$AdS_{3}$$ is the Lorentzian analogue of hyperbolic space, that is, it is the local model of Lorentzian $$3$$-manifolds with constant sectional curvature $$-1$$. In this section, we describe a geometric model of $$AdS_{3}$$ as interior of a quadric in the real projective space and illustrate some of its features. We then introduce $$AdS_{3}$$ manifolds and the notion of globally hyperbolicity. The main references for this material are [17] and [3]. Consider in $$\mathbb{R}^{4}$$ the bilinear form of signature $$(2,2)$$  ⟨x,y⟩2,2=x1y1+x2y2−x3y3−x4y4       x,y∈R4. We denote with $$Q$$ the hyperboloid   Q={x∈R4 | ⟨x,x⟩2,2=−1 }. The restriction of the bilinear form $$\langle \cdot, \cdot \rangle_{2,2}$$ to the tangent spaces of $$Q$$ induces a Lorentzian metric on $$Q$$ with constant sectional curvature $$-1$$. Geodesics and totally geodesic planes are obtained intersecting $$Q$$ with planes and hyperplanes of $$\mathbb{R}^{4}$$ through the origin. Moreover, the group of orientation and time-orientation isometry of $$Q$$ is the connected component of $$SO(2,2)$$ containing the identity. We define the Anti-de Sitter space $$AdS_{3}$$ as the image of the projection of $$Q$$ into $$\mathbb{R}\mathbb{P}^{3}$$. More precisely, if we denote with $$\pi: \mathbb{R}^{4}\setminus \{0\} \rightarrow \mathbb{R}\mathbb{P}^{3}$$ the canonical projection, we define   AdS3=π({x∈R4 | ⟨x,x⟩2,2<0 }). It can be easily verified that $$\pi: Q \rightarrow AdS_{3}$$ is a double cover, hence we can endow $$AdS_{3}$$ with the unique Lorentzian structure that makes $$\pi$$ a local isometry. It then follows by the definition that geodesic and totally geodesic planes of $$AdS_{3}$$ are obtained intersecting $$AdS_{3}$$ with projective lines and planes. It is then natural to define the boundary at infinity of $$AdS_{3}$$ as   ∂∞AdS3=π({x∈R4 | ⟨x,x⟩2,2=0 }. It can be easily verified that $$\partial_{\infty}AdS_{3}$$ coincides with the image of the Segre embedding   s:RP1×RP1→RP3, hence the boundary at infinity of Anti-de Sitter space is a double-ruled quadric homeomorphic to $$S^{1} \times S^{1}$$. We will talk about left and right ruling in order to distinguish the two rulings. This homeomorphism can be also described geometrically in the following way. Fix a totally geodesic space-like plane $$P_{0}$$ in $$AdS_{3}$$. The boundary at infinity of $$P_{0}$$ is a circle. Let $$\xi \in \partial_{\infty}AdS_{3}$$. There exists a unique line of the left ruling $$l_{\xi}$$ and a unique line of the right ruling $$r_{\xi}$$ passing through $$\xi$$. The identification between $$\partial_{\infty}AdS_{3}$$ and $$S^{1} \times S^{1}$$ induced by $$P_{0}$$ associates to $$\xi$$ the intersection points $$\pi_{l}(\xi)$$ and $$\pi_{r}(\xi)$$ between $$l_{\xi}$$ and $$r_{\xi}$$ with the boundary at infinity of $$P_{0}$$. These two maps   πl:∂∞AdS3→S1      πr:∂∞AdS3→S1 are called left and right projections. The action of an orientation and time-orientation preserving isometry of $$AdS_{3}$$ extends continuously to the boundary at infinity and it is projective on the two rulings, thus giving an identification between $$SO_{0}(2,2)$$ and $$\mathbb{P}SL(2,\mathbb{R})\times \mathbb{P}SL(2,\mathbb{R})$$. The projective duality between points and planes of $$\mathbb{R}\mathbb{P}^{3}$$ induces a duality in $$AdS_{3}$$ between points and totally geodesic space-like planes. This duality induces then a duality between smooth, space-like convex surfaces in $$AdS_{3}$$. Namely, let $$\tilde{S}\subset AdS_{3}$$ be a convex space-like surface. Denote by $$\tilde{S}^{*}$$ the set of points which are duals to the tangent planes of $$\tilde{S}$$. The relation between $$\tilde{S}$$ and $$\tilde{S}^{*}$$ is summarised in the following lemma. Lemma 2.1 (see e.g., Section 11 of [3]). Let $$\tilde{S}\subset AdS_{3}$$ be a smooth, space-like surface with curvature $$\kappa <-1$$. Then (i) the dual surface $$\tilde{S}^{*}$$ is smooth and locally strictly convex; (ii) the pull-back of the induced metric on $$\tilde{S}^{*}$$ through the duality map is the third fundamental form1 of $$\tilde{S}$$; (iii) if $$\kappa$$ is constant, the dual surface $$\tilde{S}^{*}$$ has curvature $$\kappa^{*}=-\frac{\kappa}{\kappa+1}$$. □ (We recall that the third fundamental form of an embedded surface $$S$$ with induced metric $$I$$ and shape operator $$B$$ is the bilinear form on $$TS$$ defined by $$III(X,Y)=I(BX,BY)$$ for every vector field $$X$$ and $$Y$$ on $$S$$.) An $$AdS_{3}$$ manifold $$M$$ is a $$3$$-manifold endowed with a Lorentzian metric locally isometric to $$AdS_{3}$$. We say that $$M$$ is globally hyperbolic (GH) if it contains an embedded space-like surface (called Cauchy surface) which intersects every time-like line exactly once. $$M$$ is said to be spacially compact (C) if it admits a compact Cauchy surface. This leads to the following definition: Definition 2.2. A globally hyperbolic maximal compact (GHMC) $$AdS_{3}$$ manifold is a globally hyperbolic $$AdS_{3}$$ manifold $$M$$ containing a compact Cauchy surface with the following property: if $$i: M \rightarrow M'$$ is an isometric embedding of $$M$$ into another $$GHC$$$$AdS_{3}$$ manifold $$M'$$ sending a Cauchy surface into a Cauchy surface, then $$i$$ is an isometry. □ If $$M$$ is a GHMC $$AdS_{3}$$-manifold, then its universal cover can be identified with a subset $$D$$ of $$AdS_{3}$$ which can be roughly described as follows: the closure of $$D$$ intersects the boundary at infinity of $$AdS_{3}$$ along a curve $$\rho$$ and the interior of $$D$$ is the set of points such that the boundary at infinity of the dual planes are disjoint from $$\rho$$. This subset $$D=D(\rho)$$ is called the domain of dependence of the curve $$\rho$$. Remark 2.3. The above description can be made more precise, taking into account the causal property of the curve $$\rho$$ and its regularity (see e.g., [17]) but we will not use these notions in the rest of the article. □ The duality between smooth space-like surfaces in $$AdS_{3}$$ induces a similar duality between smooth space-like surfaces in a GHMC $$AdS_{3}$$ manifold. More precisely, let $$S\subset M$$ be a smooth, space-like, strictly-convex surface in a GHMC $$AdS_{3}$$ manifold. The lift of $$S$$ to the universal cover of $$M$$ can be identified with a surface $$\tilde{S}$$ in $$AdS_{3}$$, invariant under the action of the fundamental group of $$S$$. The dual surface $$\tilde{S}^{*}$$ is also invariant, so it corresponds to a surface $$S^{*}$$ in $$M$$. Clearly, since the fundamental group of $$S$$ acts by isometries, an analogue of Lemma 2.1 holds. 3 Equivariant isometric embeddings Let $$S$$ be a connected, compact, oriented surface of genus $$\tau \geq 2$$ and let $$g$$ be a Riemannian metric on $$S$$ with curvature less than $$-1$$. An isometric equivariant embedding of $$S$$ into $$AdS_{3}$$ is given by a couple $$(f,\rho)$$, where $$f:\tilde{S}\rightarrow AdS_{3}$$ is an isometric embedding of the universal Riemannian cover of $$S$$ into $$AdS_{3}$$ and $$\rho$$ is a representation of the fundamental group of $$S$$ into $$\mathbb{P}SL(2,\mathbb{R})\times \mathbb{P}SL(2,\mathbb{R})$$ such that   f(γx)=ρ(γ)f(x)   ∀ γ∈π1(S)   ∀ x∈S~. The group $$\mathbb{P}SL(2,\mathbb{R})\times \mathbb{P}SL(2,\mathbb{R})$$ acts on a couple $$(f,\rho)$$ by post-composition on the embedding and by conjugation on the representation. We denote by $$I(S,g)$$ the set of isometric embeddings of $$S$$ into $$AdS_{3}$$ modulo the action of $$\mathbb{P}SL(2,\mathbb{R})\times \mathbb{P}SL(2,\mathbb{R})$$. Also in an Anti-de Sitter setting, an analogue of the Fundamental Theorem for surfaces in the Euclidean space holds: Theorem 3.1. There exists an isometric embedding of $$(S,g)$$ into an $$AdS_{3}$$ manifold if and only if it is possible to define a $$g$$-self-adjoint operator $$b:TS\rightarrow TS$$ satisfying    det(b)=−κ−1      Gauss equationd∇b=0      Codazzi equation Moreover, the operator $$b$$ determines the isometric embedding uniquely, up to global isometries. □ This theorem enables us to identify $$I(S,g)$$ with the space of solutions of the Gauss-Codazzi equations, which can be studied using the classical techniques of elliptic operators. Lemma 3.2. The space $$I(S,g)$$ is a manifold of dimension $$6\tau -6$$. □ Proof. We can mimic the proof of Lemma $$3.1$$ in [15]. Consider the sub-bundle $$\mathcal{F}^{g}\subset \text{Sym}(TS)$$ over $$S$$ of symmetric operators $$b:TS\rightarrow TS$$ satisfying the Gauss equation. We prove that the operator   d∇:Γ∞(Fg)→Γ∞(Λ2TS⊗TS) is elliptic of index $$6\tau-6$$, equal to the dimension of the kernel of its linearisation. Let $$J_{0}$$ be the complex structure induced by $$g$$. For every $$b\in \Gamma^{\infty}({\mathcal{F}^{g}})$$, the operator   J=J0bdet(b) defines a complex structure on $$S$$. In particular we have an isomorphism   F:Γ∞(Fg) →Ab ↦J0bdet(b) between smooth sections of the sub-bundle $$\Gamma^{\infty}(\mathcal{F}^{g})$$ and the space $$\mathcal{A}$$ of complex structures on $$S$$, with inverse   F−1:A →Γ∞(Fg)J ↦−−κ−1J0J. This allows us to identify the tangent space of $$\Gamma(\mathcal{F}^{g})$$ at $$b$$ with the tangent space of $$\mathcal{A}$$ at $$J$$, which is the vector space of operators $$\dot{J}: TS \rightarrow TS$$ such that $$\dot{J}J+J\dot{J}=0$$. Under this identification the linearisation of $$d^{\nabla}$$ is given by   L(J˙)=−J0(d∇J˙). We deduce that $$L$$ has the same symbol and the same index of the operator $$\overline{\partial}$$, sending quadratic differentials to vector fields. Thus $$L$$ is elliptic with index $$6\tau-6$$. To conclude, we need to show that its cokernel is empty, or, equivalently, that its adjoint $$L^{*}$$ is injective. If we identify $$\Lambda^{2}TS\otimes TS$$ with $$TS$$ using the metric $$g$$, the adjoint operator $$L^{*}$$ is given by (see Lemma 3.1 in [15] for the computation)   (L∗ψ)(u)=−12(∇J0uψ+J∇J0Juψ). The kernel of $$L^{*}$$ consists of all the vector fields $$\psi$$ on $$S$$ such that for every vector field $$u$$  J∇uψ=−∇J0JJ0uψ. We can interpret this equation in terms of intersection of pseudo-holomorphic curves: the Levi-Civita connection $$\nabla$$ induces a decomposition of $$T(TS)$$ into a vertical $$V$$ and a horizontal $$H$$ sub-bundle. We endow $$V$$ with the complex structure $$J$$, and $$H$$ with the complex structure $$-J_{0}JJ_{0}$$. In this way, the manifold $$TS$$ is endowed with an almost-complex structure and the graph of $$\psi$$ is a pseudo-holomorphic curve. Since pseudo-holomorphic curves have positive intersections, if the graph of $$\psi$$ did not coincide with the graph of the null section, their intersection would be positive. On the other hand, it is well-known that this intersection coincides with the Euler characteristic of $$S$$, which is negative. Hence, we conclude that $$\psi$$ is identically zero and that $$L^{*}$$ is injective. ■ Similarly, we obtain the following result: Lemma 3.3. Let $$\{g_{t}\}_{t\in [0,1]}$$ be a differentiable curve of metrics with curvature less than $$-1$$. The set   W=⋃t∈[0,1]I(S,gt) is a manifold with boundary of dimension $$6\tau-5$$. □ Proof. Again we can mimic the proof of Lemma $$3.2$$ in [15]. Consider the sub-bundle $$\mathcal{F}\subset \text{Sym}(TS)$$ over $$S\times[0,1]$$ of symmetric operators, whose fibre over a point $$(x,t)$$ consists of the operators $$b:TS \rightarrow TS$$, satisfying the Gauss equation with respect to the metric $$g_{t}$$. The same reasoning as for the previous lemma shows that   d∇:Γ∞(F)→Γ∞(Λ2TS⊗TS) is Fredholm of index $$6\tau-5$$. Since $$W=(d^{\nabla})^{-1}(0)$$, the result follows from the implicit function theorem for Fredholm operators. ■ Let $$N$$ be a GHMC $$AdS_{3}$$ manifold endowed with a time orientation, that is, a nowhere vanishing time-like vector field. Let $$S$$ be a convex embedded surface in $$N$$. We say that $$S$$ is past-convex (resp. future-convex), if its past (resp. future) is geodesically convex. We will use the convention to compute the shape operator of $$S$$ using the future-directed normal. With this choice if $$S$$ is past-convex (resp. future-convex) then it has strictly positive (resp. strictly negative) principal curvatures. Definition 3.4. We will denote with $$I(S,g)^{+}$$ and $$I(S,g)^{-}$$ the spaces of equivariant isometric embeddings of $$S$$ as a past-convex and future-convex surface, respectively. □ 4 Parameterisation of GHMC $$\boldsymbol{AdS}_{\bf 3}$$ manifolds In his pioneering work ([17]), Mess studied the geometry of GHMC $$AdS_{3}$$ manifolds, discovering many connections with Teichmüller theory. We recall here some of his results, which we are going to use further. Let $$N$$ be a GHMC $$AdS_{3}$$ spacetime, that is, $$N$$ is endowed with an orientation and a time-orientation. It contains a space-like Cauchy surface, which is a closed surface $$S$$ of genus $$\tau \geq 2$$. It follows that $$N$$ is diffeomorphic to the product $$S\times \mathbb{R}$$. It contains a convex core, that is, a minimal convex subset homotopy equivalent to $$N$$, which can be either a totally geodesic surface or a topological submanifold homeomorphic to $$S\times [0,1]$$, whose boundary components are space-like surfaces, naturally endowed with a hyperbolic metric, pleated along measured geodesic laminations. The holonomy representation $$\rho$$ of the fundamental group $$\pi_{1}(N)\cong \pi_{1}(S)$$ into the group $$\mathbb{P}SL(2,\mathbb{R})\times \mathbb{P}SL(2,\mathbb{R})$$ of orientation and time-orientation preserving isometries of $$AdS_{3}$$ induced by the $$AdS_{3}$$-structure can be split, by projecting to each factor, into two representations   ρl=p1∘ρ:π1(S)→PSL(2,R)   ρr=p2∘ρ:π1(S)→PSL(2,R) called left and right representations. Mess proved that these have Euler class $$|e(\rho_{l})|=|e(\rho_{r})|=2g-2$$ and, using Goldman’s criterion ([12]), he concluded that they are discrete and faithful representations, and thus their classes represent elements of Teichmüller space. Moreover, every couple of points in Teichmüller space can be realised uniquely in this way, thus giving a parameterisation of the set of GHMC $$AdS_{3}$$-structures on $$S\times \mathbb{R}$$ up to isotopy by the product of two copies of the Teichmüller space of $$S$$ (Prop. 19 and Prop. 20 in [17]). We say that a GHMC $$AdS_{3}$$ manifold $$N$$ is Fuchsian if its left and right representations represent the same class in Teichmüller space. This happens if and only if the convex core of $$N$$ is a totally geodesic surface. Moreover, the left and right hyperbolic metrics corresponding to the left and right representations can be constructed explicitly starting from space-like surfaces embedded in $$N$$. Mess gave a description in a non-smooth setting using the upper and lower boundary of the convex core of $$N$$ as space-like surfaces. More precisely, if $$m^{\pm}$$ are the hyperbolic metrics on the upper and lower boundary of the convex core and $$\lambda^{\pm}$$ are the measured geodesic lamination along which they are pleated, the left and right metrics $$h_{l}$$ and $$h_{r}$$ are related to $$m^{\pm}$$ by an earthquake along $$\lambda^{\pm}$$:   hl=Elλ+(m+)=Erλ−(m−)      hr=Erλ+(m+)=Elλ−(m−). Later, this description was extended ([14]), thus obtaining explicit formulas for the left and right metric, in terms of the induced metric $$I$$, the complex structure $$J$$ and the shape operator $$B$$ of any strictly negatively curved smooth space-like surface $$S$$ embedded in $$N$$. The construction goes as follows: We fix a totally geodesic space-like plane $$P_{0}$$. Let $$\tilde{S}\subset AdS_{3}$$ be the universal cover of $$S$$. Let $$\tilde{S}'\subset U^{1}AdS_{3}$$ be its lift into the unit tangent bundle of $$AdS_{3}$$ and let $$p:\tilde{S}' \rightarrow \tilde{S}$$ be the canonical projection. For any point $$(x,v)\in \tilde{S}'$$, there exists a unique space-like plane $$P$$ in $$AdS_{3}$$ orthogonal to $$v$$ and containing $$x$$. We define two natural maps $$\Pi_{\infty, l}$$ and $$\Pi_{\infty,r}$$ from $$\partial_{\infty}P$$ to $$\partial_{\infty}P_{0}$$, sending a point $$x\in \partial_{\infty}P$$ to the intersection between $$\partial_{\infty}P_{0}$$ and the unique line of the left or right foliation of $$\partial_{\infty}AdS_{3}$$ containing $$x$$. Since these maps are projective, they extend to hyperbolic isometries $$\Pi_{l}, \Pi_{r}:P \rightarrow P_{0}$$. Identifying $$P$$ with the tangent space of $$\tilde{S}$$ at the point $$x$$, the pull-backs of the hyperbolic metric on $$P_{0}$$ by $$\Pi_{l}$$ and by $$\Pi_{r}$$ define two hyperbolic metrics on $$\tilde{S}$$  hl=I((E+JB)⋅,(E+JB)⋅)    and    hr=I((E−JB)⋅,(E−JB)⋅). The isotopy classes of the corresponding metrics on $$S$$ do not depend on the choice of the space-like surface $$S$$ and their holonomies are precisely $$\rho_{l}$$ and $$\rho_{r}$$, respectively (Lemma 3.16 in [14]). This parameterisation enables us to formulate our original question about the prescription of the metrics on the boundary of a compact $$AdS_{3}$$ manifold in terms of existence of an intersection of particular subsets of $$\mathrm{Teich}(S) \times \mathrm{Teich}(S)$$. Definition 4.1. Let $$g$$ be a metric on $$S$$ with curvature $$\kappa<-1$$. We define the maps   ϕg±:I(S,g)±→Teich(S)×Teich(S)b↦(hl(g,b),hr(g,b)):=(g((E+Jb)⋅,(E+Jb)⋅),g((E−Jb)⋅,(E−Jb)⋅)) associating to every isometric embedding of $$(S,g)$$ the left and right metric of the GHMC $$AdS_{3}$$ manifold containing it. □ We recall that we use the convention to compute the shape operator using always the future-oriented normal. In this way, the above formulas hold for both future-convex and past-convex surfaces, without changing the orientation of the surface $$S$$. We will prove (in Section 8) the following fact, which is the main theorem of the article: Theorem 4.2. For every couple of metrics $$g_{+}$$ and $$g_{-}$$ on $$S$$ with curvature less than $$-1$$, we have   ϕg++(I(S,g+)+)∩ϕg−−(I(S,g−)−)≠∅. Therefore, there exists a GHMC $$AdS_{3}$$ manifold containing a past-convex space-like surface isometric to $$(S,g_{+})$$ and a future-convex space-like surface isometric to $$(S, g_{-})$$. □ Corollary 4.3. For every couple of metrics $$g_{+}$$ and $$g_{-}$$ on $$S$$ with curvature less than $$-1$$, there exists a globally hyperbolic convex compact $$AdS_{3}$$ manifold $$K\cong S\times [0,1]$$, whose induced metrics on the boundary are exactly $$g_{\pm}$$. □ If we apply the previous corollary to the dual surfaces (introduced in Section 2), we obtain an analogue result about the prescription of the third fundamental form: Corollary 4.4. For every couple of metrics $$g_{+}$$ and $$g_{-}$$ on $$S$$ with curvature less than $$-1$$, there exists a compact $$AdS_{3}$$ manifold $$K\cong S\times [0,1]$$, whose induced third fundamental forms on the boundary are exactly $$g_{\pm}$$. □ 5 Topological intersection theory As outlined in the introduction, the main tool used in the proof of the main theorem is the intersection theory of smooth maps between manifolds, which is developed for example in [13]. We recall here the basic constructions and the fundamental results. If not otherwise stated, all manifolds considered in this section are non-compact without boundary. Let $$X$$ and $$Z$$ be manifolds of dimension $$m$$ and $$n$$, respectively and let $$A$$ be a closed submanifold of $$Z$$ of codimension $$k$$. Suppose that $$m-k \geq 0$$. We say that a smooth map $$f:X \rightarrow Z$$ is transverse to $$A$$ if for every $$z\in \mathrm{Im}(f)\cap A$$ and for every $$x\in f^{-1}(z)$$ we have   df(TxX)+TzA=TzZ. Under this hypothesis, $$f^{-1}(A)$$ is a submanifold of $$X$$ of codimension $$k$$. When $$k=m$$ and $$f^{-1}(A)$$ consists of a finite number of points we define the intersection number between $$f$$ and $$A$$ as   ℑ(f,A):=|f−1(A)|       (mod 2). Remark 5.1. When $$A$$ is a point $$p\in Z$$, $$f$$ is transverse to $$p$$ if and only if $$p$$ is a regular value for $$f$$. Moreover, if $$f$$ is proper, $$f^{-1}(p)$$ consists of a finite number of points and the above definition coincides with the classical definition of degree (mod $$2$$) of a smooth and proper map. □ We say that two smooth maps $$f: X \rightarrow Z$$ and $$g: Y \rightarrow Z$$ are transverse if the map   f×g:X×Y→Z×Z is transverse to the diagonal $$\Delta \subset Z\times Z$$. If $$\mathrm{Im}(f) \cap \mathrm{Im}(g)=\emptyset$$, then $$f$$ and $$g$$ are transverse by definition. Suppose now that $$2\dim X=2\dim Y=\dim Z$$. Moreover, suppose that the maps $$f:X \rightarrow Z$$ and $$g: Y \rightarrow Z$$ are transverse and the preimage $$(f\times g)^{-1}(\Delta)$$ consists of a finite number of points. We define the intersection number between $$f$$ and $$g$$ as   ℑ(f,g):=ℑ(f×g,Δ)=|(f×g)−1(Δ)|       (mod 2). It follows by the definition that if $$\Im(f,g)\ne 0$$ then $$\mathrm{Im}(f)\cap \mathrm{Im}(g) \neq \emptyset$$. One important feature of the intersection number that we will use further is the invariance under cobordism. We say that two maps $$f_{0}:X_{0} \rightarrow Z$$ and $$f_{1}: X_{1} \rightarrow Z$$ are cobordant if there exists a manifold $$W$$ and a smooth function $$F:W \rightarrow Z$$ such that $$\partial W=X_{0}\cup X_{1}$$ and $$F_{|_{X_{i}}}=f_{i}$$. Proposition 5.2. Let $$W$$ be a non-compact manifold with boundary $$\partial W=X_{0}\cup X_{1}$$. Let $$H: W \rightarrow Z$$ be a smooth map and denote by $$h_{i}$$ the restriction of $$H$$ to the boundary component $$X_{i}$$ for $$i=0,1$$. Let $$A \subset Z$$ be a closed submanifold. Suppose that (i) $$\mathrm{codim} A=\dim X_{i}$$; (ii) $$H$$ is transverse to $$A$$; (iii) $$H^{-1}(A)$$ is compact. Then $$\Im(h_{0},A)=\Im(h_{1},A)$$. □ Proof. By hypothesis the pre-image $$H^{-1}(A)$$ is a compact, properly embedded $$1$$-manifold, that is, it is a finite disjoint union of circles and arcs with ending points on a boundary component of $$W$$. This implies that $$h_{0}^{-1}(A)$$ and $$h_{1}^{-1}(A)$$ have the same parity. ■ In particular, we deduce the following result about the intersection number of two maps: Corollary 5.3. Let $$W$$ be a non-compact manifold with boundary $$\partial W=X_{0}\cup X_{1}$$. Let $$F: W \rightarrow Z$$ be a smooth map and denote by $$f_{i}$$ the restriction of $$F$$ to the boundary component $$X_{i}$$ for $$i=0,1$$. Let $$g:Y \rightarrow Z$$ be a smooth map. Suppose that (i) $$2\dim X_{i}=2\dim Y=\dim Z$$; (ii) $$F$$ and $$g$$ are transverse; (iii) $$(F\times g)^{-1}(\Delta)$$ is compact. Then $$\Im(f_{0},g)=\Im(f_{1},g)$$. □ Proof. Apply the previous proposition to the map $$H=F\times g: W\times Y \rightarrow Z\times Z$$ and to the submanifold $$A=\Delta$$, the diagonal of $$Z\times Z$$. ■ The hypothesis of transversality in the previous propositions is not restrictive, as it is always possible to perturb the maps involved on a neighbourhood of the set on which transversality fails: Theorem 5.4 (Theorem p.72 in [13]). Let $$h: W \rightarrow Z$$ be a smooth map between manifolds, where only $$W$$ has boundary. Let $$A$$ be a closed submanifold of $$Z$$. Suppose that $$h$$ is transverse to $$A$$ on a closed set $$C\subset W$$. Then there exists a smooth map $$\tilde{h}:W \rightarrow Z$$ homotopic to $$h$$ such that $$\tilde{h}$$ is transverse to $$A$$ and $$\tilde{h}$$ agrees with $$h$$ on a neighbourhood of $$C$$. □ Now the question arises whether the intersection number depends on the particular perturbation of the map that we obtain when applying Theorem 5.4. Proposition 5.5. Let $$h:X\rightarrow Z$$ be a smooth map between manifolds. Let $$A$$ be a submanifold of $$Z$$, whose codimension equals the dimension of $$X$$. Suppose that $$h^{-1}(A)$$ is compact. Let $$\tilde{h}$$ and $$\tilde{h}'$$ be perturbations of $$h$$, which are transverse to $$A$$ and coincide with $$h$$ outside the interior part of a compact set $$B$$ containing $$h^{-1}(A)$$. Then   ℑ(h~,A)=ℑ(h~′,A). □ Proof. Let $$\tilde{H}:W=X\times [0,1]\rightarrow Y$$ be an homotopy between $$\tilde{h}$$ and $$\tilde{h}'$$ such that for every $$x\in (X\setminus B)\times [0,1]$$ we have $$\tilde{H}(x,t)=h(x)$$. Notice that $$\tilde{H}^{-1}(A)$$ is compact. Up to applying Theorem 5.4 to the closed set $$C=(X\setminus B)\times [0,1]\cup \partial W$$, we can suppose that $$\tilde{H}$$ is transverse to $$A$$. By Proposition 5.2, we have that $$\Im(\tilde{h}_{0},A)=\Im(\tilde{h}_{1},A)$$ as claimed. ■ Moreover, in particular circumstances, we can actually obtain a $$1-1$$ correspondence between the points of $$h_{0}^{-1}(A)$$ and $$h_{1}^{-1}(A)$$. The following proposition will not be used for the proof of the main result of the article, but it might be a useful tool to prove the uniqueness part of the question addressed in this article, as explained in Remark 8.3. Proposition 5.6. Under the same hypothesis as Proposition 5.2, suppose that the cobordism $$(W, H)$$ between $$h_{0}$$ and $$h_{1}$$ satisfies the following additional properties: (i) $$W$$ fibres over the interval $$[0,1]$$ with fibre $$X_{t}$$; (i) the restriction $$h_{t}$$ of $$H$$ at each fibre is tranverse to $$A$$. Then $$|h_{0}^{-1}(A)|=|h_{1}^{-1}(A)|$$. □ Proof. It is sufficient to show that in $$H^{-1}(A)$$ there are no arcs with ending points in the same boundary component. By contradiction, let $$\gamma$$ be an arc with ending point in $$X_{0}$$. Define   t0=sup{t∈[0,1] | γ∩Xt≠∅}. A tangent vector $$\dot{\gamma}$$ at a point $$p \in X_{t_{0}}\cap \gamma$$ is in the kernel of the map   dpH:T(p,t0)W→Tq(Z×Z)/Tq(A), where $$q=H(p)$$. The contradiction follows by noticing that on the one hand $$\dot{\gamma}$$ is contained in the tangent space $$T_{p}X_{t_{0}}$$ by construction but on the other hand $$d_{p}h_{t_{0}}: T_{p}X_{t_{0}} \rightarrow T_{q}(Z\times Z)/T_{q}(A)$$ is an isomorphism by transversality. A similar reasoning works when $$\gamma$$ has ending points in $$X_{1}$$. ■ 6 Some properties of the maps $$\boldsymbol{\phi}^{\boldsymbol{\pm}}$$ This section contains the most technical part of the article. We summarise here briefly, for the convenience of the reader, what the main results of this section are? For every metric $$g$$ on $$S$$ with curvature less than $$-1$$ we have defined in Section 4 the maps $$\phi_{g}^{\pm}$$ which associate to every isometric embedding of $$(S,g)$$ into a GHMC $$AdS_{3}$$ manifold $$M$$ the class in Teichmüller space of the left and right metrics of $$M$$. It follows easily from Lemma 3.3 that for any couple of metrics $$g$$ and $$g'$$ with curvature less than $$-1$$ the maps $$\phi^{\pm}_{g}$$ and $$\phi^{\pm}_{g'}$$ are cobordant through a map $$\Phi^{\pm}$$. In this section, we will define the maps $$\Phi^{\pm}$$ and will study some of its properties, which will enable us to apply the topological intersection theory described in the previous section. More precisely, the first step will consist of proving that the all the maps involved are smooth. This is the content of Proposition 6.1 and the proof will rely on the fact that the holonomy representation of a hyperbolic metric depends smoothly on the metric. Then we will deal with the properness of the maps $$\Phi^{\pm}$$ (Corollary 6.8) that will follow from a compacteness result of isometric embeddings (Corollary 6.5). This will allow us also to have a control on the space where two maps $$\phi_{g}$$ and $$\phi_{g'}$$ intersect: when we deform one of the two metrics the intersection remains contained in a compact set (Proposition 6.13). Recall that given a smooth path of metrics $$\{g_{t}\}_{t \in [0,1]}$$ on $$S$$ with curvature less than $$-1$$, the set   W±=⋃t∈[0,1]I±(S,gt) is a manifold with boundary $$\partial W^{\pm}=I(S, g_{0})^{\pm} \cup I(S, g_{1})^{\pm}$$ of dimension $$6\tau-5$$ (Lemma 3.3). We define the maps   Φ±:W± →Teich(S)×Teich(S)bt ↦(hl(gt,bt),hr(gt,bt)):=(gt((E+Jbt)⋅,(E+Jbt)⋅),gt((E−Jbt)⋅,(E−Jbt)⋅)) associating to an equivariant isometric embedding (identified with its Codazzi operator $$b_{t}$$) of $$(S, g_{t})$$ into $$AdS_{3}$$ the class in Teichmüller space of the left and right metrics of the GHMC $$AdS_{3}$$ manifold containing it. We remark that the restrictions of $$\Phi^{\pm}$$ to the boundary coincide with the maps $$\phi^{\pm}_{g_{0}}$$ and $$\phi^{\pm}_{g_{1}}$$ defined in Section 4. We deal first with the regularity of the maps. Proposition 6.1. The functions $$\Phi^{\pm}:W^{\pm} \rightarrow \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$ are smooth. □ Proof. Let $$\mathcal{M}_{S}$$ be the set of hyperbolic metrics on $$S$$. We can factorise the map $$\Phi^{\pm}$$ as follows:   W±→Φ′±MS×MS→πTeich(S)×Teich(S) where $$\Phi'^{\pm}$$ associates to an isometric embedding of $$(S, g_{t})$$ (determined by an operator $$b_{t}$$ satisfying the Gauss-Codazzi equation) the couple of hyperbolic metrics $$(g_{t}((E+J_{t}b_{t})\cdot, (E+J_{t}b_{t})\cdot), g_{t}((E-J_{t}b_{t})\cdot, (E-J_{t}b_{t})\cdot))$$, and $$\pi$$ is the projection to the corresponding isotopy class, or, equivalently, the map which associates to a hyperbolic metric its holonomy representation. Since the maps $$\Phi'^{\pm}$$ are clearly smooth by definition, we just need to prove that the holonomy representation depends smoothly on the metric. Let $$h$$ be a hyperbolic metric on $$S$$. Fix a point $$p \in S$$ and a unitary frame $$\{v_{1}, v_{2}\}$$ of the tangent space $$T_{p}S$$. We consider the ball model for the hyperbolic plane and we fix a unitary frame $$\{w_{1}, w_{2}\}$$ of $$T_{0}\mathbb{H}^{2}$$. We can realise every element of the fundamental group of $$S$$ as a closed path passing through $$p$$. Let $$\gamma$$ be a path passing through $$p$$ and let $$\{U_{i}\}_{i=0, \dots n}$$ be a finite covering of $$\gamma$$ such that every $$U_{i}$$ is homeomorphic to a ball. We know that there exists a unique map $$f_{0}: U_{0} \rightarrow B_{0}\subset \mathbb{H}^{2}$$ such that   {f0(p)=0dpf0(vi)=wif0∗gH2=h.  Then, for every $$i\geq 1$$ there exists a unique isometry $$f_{i}: U_{i} \rightarrow B_{i}\subset \mathbb{H}^{2}$$ which coincides with $$f_{i-1}$$ on the intersection $$U_{i} \cap U_{i-1}$$. Let $$q=f_{n}(p) \in \mathbb{H}^{2}$$. The holonomy representation sends the homotopy class of the path $$\gamma$$ to the isometry $$I_{q}: \mathbb{H}^{2}\rightarrow \mathbb{H}^{2}$$ such that $$I_{q}(q)=0$$. Moreover, its differential maps the frame $$\{u_{i}=df_{n}(v_{i})\}$$ to the frame $$w_{i}$$. The isometry $$I_{q}$$ depends smoothly on $$q$$ and on the frame $$u_{i}$$, which depend smoothly on the metric because each $$f_{i}$$ does. ■ The next step is about the properness of the maps $$\Phi^{\pm}$$. This will involve the study of sequences of isometric embeddings of a disc into a simply-connected spacetime, which have been extensively and profitably analysed in [18]. In particular, the author proved that, under reasonable hypothesis, a sequence of isometric embeddings of a disc into a simply-connected spacetime has only two possible behaviours: it converges $$C^{\infty}$$, up to subsequences, to an isometric embedding, or it is degenerate in a precise sense: Theorem 6.2 (Theorem 5.6 in [18]). Let $$\tilde{f}_{n}: D\rightarrow X$$ be a sequence of isometric immersions of a disc $$D$$ in a simply connected Lorentzian spacetime $$(X, \tilde{g})$$ such that the corresponding shape operators have uniformly positive determinant. Assume that the metrics $$\tilde{f}_{n}^{*}\tilde{g}$$ converge $$C^{\infty}$$ towards a Riemannian metric $$\tilde{g}_{\infty}$$ on $$D$$ and that there exists a point $$x \in D$$ such that the sequence of the $$1$$-jets $$j^{1}\tilde{f}_{n}(x)$$ converges. If the sequence $$\tilde{f}_{n}$$ does not converge in the $$C^{\infty}$$ topology in a neighbourhood of $$x$$, then there exists a maximal geodesic $$\gamma$$ of $$(D, \tilde{g}_{\infty})$$ and a geodesic arc $$\Gamma$$ of $$(X, \tilde{g})$$ such that the sequence $$(\tilde{f}_{n})_{|_{\gamma}}$$ converges towards an isometry $$\tilde{f}_{\infty}: \gamma \rightarrow \Gamma$$. □ We start with a straightforward application of the Maximum Principle, which we recall here in the form useful for our purposes (see e.g., Proposition 4.6 in [3]). Proposition 6.3 (Maximum Principle). Let $$\Sigma_{1}$$ and $$\Sigma_{2}$$ two future-convex space-like surfaces embedded in a GHMC $$AdS_{3}$$ manifold $$M$$. If they intersect in a point $$x$$ and $$\Sigma_{1}$$ is in the future of $$\Sigma_{2}$$ then the product of the principal curvatures of $$\Sigma_{2}$$ is larger than the product of the principal curvatures of $$\Sigma_{1}$$. □ Proposition 6.4. Let $$\Sigma$$ be a future-convex space-like surface embedded into a GHMC $$AdS_{3}$$ manifold $$M$$. Suppose that the Gaussian curvature of $$\Sigma$$ is bounded between $$-\infty < \kappa_{min}\leq \kappa_{max} < -1$$. Denote with $$S_{min}$$ and $$S_{max}$$ the unique future-convex space-like surfaces with constant curvature $$\kappa_{min}$$ and $$\kappa_{max}$$ embedded in $$M$$ (Corollary 4.7 in [3]). Then $$\Sigma$$ is in the past of $$S_{max}$$ and in the future of $$S_{min}$$. □ Proof. Consider the unique (Corollary 4.7 in [3]) $$\kappa$$-time   T:I−(∂−C(M))→(−∞,−1), that is, the unique function defined on the past of the convex core of $$M$$ such that the level sets $$T^{-1}(\kappa)$$ are future-convex space-like surfaces of constant curvature $$\kappa$$. The restriction of $$T$$ to $$\Sigma$$ has a maximum $$t_{max}$$ and a minimum $$t_{min}$$. Consider the level sets $$L_{min}=T^{-1}(t_{min})$$ and $$L_{max}=T^{-1}(t_{max})$$. By construction $$\Sigma$$ is in the future of $$L_{min}$$ and they intersect in a point $$x$$, hence, by the Maximum Principle and the Gauss equation, we obtain the following inequality for the Gaussian curvature of $$\Sigma$$ at the point $$x$$:   tmin≥κ(x)≥κmin. Similarly, we obtain that $$t_{max}\leq \kappa(y)\leq \kappa_{max}$$, where $$y$$ is the point of intersection between $$L_{max}$$ and $$\Sigma$$. But this implies that $$\Sigma$$ is in the past of the level set $$T^{-1}(\kappa_{max})$$ and in the future of the level set $$T^{-1}(\kappa_{min})$$, which correspond respectively to the surfaces $$S_{max}$$ and $$S_{min}$$ by uniqueness. ■ Corollary 6.5. Let $$g_{n}$$ be a compact family of metrics in the $$C^{\infty}$$ topology with curvatures $$\kappa < -1$$ on a surface $$S$$. Let $$f_{n}: (S, g_{n})\rightarrow M_{n}=(S\times \mathbb{R}, h_{n})$$ be a sequence of isometric embeddings of $$(S, g_{n})$$ as future-convex space-like surfaces into GHMC $$AdS_{3}$$ manifolds. If the sequence $$h_{n}$$ converges to an $$AdS$$ metric $$h_{\infty}$$ in the $$C^{\infty}$$-topology, then $$f_{n}$$ converges $$C^{\infty}$$, up to subsequences, to an isometric embedding into $$M_{\infty}=(S\times \mathbb{R}, h_{\infty})$$. □ Proof. Consider the equivariant isometric embeddings $$\tilde{f}_{n}: (\tilde{S}, \tilde{g}_{n}) \rightarrow \tilde{AdS}_{3}$$ obtained by lifting $$f_{n}$$ to the universal cover. We denote with $$\tilde{S}_{n}$$ the images of the disc $$\tilde{S}$$ under the map $$\tilde{f}_{n}$$ and let $$\tilde{h}_{n}$$ be the lift of the Lorentzian metrics $$h_{n}$$ on $$\tilde{AdS}_{3}$$. By hypothesis $$\tilde{f}_{n}^{*}\tilde{h}_{n}=\tilde{g}_{n}$$ admits a subsequence converging to $$\tilde{g}_{\infty}$$. Fix a point $$x \in \tilde{S}$$. Since the isometry group of $$\tilde{AdS}_{3}$$ acts transitively on points and frames, we can suppose that $$\tilde{f}_{n}(x)=y \in \tilde{AdS}_{3}$$ and $$j^{1}\tilde{f}_{n}(x)=z$$ for every $$n \in \mathbb{N}$$. Moreover, the condition on the curvature of the metrics $$g_{n}$$ guarantees that the sequence $$\tilde{f}_{n}$$ is uniformly elliptic. Therefore, we are under the hypothesis of Theorem 6.2. The previous proposition allows us to determine precisely in which region of $$M_{n}$$ each surface $$f_{n}(S)$$ lies. Since the family of metrics $$g_{n}$$ is compact, the curvatures $$\kappa_{n}$$ of the surfaces $$f_{n}(S)$$ in $$M_{n}$$ are uniformly bounded $$\kappa^{min} \leq \kappa_{n} \leq \kappa^{max}\leq -1-3\epsilon$$ for some $$\epsilon>0$$. By the previous proposition each surface $$f_{n}(S)$$ is in the past of $$\Sigma_{n}^{max}$$ and in the future of $$\Sigma_{n}^{min}$$, where $$\Sigma_{n}^{min}$$ and $$\Sigma_{n}^{max}$$ are the unique future-convex space-like surfaces of $$M_{n}$$ with constant curvature $$\kappa^{min}$$ and $$\kappa^{max}$$. Let $$\Sigma_{\epsilon}$$ be the unique future-convex space-like surface in $$M_{\infty}$$ with constant curvature $$-1-2\epsilon$$. We think of $$\Sigma$$ as a fixed surface embedded in $$S\times \mathbb{R}$$ and we change the Lorentzian metric of the ambient space. Since $$h_{n}$$ converges to $$h_{\infty}$$, the metrics induced on $$\Sigma_{\epsilon}$$ by $$h_{n}$$ converge to the metric induced on $$\Sigma_{\epsilon}$$ by $$h_{\infty}$$. In particular, for $$n$$ sufficiently large the curvature of $$\Sigma_{\epsilon}$$ as surface embedded in $$M_{n}=(S \times \mathbb{R}, h_{n})$$ is bounded between $$-1-3\epsilon$$ and $$-1-\epsilon$$. Therefore, $$\Sigma_{\epsilon}$$ is convex in $$M_{n}$$ and by the previous proposition $$\Sigma_{\epsilon}$$ is in the future of $$\Sigma_{n}^{max}$$ for every $$n$$ sufficiently big. This implies that each surface $$f_{n}(S)$$ is in the past of the surface $$\Sigma_{\epsilon}$$. We can now conclude that the sequence $$f_{n}$$ must converge to an isometric embedding. Suppose by contradiction that the sequence $$\tilde{f}_{n}$$ is not convergent in the $$C^{\infty}$$ topology in a neighbourhood of $$x$$, then there exists a maximal geodesic $$\tilde{\gamma}$$ of $$(\tilde{S}, \tilde{g}_{\infty})$$ and a geodesic segment $$\tilde{\Gamma}$$ in $$\tilde{AdS}_{3}$$ such that $$(\tilde{f}_{n})_{|_{\tilde{\gamma}}}$$ converges to an isometry $$\tilde{f}_{\infty}: \tilde{\gamma} \rightarrow \tilde{\Gamma}$$. This implies that $$\tilde{\Gamma}$$ has infinite length. The projection of $$\tilde{\Gamma}$$ must be contained in the past of $$\Sigma_{\epsilon}$$, because each $$f_{n}(S)$$ is contained there for $$n$$ sufficiently large. But the past of $$\Sigma_{\epsilon}$$ is disjoint from the convex core of $$M_{\infty}$$ and this contradicts the following lemma. ■ Lemma 6.6. In a GHMC $$AdS_{3}$$-manifold every complete space-like geodesic is contained in the convex core. □ Proof. Let $$c$$ be a complete space-like geodesic in a GHMC $$AdS_{3}$$ manifold $$M$$. By a result of Mess ([17]), we can realise $$M$$ as the quotient of the domain of dependence $$D(\rho)\subset AdS_{3}$$ of a curve $$\rho$$ on the boundary at infinity by the action of the fundamental group of $$S$$. The lift $$\bar{c}$$ of $$c$$ has ending points on the curve $$\rho$$, hence $$\bar{c}$$ is contained in the convex hull of $$\rho$$ into $$AdS_{3}$$ and its projection is contained in the convex core of $$M$$. ■ Remark 6.7. Clearly, the same result holds for equivariant isometric embeddings of past-convex space-like surfaces, as it is sufficient to reverse the time-orientation. □ Corollary 6.8. The functions $$\Phi^{\pm}:W^{\pm} \rightarrow \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$ are proper. □ Proof. We prove the claim for the function $$\Phi^{-}$$, the other case being analogous. Let $$(h_{l}(g_{t_{n}}, b_{t_{n}}), h_{r}(g_{t_{n}}, b_{t_{n}}))\in \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$ be a convergent sequence in the image of the map $$\Phi^{-}$$. This means that the sequence of GHMC $$AdS_{3}$$ manifolds $$M_{n}$$ parametrised by $$(h_{l}(g_{t_{n}}, b_{t_{n}}), h_{r}(g_{t_{n}}, b_{t_{n}}))$$ is convergent. By definition of the map $$\Phi^{-}$$, each $$M_{n}$$ contains an embedded future-convex, space-like surface isometric to $$(S, g_{t_{n}})$$, whose immersion $$f_{n}$$ into $$M_{n}$$ is represented by the Codazzi operator $$b_{t_{n}}$$. By Corollary 6.5, the sequence of isometric immersions $$f_{n}$$ is convergent up to subsequences, thus $$\Phi^{-}$$ is proper. ■ This allows us to show that for every metric $$g_{-}$$ and for every smooth path of metrics $$\{g_{t}^{+}\}_{t \in [0,1]}$$ on $$S$$ with curvature $$\kappa <-1$$ the intersection between $$\Phi^{+}(W^{+})$$ and $$\phi^{-}_{g_{-}}(I(S, g_{-})^{-})$$ is compact. This will follow combining some technical results about the geometry of $$AdS_{3}$$ manifolds and length-spectrum comparisons. Definition 6.9. Let $$g$$ be a metric with negative curvature on $$S$$. We define the length function   ℓg:π1(S)→R+ which associates to every homotopy non-trivial loop on $$S$$, the length of its $$g$$-geodesic representative. □ We recall that when $$g$$ is a hyperbolic metric, Thurston proved (see e.g. [9]) that the length function can be extended uniquely to a function on the space of measured geodesic laminations on $$S$$, which we still denote with $$\ell_{g}$$. We will need the following technical results: Lemma 6.10 (Lemma 9.6 in [5]). Let $$N$$ be a globally hyperbolic compact $$AdS_{3}$$-manifold foliated by future-convex space-like surfaces. Then, the sequence of metrics induced on each surface decreases when moving towards the past. In particular, if $$\Sigma_{1}$$ and $$\Sigma_{2}$$ are two future-convex space-like surfaces with $$\Sigma_{1}$$ in the future of $$\Sigma_{2}$$, then for every closed geodesic $$\gamma$$ in $$\Sigma_{1}$$ we have   ℓg2(γ′)≤ℓg1(γ), where $$\gamma'$$ is the closed geodesic on $$\Sigma_{2}$$ homotopic to $$\gamma$$ and $$g_{1}$$ and $$g_{2}$$ are the induced metric on $$\Sigma_{1}$$ and $$\Sigma_{2}$$, respectively. □ Lemma 6.11. Let $$g_{n}$$ be a compact family of smooth metrics on $$S$$ with curvature less than $$-1$$. Let $$m_{n}$$ be a family of hyperbolic metrics such that   ℓgn(γ)≤ℓmn(γ) for every $$\gamma \in \pi_{1}(S)$$. Then $$m_{n}$$ lies in a compact subset of the Teichmüller space of $$S$$. □ Proof. The idea is to use Thurston asymmetric metric on Teichmüller space. To this aim, we will deduce from the hypothesis a comparison between the length spectrum of $$m_{n}$$ and that of the hyperbolic metrics $$h_{n}$$ in the conformal class of $$g_{n}$$. Let $$\kappa<-1$$ be the infimum of the curvatures of the family $$g_{n}$$. Since $$g_{n}$$ is a compact family, $$\kappa> -\infty$$. Let $$\bar{g}_{n}=-\frac{1}{\kappa}h_{n}$$ be the metrics of constant curvature $$\kappa$$ in the conformal class of $$g_{n}$$. We claim that   ℓhn(γ)≤|κ|ℓmn(γ) for every $$\gamma \in \pi_{1}(S)$$. For instance, if we write $$\bar{g}_{n}=e^{2u_{n}}g_{n}$$, the smooth function $$u_{n}: S \rightarrow \mathbb{R}$$ satisfies the differential equation   e2un(x)κ=κgn(x)+Δgnun(x), where $$\kappa_{g_{n}}$$ is the curvature of $$g_{n}$$. Since $$\kappa_{g_{n}}\geq \kappa$$, $$\Delta_{g_{n}}u_{n}$$ is positive at the point of maximum of $$u_{n}$$ and $$\kappa<-1$$, we deduce that $$e^{2u_{n}}\leq 1$$, hence   ℓg¯n(γ)≤ℓgn(γ) for every $$\gamma \in \pi_{1}(S)$$. It is then clear that   ℓg¯n(γ)=1κℓhn(γ) for every $$\gamma \in \pi_{1}(S)$$ and the claim follows. Moreover, by the inequality   ℓhn(γ)≤|κ|ℓgn(γ)    ∀ γ∈π1(S) we deduce that $$h_{n}$$ is contained in a compact set of Teichmüller space: if that were not the case, there would exists a curve $$\gamma$$ such that $$\ell_{h_{n}}(\gamma)\xrightarrow{n \to \infty} +\infty$$, which is impossible because $$g_{n}$$ is a compact family. We can conclude now using Thurston asymmetric metric: given two hyperbolic metrics $$h$$ and $$h'$$, Thurston asymmetric distance between $$h$$ and $$h'$$ is defined as   dTh(h,h′)=supγ∈π1(S)log⁡(ℓh(γ)ℓh′(γ)). It is well-known ([20]) that if $$h'_{n}$$ is a divergent sequence than $$d_{Th}(K, h_{n}) \to + \infty$$, where $$K$$ is any compact set in Teichmüller space. Now, by the length spectrum comparison   ℓhn(γ)≤|κ|ℓmn(γ)    ∀ γ∈π1(S), we deduce that $$d_{Th}(h_{n}, m_{n}) \leq \log(\sqrt{|\kappa|})< +\infty$$, hence $$m_{n}$$ must be contained in a compact set. ■ We will need also the following fact about the geometry of the convex core of a GHMC $$AdS_{3}$$ manifold. Lemma 6.12 (Prop. 5 in [7]). Let $$M$$ be a GHMC $$AdS_{3}$$ manifold. Denote by $$m^{+}$$ and $$m^{-}$$ the hyperbolic metrics on the upper and lower boundary of the convex core of $$M$$. Let $$\lambda^{+}$$ and $$\lambda^{-}$$ be the measured geodesic laminations on the upper and lower boundary of the convex core of $$M$$. For all $$\epsilon>0$$, there exists some $$A>0$$ such that, if $$m^{+}$$ is contained in a compact set and $$\ell_{m^{+}}(\lambda^{+}) \geq A$$, then $$\ell_{m^{-}}(\lambda^{+})\leq \epsilon \ell_{m^{+}}(\lambda^{+})$$. □ Proposition 6.13. For every metric $$g^{-}$$ and for every smooth path of metrics $$\{g_{t}^{+}\}_{t \in [0,1]}$$ on $$S$$ with curvature $$\kappa < -1$$, the set $$(\Phi^{+}\times \phi^{-}_{g^{-}})^{-1}(\Delta)$$ is compact. □ Proof. We need to prove that every sequence of isometric embeddings $$(b^{+}_{t_{n}}, b^{-}_{n})$$ in $$(\Phi^{+} \times \phi^{-}_{g^{-}})^{-1}(\Delta)$$ admits a convergent subsequence. By definition, for every $$n \in \mathbb{N}$$, there exists a GHMC $$AdS_{3}$$ manifold $$M_{n}$$ containing a past-convex surface isometric to $$(S, g^{+}_{t_{n}})$$ with shape operator $$b^{+}_{t_{n}}$$ and a future-convex surface isometric to $$(S, g^{-})$$ with shape operator $$b^{-}_{n}$$. By Lemma 6.10 and Lemma 6.11, the metrics $$m_{n}^{+}$$ and $$m_{n}^{-}$$ on the upper and lower boundary of the convex core of $$M_{n}$$ are contained in a compact set of $$\mathrm{Teich}(S)$$. We are going to prove now that the sequences of left and right metrics of $$M_{n}$$ are contained in a compact set of Teichmüller space, as well. Suppose by contradiction that the sequence of left metric $$h_{l_{n}}$$ of $$M_{n}$$ is not contained in a compact set. By Mess parameterisation (see Section 4, or [17]), the left metrics are related to the metrics $$m_{n}^{+}$$ and to the measured geodesic laminations $$\lambda_{n}^{+}$$ of the upper-boundary of the convex core by an earthquake:   hln=Eλn+l(mn+). Since $$h_{l_{n}}$$ is divergent, the sequence of measured laminations $$\lambda_{n}^{+}$$ is divergent, as well. In particular, this implies that $$\ell_{m_{n}^{+}}(\lambda_{n}^{+})$$ goes to infinity. Therefore, by Lemma 6.12, for every $$\epsilon>0$$ there exists $$n_{0}$$ such that the inequality $$\ell_{m_{n}^{-}}(\lambda^{+}_{n})\leq \epsilon\ell_{m_{n}^{+}}(\lambda^{+}_{n})$$ holds for $$n\geq n_{0}$$. From this we deduce a contradiction, because we prove that the inequality   ℓmn−(λn+)≤ϵℓmn+(λn+)     ∀ n≥n0 implies that the sequence $$m_{n}^{-}$$ is divergent, which contradicts what we proved in the previous paragraph. For instance, if $$m_{n}^{-}$$ were contained in a compact set of Teichmüller space, there would exist (using again Thurston’s asymmetric metric) a constant $$C>1$$ such that   ℓmn+(γ)ℓmn−(γ)≤C    ∀ n≥n0. By density this inequality must hold also for every measured geodesic lamination on $$S$$. But we have seen that for every $$\epsilon >0$$ we can find $$n_{0}$$ such that for every $$n \geq n_{0}$$, we have   ℓmn+(λn+)ℓmn−(λn+)≥1ϵ, thus obtaining a contradiction. A similar argument proves that also the sequence of right metrics $$h_{r_{n}}$$ must be contained in a compact set of $$\mathrm{Teich}(S)$$. Since the sequences of left and right metrics of $$M_{n}$$ converge, up to subsequence, we can concretely realise the corresponding subsequence $$M_{n}$$ as $$(S \times \mathbb{R}, h_{n})$$ such that $$h_{n}$$ converges in the $$C^{\infty}$$-topology to an Anti-de Sitter metric $$h_{\infty}$$ and each $$M_{n}$$ contains a future-convex space-like surface with embedding data $$(g^{-}, b^{-}_{n})$$ and a past-convex space-like surface with embedding data $$(g^{+}_{t_{n}}, b_{t_{n}}^{+})$$. The proof is then completed applying Corollary 6.5. ■ 7 Prescription of an isometric embedding and half-holonomy representation This section is dedicated to the proof of the following result about the existence of an $$AdS_{3}$$ manifold with prescribed left metric containing a convex space-like surface with prescribed induced metric: Proposition 7.1. Let $$g$$ be a metric on $$S$$ with curvature less than $$-1$$ and let $$h$$ be a hyperbolic metric on $$S$$. There exists a GHMC $$AdS_{3}$$ manifold $$M$$ with left metric isotopic to $$h$$ containing a past-convex space-like surface isometric to $$(S,g)$$. □ If we denote with   p1:Teich(S)×Teich(S)→Teich(S) the projection on to the left factor, Propostition 7.1 is equivalent to proving that the map $$p_{1}\circ\phi^{+}_{g}: I(S,g)^{+}\rightarrow \mathrm{Teich}(S)$$ is surjective. After showing that $$p_{1}\circ\phi^{+}_{g}$$ is proper (Corollary 7.4), this will follow from the fact that its degree (mod $$2$$) is non-zero. In order to prove properness of the map $$p_{1}\circ\phi^{+}_{g}$$, we will need the following well-known result about the behaviour of the length function while performing an earthquake. Lemma 7.2 (Lemma 7.1 in [6]). Given a geodesic lamination $$\lambda\in \mathcal{M}\mathcal{L}(S)$$ and a hyperbolic metric $$g\in \mathrm{Teich}(S)$$, let $$g'=E_{l}^{\lambda}(g)$$. Then for every closed geodesic $$\gamma$$ in $$S$$ the following estimate holds   ℓg(γ)+ℓg′(γ)≥λ(γ). □ Proposition 7.3. For every path of metrics $$\{g_{t}\}_{t\in [0,1]}$$ with curvature less than $$-1$$, the projection $$p_{1}: \Phi^{+}(W^{+}) \rightarrow \mathrm{Teich}(S)$$ is proper. □ Proof. Let $$h_{l}(g_{t_{n}}, b_{t_{n}})$$ be a convergent sequence of left metrics. We need to prove that the corresponding sequence of right metrics $$h_{r}(g_{t_{n}}, b_{t_{n}})$$ is convergent, as well. By hypothesis, $$(S,g_{t_{n}})$$ is isometrically embedded as past-convex space-like surface in each GHMC $$AdS_{3}$$ manifold $$M_{n}$$ parametrised by $$(h_{l}(g_{t_{n}}, b_{t_{n}}), h_{r}(g_{t_{n}}, b_{t_{n}}))$$. By Lemma 6.10 and Lemma 6.11, the metrics $$m_{n}^{+}$$ on the past-convex boundary of the convex core of $$M_{n}$$ are contained in a compact set of $$\mathrm{Teich}(S)$$. Moreover, by a result of Mess ([17]), the left metrics $$h_{l}(g_{t_{n}}, b_{t_{n}})$$, the metrics $$m_{n}^{+}$$ and the measured laminations on the convex core $$\lambda^{+}_{n}$$ are related by an earthquake   hl(gtn,btn)=Elλn+(mn+). Since $$h_{l}(g_{t_{n}}, b_{t_{n}})$$ is convergent, by Lemma 7.2, the sequence of measured laminations $$\lambda_{n}^{+}$$ must be contained in a compact set. Therefore, by continuity of the right earthquake   Er:Teich(S) ×ML(S)→Teich(S)(h,λ) ↦Erλ(h) the sequence   hr(gtn,btn)=Erλn+(mn+) is convergent, up to subsequences. ■ In particular, considering a constant path of metrics, we obtain the following: Corollary 7.4. The projection $$p_{1}:\phi^{+}_{g}(I(S,g)^{+}) \rightarrow \mathrm{Teich}(S)$$ is proper. □ Proposition 7.5. For every metric $$g$$ of curvature $$\kappa<-1$$, the map   p1∘ϕg+:I(S,g)+→Teich(S) is proper of degree $$1$$ mod $$2$$. □ Proof. Consider a path of metrics $$(g_{t})_{t\in [0,1]}$$ with curvature less than $$-1$$ connecting $$g=g_{0}$$ with a metric of constant curvature $$g_{1}$$. By Corollary 6.8 and Corollary 7.4, the maps $$p_{1}\circ\phi_{g_{0}}^{+}:I(S,g_{0})^{+}\rightarrow \mathrm{Teich}(S)$$ and $$p_{1}\circ\phi^{+}_{g_{1}}:I(S,g_{1})^{+}\rightarrow \mathrm{Teich}(S)$$ are proper and cobordant, hence they have the same degree (mod $$2$$). (This follows from Remark 5.1, Propositions 5.2 and 7.3). Thus, we can suppose that $$g$$ has constant curvature $$\kappa<-1$$. There exists a unique element in $$I(S,g)^{+}$$ such that $$h_{l}(g, b)=-\kappa g$$: a direct computation shows that $$b=\sqrt{-\kappa-1}E$$ works and uniqueness follows by the theory of landslides developed in [4]. We sketch here the argument and we invite the interested reader to consult the aforementioned article for more details. Pick $$\theta \in (0,\pi)$$ such that $$\kappa=-\frac{1}{\cos^{2}(\theta/2)}$$. The landslide   Leiθ1:Teich(S)×Teich(S) →Teich(S)(h,h∗) ↦h′ associates to a couple of hyperbolic metrics $$(h,h^{*})$$, the left metric of a GHMC $$AdS_{3}$$ manifold containing a space-like embedded surface with induced first fundamental form $$I=\cos^{2}(\theta/2)h$$ and third fundamental form $$III=\sin^{2}(\theta/2)h^{*}$$. It has been proved (Theorem 1.14 in [4]) that for every $$(h,h')\in \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$, there exists a unique $$h^{*}$$ such that $$L_{e^{i\theta}}^{1}(h,h^{*})=h'$$. Moreover, the shape operator $$b$$ of the embedded surface can be recovered by the formula (Lemma 1.9 in [4])   b=tan⁡(θ/2)B where $$B:TS \rightarrow TS$$ is the unique $$h$$-self-adjoint operator such that $$h^{*}=h(B\cdot, B\cdot)$$. Therefore, if we choose $$h=h'=-\kappa g$$, the uniqueness of the operator $$b$$ follows by the uniqueness of $$h^{*}$$ and $$B$$. Hence, the degree (mod $$2$$) of the map is $$1$$, provided $$-\kappa g$$ is a regular value. Let $$\dot{b}\in T_{b}I(S,g)^{+}$$ be a non-trivial tangent vector. We remark that, since elements of $$I(S,g)^{+}$$ are $$g$$-self-adjoint, Codazzi tensor of determinant $$-1-\kappa$$, the tangent space $$T_{b}I(S,g)^{+}$$ can be identified with the space of traceless, Codazzi, $$g$$-self-adjoint tensors. We are going to prove that the deformation induced on the left metric is non-trivial, as well. Let $$b_{t}$$ be a path in $$I(S,g)^{+}$$ such that $$b_{0}=b=\sqrt{-\kappa-1}E$$ and $$\frac{d}{dt}b_{t}=\dot{b}$$ at $$t=0$$. The complex structures induced on $$S$$ by the metrics $$h_{l}(g, b_{t})$$ are   Jt=(E+Jbt)−1J(E+Jbt) where $$J$$ is the complex structure induced by $$g$$. Taking the derivative of this expression at $$t=0$$ we get   J˙=2κ[E−−κ−1J]b˙ which is non-trivial in $$T_{-\kappa g}\mathrm{Teich}(S)$$ because, as explained in Theorem 1.2 of [10], the space of traceless and Codazzi operators in $$T_{J}\mathcal{A}$$ has trivial intersection with the kernel of the differential of the projection $$\pi: \mathcal{A} \rightarrow \mathrm{Teich}(S)$$, which sends a complex structure $$J$$ to its isotopy class. ■ In particular, for every smooth metric $$g$$ on $$S$$ with curvature less than $$-1$$, the map $$p_{1}\circ\phi^{+}_{g}: I(S,g)^{+}\rightarrow \mathrm{Teich}(S)$$ is surjective (a proper, non-surjective map has vanishing degree (mod $$2$$)) and we deduce Proposition 7.1. 8 Proof of the main result We have now all the ingredients to prove Theorem 4.2. As outlined in the Introduction, the first step consists of verifying that in one particular case, i.e. when we choose the metrics $$g'_{+}=-\frac{1}{\kappa}h$$ and $$g'_{-}=-\frac{1}{\kappa^{*}}h$$, where $$h$$ is any hyperbolic metric and $$\kappa^{*}=-\frac{\kappa}{\kappa+1}=\kappa=-2$$, the maps $$\phi^{+}_{g'_{+}}$$ and $$\phi^{-}_{g'_{-}}$$ have a unique transverse intersection. It is a standard computation to verify that $$b^{+}=E$$ and $$b^{-}=-E$$ are Codazzi operators corresponding to an isometric embedding of $$(S, g'_{+})$$ as a past-convex space-like surface and to an isometric embedding of $$(S, g'_{-})$$ as a future-convex space-like surface respectively into the GHMC $$AdS_{3}$$ manifold $$M$$ parametrised by $$(h,h) \in \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$. This manifold $$M$$ is unique due to the following: Theorem 8.1 (Theorem 1.15 in [5]). Let $$h_{+}$$ and $$h'_{-}$$ be hyperbolic metrics and let $$\kappa_{+}$$ and $$\kappa_{-}$$ be real numbers less than $$-1$$. There exists a GHMC $$AdS_{3}$$ manifold $$M$$ which contains an embedded future-convex space-like surface with induced metric $$\frac{1}{|\kappa_{-}|}h_{-}$$ and an embedded past-convex space-like surface with induced metric $$\frac{1}{|\kappa_{+}|}h_{+}$$. Moreover, if $$\kappa_{+}=-\frac{\kappa_{-}}{\kappa_{-}+1}$$, then $$M$$ is unique. □ We notice that $$M$$ is Fuchsian, that is, it is parametrised by a couple of isotopic metrics in Teichmüller space. A priori, there might be other isometric embeddings of $$(S, g'_{+})$$ as a past-convex space-like surface and of $$(S, g'_{-})$$ as a future-convex space-like surface into $$M$$ not equivalent to the ones found before. Actually, this is not the case due to the following result about isometric embeddings of convex surfaces into Fuchsian Lorentzian manifolds: Theorem 8.2 (Theorem 1.1 in [16]). Let $$(S,g)$$ be a Riemannian surface of genus $$\tau\geq 2$$ with curvature strictly smaller than $$-1$$. Let $$x_{0} \in \tilde{AdS}_{3}$$ be a fixed point. There exists an equivariant isometric embedding $$(f, \rho)$$ of $$(S,g)$$ into $$\tilde{AdS}_{3}$$ such that $$\rho$$ is a representation of the fundamental group of $$S$$ into the group $$\mathrm{Isom}(\tilde{AdS}_{3}, x_{0})$$ of isometries of $$\tilde{AdS}_{3}$$ fixing $$x_{0}$$. Such an embedding is unique modulo $$\mathrm{Isom}(\tilde{AdS}_{3}, x_{0})$$. □ As a consequence, if we denote with $$\Delta$$ the diagonal of $$\mathrm{Teich}(S)^{2}\times \mathrm{Teich}(S)^{2}$$, we have proved that   (ϕg+′+×ϕg−′−)−1(Δ)=(E,−E)∈I(S,g+′)+×I(S,g−′)−. We need to verify next that at this point the intersection   ϕg+′+(I(S,g+′)+)∩ϕg−′−(I(S,g−′)−) is transverse. Suppose by contradiction that the intersection is not transverse, then there exists a non-trivial tangent vector $$\dot{b}^{+} \in T_{E}I(S,g'_{+})^{+}$$ and a non-trivial tangent vector $$\dot{b}^{-} \in T_{-E}I(S,g'_{-})^{-}$$ such that   dϕg+′+(b˙+)=dϕg−′−(b˙−)∈ThTeich(S)×ThTeich(S). We recall that elements of $$T_{E}I(S,g'_{+})^{+}$$ can be represented by traceless, $$g'_{+}$$-self-adjoint, Codazzi operators. With this in mind, let us compute explicitly $$d\phi^{+}_{g'_{+}}(\dot{b}^{+})$$. Let $$b_{t}^{+}$$ be a smooth path in $$I(S,g'_{+})^{+}$$ such that $$b^{+}_{0}=E$$ and $$\frac{d}{dt}_{|_{t=0}}b^{+}_{t}=\dot{b}^{+}\ne 0$$. The complex structures induced on $$S$$ by the left metrics $$h_{l}(b_{t})$$ are   Jl+=(E+Jbt+)−1J(E+Jbt+), where $$J$$ is the complex structure of $$(S, g'_{+})$$. We compute now the derivative of this expression at $$t=0$$. Since the operators $$b_{t}$$ are $$g_{+}'$$-self-adjoint, $$Jb_{t}^{+}$$ is traceless, hence the Hamilton-Cayley equation reduces to $$(Jb_{t}^{+})^{2}+\det(Jb_{t}^{+})E=(Jb_{t}^{+})^{2}+E=0$$. We deduce that   (E+Jbt+)(E−Jbt+)=2E. Therefore, the variation of the complex structures induced by the left metrics is   J˙l+ =ddt|t=0Jl+=ddt|t=012(E−Jbt+)J(E+Jbt+) =12(−Jb˙+)J(E+J)+12(E−J)J2b˙+ =−(E−J)b˙+ where, in the last passage we used the fact that, since $$\dot{b}^{+}$$ is traceless and symmetric, the relation $$J\dot{b}^{+}=-\dot{b}^{+}J$$ holds. With a similar procedure we compute the variation of the complex structures of the right metrics and we obtain   J˙r+=(E+J)b˙+∈TJA. Noticing that $$\dot{J}_{l}^{+}$$ and $$\dot{J}_{r}^{+}$$ are both traceless Codazzi operators, the image of $$\dot{b}^{+}$$ under the differential $$d\phi^{+}_{g'_{+}}$$ is simply   dϕg+′+(b˙+)=(−(E−J)b˙+,(E+J)b˙+)∈ThTeich(S)×ThTeich(S) because, as explained in Theorem 1.2 of [10], the space of traceless and Codazzi operators in $$T_{J}\mathcal{A}$$ is in direct sum with the kernel of the differential of the projection $$\pi: \mathcal{A} \rightarrow \mathrm{Teich}(S)$$, which sends a complex structure $$J$$ to its isotopy class and gives an isomorphism between the space of traceless, Codazzi, self-adjoint tensors and $$T_{h}\mathrm{Teich}(S)$$. With a similar reasoning we obtain that   dϕg−′−(b˙−)=(−(E+J)b˙−,(E−J)b˙−)∈ThTeich(S)×ThTeich(S). By imposing that $$d\phi^{+}_{g'_{+}}(\dot{b}^{+})=d\phi^{-}_{g'_{-}}(\dot{b}^{-})$$ we obtain the linear system   {(−E+J)b˙+=−(E+J)b˙−(E+J)b˙+=(E−J)b˙−  which has solutions if and only if $$\dot{b}^{+}=\dot{b}^{-}=0$$. Therefore, the intersection is transverse and we can finally state that   ℑ(ϕg+′+,ϕg−′−)=1. Now we use the theory described in Section 5 to prove that an intersection persists under a deformation of one metric that fixes the other. Let $$g_{+}$$ and $$g_{-}$$ be two arbitrary metrics on $$S$$ with curvature less than $$-1$$. We will still denote with $$g'_{+}$$ and with $$g'_{-}$$ the metrics introduced in the previous paragraph with self-dual constant curvature and in the same conformal class. Consider two paths of metrics $$\{g_{+}^{t}\}_{t \in [0,1]}$$ and $$\{g_{-}^{t}\}_{t \in [0,1]}$$ with curvature less than $$-1$$ such that $$g_{+}^{0}=g_{+}$$, $$g_{+}^{1}=g'_{+}$$, $$g_{-}^{0}=g_{-}$$ and $$g_{-}^{1}=g'_{-}$$. We will first prove that   ϕg++(I(S,g+)+)∩ϕg−′−(I(S,g−′)−)≠∅. Suppose by contradiction that this intersection is empty. Then the map $$\phi_{g_{+}}^{+} \times \phi_{g'_{-}}^{-}$$ is trivially transverse to $$\Delta$$. Consider the manifold   W+=⋃t∈[0,1]I(S,gt+)+ and the map   Φ+×ϕg−′−:X=W+×I(S,g−′)−→(Teich(S))4=Y as defined in Section 6. By assumption the restriction of $$\Phi^{+}\times \phi_{g'_{-}}^{-}$$ to the boundary is transverse to $$\Delta$$ and by Proposition 6.13, the set $$D=(\Phi^{+}\times \phi_{g'_{-}}^{-})^{-1}(\Delta)$$ is compact. Let $$B$$ be the interior of a compact set containing $$D$$ and let $$C=(X \setminus B)\cup \partial X$$. By construction, $$\Phi^{+}\times \phi_{g_{-}}^{-}$$ is transverse to $$\Delta$$ along the closed set $$C$$. Applying Theorem 5.4, there exists a smooth map $$\Psi: X \rightarrow Y$$ which is transverse to $$\Delta$$ and which coincides with $$\Phi^{+}\times \phi_{g'_{-}}^{-}$$ on $$C$$. In particular, the value on the boundary remains unchanged and $$\Psi^{-1}(\Delta)$$ is still a compact set. By Proposition 5.3, the intersection number of the maps   ϕg+′+:I(S,g+′)+→Teich(S)×Teich(S)  and  ϕg++:I(S,g+)+→Teich(S)×Teich(S) with the map   ϕg−′−:I(S,g−)−→Teich(S)×Teich(S), as defined in Section 5, must be the same. This gives a contradiction, because   0=ℑ(ϕg++,ϕg−′−)≠ℑ(ϕg+′+,ϕg−′−)=1. So we have proved that $$\phi_{g_{+}}^{+}(I(S,g_{+})^{+}) \cap \phi^{-}_{g'_{-}}(I(S,g'_{-})^{-}) \ne \emptyset$$, but we do not know if the intersection is transverse. Repeating the above argument choosing the closed set $$C=(X\setminus B)\cup I(S,g'_{+})^{+}$$, we obtain that a perturbation $$\psi$$ of $$\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}}$$ which is transverse to $$\Delta$$ and coincides with $$\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}}$$ outside the interior of a compact set containing $$(\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}})^{-1}(\Delta)$$ has intersection number $$\Im(\psi,\Delta)=1$$. By Proposition 5.5, every perturbation of the map $$\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}}$$ obtained in this way has intersection number with $$\Delta$$ equal to $$1$$. This enables us to deform the metric $$g_{-}$$ without losing the intersection, by repeating a similar argument. Suppose by contradiction that   ϕg++(I(S,g+)+)∩ϕg−−(I(S,g−)−)=∅. Consider the manifold   W−=⋃t∈[0,1]I(S,gt−)− and the map   Φ−×ϕg++:X=W−×I(S,g+)+→(Teich(S))4=Y. By assumption the restriction of the map $$\Phi^{-}\times \phi_{g_{+}}^{+}$$ to the first boundary component $$X_{0}=I(S,g_{+})^{+}\times I(S,g_{-})^{-}$$ is transverse to $$\Delta$$ and by Proposition 6.13, the pre-image $$D=(\Phi^{-}\times \phi_{g_{+}}^{+})^{-1}(\Delta)$$ is a compact set. Let $$B$$ be the interior of a compact set containing $$D$$ and let $$C=(X \setminus B)\cup X_{0}$$. By construction, $$\Phi^{-}\times \phi_{g_{+}}^{+}$$ is transverse to $$\Delta$$ along the closed set $$C$$. Applying Theorem 5.4, there exists a smooth map $$\Psi: X \rightarrow Y$$ which is transverse to $$\Delta$$ and which coincides with $$\Phi^{-}\times \phi_{g_{+}}^{+}$$ on $$C$$. In particular, the value on the boundary $$X_{0}$$ remains unchanged and $$\Psi^{-1}(\Delta)$$ is still a compact set. Moreover, the value of $$\Psi$$ on the other boundary component is a perturbation of $$\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}}$$ which is transverse to $$\Delta$$ and coincides with $$\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}}$$ outside the interior of a compact set containing $$(\phi_{g_{+}}^{+}\times \phi^{-}_{g'_{-}})^{-1}(\Delta)$$. Hence, by Proposition 5.5 and by Proposition 5.3, the intersection number $$\Im(\phi^{+}_{g_{+}}, \phi_{g_{-}}^{-})$$ must be equal to $$1$$, thus giving a contradiction. Remark 8.3. It might be possible to prove the uniqueness of this intersection by applying Proposition 5.6. To this aim, it would be necessary to show that for every couple of metrics $$g_{+}$$ and $$g_{-}$$ with curvature strictly smaller than $$-1$$, the functions $$\phi_{g_{+}}^{+}: I(S,g_{+})^{+} \rightarrow \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$ and $$\phi_{g_{-}}^{-}: I(S,g_{-})^{-} \rightarrow \mathrm{Teich}(S) \times \mathrm{Teich}(S)$$ are transverse. □ Funding The project benefited from the fundings of the Doctoral Program of the University of Luxembourg. Acknowledgements This is a part of a PhD project I have been doing under the supervision of Prof. Jean-Marc Schlenker. I would like to thank him for his patient guidance and invaluable support throughout this work. I am also grateful to Nicolas Tholozan and Daniel Monclair for our fruitful conversations about the subject. Communicated by Prof. Igor Rivin References [1] Barbot T. Bonsante F. Danciger J. Goldman W. M. Guéritaud F. Kassel F. Krasnov K. Schlenker J.-M. and Zeghib. A. Some Questions on Anti-De Sitter Geometry , 2012, arXiv:1205.6103. [2] Barbot T. Béguin F. and Zeghib. A. “Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on $$AdS_{3.}$$” Geometriae Dedicata  126 ( 2007): 71– 129. Google Scholar CrossRef Search ADS   [3] Barbot T. Béguin F. and Zeghib. A. “Prescribing Gauss curvature of surfaces in $$3$$-dimensional spacetimes. Application to the Minkowski problem in the Minkowsky space.” Annales de l’Insitut Fourier  61, no. 2 ( 2011): 511– 91. Google Scholar CrossRef Search ADS   [4] Bonsante F. Mondello G. and Schlenker. J.-M. “A cyclic extension of the earthquake flow I.” Geometry and Topology  17 ( 2013): 157– 234. Google Scholar CrossRef Search ADS   [5] Bonsante F. Mondello G. and Schlenker. J.-M. “A cyclic extension of the earthquake flow II.” Annales Scientifiques de l’ENS  48 ( 2015): 811– 59. [6] Bonsante F. and Schlenker. J.-M. “AdS manifolds with particles and earthquakes on singular surfaces.” Geometric and Functional Analysis  19, no. 1 ( 2009): 41– 82. Google Scholar CrossRef Search ADS   [7] Diallo B. “Prescribing metrics on the boundary of convex cores of globally hyperbolic maximal compact ADS $$3$$-manifolds.” 2013, arXiv:1303.7406. [8] Epstein D. B. A. and Marden. A. Convex Hulls in Hyperbolic Spaces, a Theorem of Sullivan, and Measured Pleated Surfaces.  Analytical and geometric aspects of hyperbolic space, volume 111 of L.M.S Lecture Notes Series, Warwick and Durham, 1986. [9] Fathi A. Laudenbach F. and Poenaru. V. Travaux de Thurston sur les Surfaces.  Astérisque 66– 7, Société Mathématique de France, 1979. [10] Fischer A. E. and Tromba. A. J. “A new proof that Teichmüller space is a cell.” Transactions of the American Mathematical Society  303, no. 1 ( 1987): 257– 62. [11] Gerhardt C. “H-surfaces in Lorentzian manifolds.” Communications in Mathematical Physics  89, no. 4 ( 1983): 523– 53. Google Scholar CrossRef Search ADS   [12] Goldman W. “Topological components of spaces of representations.” Inventiones Mathematicae  93, no. 3 ( 1988): 557– 607. Google Scholar CrossRef Search ADS   [13] Guillemin V. and Pollack. A. Differential Topology.  Prentice-Hall, Englewood Cliffs, New Jersey, 1974. [14] Krasnov K. and Schlenker. J.-M. “Minimal surfaces and particles in $$3$$-manifolds.” Geometriae Dedicata  126 ( 2007): 187– 254. Google Scholar CrossRef Search ADS   [15] Labourie F. “Métriques prescrites sur le bord des variété hyperboliques de dimension 3.” Journal of Differential Geometry  35, no. 3 ( 1992): 609– 26. Google Scholar CrossRef Search ADS   [16] Labourie F. and Schlenker. J.-M. “Surfaces convexes fuchsiennes dans les espaces lorentziens àcourbure constante.” Mathematische Annalen  316 ( 2000): 465– 83. Google Scholar CrossRef Search ADS   [17] Mess G. “Lorentz spacetimes of constant curvature.” Geometriae Dedicata  126 ( 2007): 3– 45. Google Scholar CrossRef Search ADS   [18] Schlenker J.-M. “Surfaces convexes dans des espaces lorentziens àcourbure constante.” Communications in Analysis and Geometry  4, no. 1–2 ( 1996): 285– 331. Google Scholar CrossRef Search ADS   [19] Schlenker J.-M. “Hyperbolic manifolds with convex boundary.” Inventiones Mathematicae  163 ( 2006): 109– 69. Google Scholar CrossRef Search ADS   [20] Thurston W. P. “Minimal stretch maps between hyperbolic surfaces.” 1998, arXiv:math/9801039. © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]

Lê, Thang T Q

doi: 10.1093/imrn/rnw280pmid: N/A

Abstract We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces. 1 Introduction 1.1 Positivity of Kauffman bracket skein algebras Let $${\mathbb Z[q^{\pm 1}]}$$ be the ring of Laurent polynomial in an indeterminate $$q$$ with integer coefficients. Suppose $$\Sigma$$ is an oriented surface. The Kauffman bracket skein algebra $$\mathscr S_{\mathbb Z[q^{\pm 1}]}(\Sigma)$$ is a $${\mathbb Z[q^{\pm 1}]}$$-algebra generated by unoriented framed links in $$\Sigma \times [0,1]$$ subject the Kauffman skein relations [6] which are recalled in Section 2. The skein algebras were introduced by Przyticki [12] and independently Turaev [15] in an attempt to generalize the Jones polynomial to links in general three-manifolds, and have connections and applications to many interesting objects like character varieties, topological quantum field theories, quantum invariants, quantum Teichmüller spaces, and many others (see, e.g., [1, 3, 9, 10, 12–14]). The quotient $$\mathscr S_\mathbb Z(\Sigma)= \mathscr S_{\mathbb Z[q^{\pm 1}]}(\Sigma)/(q-1)$$, which is a $$\mathbb Z$$-algebra, is called the skein algebra at $$q=1$$. For $$R=\mathbb Z$$ let $$R_+ = \mathbb Z_+$$, the set of non-negative integers, and for $$R={\mathbb Z[q^{\pm 1}]}$$ let $$R_+= \mathbb Z_+[q^{\pm1}]$$, the set of Laurent polynomials with non-negative coefficients, that is, the set of $$\mathbb Z_+$$-linear combinations of integer powers of $$q$$. An $$R$$-algebra, where $$R={\mathbb Z[q^{\pm 1}]}$$ or $$R=\mathbb Z$$, is said to be positive if it is free as an $$R$$-module and has a basis $$\mathcal B$$ in which the structure constants are in $$R_+$$, that is, for any $$b,b', b''\in \mathcal B$$, the $$b$$-coefficient of the product $$b'b''$$ is in $$R_+$$. Such a basis $$\mathcal B$$ is called a positive basis of the algebra. The important positivity conjecture of Fock and Goncharov [5] states that $$\mathscr S_R(\Sigma)$$ is positive for the case $$R={\mathbb Z[q^{\pm 1}]}$$ and $$R=\mathbb Z$$. The first case obviously implies the second. The second case, on the positivity over $$\mathbb Z$$, was proved by Thurston [14]. Moreover, Thurston makes precise the positivity conjecture by specifying the positive basis as follows. A normalized sequence of polynomials over $$R$$ is a sequence $${\mathbf P}=(P_n(t)_{n\ge 0})$$, such that for each $$n$$, $$P_n(t)\in R[t]$$ is a monic polynomial of degree $$n$$. Every normalized sequence $${\mathbf P}$$ over $$R$$ gives rise in a canonical way a basis $$\mathcal B_{\mathbf P}$$ of the $$R$$-module $$\mathscr S_R(\Sigma)$$, see Section 2. Definition 1. A sequence $${\mathbf P}=(P_n(t)_{n\ge 0})$$ of polynomials in $$R[t]$$ is positive over $$R$$ if it is normalized and the basis $$\mathcal B_{\mathbf P}$$ is a positive basis of $$\mathscr S_R(\Sigma)$$, for any oriented surface $$\Sigma$$. □ Then Thurston proved the following. Theorem 1.1. [14] The sequence $$(T_n)$$ of Chebyshev’s polynomials is positive over $$R=\mathbb Z$$. □ Here in this paper, Chebyshev’s polynomials $$T_n(t)$$ are defined recursively by   T0(t)=1, T1(t)=t,T2(t)=t2−2,Tn(t)=tTn−1(t)−Tn−2(t)for  n≥3. If $$T_0$$ is the constant polynomial 2, then the above polynomials are Chebyshev’s polynomials of type 1. Here we have to set $$T_0=1$$ since by default, all normalized sequence begins with 1. Thurston suggested the following conjecture, making precise the positivity conjecture. Conjecture 1. The sequence $$(T_n)$$ of Chebyshev’s polynomials is positive over $${\mathbb Z[q^{\pm 1}]}$$. □ We will show Theorem 1.2. Let $$R=\mathbb Z$$ or $$R={\mathbb Z[q^{\pm 1}]}$$ and $${\mathbf P}= (P_n(t)_{n\ge 0})$$ be a sequence of polynomials in $$R[t]$$. If $${\mathbf P}$$ is positive over $$R$$, then $$P_n(t)$$ is an $$R_+$$-linear combination of $$T_0(t), T_1(t), \dots, T_n(t)$$. Besides, $$P_1(t)=T_1(t)=t$$. □ For the case $$R=\mathbb Z$$, this result complements well Theorem 1.1, as they together claim that the sequence of Chebyshev polynomials is the minimal one in the set of positive sequence over $$\mathbb Z$$. For $$R={\mathbb Z[q^{\pm 1}]}$$, our result says that the sequence of Chebyshev polynomial should be the minimal positive sequence. A stronger version of Theorem 1.2 is proved in Section 3. Remark 1.1. The positivity conjecture considered here is different from the one discussed in [11], which claims that every element of a certain basis is a $$\mathbb Z_+$$-linear combination of monomials of cluster variables. □ 1.2 Marked surfaces A marked surface is a pair $$(\Sigma,\mathcal P)$$, where $$\Sigma$$ is a compact-oriented surface with (possibly empty) boundary $$\partial \Sigma$$ and $$\mathcal P$$ is a finite set in $$\partial \Sigma$$. Muller [10] defined the skein algebra $$\mathscr S_R(\Sigma,\mathcal P)$$, extending the definition from surfaces to marked surfaces. When $$R=\mathbb Z$$, this algebra had been known earlier, and actually, the above-mentioned result of Thurston (Theorem 1.1) was proved also for the case of marked surfaces. However, there are two types of basis generators of the skein algebras, namely loops and arcs, and one needs two normalized sequences of polynomials $${\mathbf P}$$ and $${\mathbf Q}$$ to define an $$R$$-basis of the skein algebra $$\mathscr S_R(\Sigma,\mathcal P)$$. Here $${\mathbf P}$$ is applicable to loops, and $${\mathbf Q}$$ is applicable to arcs, see Section 4. A pair of sequences of polynomials $$({\mathbf P},{\mathbf Q})$$ are positive over $$R$$ if they are normalized and the basis they generate is positive for any marked surface. Thurston result says that with $${\mathbf P}= (T_n)$$, the sequence of Chebyshev polynomials, and $${\mathbf Q}=(Q_n)$$ defined by $$Q_n(t)=t^n$$, the pair $$({\mathbf P},{\mathbf Q})$$ are positive over $$\mathbb Z$$. We obtained also an extension of Theorem 1.2 to the case of marked surface as follows. Theorem 1.3. Let $$R=\mathbb Z$$ or $$R={\mathbb Z[q^{\pm 1}]}$$. Suppose a pair $$({\mathbf P},{\mathbf Q})$$ of sequences of polynomials in $$R[t]$$, $${\mathbf P}= (P_n(t)_{n\ge 0})$$ and $${\mathbf Q}= (Q_n(t)_{n\ge 0})$$, are positive over $$R$$. Then $$P_n(t)$$ is an $$R_+$$-linear combination of $$T_0(t), T_1(t), \dots, T_n(t)$$ and $$Q_n(t)$$ is an $$R_+$$-linear combination of $$1, t, \dots, t^n$$. Moreover, $$P_1(t)=Q_1(t)=t$$. □ 1.3 Plan of the paper In Section 2 we recall basic facts about the Kauffman bracket skein algebras of surfaces. We present the proofs of Theorems 1.2 and 1.3 in, respectively, Sections 3 and 4. 2 Skein Algebras We recall here basic notions concerning the Kauffman bracket skein algebra of a surface. 2.1 Ground ring Throughout the paper we work with a ground ring $$R$$ which is more general than $$\mathbb Z$$ and $${\mathbb Z[q^{\pm 1}]}$$. We will assume that $$R$$ is a commutative domain over $$\mathbb Z$$ containing an invertible element $$q$$ and a subset $$R_+$$ such that $$R_+$$ is closed under addition and multiplication, $$q, q^{-1} \in R_+$$, and $$R_+ \cap (- R_+)= \{0\}$$. For example, we can take $$R= \mathbb Z$$ with $$q=1$$ and $$R_+= \mathbb Z_+$$, or $$R={\mathbb Z[q^{\pm 1}]}$$ with $$R_+= \mathbb Z_+[q^{\pm 1}]$$. The reader might think of $$R$$ as one of these two rings. 2.2 Kauffman bracket skein algebra Suppose $$\Sigma$$ is an oriented surface. The Kauffman bracket skein module$$\mathscr S_R(\Sigma)$$ is the $$R$$-module freely spanned by isotopy classes of non-oriented framed links in $$\Sigma \times [0,1]$$ modulo the skein relation and the trivial loop relation described in Figure 1. In all figures, framed links are drawn with blackboard framing. More precisely, the trivial loop relation says if $$L$$ is a loop bounding a disk in $$\Sigma\times[0,1]$$ with framing perpendicular to the disk, then $$L= -q^2 -q^{-2}.$$ And the skein relation says   L=qL++q−1L− if $$L, L_+, L_-$$ are identical except in a ball in which they look like in Figure 2. Fig. 1. View largeDownload slide Skein relation (left) and trivial loop relation (right). Fig. 1. View largeDownload slide Skein relation (left) and trivial loop relation (right). Fig. 2. View largeDownload slide From left to right: $$L, L_+, L_-$$. Fig. 2. View largeDownload slide From left to right: $$L, L_+, L_-$$. For future reference, we say that the diagram $$L_+$$ (resp. $$L_-$$) in Figure 2 is obtained from $$L$$ by the positive (resp. negative) resolution of the crossing. The $$R$$-module $$\mathscr S_R(\Sigma)$$ has an algebra structure where the product of two links $$\alpha_1$$ and $$\alpha_2$$ is the result of stacking $$\alpha_1$$ atop $$\alpha_2$$ using the cylinder structure of $$\Sigma \times [0,1]$$. Over $$R=\mathbb Z$$, the skein algebra $$\mathscr S_R(\Sigma)$$ is commutative and is closely related to the $$SL_2(\mathbb C)$$-character variety of $$\Sigma$$ (see [1, 2, 13, 15]). Over $$R={\mathbb Z[q^{\pm 1}]}$$, $$\mathscr S_R(\Sigma)$$ is not commutative in general and is closely related to quantum Teichmüller spaces [4]. 2.3 Bases We will consider $$\Sigma$$ as a subset of $$\Sigma\times [0,1]$$ by identifying $$\Sigma$$ with $$\Sigma \times \{1/2\}$$. As usual, links in $$\Sigma$$, which are closed one-dimensional non-oriented submanifolds of $$\Sigma$$, are considered up to ambient isotopies of $$\Sigma$$. A link $$\alpha$$ in $$\Sigma$$ is an essential if it has no trivial component, that is, a component bounding a disk in $$\Sigma$$. By convention, the empty set is considered an essential link. The framing of a link is vertical if at every point $$x$$, the framing is parallel to $$x \times [0,1]$$, with the direction equal to the positive direction of $$[0,1]$$. By [13, Theorem 5.2], $$\mathscr S_R(\Sigma)$$ is free over $$R$$ with basis the set of all isotopy classes of essential links in $$\Sigma$$ with vertical framing. This basis can be parameterized as follows. An integer lamination of $$\Sigma$$ is an unordered collection $$\mu=(n_i,C_i)_{i=1}^m$$, where each $$n_i$$ is a positive integer, each $$C_i$$ is a non-trivial knot in $$\Sigma$$, no two $$C_i$$ intersect, no two $$C_i$$ are ambient isotopic. For each integer lamination $$\mu$$, define an element $$b_\mu\in \mathscr S_R(\Sigma)$$ by   bμ=∏i(Ci)ni. Then the set of all $$b_\mu$$, where $$\mu$$ runs the set of all integer laminations including the empty one, is the above mentioned basis of $$\mathscr S_R(\Sigma)$$. Suppose $${\mathbf P}=(P_n(t)_{n\ge 0})$$ is a normalized sequence of polynomials in $$R[t]$$. Then we can twist the basis element $$b_\mu$$ by $${\mathbf P}$$ as follows. Let   bμ,P:=∏iPni(Ci). As $$\{ P_n(t)\} \mid n \in \mathbb Z_+ \}$$, just like $$\{ z^n\}$$, is a basis of $$R[t]$$, the set $$\mathcal B_{\mathbf P}$$ of all $$b_{\mu,{\mathbf P}}$$, when $$\mu$$ runs the set of all integer laminations, is a free $$R$$-basis of $$\mathscr S_R(\Sigma)$$. 3 Theorem 1.2 and Its Stronger Version We present here the proof of a stronger version of Theorem 1.2, using a result from [7] which we recall first. 3.1 Skein algebra of the annulus Let $${\mathbb A} \subset \mathbb R^2$$ be the annulus $${\mathbb A}= \{ x \in \mathbb R^2, 1 \le | x| \le 2\}$$. Let $$z$$ be the core of the annulus defined by $$z = \{ {x}, |x | = 3/2\}$$. It is easy to show that, as an algebra, $$ \mathscr S_R({\mathbb A}) = R[z]$$. Let $$p_1= (0,1)\in \mathbb R^2$$ and $$p_2=(0,2)\in \mathbb R^2$$, which are points in $$\partial {\mathbb A}$$. Then $${\mathbb A_{io}}=({\mathbb A}, \{p_1, p_2\})$$ is an example of a marked surface. See Figure 3, which also depicts the arcs $$\theta_0$$, $$\theta_{-1}$$, $$\theta_{2}$$. Here $$\theta_0$$ is the straight segment $$p_1p_2$$, and $$\theta_n$$, for $$n\in \mathbb Z$$, is an arc properly embedded in $${\mathbb A}$$ beginning at $$p_1$$ and winding clockwise (resp. counterclockwise) $$|n|$$ times if $$n\ge 0$$ (resp. $$n < 0$$) before getting to $$p_2$$. Fig. 3. View largeDownload slide Marked annulus $${\mathbb A_{io}}$$, arcs $$\theta_0$$, $$\theta_{-1}$$, $$\theta_{2}$$, and element $$\theta_0\bullet z.$$ Fig. 3. View largeDownload slide Marked annulus $${\mathbb A_{io}}$$, arcs $$\theta_0$$, $$\theta_{-1}$$, $$\theta_{2}$$, and element $$\theta_0\bullet z.$$ A $${\mathbb A_{io}}$$-arc is a proper embedding of the interval $$[0,1]$$ in $${\mathbb A} \times [0,1]$$ equipped with a framing such that one end point is in $$p_1 \times [0,1]$$ and the other is in $$p_2 \times [0,1]$$, and the framing is vertical at both end points. A $${\mathbb A_{io}}$$-tangle is a disjoint union of a $${\mathbb A_{io}}$$-arc and a (possibly empty) framed link in $${\mathbb A} \times [0,1]$$. Isotopy of $${\mathbb A_{io}}$$-tangles is considered in the class of $${\mathbb A_{io}}$$-tangles. Let $$\mathscr S'_R({\mathbb A_{io}})$$ be the $$R$$-module spanned by isotopy classes of $${\mathbb A_{io}}$$-tangles modulo the usual skein relation and the trivial knot relation. As usual, each $$\theta_n$$ is equipped with the vertical framing, and is considered as an element of $$\mathscr S'_R({\mathbb A_{io}})$$. For $$\alpha\in \mathscr S_R({\mathbb A})$$ let $$\theta_0 \bullet \alpha\in \mathscr S'_R({\mathbb A_{io}})$$ be the element obtained by placing $$\theta_0$$ on top of $$\alpha$$, see Figure 3 for an example. In [7] we proved the following. Proposition 3.1. For any integer $$n \ge 1$$, we have   θ0∙Tn(z)=qnθn+q−nθ−n. (1) □ 3.2 Stronger version of Theorem 3.2 We say that a sequence $${\mathbf P}=(P_n)$$ of normalized polynomials in $$R[t]$$ is positive for $$\Sigma$$ over $$R$$ if $$\mathcal B_{\mathbf P}$$ is a positive basis for $$\mathscr S_R(\Sigma)$$. Thus, $${\mathbf P}=(P_n)$$ is positive if and only if it is positive for any oriented surface. Theorem 1.2 follows from the following stronger statement. Theorem 3.2. Suppose $${\mathbf P}=(P_n(t))$$ is positive for a non-planar oriented surface $$\Sigma$$. Then for every $$n \ge 0$$, $$P_n(t)$$ is an $$R_+$$-linear combination of $$T_k(t)$$ with $$k \le n$$. Besides, $$P_1(t)=t$$. □ Proof. Since $$\Sigma$$ is non-planar, there are two simple closed curves $$z,z'\subset \Sigma$$ which intersect transversally at one point. We identify a small tubular neighborhood of $$z$$ with the annulus $${\mathbb A}$$ such that $$z'\cap {\mathbb A}$$ is the segment $$p_1 p_2$$, see Figure 4. Note that $$z$$ and $$z'$$, as homology classes in $$H_1(\Sigma,\mathbb Z)$$, are linearly independent over $$\mathbb Z$$. By definition $$P_1(t)=t+a$$, $$a \in R$$. Fix an integer $$n\ge 1$$. There are $$c_k \in R$$ such that   Pn(t)=∑k=0nckTk(t). Let $$z_{1,k}$$ be the curve $$(z' \setminus {\mathbb A})\cup \theta_k$$. We have   P1(z′)Pn(z)=aPn(z)+z′∑k=0nckTk(z)=aPn(z)+c0z′+∑k=1nckz′Tk(z)=aPn(z)+c0z′+∑k=1nck(qkz1,k+ckq−kz1,−k), (2) where the last equality follows from Proposition 3.1. Using homology, one sees that in the collection $$\{ z, z', z_{1,k}, z_{1,-k} \mid k \ge 1\}$$ every curve is a non-trivial knot in $$\Sigma$$, and any two of them are non-isotopic. Rewriting $$t = P_1(t) -a$$, we have   P1(z′)Pn(z)=aPn(z)+c0(P1(z′)−a)+∑k=1nck[qk(P1(z1,k)−a)+q−k(P1(z1,−k)−a)]=aPn(z)+c0P1(z′)+∑k=1nck[qk(P1(z1,k))+q−k(P1(z1,−k))]−ac0−∑k=1nack(qk+q−k). By the positivity, the coefficients of $$P_1(z')$$, $$P_n(z)$$, $$P_1(z_{1,k})$$ (for every $$k \ge 1$$) and the constant coefficient are in $$R_+$$. This shows that $$a, c_k \in R_+$$ for every $$k$$, and   d:=−ac0−∑k=1nack(qk+q−k)∈R+. Note that $$-d = a( c_0 + \sum_{k=1}^n c_k (q^k + q^{-k})) \in R_+$$, since $$a, c_k \in R_+$$. Thus, both $$d$$ and $$-d$$ are in $$R_+$$. Then $$d=0$$ by the assumption on $$R$$. It follows that   a(c0+∑k=1nck(qk+q−k))=0. (3) Claim. Suppose $$x,y\in R_+$$ such that $$x+y=0$$. Then $$x=y=0$$. Proof of Claim. We have $$x=-y$$. Thus, $$y$$ and $$-y=x$$ are in $$ R_+$$, implying $$y=0$$. Then $$x=0$$. The claim means that if $$x,y \in R_+$$ and $$x\neq 0$$, then $$x+y \neq 0$$. In particular, $$q^n+q^{-n} \neq 0$$. Note that $$c_n=1$$ because $$P_n(t)$$ is a monic polynomial. All the summands in the sum   c0+∑k=1nck(qk+q−k) are in $$R_+$$, and the summand with $$k=n$$ is non-zero. Hence the above sum is non-zero. Since $$R$$ is a domain, from (3) we conclude that $$a=0$$. This completes the proof of the theorem. ■ Fig. 4. View largeDownload slide The union $${\mathbb A} \cup z'$$. The bold arc is $$z'\setminus {\mathbb A}$$. Fig. 4. View largeDownload slide The union $${\mathbb A} \cup z'$$. The bold arc is $$z'\setminus {\mathbb A}$$. 4 Marked Surfaces 4.1 Skein algebra of marked surfaces Suppose $$(\Sigma,\mathcal P)$$ is a marked surface, that is, $$\mathcal P\subset \partial \Sigma$$ is a finite set. A framed 3D $$\mathcal P$$-tangle $$\alpha$$ in $$\Sigma\times [0,1]$$ is a framed proper embedding of a one-dimensional non-oriented compact manifold into $$\Sigma\times [0,1]$$ such that $$\partial \alpha \subset \mathcal P \times [0,1]$$, and the framing at every boundary point of $$\alpha$$ is vertical. Two framed 3D $$\mathcal P$$-tangles are isotopic if they are isotopic through the class of framed 3D $$\mathcal P$$-tangles. Just like the link case, a framed 3D $$\mathcal P$$-tangle $$\alpha$$ is depicted by its diagram on $$\Sigma$$, with vertical framing on the diagram. Here a diagram of $$\alpha$$ is a projection of $$\alpha$$ on to $$\Sigma$$ in general position, with an order of strands at every crossing. Crossings come in two types. If the crossing is in $$\Sigma \setminus \mathcal P$$, then it is a usual double point (with usual over/under information). If the crossing is a point in $$\mathcal P$$, there may be two or more number of strands, which are ordered. The order indicates which strand is above which. When there are two strands, the lower one is depicted by a broken line, see, for example, Figure 5. Fig. 5. View largeDownload slide Trivial arc relation: if a framed 3D $$\mathcal P$$-tangle $$\alpha$$ has a trivial arc, then $$\alpha=0$$. Fig. 5. View largeDownload slide Trivial arc relation: if a framed 3D $$\mathcal P$$-tangle $$\alpha$$ has a trivial arc, then $$\alpha=0$$. Let $$\mathscr S_R(\Sigma,\mathcal P)$$ be the $$R$$-module spanned by the set of isotopy classes of framed 3D $$\mathcal P$$-tangles in $$\Sigma\times [0,1]$$ modulo the skein relation, the trivial knot relation (Figure 1), and the trivial arc relation of Figure 5. The following relation holds, see [8]. Proposition 4.1. In $$\mathscr S(\Sigma,\mathcal P)$$, the reordering relation depicted in Figure 6 holds. □ Fig. 6. View largeDownload slide Reordering relation. Fig. 6. View largeDownload slide Reordering relation. Again one defines the product $$\alpha_1 \alpha_2$$ of two skein elements $$\alpha_1$$ and $$\alpha_2$$ by stacking $$\alpha_1$$ above $$\alpha_2$$. This makes $$\mathscr S_R(\Sigma,\mathcal P)$$ an $$R$$-algebra, which was first defined by Muller [10]. 4.2 Basis for $$\mathscr S_R(\Sigma,\mathcal P)$$ A $$\mathcal P$$-arc is an immersion $$\alpha: [0,1]\to \Sigma$$ such that $$\alpha(0), \alpha(1)\in \mathcal P$$ and the restriction of $$\alpha$$ on to $$(0,1)$$ is an embedding into $$\Sigma \setminus \mathcal P$$. A $$\mathcal P$$-knot is an embedding of $$S^1$$ into $$\Sigma \setminus \mathcal P$$. A $$\mathcal P$$-knot or a $$\mathcal P$$-arc is trivial if it bounds a disk in $$\Sigma$$. Two $$\mathcal P$$-arcs (resp. $$\mathcal P$$-knots) are $$\mathcal P$$-isotopic if they are isotopic in the class of $$\mathcal P$$-arcs (resp. $$\mathcal P$$-knots). A $$\mathcal P$$-arc is called boundary if it can be isotoped into $$\partial \Sigma$$, otherwise it is called inner. We consider every $$\mathcal P$$-arc and every $$\mathcal P$$-knot as an element of $$\mathscr S_R(\Sigma,\mathcal P)$$ by equipping it with the vertical framing. In the case when the two ends of a $$\mathcal P$$-arc are the same point $$p\in\mathcal P$$, we order the left strand to be above the right one. An integer $$\mathcal P$$-lamination of $$\Sigma$$ is an unordered collection $$\mu=(n_i,C_i)_{i=1}^m$$, where each $$n_i$$ is a positive integer, each $$C_i$$, called a component of $$\mu$$, is a non-trivial $$\mathcal P$$-knot or a non-trivial $$\mathcal P$$-arc, no two $$C_i$$ intersect in $$\Sigma\setminus \mathcal P$$, no two $$C_i$$ are $$\mathcal P$$-isotopic. In an integer $$\mathcal P$$-lamination $$\mu=(n_i,C_i)_{i=1}^m$$, a knot component commutes with any other component (in the algebra $$\mathscr S_R(\Sigma,\mathcal P)$$), while a $$\mathcal P$$-arc component $$C$$$$q$$-commutes with any other component $$C'$$ in the sense that $$CC'= q^k C'C$$ for a certain $$k\in \mathbb Z$$. The $$q$$-commutativity follows from Proposition 4.1. Hence the element   bμ=∏i(Ci)ni (4) is defined up to a factor $$q^k, k\in \mathbb Z$$. To make $$b_\mu$$ really well-defined, we will fix once and for all a total order on the set of all $$\mathcal P$$-arcs in $$\Sigma$$. Now define $$b_\mu$$ by (4), where the product is taken in this order. To simplify some proofs, we further assume that in the total order any boundary $$\mathcal P$$-arc comes before any inner $$\mathcal P$$-arc, although this is not necessary. It follows from [10, Lemma 4.1] that the set of all $$b_\mu$$, where $$\mu$$ runs the set of all integer $$\mathcal P$$-laminations including the empty one, is the a free $$R$$-basis of $$\mathscr S_R(\Sigma,\mathcal P)$$. Suppose $${{\mathbf P}}=(P_n)$$ and $${\mathbf Q}=(Q_n)$$ are normalized sequences of polynomials in $$R[t]$$. Define   bμ,P,Q=∏iFni(Ci), (5) where the product is taken in the above-mentioned order, and $$F_{n_i}=P_{n_i} $$ if $$C_i$$ is a knot, $$F_{n_i}= Q_{n_i}$$ if $$C_i$$ is an inner $$\mathcal P$$-arc, $$F_{n_i}(C_i) = (C_i)^{n_i}$$ if $$C_i$$ is a boundary $$\mathcal P$$-arc. Then the set $$\mathcal B_{{{\mathbf P}}, {\mathbf Q}}$$ of all $$b_{\mu,{{\mathbf P}}, {\mathbf Q}}$$, where $$\mu$$ runs the set of all integer $$\mathcal P$$-laminations, is a free $$R$$-basis of $$\mathscr S_R(\Sigma,\mathcal P)$$. A pair of sequences of polynomials $$({{\mathbf P}},{\mathbf Q})$$ are positive over $$R$$ if they are normalized and the basis $$\mathcal B_{{{\mathbf P}}, {\mathbf Q}}$$ is positive for any marked surface. 4.3 Positive basis in quotients The following follows right away from the definition. Lemma 4.2. Let $$\mathcal B$$ be a positive basis of an $$R$$-algebra $$A$$ and $$I\subset A$$ be an ideal of $$A$$, with $$\pi: A \to A/I$$ the natural projection. Assume that $$I$$ respects the base $$\mathcal B$$ in the sense that $$I$$ is freely $$R$$-spanned by $$I \cap \mathcal B$$. Then $$\pi(\mathcal B\setminus I)$$ is a positive basis of $$A/I$$. □ 4.4 Ideal generated by boundary arcs Lemma 4.3. Suppose $$\alpha_1,\dots,\alpha_l$$ are boundary $$\mathcal P$$-arcs, and $$I=\sum_{i=1}^l \alpha_i \mathscr S_R(\Sigma,\mathcal P)$$. (a) The set $$I$$ is a two-sided ideal of $$\mathscr S_R(\Sigma,\mathcal P)$$. (b) For any normalized sequences $${{\mathbf P}}, {\mathbf Q}$$ of polynomials in $$R[t]$$, $$I$$ respects the basis $$\mathcal B_{{{\mathbf P}},{\mathbf Q}}$$. □ Proof. (a) Suppose $$\alpha$$ is a boundary $$\mathcal P$$-arc. Since $$\alpha$$$$q$$-commutes with any basis element $$b_\mu$$ defined by (4), $$\alpha\mathscr S_R(\Sigma,\mathcal P)$$ is a two-sided ideal of $$\mathscr S_R(\Sigma,\mathcal P)$$. Hence $$I$$ is also a two-sided ideal. (b) In the chosen order, the boundary $$\mathcal P$$-arcs come before any inner arcs $$\mathcal P$$-arc. Hence $$I$$ is freely $$R$$-spanned by all $$b_{\mu,{{\mathbf P}},{\mathbf Q}}$$ such that $$\mu$$ has one of the $$\alpha_i$$ as a component. Thus, $$I$$ respects $$\mathcal B_{{{\mathbf P}},{\mathbf Q}}$$. ■ 4.5 Proof of Theorem 1.3 We will prove the following stronger version of Theorem 1.3. Theorem 4.4. Suppose a pair $$({{\mathbf P}},{\mathbf Q})$$ of sequences of polynomials in $$R[t]$$, $${{\mathbf P}}= (P_n(t)_{n\ge 0})$$ and $${\mathbf Q}= (Q_n(t)_{n\ge 0})$$, are positive over $$R$$. Then $$P_n(t)$$ is an $$R_+$$-linear combination of $$T_0(t), T_1(t), \dots, T_n(t)$$ and $$Q_n(t)$$ is an $$R_+$$-linear combination of $$1, t, \dots, t^n$$. Moreover, $$P_1(t)=Q_1(t)=t$$. □ Lemma 4.5. One has $$Q_1(t)= t$$. □ Proof. Consider the marked surface $$\mathbb D_1$$, which is a disk with four marked points $$p_0, p_1, p_2$$, and $$q_1$$ as in Figure 7. Let $$x$$ be the arc $$p_0p_2$$ and $$y$$ the arc $$ p_1 q_1$$. Suppose $$Q_1(t)= t + a$$, where $$a\in R$$. Using the skein relation to resolve the only crossing of $$xy$$, we get   Q1(x)Q1(y)=aQ1(x)+aQ1(y)−a2modI∂. (6) Here $$I_\partial$$ is the ideal of $$\mathscr S_R(\mathbb D_1)$$ generated by all the boundary $$\mathcal P$$-arcs, which respects $$\mathcal B_{{{\mathbf P}},{\mathbf Q}}$$ by Lemma 4.3. Lemma 4.2 and Equation (6) show that $$ a \in R_+$$ and $$- a^2 \in R_+$$. Hence both $$a^2$$ and $$-a^2$$ are $$R_+$$, which implies $$a=0$$. ■ Fig. 7. View largeDownload slide Left: $$\mathbb D_1$$, with product $$xy$$. Middle: $$\mathbb D_5$$, with product $$x^2 y_5$$. Right: Element $$z_{2,5.}$$ Fig. 7. View largeDownload slide Left: $$\mathbb D_1$$, with product $$xy$$. Middle: $$\mathbb D_5$$, with product $$x^2 y_5$$. Right: Element $$z_{2,5.}$$ Now fix an integer number $$n \ge 2$$. Let $$\mathbb D_n$$ be the marked surface, which is a disk with $$2n+2$$ marked points which in clockwise orders are $$p_0,p_1,\dots,p_{n+1}, q_n,\dots,q_1$$. We draw $$\mathbb D_n$$ in the standard plane so that the straight segment $$p_0p_{n+1}$$, denoted by $$x$$, is vertical with $$p_0$$ being lower than $$p_{n+1}$$, and each straight segment $$p_i q_i$$ is horizontal. See Figure 7 for an example of $$\mathbb D_5$$. Let $$y_n$$ be the union of the $$n$$ straight segments $$p_i q_i$$, $$i=1,\dots,n$$. Considering $$x$$ as an element of $$\mathscr S_R(\mathbb D_n)$$, we will present $$x^k$$ by $$k$$ arcs, each going from $$p_0$$ monotonously upwards to $$p_{n+1}$$. Any two of these $$k$$ arcs do not have intersection in the interior of $$\mathbb D_n$$, and the left one is above the right one. See an example in Figure 7 where the diagram of $$x^2 y_5$$ is drawn. Then the diagram of $$x^k y_n$$ has exactly $$kn$$ double points. Let $$z_{k,n}$$ be obtained from this diagram of $$x^k y_n$$ by negatively resolving all the crossings, again see Figure 7. For each $$i=0,\dots,n-1$$ let $$\gamma_i$$ be the arc $$p_i p_{i+1}$$, which is a boundary $$\mathcal P$$-arc. By Lemma 4.3, the set   I:=∑i=0n−1γiS(Σ,P) is a two-sided ideal of $$\mathscr S(\Sigma,\mathcal P)$$ respecting $$\mathcal B_{{{\mathbf P}},{\mathbf Q}}$$. It is important that the arc $$p_n p_{n+1}$$ is not in $$I$$. Lemma 4.6. For every $$k \le n$$, one has   xkyn=q−knzk,nmodI. (7) □ Proof. Label the $$k$$ arcs of the diagram of $$x^k$$ from left to right by $$1, \dots, k$$. There are $$kn$$ crossings in the diagram of $$x^k y_n$$, and denote by $$E_{l,m}$$ the intersection of the $$l$$th arc of $$x^k$$ and the arc $$p_m q_m$$ of $$y_n$$. Order the set of all $$kn$$ crossings $$E_{lm}$$ by the lexicographic pair $$(l+m,l)$$. Suppose $$\tau$$ is one of the $$2^{kn}$$ ways of resolutions of all the $$kn$$ crossings. Let $$D_\tau$$ be the result of the resolution $$\tau$$, which is a diagram without crossing. Assume $$\tau$$ has at least one positive resolution. We will prove that $$D_\tau\in I$$. Let $$E_{l,m}$$ be the smallest crossing at which the resolution is positive. In a small neighborhood of $$E_{l,m}$$, $$D_\tau$$ has two arcs, with the lower left one denoted by $$\delta$$, see Figure 8. The resolution at any $$E_{l', m'} < E_{l,m}$$ is negative. These data are enough to determine the arc $$\mathfrak u$$ of $$D_\tau$$ containing $$\delta$$, see Figure 9. Namely, if $$m \ge l$$ then $$\mathfrak u$$ is $$\gamma_{m-l}$$, and if $$m < l$$ then $$\mathfrak u$$ is an arc whose two end points are $$p_0$$, which is 0. See Figure 10 for an example. Either case, $$D_\tau \in I$$. Hence, modulo $$I$$, the only element obtained by resolving all the crossings of $$x^k y_n$$ is the all-negative resolution one, which is $$z_{k,n}$$. The corresponding factor coming from the skein relation is $$q^{-kn}$$. This proves Identity (7). ■ Fig. 8. View largeDownload slide Left: Crossing $$E_{l,m}$$. Right: Its positive resolution, and the arc $$\delta$$. Fig. 8. View largeDownload slide Left: Crossing $$E_{l,m}$$. Right: Its positive resolution, and the arc $$\delta$$. Fig. 9. View largeDownload slide The top right corner crossing is $$E_{l,m}$$, with positive resolution. The arc of $$D_\tau$$ containing $$\delta$$ keeps going down the south-west direction. Fig. 9. View largeDownload slide The top right corner crossing is $$E_{l,m}$$, with positive resolution. The arc of $$D_\tau$$ containing $$\delta$$ keeps going down the south-west direction. Fig. 10. View largeDownload slide Left: The arc $$\mathfrak u$$ with $$l=4, m=5$$. In this case $$\mathfrak u=\gamma_1$$. Right: The arc $$\mathfrak u$$ with $$l=4, m=3$$. In this case, both end points of $$\mathfrak u$$ are $$p_0$$. Fig. 10. View largeDownload slide Left: The arc $$\mathfrak u$$ with $$l=4, m=5$$. In this case $$\mathfrak u=\gamma_1$$. Right: The arc $$\mathfrak u$$ with $$l=4, m=3$$. In this case, both end points of $$\mathfrak u$$ are $$p_0$$. Proof of Theorem 4.4. Theorem 3.2 implies that $$P_n$$ is an $$R_+$$-linear combination of $$T_k$$ with $$k \le n$$. Since $$Q_1(t)=t$$, each of $$z_{k,n}$$, $$y_n$$ in Lemma 4.6 is an element of the basis $$\mathcal B_{{{\mathbf P}},{\mathbf Q}}$$. Suppose $$Q_n(t)= \sum_{k=0}^n c_k t^k$$ with $$c_k \in R$$. From (7), we have   Qn(x)yn=(∑k=0nckxk)yn=∑k=0nckq−knzk,n+I. Note that $$z_{k,n}\not \in I$$. Since $$I$$ respects the basis $$\mathcal B_{{{\mathbf P}},{\mathbf Q}}$$, Lemma 4.2 shows that $$c_k \in R_+$$ for all $$k$$. This completes the proof of Theorem 4.4. ■ Funding This work was supported in part by the National Science Foundation grant DMS-14-06419. Acknowledgments The author would like to thank D. Thurston for valuable discussions. He also thanks the anonymous referees for useful comments which in particular lead to the current, stronger formulation of Theorem 3.2. References [1] Bullock D. “Rings of $$Sl_2(\mathbb C)$$-characters and the Kauffman bracket skein module.” Commentarii Mathematici Helvetici  72, no. 4 (1997): 521– 42. Google Scholar CrossRef Search ADS   [2] Bullock D. Frohman C. and Kania-Bartoszynska. J. “Understanding the Kauffman bracket skein module.” Journal of Knot Theory and Its Ramifications  8, no. 3 (1999): 265– 77. Google Scholar CrossRef Search ADS   [3] Bonahon F. and Wong. H. “Quantum traces for representations of surface groups in $$SL_2$$(C).” Geometry & Topology  15, no. 3 (2011): 1569– 615. Google Scholar CrossRef Search ADS   [4] Chekhov L. and Fock. V. “Quantum Teichmüller spaces” (Russian) Teoreticheskaya i Matematicheskaya Fizika  120, no. 3 (1999): 511– 28; translation in Theoretical and Mathematical Physics 120, no. 3 (1999): 1245– 59. Google Scholar CrossRef Search ADS   [5] Fock V. and Goncharov. A. “Moduli spaces of local systems and higher Teichmüller theory.” Publications Mathématiques. Institut de Hautes études Scientifiques  no. 103 (2006): 1– 211. [6] Kauffman L. “States models and the Jones polynomial.” Topology  26 (1987): 395– 407. Google Scholar CrossRef Search ADS   [7] Lê T. T. Q. “On Kauffman bracket skein modules at roots of unity.” Algebraic & Geometric Topology  15 (2015): 1093– 117. Google Scholar CrossRef Search ADS   [8] Lê T. T. Q. Quantum Teichmuller spaces and quantum trace map . Journal of the Institute of Mathematics of Jussieu (2015): preprint arXiv:1511.06054 (in press). [9] Lê T. T. Q. and Tran. A. “On the AJ conjecture for knots.” Indiana University Mathematics Journal  64 (2015): 1103– 51. Google Scholar CrossRef Search ADS   [10] Muller G. “Skein algebras and cluster algebras of marked surfaces.” Quantum Topology , (2012): preprint arXiv:1204.0020 (in press). [11] Musiker G. Schiffler R. and Williams. L. “Bases for cluster algebras from surfaces.” Compositio Mathematica  149 (2013): 217– 63. Google Scholar CrossRef Search ADS   [12] Przytycki J. “Fundamentals of Kauffman bracket skein modules.” Kobe journal of mathematics  16 (1999): 45– 66. [13] Przytycki J. and Sikora. A. “On the skein algebras and $$Sl_2(\mathbb C)$$-character varieties.” Topology  39 (2000): 115– 48. Google Scholar CrossRef Search ADS   [14] Thurston D. “Positive basis for surface skein algebras.” Proceedings of the National Academy of Sciences of the United States of America  111 (2014): 9725– 32. Google Scholar CrossRef Search ADS PubMed  [15] Turaev V. “Skein quantization of Poisson algebras of loops on surfaces.” Annales Scientifiques de l’école Normale Supérieure (4)  24, no. 6 (1991): 635– 704. Google Scholar CrossRef Search ADS   © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]

Baykur, R İnanç;Van Horn-Morris, Jeremy

doi: 10.1093/imrn/rnw281pmid: N/A

Abstract We show that there are contact $$3$$–manifolds of support genus one which admit infinitely many Stein fillings but do not admit arbitrarily large ones. These Stein fillings arise from genus–$$1$$ allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact $$3$$–manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact $$3$$–manifold with support genus zero. We also show that there are $$4$$–strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the $$4$$–ball bounded by the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus–$$1$$ mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations. 1 Introduction Open books have gained a prominent role in contact geometry and low-dimensional topology ever since Giroux established a striking correspondence between contact structures and open books on $$3$$–manifolds [21]. It is desirable to deduce contact geometric information from coarse topological invariants of supporting open books. The minimal genus for a supporting open book, called the support genus of a contact $$3$$–manifold $$(Y, \xi)$$ [17], is such an invariant. Thanks to works of Etnyre, Wendl, and several others, quite a lot is now known about contact $$3$$–manifolds with support genus zero, as well as their symplectic and Stein fillings (e.g., [16, 22, 31, 35]). In particular, it is known that not all contact structures can be supported by planar open books [16]. However, 15 years after Giroux, it is still an open question whether there are contact structures which cannot be supported by genus–$$1$$ open books. Naturally, one would like to see if there are any distinguishing properties for contact $$3$$–manifolds that can be supported by genus–$$1$$ open books, perhaps similar to those known to hold for planar ones. There are a few intimately related aspects we will consider in this note: to date, there are no known examples of planar open books (respectively, planar contact structures) that can be filled by infinitely many allowable Lefschetz fibrations (respectively, Stein surfaces), whereas there are many examples of higher $$g \geq 2$$ open books with such fillings (e.g., [9, 12, 29]). On the other hand, no genus–$$0$$ or genus–$$1$$ open book can be filled by allowable Lefschetz fibrations with arbitrarily large Euler characteristic (see e.g., [9]), whereas many examples have been found in recent years again for $$g \geq 2$$ open books [7-9, 12]. Notably, the Euler characteristic of fillings of a contact $$3$$–manifold with support genus zero is also bounded [22, 30], but this property is not known to extend to support genus one. (If it did, this would provide an obstruction for contact $$3$$–manifolds with arbitrarily large Stein fillings to admit genus–$$1$$ open books.) In this note, we will explore each of these types of fillings. We prove the following: Theorem A. There are (infinitely many) contact $$3$$–manifolds with support genus one, each one of which admits infinitely many homotopy inequivalent Stein fillings but do not admit arbitrarily large ones. These are all supported by genus–$$1$$ open books, each bounding infinitely many distinct genus–$$1$$ allowable Lefschetz fibrations on their Stein fillings with infinitely many distinct homology groups. □ Many earlier examples of contact $$3$$–manifolds with infinitely many Stein fillings (e.g., [2, 29]) have used higher genera open books, which also admit arbitrarily large Stein fillings by [9, 12]. Examples of contact $$3$$–manifolds with support genus one and admitting infinitely many Stein fillings were given by Yasui in [36], where logarithmic transforms were used to produce exotic fillings (Remark 8). Our theorem provides the first examples which illustrate that there are (support genus one) contact $$3$$–manifolds which do not admit arbitrarily large Stein fillings. In the course of proving the above theorem, we will observe that fillings of any given planar contact $$3$$–manifold not only have restricted Euler characteristics but also they can have only finitely many homology groups (Proposition 4). In contrast, we will produce infinitely many factorizations of genus–$$1$$ mapping classes into only four positive Dehn twists, which induce homologically distinct allowable Lefschetz fibrations (Lemma 5). Two ingredients in this simple construction are suitable partial conjugations which effectively change the homology (e.g., [5, 6, 29]), and the explicit monodromy of a genus–$$1$$ open book with three binding components supporting the standard Stein fillable contact structure on $$T^3$$ (Lemma 1 and Proposition 2; also [34]). The small topology of these fillings will then allow us to show that many of these contact $$3$$–manifolds admit symplectic Calabi-Yau caps introduced in [24], so as to conclude that they can not admit arbitrarily large Stein fillings (Theorem 7). The second part of our article concerns fillings of quasipositive links, which are closures of quasipositive braids. By the pioneering works of Rudolph [32] and Boileau and Orevkov [10], these links are characterized as oriented boundaries of smooth pieces of complex analytic curves in the unit $$4$$–ball –-which can be realized as braided surfaces. An intimate connection between fillings of contact $$3$$–manifolds and quasipositive braids is provided by Loi and Piergallini [26], who showed that every Stein filling of a contact $$3$$–manifold is a branched cover of the Stein $$4$$–ball along a braided surface. We will provide infinitely many complex curves in the $$4$$–ball filling the same transverse links in the boundary $$3$$–sphere: Theorem B. There are (infinitely many) elements in the $$4$$–strand braid group, each of which admits infinitely many quasipositive braid factorizations up to Hurwitz equivalence. A particular family gives infinitely many complex analytic annuli in the $$4$$–ball with different fundamental group complements, all filling the same $$2$$-component transverse link. □ We will derive these examples from a refined construction of positive factorizations of elements in the mapping class group of a torus with two boundary components. These factorizations will commute with the hyperelliptic involution exchanging the two boundary circles, so they descend to quasipositive factorizations of certain $$4$$–braids (Theorem 10). In turn, the quasipositive factorizations prescribe braided surfaces, which by the work of Rudolph, can be made complex analytic. The frugality of our factorizations makes it possible to strike remarkably small topology among these examples, such as complex annuli filling in the same $$2$$–component transverse link. The small infinite families we obtain drastically improve the existing literature on fillings of genus–$$1$$ open books and $$4$$–braids. Our examples demonstrate that for each $$k \geq 2$$, there are elements in the mapping class group of a torus with $$k$$ boundary components which admit infinitely many distinct factorizations into just four positive Dehn twists, distinguished by their homologies. (For $$k\geq 3$$, the same type of examples with slightly larger number of Dehn twists can be derived from Yasui’s work in [36].) For $$k=1$$, examples admitting a few distinct factorizations were obtained by Auroux in [3] using calculations in $$SL(2, \mathbb{Z})$$, whereas examples of quasipositive factorizations of some $$3$$– and $$4$$–braids yielding two distinct braided surfaces were obtained in [3, 4, 19] (see Remarks 11 and 12). Notably, recent work of Orevkov [28] shows that any $$3$$–braid admits at most finitely many quasipositive factorizations up to Hurwitz equivalence, and the same goes for positive factorizations of a mapping class on a torus with one boundary [11]. Hence our examples of infinite families of fillings are optimal in terms of the number of boundary components for genus–$$1$$ mapping classes and the number of strands for braids. 2 Background All manifolds in this article are assumed to be smooth, compact, and oriented. We denote a genus $$g$$ surface with $$n$$ boundary components by $$\Sigma_g^n$$, and its mapping class group by $$\Gamma_g^n$$. This is the group which consists of orientation-preserving homeomorphisms of $$\Sigma_g^n$$ that restrict to identity along $$\partial \Sigma_g^n$$, modulo isotopies of the same type. We denote by $$t_c \in \Gamma_g^n$$, the positive (right-handed) Dehn twist along the simple closed curve $$c \subset \Sigma_g^n$$. A factorization in $$\Gamma_g^n$$  ϕ=tcl⋯tc1 is called a positive factorization$$P$$ of $$\phi$$. Of particular interest to us here are the factorizations where the Dehn twist curves $$c_j$$ are all homologically essential on $$\Sigma_g^n$$. Here, we use the functorial notation for the product of mapping classes; for example, a given factorization $$P=t_{c_l} \cdots t_{c_1}$$ acts on a curve $$a$$ on $$\Sigma_g^n$$ first by $$t_{c_1}$$, then $$t_{c_2}$$, and so on. The notation $$\phi^{\eta}$$ will stand for the conjugate element $$\eta \, \phi \, \eta^{-1}$$. Lastly, we note two elementary facts regarding Dehn twists we will repeatedly use: $$t_c^{\eta}=t_{\eta (c)}$$ and if $$a$$ and $$b$$ are disjoint curves, $$t_a \, t_b= t_b \, t_a$$. Let $$\mathcal{B}_n$$ denote the $$n$$-strand braid group, which consists of orientation-preserving homeomorphism of the unit disk $$D^2$$ fixing setwise $$n$$ distinguished marked points in the interior, modulo isotopies of the same type. We denote by $$\tau_{\alpha_j}$$ the positive (right-handed) half-twist along an arc $$\alpha_j$$ between two marked points on $$D^2$$, avoiding the others. A factorization in $$\mathcal{B}_n$$  b=ταl⋯τα1 is then called a quasipositive factorization of the braid $$b$$. A link in the $$3$$–sphere is said to be quasipositive if it can be realized as the closure of a quasipositive braid. 2.1 Contact $$3$$–manifolds and supporting open books A (co-oriented) contact structure on a $$3$$–manifold $$Y$$ is a plane field $$\xi$$, which can be globally written as the kernel of a contact form$$\alpha \in \Omega^1(Y)$$ with $$\alpha \wedge d\alpha\neq 0$$. An open book on $$Y$$ consists of $$L$$, an $$n$$-component oriented link in $$Y$$ which is fibered with bundle map $$h\mskip0.5mu\colon\thinspace Y \setminus L \to S^1$$ such that $$\partial F_t = L$$ for all $$F_t=h^{-1}(t)$$. Here $$L$$ is called the binding, the surface $$F_t \cong \Sigma_g^n$$, for any $$t$$, is called the page of the open book, and in this case $$h$$ is called a genus–$$g$$ open book. An open book is determined up to isomorphism by an element $$\phi \in \Gamma_g^n$$, called the monodromy, which prescribes the return map of a flow transverse to the pages and meridional near the binding. A contact $$3$$–manifold$$(Y, \xi)$$ is said to be supported by or compatible with an open book $$h$$ if $$\xi$$ is isotopic to a contact structure given by a $$1$$–form $$\alpha$$ satisfying $$\alpha>0$$ on the positively oriented tangent planes to $$L$$ and $$d\alpha$$ is a volume form on every page. By the works of Thurston and Winkelnkemper [33] and Giroux [21], every open book on $$Y$$ supports some contact structure, and conversely, there are (infinitely many) open books supporting a given contact structure on $$Y$$. The support genus of a contact $$3$$–manifold $$(Y, \xi)$$ is then defined to be the smallest possible genus for an open book supporting it. 2.2 Stein fillings, allowable Lefschetz fibrations, and braided surfaces A special class of contact $$3$$–manifolds arise as the boundaries of affine complex $$4$$–manifolds. A Stein filling of a contact $$3$$–manifold $$(Y, \xi)$$ is a $$4$$–manifold $$(X,J)$$ equipped with an affine complex structure $$J$$ in its interior, where the maximal complex distribution along $$\partial X = Y$$ is $$\xi$$. A Lefschetz fibration on a $$4$$–manifold $$X$$ is a map $$f\mskip0.5mu\colon\thinspace X \to D^2$$, where each critical point of $$f$$ lies in the interior of $$X$$ and conforms to the local complex model $$f(z_1,z_2)=z_1 z_2$$ under orientation preserving charts. These singularities are obtained by attaching $$2$$–handles to a regular fiber along simple-closed curves $$c_j$$, called the vanishing cycles. A Lefschetz fibration is said to be allowable if the fiber has non-empty boundary and all its vanishing cycles are homologically nontrivial on $$F \cong \Sigma_g^n$$, $$n>0$$. Given an allowable Lefschetz fibration $$f\mskip0.5mu\colon\thinspace X \to D^2$$, if we let $$p$$ be a regular value in the interior of the base $$D^2$$, then composing $$f$$ with the radial projection $$D^2 \setminus\{p\} \to \partial D^2$$ we obtain an open book $$h\mskip0.5mu\colon\thinspace \partial X \setminus \partial f^{-1}(p) \to S^1$$. Importantly, any positive factorization $$P$$ of an open book monodromy $$\phi = t_{c_l} \cdots t_{c_1}$$ in $$\Gamma_g^n$$ prescribes an allowable Lefschetz fibration $$f$$ on a Stein filling $$(X, J)$$ of $$(Y, \xi)$$ supported by this open book. A braided surface in the $$4$$–ball is an embedded compact surface $$S$$, along which the canonical projection $$\pi: D^4 \cong D^2 \times D^2 \to D^2$$ restricts to a simple positive branched covering, where $$S$$ and $$\pi|_S$$ conform to the local complex models $$w =z^2$$ and $$\pi(w, z)=z$$ around each branch point. Any quasipositive factorization of a braid $$b=\tau_{\alpha_l} \cdots \tau_{\alpha_1}$$ in $$\mathcal{B}_n$$ prescribes a braided surface $$S$$. All these geometric objects come together in a beautiful theorem of Loi and Piergallini. Building on Eliashberg’s topological characterization of Stein fillings and the work of Rudolph on braided surfaces, they showed that every Stein filling comes from an allowable Lefschetz fibration on $$X$$ [26] (also see [1] and [7] for a further generalization to Lefschetz fibrations over arbitrary compact surfaces with non-empty boundaries), which arises as a branched covering of the Stein $$4$$–ball along a braided surface $$S$$. On the other hand, the works of Rudolph [32] and Boileau and Orevkov [10] established that quasipositive links are precisely those which are oriented boundaries of smooth pieces of complex analytic curves in the unit $$4$$–ball, realized as braided surfaces. The algebraic topology of a Stein filling $$X$$ equipped with an allowable Lefschetz fibration is easy to read off from the corresponding positive factorization $$P$$. In particular, we have   π1(X)≅π1(Σgn)/N(c1,…,cl), where $$N(c_1, \ldots, c_l)$$ is the subgroup of $$\pi_1(\Sigma_g^n)$$ generated normally by $$\{c_i\}$$. For $$\{a_j\}$$ chosen generators of $$\pi_1(\Sigma_g^n) \cong \mathbb{Z}^{2g+n-1}$$, we therefore get   π1(X)≅⟨a1,…,a2g+n−1|R1,…,Rl⟩, where each $$R_i$$ is a relation obtained by expressing $$c_i$$ in $$\{a_j\}$$. 3 Preliminary results We begin with a relation in the genus–$$1$$ mapping class group, which will play a key role in our constructions to follow. Lemma 1. The following relation holds in the mapping class group $$\Gamma_1^3$$:   tb1tb2tb3=ta1−3ta2−3ta3−3tδ1tδ2tδ3, where the curves $$a_i, b_i, \delta_i$$, $$i=1, 2, 3$$, are as shown in Figure 1. □ Fig. 1. View largeDownload slide Dehn twist curves $$a_i$$, $$b_i$$, $$\delta_i$$ on $$\Sigma_1^3$$. Fig. 1. View largeDownload slide Dehn twist curves $$a_i$$, $$b_i$$, $$\delta_i$$ on $$\Sigma_1^3$$. Proof. This follows from the following relation in $$\Gamma_1^3$$, known as the star relation [20]:   (ta1ta2ta3tb2)3=tδ1tδ2tδ3, which can be derived by applying the lantern relation and braid relations to the well-known $$3$$-chain relation [20, 23]. For $$\Delta= t_{\delta_1} t_{\delta_2} t_{\delta_3}$$ the boundary multitwist, and $$\eta= t_{a_1} t_{a_2} t_{a_3}$$, we can rewrite it as   ηtb2ηtb2ηtb2 =Δ(η−1tb2η)tb2(ηtb2η−1) =η−3Δtη−1(b2)tb2tη(b2) =η−3Δtb1tb2tb3 =η−3Δ noting that $$\Delta$$ commutes with $$\eta$$. Since $$a_1, a_2, a_3$$ are all disjoint, $$\eta^{-3}= t_{a_1}^{-3} t_{a_2}^{-3} t_{a_3}^{-3}$$, which gives us the desired relation. ■ As we show below, the mapping class above prescribes an important open book. A detailed proof of the next proposition can be found in [34]. Here we will give an alternate argument. Proposition 2. The open book with monodromy $$\psi = t_{b_1} t_{b_2} t_{b_3}$$ supports $$(T^3, \xi_{\rm{can}})$$, where $$\xi_{\rm{can}}$$ is the canonical contact structure on the $$3$$–torus $$T^3$$, the unit cotangent bundle of the $$2$$–torus $$T^2$$. □ Proof. The positive factorization $$t_{b_1} t_{b_2} t_{b_3}$$ prescribes an allowable Lefschetz fibration whose boundary is the open book with monodromy $$\psi$$, which implies that the boundary contact $$3$$–manifold is Stein fillable. The handlebody decomposition for this fibration yields the Kirby diagram in Figure 2. It is now a Kirby calculus exercise, which we will leave the details of to the reader. Sliding two of the $$(-1)$$–framed $$2$$–handles over the third one, we obtain two pairs of canceling $$1$$– and $$2$$–handle pairs we can remove. The resulting diagram is easily seen to be the standard diagram for $$D^2 \times T^2$$ with two $$1$$–handles and one $$2$$–handle, whose boundary is $$T^3$$. Or instead, one can obtain a surgery diagram for the boundary $$3$$–manifold after trading the $$1$$–handles with $$0$$–framed unknots in the first picture, and perform the similar link calculus to arrive at the surgery diagram for $$T^3$$ given on the right hand side of Figure 2. Now by Eliashberg [14], the only contact structure on $$T^3$$ admitting a Stein filling is the canonical structure $$\xi_{\rm{can}}$$. ■ Fig. 2. View largeDownload slide Kirby diagram for the Lefschetz fibration filling $$Y$$, whose boundary is diffeomorphic to $$T^3$$. Fig. 2. View largeDownload slide Kirby diagram for the Lefschetz fibration filling $$Y$$, whose boundary is diffeomorphic to $$T^3$$. Remark 3. The star relation prescribes a genus–$$1$$ elliptic fibration with three $$(-1)$$–sections [23]. The lemma therefore implies that we can construct the Stein filling of $$T^3$$ as the complement of the union of an $$I_9$$ fiber (which one gets by clustering the nodal singularities induced by all the vanishing cycles $$a_i$$) and three $$(-1)$$–sections. □ Before we move on to our construction, we record the following result, which might be of particular interest. Proposition 4. Let $$P$$ be a positive factorization of a mapping class $$\psi \in \Gamma_0^n$$ into Dehn twists along homologically essential curves. Let $$(X_P, J_P)$$ denote the corresponding genus-$$0$$ allowable Lefschetz fibration. Then $$\{ H_i(X_P) \, | \, \psi=P \}$$ is a finite set. It follows that finitely many groups arise as homology groups of all possible Stein fillings (indeed, all minimal symplectic fillings) of a fixed contact $$3$$–manifold supported by a planar open book. □ Proof. For $$\psi = P$$ in $$\Gamma_0^n$$, $$X=X_P$$ admits a handle decomposition with one $$0$$–handle, $$n$$$$1$$–handles, and $$\ell$$$$2$$–handles, where $$\ell$$ is equal to the number of Dehn twists in $$P$$. We thus have $$\widetilde{H}_i(X ; \mathbb{Z})=0$$ for $$i \neq 1, 2$$. From [30], proof of Theorem 2.2, (cf. [22], proof of Theorem 1.1), we see that for a planar mapping class element $$\psi$$, the number of Dehn twists in any positive factorization of $$\psi$$ (i.e., $$\ell$$) is uniformly bounded. This shows that $$H_2(X; \mathbb{Z})$$ has a finite number of possibilities. Further, the collection of curves $$\{c_i\}$$ in $$P$$ gives a presentation of $$H_1(X; \mathbb{Z})$$ with $$n$$ generators and $$\ell$$ relations, each of the type   mj,1a1+…+mj,nan=0 Each relator is given by writing the homology class of the curve $$c_j$$ in the presentation $$P$$ in the basis $$\{a_i\}$$, where $$a_i$$ is the corresponding homology generator in $$H_1(X; \mathbb{Z})$$ for each boundary component $$\delta_i$$, and more importantly where $$0 \leq | m_{j,i} | \leq 1$$. Therefore, there are only finitely many possible relations in a presentation of $$H_1(X; \mathbb{Z})$$ for a Stein filling $$X$$ of $$\xi$$. The remainder of the proposition follows from work of Wendl [35] and Niederkrüger–Wendl [27] equating minimal strong symplectic fillings with positive factorizations for planar contact manifolds. Namely, any such filling of a contact $$3$$–manifold with a given planar supporting open book admits an allowable Lefschetz fibration (with planar fibers) filling this open book. ■ 4 Constructions of infinitely many fillings The contact $$3$$–manifold $$(Y, \xi)$$ we are interested is the one derived from $$(T^3, \xi_{\rm{can}})$$ by a Legendrian surgery along $$b_2$$, which by [18] is supported by the open book with monodromy $$\phi= \psi \, t_{b_2}= t_{b_1} t_{b_2} t_{b_3} t_{b_2}$$. Moreover, by [35], the Stein filling prescribed by the new positive factorization $$ t_{b_1} t_{b_2} t_{b_3} t_{b_2}$$ is obtained by a Weinstein handle attachment to the unique Stein filling $$(D^2 \times T^2, J_{\rm{can}})$$ of $$(T^3, \xi_{\rm{can}})$$. Next is the construction of Stein fillings of $$(Y, \xi)$$ with distinct homologies: Lemma 5. Let $$(Y, \xi)$$ be the contact $$3$$–manifold supported by the genus–$$1$$ open book with three boundary components, whose monodromy is $$\phi= t_{b_1} t_{b_2} t_{b_3} t_{b_2}$$. For any integer $$n$$, the mapping class $$\phi$$ admits a positive factorization   Pn=(tb1tb2tb3tta1n(b2))ta1−n=tta1−n(b1)tta1−n(b2)tta1−n(b3)tb2. Let $$(X_n, J_n)$$ be the Stein filling which is the total space of the allowable Lefschetz fibration prescribed by $$P_n$$. The family $$ \{ \, (X_n, J_n) \, | \, n \in \mathbb{N} \}$$ consists of distinct Stein fillings of the contact $$3$$–manifold $$(Y, \xi)$$, distinguished by their first homology $$H_1(X_n; \mathbb{Z})= \mathbb{Z} \oplus \, \mathbb{Z} / n\mathbb{Z}$$. □ Proof. In the mapping class group $$\Gamma_1^3$$, we have   tb1tb2tb3tb2ta1n=ta1n((tb1tb2tb3)ta1−ntb2)ta1−n=ta1n(tb1tb2tb3tb2)ta1−n. The mapping class $$t_{b_1} t_{b_2} t_{b_3} = t_{a_1}^{-3} t_{a_2}^{-3} t_{a_3}^{-3} t_{\delta_1} t_{\delta_2} t_{\delta_3}$$ commutes with $$t_{a_1}^n$$, and thus it stays fixed under the conjugation by $$t_{a_1}^{-n}$$. We conclude that each $$P_n$$ is a positive factorization of $$\phi = t_{b_1} t_{b_2} t_{b_3} t_{b_2}$$. We can calculate $$H_1(X_n; \mathbb{Z})$$ using the positive factorization $$P_n$$, or the conjugate factorization $$P_n ^{t_{a_1}^{n}}$$ (which prescribes an isomorphic Lefschetz fibration), that is,   H1(Xn;Z)≅H1(Σ13;Z)/N, where $$N$$ is normally generated by the vanishing cycles $$b_1, b_2, b_3,$$ and $$t_{a_1}^n(b_2)$$. The right-hand side is generated by the homology classes of the curves $$a_1, a_2, a_3, b_2$$ on $$\Sigma_1^3$$ (which we will denote by the same letters), with the relations   b2−a1−a2−a3=0, b2=0, b2+a1+a2+a3=0, b2+na1=0 induced by the vanishing cycles $$b_1, b_2, b_3,$$ and $$t_{a_1}^{n}(b_2)$$, where the last one is easily calculated by the Picard–Lefschetz formula. For $$n \in \mathbb{N}$$, from $$b_2=0$$, $$a_3=-a_1-a_2$$ and $$n \, a_1=0$$, we get $$H_1(X_n; \mathbb{Z}) = \mathbb{Z} \oplus \, \mathbb{Z} / n\mathbb{Z}$$, generated by $$a_2$$ and $$a_1$$. Lastly, our claim on the support genus of $$(Y, \xi)$$ follows from Proposition 4. ■ Remark 6. The contact manifold $$(Y_0, \xi_0)$$ with monodromy $$t_{b_1} t_{b_2} t_{b_3} t_{b_2}$$ is obtained from that with monodromy $$t_{b_1} t_{b_2} t_{b_3}$$ by Legendrian surgery on $$b_2$$. As we noted, the monodromy $$t_{b_1} t_{b_2} t_{b_3}$$ yields $$T^3$$ and the embedding of this open book maps $$b_2$$ to a curve isotopic to $$S^1 \times \{\text{pt}\}$$ inside $$T^3=S^1 \times T^2$$. This curve is Legendrian and traverses the direction of the twisting of the contact planes and has $$tb = -1$$. Surgery on $$b_2$$ yields a Seifert fibered $$3$$–manifold over $$T^2$$ with a single singular fiber of order $$2$$. □ Next is our first main result, which will build on the example given in Lemma 5. We will then revamp our examples to an infinite family simply by combining them with Stein fillable planar open books. Theorem 7. There are (infinitely many) contact $$3$$–manifolds with support genus one, each one of which admits infinitely many homotopy inequivalent Stein fillings, but do not admit arbitrarily large ones. These are all supported by genus–$$1$$ open books bounding genus–$$1$$ allowable Lefschetz fibrations on their Stein fillings with infinitely many distinct homology groups. Proof. We will construct a countable family of such examples, $$(Y_m, \xi_m)$$, $$m \in \mathbb{N}$$, each admitting a countable family of distinct Stein fillings $$(X_{n,m}, J_{n,m})$$, $$n\in \mathbb{N}$$. All will be constructed from our principal example by taking a connected sum with some planar contact 3-manifold. Let us begin with $$(Y_0, \xi_0)= (Y, \xi)$$. Capping all three boundaries of the genus–$$1$$ open book on $$(Y, \xi)$$ with monodromy $$\phi$$ we get a symplectic cobordism from $$(Y, \xi)$$ to a symplectic $$T^2$$–fibration [15] with the induced monodromy $$t_{b'_1} t_{b'_2} t_{b'_3} t_{b'_2}$$, where $$b'_i$$ are the images of the curves $$b_i$$ on the closed genus–$$1$$ surface. By Remark 3, we already know that $$t_{b'_1} t_{b'_2} t_{b'_3}$$ can be completed to a positive factorization of the elliptic fibration on $$E(1)$$. So the factorization $$t_{b'_1} t_{b'_2} t_{b'_3} t_{b'_2}$$ can be completed to a symplectic elliptic fibration on the $${\rm K3}$$ surface. (This means that we can symplectically embed all our Stein fillings $$(X_n, J_n)$$ into $${\rm K3}$$ surface.) That is, the contact $$3$$–manifold $$(Y, \xi)$$ admits a Calabi–Yau cap, which by [24] implies that the Betti numbers of Stein fillings of $$(Y, \xi)$$ is finite. In particular, it cannot admit arbitrarily large Stein fillings. Now let $$(Y', \xi')$$, where $$Y' \neq S^3$$, be any contact $$3$$–manifold which admits a planar supporting open book. We simply take $$(Y', \xi')$$ to be $$S^1 \times S^2$$ with the canonical contact structure supported by an annulus open book with trivial monodromy $$\phi'$$, which is filled by the trivial allowable Lefschetz fibration with annulus fibers on $$(X', J')$$, the unique Stein filling $$(S^1 \times D^3, J')$$ of $$(Y', \xi)$$ [13]. By taking certain Murasugi sums of the open book $$\phi$$ with $$m$$ copies of $$\phi'$$, we can get a new genus–$$1$$ open book $$\phi_m$$ supporting a contact $$3$$–manifold $$(Y_m, \xi_m)$$, where we set $$Y_m=Y \# \, m Y'$$. By our assumption on $$Y'$$, we see that $$\pi_1(Y_m)$$ are all different; for $$Y'= S^1 \times S^2$$, we have $$\pi_1(Y_m )=\pi_1(Y) * \mathbb{Z}^m$$. So $$\{(Y_m, \xi_m) \, | \, m \in \mathbb{N}\}$$ consists of infinitely many distinct elements. This Murasugi sum can be extended to the filling allowable Lefschetz fibrations, giving us a family of genus–$$1$$ allowable Lefschetz fibrations on the Stein fillings $$(X_{n,m}, J_{n,m})$$ of each $$(Y_m, \xi_m)$$, for $$n \in \mathbb{N}$$. Since   H1(Xn,m)=H1(Xn)⊕H1(X′)=Z1+m⊕(Z/nZ), these fillings are all distinct. Moreover, once again by Proposition 4, all $$(Y_m, \xi_m)$$ have support genus $$1$$. However, $$(Y_m, \xi_m)$$ does not admit arbitrarily large Stein fillings: if it did, by [13] these would split as boundary connected sums of Stein fillings of $$(Y_m, \xi_m)$$ and copies of $$(Y', \xi')$$, respectively. By our pick of $$(Y', \xi')$$, this means $$(Y_m, \xi_m)$$ admits arbitrarily large Stein fillings, contradicting the observation we have made above. ■ Remark 8. The first examples of contact $$3$$–manifolds with support genus one admitting infinitely many Stein fillings were given by Yasui in [36]. Yasui uses the same open book description for the standard contact structure on $$T^3$$ from the second author’s thesis [34] to perform logarithmic transforms and produce infinitely many pairwise exotic Stein fillings, as well as homologically distinct ones. These also admit genus–$$1$$ allowable Lefschetz fibrations; however, their monodromies are different than ours and have many more Dehn twists. Unlike the small fillings from Lemma 5, we tailored for the proof of the above theorem, most examples with larger topology (e.g., the fillings with distinct homologies in [36]) do not seem to embed into the $${\rm K3}$$ surface. It seems plausible that some of the applications in [36] can be modified to produce similar examples to ours; however, the rather subtle partial conjugation employed in our construction of distinct positive factorizations arguably yields the most direct and explicit proof. □ Remark 9. (Uniruled and Calabi–Yau caps). As illustrated by our examples, admitting a Calabi–Yau cap [24] does not impose finiteness on possible homology groups of Stein or minimal symplectic fillings of a contact $$3$$–manifold. However, admitting a symplectic cap into a rational surface (as a uniruled cap in [24]) might be sufficient for this, which would extend the case of contact $$3$$–manifolds with support genus zero we covered in Proposition 4. □ Our second main result will also build on a small variation of Lemma 5, this time to provide infinitely many fillings of quasipositive braids. Theorem 10. There are (infinitely many) elements in the $$4$$–strand braid group, each of which admits infinitely many quasipositive braid factorizations up to Hurwitz equivalence. A particular family gives infinitely many complex analytic annuli in the $$4$$–ball with different fundamental group complements, all filling the same $$2$$-component transverse link. □ Proof. Recall the positive factorizations for the open book monodromy we had in Lemma 5. Capping off the boundary component $$\delta_3$$ of $$\Sigma_1^3$$, we obtain another genus–$$1$$ open book $$\widehat{\phi}$$ on a new contact $$3$$–manifold $$(\widehat{Y}, \hat{\xi})$$. It bounds allowable Lefschetz fibrations on $$\widehat{X}_n$$ with four singular fibers, coming from positive factorizations   P^n=(tb^1tb^2tb^3tta^1n(b^2))ta^1−n of $$\widehat{\phi}$$ in $$\Gamma_1^2$$. Here $$\hat{a}_i, \hat{b}_i, \delta_i$$ denote the images of $$a_i$$, $$b_i, \delta_i$$ in the new page $$\Sigma_1^2$$; in particular, $$\hat{a}_1=\hat{a}_3$$. The same homology arguments show that $$\widehat{X}_n$$ can be distinguished by their first homology, $$H_1(\widehat{X}_n; \mathbb{Z})= \mathbb{Z} \, / n \, \mathbb{Z}$$. For, $$H_1(\widehat{X}_n; \mathbb{Z})$$ is now generated by the homology classes of $$\hat{a}_1, \hat{a}_2, \hat{b}_2$$, with the relations   b^2−2a^1−a^2=0, b^2=0, b^2+2a^1+a^2=0, b^2+na^1=0 induced by the vanishing cycles $$\hat{b}_1, \hat{b}_2, \hat{b}_3$$ and $$t_{\hat{a}_1}^{n}(\hat{b}_2)$$. So $$\hat{a}_1$$ generates the whole group and is of order $$n$$. One perk of the factorizations above is that they commute with a hyperelliptic involution $$\iota$$ on $$\Sigma_1^2$$. So each positive factorization $$\widehat{P}_n$$ descends to a four factor quasipositive factorization of the same element in the $$4$$-strand braid group $$\mathcal{B}_4$$. Figure 4 shows the arcs between the four marked points on the $$2$$-disk, twists along which correspond to the Dehn twists along curves on $$\Sigma_1^2$$ featured in $$\widehat{P}_n$$. Denoting by $$\tau_1, \tau_2,$$ and $$\tau_3$$, the standard Artin generators corresponding to arc twists along arcs $$\alpha_1, \beta_2$$, and $$\alpha_2$$ in Figure 4, we can pull the factorizations of $$\widehat{P}_n$$ to the braid group. For convenience, we conjugate this factorization to get the factorizations   ((τ12τ3)−1τ2(τ12τ3)) τ2 ((τ12τ3)τ2(τ12τ3)−1) (τ1nτ2τ1−n) of the braid (using functional notation for the product in the braid group)   τ3−3τ1−6(τ3τ2τ1)4(τ1nτ2τ1−n). Each of these braids is conjugate to each other and in particular to the braid with $$n=0$$. The closure of this braid is a two component link of the unknot and the mirror of the twist knot $$5_2$$. Topologically, the allowable Lefschetz fibrations on $$\widehat{X}_n$$ arise as double-branched coverings along positively braided surfaces$$S_n$$ in the $$4$$–ball, all filling the same $$4$$–strand braid on the boundary. These surfaces arise as quasipositive factorizations of the corresponding braid quotient. By the work of Rudolph [32], all $$S_n$$ are complex analytic curves. The braided surfaces $$S_n$$ are made of four horizontal disks corresponding to the four braid strands and half-twisted bands connecting the disks corresponding to each arc-twist in the factorization (connecting disks corresponding to the endpoints of the arc in question). Thus, we easily see that for $$n$$ odd, half-twists $$\beta_1 = \left( \left( \tau_1^2 \tau_3 \right)^{-1} \tau_2 \left( \tau_1^2 \tau_3 \right) \right),$$$$\beta_2 = \tau_2,$$$$\beta_3 = \left( \left( \tau_1^2 \tau_3 \right) \tau_2 \left( \tau_1^2 \tau_3 \right)^{-1} \right),$$ and $$t_{\alpha_1}^n(\beta_2)$$ yield a connected surface, whereas for $$n,$$ even there are two connected components in $$S_n$$. The Euler characteristic $$\chi(S_n)= 4 - 4 = 0$$ (the number of sheets minus the number of bands), so $$S_n$$ is an annulus when $$n$$ is odd and is a link of punctured torus and a disk when $$n$$ is even. For either family of branched covers parametrized by odd $$n$$ or even $$n,$$ we necessarily have distinct surfaces in $$D^4$$. Up to a finite ambiguity when $$n$$ is even, the groups $$\pi_1(D^4 \setminus S_n)$$ are distinct. To see this, note that by the previous calculations, the double branched covers $$\widehat{X}_n$$ are all branched along homeomorphic surfaces $$S_n$$ in $$D^4$$ but have different homologies. Additionally, both $$\pi_1$$ (and so $$H_1$$) of the double branched cover are calculated directly from the group $$\pi_1(D^4 \setminus S_n).$$ The fundamental group of the double cover of the complement of $$S_n$$ is the kernel of a surjective map from $$H_1$$ to $$\mathbb{Z}/2\mathbb{Z}$$. This map is unique if $$S_n$$ is connected, as $$H_1 = \mathbb{Z}$$. When $$S_n$$ is disconnected, $$H_1$$ of the complement is $$\mathbb{Z}^2$$. There are three nontrivial maps, then, of this fundamental group, and the one that calculates $$\pi_1(\widehat{X}_n)$$ sends the meridional generators both to $$1$$. Thus it is possible that there could be two surfaces with isomorphic knot groups but which are distinguished by their “peripheral structure.” Lastly, we note that we could get more elements in $$\mathcal{B}_4$$ with infinitely many factorizations with similarly distinct $$\pi_1$$ complements by simply adding more $$t_{\hat{b}_2}$$ factors to $$\widehat{P}_n$$. This would have no effect on the homology calculation in the cover as the collection of vanishing cycles will be the same. It will, however, change the topology of $$S_n$$ and yield higher genera knotted complex surfaces filling in these links. ■ Fig. 3. View largeDownload slide Dehn twist curves $$\hat{a}_i$$, $$\hat{b}_i$$, $$\delta_i$$ on $$\Sigma_1^2$$, symmetric under the hyperelliptic involution with four fixed points. Fig. 3. View largeDownload slide Dehn twist curves $$\hat{a}_i$$, $$\hat{b}_i$$, $$\delta_i$$ on $$\Sigma_1^2$$, symmetric under the hyperelliptic involution with four fixed points. Fig. 4. View largeDownload slide Arcs $$\alpha_i, \beta_j$$, and the boundary $$\delta$$ of the disk are doubly covered by the simple closed curves $$\hat{a}_i, \hat{b}_j$$ and the boundary $$\delta_1 \sqcup \delta_2$$ of $$\Sigma_1^2$$. Here $$\tau_{\alpha_1}(\beta_2)$$ descends from $$\tau_{\hat{a}_1}(\hat{b}_2)$$ (illustrating $$n=1$$ case). Fig. 4. View largeDownload slide Arcs $$\alpha_i, \beta_j$$, and the boundary $$\delta$$ of the disk are doubly covered by the simple closed curves $$\hat{a}_i, \hat{b}_j$$ and the boundary $$\delta_1 \sqcup \delta_2$$ of $$\Sigma_1^2$$. Here $$\tau_{\alpha_1}(\beta_2)$$ descends from $$\tau_{\hat{a}_1}(\hat{b}_2)$$ (illustrating $$n=1$$ case). Remark 11 (Distinct positive factorizations in $$\Gamma_1^k$$). The examples given in the proofs of Theorems 7 and 10 demonstrate that there are genus–$$1$$ mapping classes in $$\Gamma_1^k$$ for each $$k \geq 2$$, each one of which admits infinitely many inequivalent positive factorizations with four positive Dehn twists, all with distinct homologies. If we cap off one more boundary component, our induced factorizations in $$\Gamma_1^1$$ yield two distinct fundamental groups; $$\pi_1=1$$ or $$\mathbb{Z}_3$$. These complement the work of Auroux in [3], where pairs of distinct factorizations into three or four positive Dehn twists, some with distinct homologies, were detected for a few other mapping classes in $$\Gamma_1^1$$. □ Remark 12 (Quasipositive factorizations for low order braids). There are examples of low order braids filled by different braided surfaces in the literature. The articles [4] and [3] give examples of pairs of braided surfaces with distinct fundamental group complements, filling the same $$3$$-braids. These fillings have distinct topology–-one surface is connected and the other one is not. Examples of pairs of connected braided surfaces filling a $$4$$-braid were obtained by Geng in [19]. Given Theorem 10, one should ask whether an infinite family of braided surfaces with distinct fundamental group complements can possibly fill a $$3$$-braid. □ Orevkov [28] has recently shown that any $$3$$–braid admits at most finitely many quasipositive factorizations up to Hurwitz equivalence [Corollary 2]. Since $$\Gamma_1^1$$ is isomorphic to $$\mathcal{B}_3$$, where positive Dehn twists correspond to positive half-twists, any mapping class in $$\Gamma_1^1$$ also admits at most finitely many positive factorizations. (Also see [11].) We conclude that our examples of genus–$$1$$ mapping classes and braids with infinitely many positive and quasipositive factorizations realize the smallest possible number of boundary components and number of strands, respectively. Remark 13 (Fillings of planar spinal open books). An open question regarding contact $$3$$–manifolds supported by planar open books is if they admit at most finitely many Stein fillings. One can deduce a negative answer for contact $$3$$–manifolds supported by rational planar spinal open books, even though it is shown in [25] that in the integral case, these behave very similarly to planar open books in general. For example, the minimal symplectic fillings of integral planar spinal open books also admit planar Lefschetz fibrations, though over more general compact surfaces (the topology of which are determined by the spine of the open book). Additionally, these all admit compatible Stein structures. (The reader can turn to the Appendix of [7] for an overview of spinal open books introduced in [25].) In turn, our homology arguments in [9] [Proposition 1] (cf. [22, 31]) work the same for the induced positive factorizations of integral spinal open books, since the commutator elements in the mapping class description of a Lefschetz fibration over a non-planar base surface die in the first homology of the corresponding mapping class group. We therefore conclude that contact $$3$$–manifolds supported by integral planar spinal open books, just like in the case of planar open books, will not admit arbitrarily large Stein fillings. Here is a sketch of the construction of these examples: consider the simplest monodromy we had for the annuli fillings in Theorem 10. If we take the third power of it, we get a pure $$4$$–braid, whereas the third power of the positive factorizations will still be distinct: the collection of Dehn twists presenting the Lefschetz fibrations in the double branched covers is the same (as the third root), so the fillings can still be distinguished by the same $$\pi_1$$/homology calculation. If we now drill out a fibered tubular neighborhood of the new braided surfaces, the boundary contact 3-manifold is compatible with a rational planar spinal open book, whereas the interior is the complement of a complex analytic genus $$3$$ curve (filling the pure braid), which will yield a minimal symplectic filling. However, these examples fail to be “Lefschetz-amenable” in the terminology of [25] and a classification is unknown. □ Funding This work was partially supported by the NSF Grant [DMS-1510395 to R.I.B] and also supported by a grant from the Simons Foundation [279342, J.V.H.M.]. Acknowledgments We would like to thank Stepan Orevkov for pointing out his result in [28]. References [1] Akbulut S. and Ozbagci. B. “Lefschetz Fibrations on Compact Stein Surfaces.” Geometry & Topology  5 ( 2001): 319– 34. Google Scholar CrossRef Search ADS   [2] Akhmedov A. Etnyre J. B. Mark T. and Smith. I. “A Note on Stein Fillings of Contact Manifolds.” Mathematical Research Letters  15, no. 6 ( 2008): 1127– 32. Google Scholar CrossRef Search ADS   [3] Auroux D. “Factorizations in $$\rm{SL}(2,\mathbb{Z})$$ and simple examples of inequivalent Stein fillings.” Journal of Symplectic Geometry  13, no. 2 ( 2015): 261– 77. Google Scholar CrossRef Search ADS   [4] Auroux D. Kulikov V. and Shevchishin. V. “Regular homotopy of Hurwitz curves.” Izvestiya: Mathematics  68 ( 2004): 521– 42. Google Scholar CrossRef Search ADS   [5] Baykur R. I. “Small symplectic Calabi-Yau surfaces and exotic $$4$$–manifolds via genus–$$3$$ pencils.” preprint; http://arxiv.org/abs/1511.05951. [6] Baykur R. I. and Korkmaz. M. “Small Lefschetz fibrations and exotic $$4$$–manifolds.” to appear in Mathematische Annalen , ( 2016), https://doi.org/10.1007/s00208-016-1466-2. Google Scholar CrossRef Search ADS   [7] Baykur R. I. and Van Horn-Morris. J. “Families of contact $$3$$–manifolds with arbitrarily large Stein fillings.” with an appendix by Lisi S. and Wendl C. Journal of Differential Geometry  101, no. 3 ( 2015): 423– 65. Google Scholar CrossRef Search ADS   [8] Baykur R. I. and Van Horn-Morris. J. “Topological complexity of symplectic $$4$$–manifolds and Stein fillings.” Journal of Symplectic Geometry  14, no. 1 ( 2016): 171– 202. Google Scholar CrossRef Search ADS   [9] Baykur R. I. Monden N. and Van Horn-Morris. J. “Postive factorizations of mapping classes.” to appear in Algebraic & Geometric Topology;  http://arxiv.org/abs/1412.0352. [10] Boileau M. and Orevkov. S. “Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe.” Comptes Rendus de l’Académie des Sciences - Series I - Mathematics  332, no. 9 ( 2001): 825– 30. Google Scholar CrossRef Search ADS   [11] Cadavid C. Moreno L. and Vélez J. D. Hurwitz complete sets of factorizations in the modular group and the classification of Lefschetz elliptc fibrations over the disk,  preprint http://arxiv.org/abs/1309.5871. [12] Dalyan E. Korkmaz M. and Pamuk. M. “Arbitrarily long factorizations in mapping class groups.” International Mathematics Research Notices  no. 19 ( 2015): 9400– 14. [13] Eliashberg Y. “Filling by holomorphic discs and its applications.” London Mathematical Society Lecture Note Series , Cambridge Univ. Press, Cambridge, 151 ( 1991): 45– 67. Google Scholar CrossRef Search ADS   [14] Eliashberg Y. “Unique holomorphically fillable contact structure on the $$3$$–torus.” International Mathematics Research Notices  no. 2 ( 1996): 77– 82. [15] Eliashberg Y. “A few remarks about symplectic filling.” Geometry & Topology  8 ( 2004): 277– 93. Google Scholar CrossRef Search ADS   [16] Etnyre J. B. “Planar open book decompositions and contact structures.” International Mathematics Research Notices  no. 79 ( 2004): 4255– 67. [17] Etnyre J. B. and Ozbagci. B. “Invariants of contact structures from open books.” Transactions of the American Mathematical Society  360, no. 6 ( 2008): 3133– 51. Google Scholar CrossRef Search ADS   [18] Gay D. “Explicit concave fillings of contact three-manifolds.” Mathematical Proceedings of the Cambridge Philosophical Society  133, no. 3 ( 2002): 431– 41. Google Scholar CrossRef Search ADS   [19] Geng A. “Two surfaces in $$D^4$$ bounded by the same knot.” Journal of Symplectic Geometry  9, no. 2 ( 2011): 119– 22. Google Scholar CrossRef Search ADS   [20] Gervais S. “A finite presentation of the mapping class group of a punctured surface.” Topology  40 ( 2001): 703– 25. Google Scholar CrossRef Search ADS   [21] Giroux E. Géométrie De Contact: De La Dimension Trois Vers Les Dimensions Supérieures.  Proceedings of the International Congress of Mathematicians, vol. II ( Beijing, 2002), 405– 14, Higher Ed. Press, Beijing, 2002. [22] Kaloti A. Stein fillings of planar open books , preprint http://arXiv.org/abs/1311.0208. [23] Korkmaz M. and Ozbagci. B. “On sections of elliptic fibrations.” Michigan Mathematical Journal  56 ( 2008): 77– 87. Google Scholar CrossRef Search ADS   [24] Li T.-J. Mak C. Y. and Yasui. K. Calabi-Yau Caps and Uniruled Caps,  preprint http://arxiv.org/abs/1412.3208. [25] Lisi S. Van Horn-Morris J. and Wendl C. On symplectic fillings of spinal open book decompositions , preprint in preparation. [26] Loi A. and Piergallini. R. “Compact Stein surfaces with boundary as branched covers of B4.” Inventiones Mathematicae  143 ( 2001): 325– 48. Google Scholar CrossRef Search ADS   [27] Niederkrüger K. and Wendl. C. “Weak symplectic fillings and holomorphic curves.” Annales Scientifiques de l’école Normale Supérieure (4)  44, no. 5 ( 2011): 801– 53. Google Scholar CrossRef Search ADS   [28] Orevkov S. “On the Hurwitz action on quasipositive factorizations of $$3$$–braids.” Doklady Akademii Nauk  461, no. 3 ( 2015): 263– 7; English translation in Doklady Mathematics 91, no. 2 ( 2015): 173– 7. [29] Ozbagci B. and Stipsicz. A. “Contact $$3$$–manifolds with infinitely many Stein fillings.” Proceedings of the American Mathematical Society  132, no. 5 ( 2004): 1549– 58. Google Scholar CrossRef Search ADS   [30] Plamenevskaya O. “On Legendrian surgeries between lens spaces.” Journal of Symplectic Geometry  10, no. 2 ( 2012): 165– 81. Google Scholar CrossRef Search ADS   [31] Plamenevskaya O. and Van Horn-Morris. J. “Planar open books, monodromy factorizations, and symplectic fillings.” Geometry & Topology  14, no. 4 ( 2010): 2077– 101. [32] Rudolph L. “Algebraic functions and closed braids.” Topology  22, no. 2 ( 1983): 191– 202. Google Scholar CrossRef Search ADS   [33] Thurston W. P. and Winkelnkemper. H. E. “On the existence of contact forms.” Proceedings of the American Mathematical Society  52 ( 1975): 345– 7. Google Scholar CrossRef Search ADS   [34] Van Horn-Morris J. Constructions of Open Book Decompositions. PhD Dissertation , UT Austin, 2007. [35] Wendl C. “Strongly fillable contact manifolds and J -holomorphic foliations.” Duke Mathematical Journal  151, no. 3 ( 2010): 337– 84. Google Scholar CrossRef Search ADS   [36] Yasui K. Partial twists and exotic Stein fillings,  preprint http://arxiv.org/abs/1406.0050. © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]

Juhnke-Kubitzke, Martina;Van Le, Dinh

doi: 10.1093/imrn/rnw284pmid: N/A

Abstract One of the major open questions in matroid theory asks whether the $$h$$-vector $$(h_0,h_1,\ldots,h_s)$$ of the broken circuit complex of a matroid $$M$$ satisfies the following inequalities:   h0≤h1≤⋯≤h⌊s/2⌋andhi≤hs−i  for  0≤i≤⌊s/2⌋. This article affirmatively answers the question for matroids that are representable over a field of characteristic zero. 1 Introduction The notion of broken circuit complexes goes back to Whitney [39], who used his broken circuit idea to interpret the coefficients of the chromatic polynomial of a graph. This notion was later extended to matroids by Rota [27] and Brylawski [6]. Given a loopless matroid $$M$$ on ground set $$E$$, which is endowed with a linear ordering $$<$$, a broken circuit of $$(M,<)$$ is a circuit of $$M$$ with its least element removed. The broken circuit complex of $$(M,<)$$, denoted by $$BC_<(M)$$ (or briefly $$BC(M)$$ if no confusion may arise), is defined by   BC(M):={F⊆E:F contains no broken circuit}. Broken circuit complexes have shown to be important in multiple ways. From the algebraic point of view, they play an interesting role in the study of hyperplane arrangements. In particular, the broken circuit idea was used to construct bases for two fundamental algebraic objects associated with a hyperplane arrangement, namely, the Orlik–Solomon algebra and the Orlik–Terao algebra [2, 25]. Through these constructions, broken circuit complexes have been an essential tool for studying important algebraic and homological properties of those algebras [11, 12, 17, 18, 21]. From the combinatorial point of view, $$f$$-vectors and $$h$$-vectors of broken circuit complexes encode very useful information about the underlying matroids. Recall that the characteristic polynomial of a matroid $$M$$ is defined as $$\chi(M;t):=\sum_{X\subseteq E}(-1)^{|X|}t^{r(M)-r(X)}$$, where $$r(\cdot)$$ denotes the rank function of $$M$$. This polynomial, which was introduced by Rota [27] as a generalization of the chromatic polynomial of a graph, plays a prominent role in the study of many combinatorial problems; see, for example, [8, 41]. A fascinating property of $$f$$-vectors of broken circuit complexes, which primarily makes these complexes important, is the following formula due to Whitney [39] and Rota [27]:   χ(M;t)=∑i=0r(−1)ifitr−i, (1) where $$f_i$$ denotes the number of faces of $$BC(M)$$ of cardinality $$i$$. The $$h$$-vector of $$BC(M)$$, on the other hand, encodes the shelling polynomial of $$BC(M)$$ [2]. Furthermore, several properties of $$M$$ (such as connectivity [10] or being a series–parallel network [5]) and of $$BC(M)$$ (such as Gorensteinness or being a complete intersection [18]) are determined by the $$h$$-vector of $$BC(M)$$. For these reasons, $$f$$-vectors and $$h$$-vectors of broken circuit complexes are among the most interesting numerical invariants in matroid theory. Recently, great advances have been made in the study of $$f$$-vectors and $$h$$-vectors of broken circuit complexes. In particular, the long-standing conjectures of Rota–Heron [13, 28] and Welsh [38] on the unimodality and log-concavity of the $$f$$-vector of $$BC(M)$$ have been resolved by Adiprasito, Huh and Katz [1]. Additionally, Huh [16] proved that the $$h$$-vector of $$BC(M)$$ is log-concave if $$M$$ is representable over a field of characteristic zero. Recall that a sequence $$(a_0,a_1,\ldots,a_n)$$ of real numbers is said to be log-concave if $$a_j^2\ge a_{j-1}a_{j+1}$$ for all $$1\le j\le n-1$$. Also, this sequence is called unimodal if there exists $$0\leq p\leq n$$ such that $$a_0\leq a_1\leq \cdots \leq a_p\geq a_{p+1}\geq \cdots \geq a_n$$. Observe that if a sequence of positive numbers is log-concave, then it is unimodal. Despite the significant advances mentioned above, $$f$$-vectors and $$h$$-vectors of broken circuit complexes are still rather mysterious. In fact, the problem of characterizing these vectors is widely regarded as out of reach at the moment. A more realistic problem would be to find as many restrictions on these vectors as possible. Such restrictions are predicted by the following conjecture, which is in the focus of this article: Conjecture 1.1. Let $$M$$ be a loopless matroid. Let $$(h_0,h_1,\ldots,h_s)$$ be the $$h$$-vector of $$BC(M)$$, where $$s$$ is the largest index $$j$$ with $$h_j\ne0$$. Then the following inequalities hold:   h0≤h1≤⋯≤h⌊s/2⌋andhi≤hs−i  for  0≤i≤⌊s/2⌋. □ A sequence $$(h_0,h_1,\ldots,h_s)$$ of real numbers that satisfies the inequalities in the above conjecture is called strongly flawless, and it is called flawless if $$h_i\le h_{s-i}\ \text{ for }\ 0\leq i \leq \lfloor s/2\rfloor$$. Clearly, the strongly flawless condition can be rephrased as $$h_i\le h_j$$ for $$0\le i\le j\le s-i$$. Moreover, for a unimodal sequence, being flawless is equivalent to being strongly flawless. Conjecture 1.1 goes back to a still wide open conjecture of Stanley [32], which anticipates that the $$h$$-vector of the independence complex$$IN(M)$$ of a matroid $$M$$ is a pure $$O$$-sequence. The reader is referred to [3] for the definition of pure $$O$$-sequences as well as recent developments in the study of these interesting objects. Recall that $$IN(M)$$ is the collection of all independent sets in $$M$$, and that it contains $$BC(M)$$ as a subcomplex. In [14], Hibi showed that a pure $$O$$-sequence is strongly flawless. Inspired by this result, he proposed a weaker version of Stanley’s conjecture in [15], predicting that the $$h$$-vector of $$IN(M)$$ must be strongly flawless. This conjecture was resolved by Chari [9], who proved that $$IN(M)$$ has a convex ear decomposition. Subsequently, an algebraic version of Chari’s proof, which shows the existence of $$g$$-elements for a general Artinian reduction of the Stanley–Reisner ring of $$IN(M)$$, was given by Swartz in [34]. Therein, Conjecture 1.1 was also mentioned implicitly. As the set of $$h$$-vectors of independence complexes is strictly contained in the set of $$h$$-vectors of broken circuit complexes (see [6]), Conjecture 1.1 is stronger than and, in particular, implies Hibi’s conjecture. It is worth emphasizing that the techniques of Chari and Swartz for proving Hibi’s conjecture do not work in the case of broken circuit complexes, and thus cannot be used to establish Conjecture 1.1. Indeed, Swartz [34] provided examples of matroids whose broken circuit complexes do not admit $$g$$-elements and hence also fail to have a convex ear decomposition. The main goal of this article is to verify Conjecture 1.1 for matroids representable over a field of characteristic zero. In fact, we prove a somewhat stronger result. We say that a class of matroids $$\mathscr{M}$$ has a certain property (such as unimodal or strongly flawless) if the $$h$$-vector of the broken circuit complex of every matroid in $$\mathscr{M}$$ has that property. The main result of this article is as follows. Theorem 1.2. Let $$\mathscr{M}$$ be a minor-closed class of matroids. If $$\mathscr{M}$$ is unimodal, then it is strongly flawless. □ This theorem implies Conjecture 1.1 for matroids representable over a field of characteristic zero, by virtue of Huh’s log-concavity result [16] (see Corollary 3.5). Let us briefly outline how the proof of Theorem 1.2 proceeds. As mentioned before, a unimodal, flawless sequence is also strongly flawless. So it suffices to show that the $$h$$-vector of $$BC(M)$$ is flawless for every matroid $$M\in\mathscr{M}$$. To this end, we first reduce the proof to the case where $$M$$ is minimally connected (see Lemma 3.1). In this case, $$M$$ contains a removable series class $$S$$ (see Lemma 2.2). We then find two different ways to relate the $$h$$-vector of $$BC(M)$$ to the $$h$$-vector of $$BC(M/S)$$ (see Lemmas 3.2 and 3.3). Combining these comparisons, the flawlessness of the $$h$$-vector of $$BC(M)$$ will follow by induction and the unimodality of the $$h$$-vector of $$BC(M/S)$$. This article is organized as follows. In the next section, we review the basic notions of matroids and broken circuit complexes. Section 3 contains the proof of Theorem 1.2 and its immediate application to Orlik–Terao algebras. Finally, some questions related to our work are discussed in Section 4. 2 Preliminaries 2.1 Matroids The notion of matroids was introduced by Whitney [40] as a common generalization of dependence in linear algebra and graph theory. Since then a rich theory of matroids has been developed which provides a framework for approaching many combinatorial problems. In the following, we collect the needed facts and definitions from matroid theory, referring to the seminal book by Oxley [24] for more details. Definition 2.1. A matroid$$M=(E,\mathscr{I})$$ consists of a finite ground set $$E$$ and a nonempty collection $$\mathscr{I}$$ of subsets of $$E$$, called independent sets, satisfying the following conditions: (i) If $$I\in\mathscr{I}$$ and $$J\subseteq I$$, then $$J\in\mathscr{I}$$. (ii) If $$I,I'\in\mathscr{I}$$ and $$|I|<|I'|$$, then there exists $$e\in I'-I$$ such that $$I\cup e\in\mathscr{I}$$. □ In a matroid $$M=(E,\mathscr{I})$$, a basis is a maximal independent set. A subset of $$E$$ is called dependent if it is not a member of $$\mathscr{I}$$. A circuit is a minimal dependent set, and an $$m$$-circuit is a circuit of cardinality $$m$$. For any set $$X\subseteq E$$, all maximal independent subsets of $$X$$ have the same size, which is called the rank$$r(X)$$ of $$X$$. In particular, the rank of $$E$$, which is the common cardinality of all the bases of $$M$$, is also called the rank of $$M$$ and denoted by $$r(M)$$. A matroid can be specified by either its collection of bases, its collection of circuits, or its rank function. In fact, there are equivalent definitions of matroids in terms of bases, circuits, and rank functions. Two matroids $$M=(E,\mathscr{I})$$ and $$M'=(E',\mathscr{I}')$$ are isomorphic if there exists a bijection $$\varphi:E\to E'$$ such that for every subset $$X$$ of $$E$$, $$X\in\mathscr{I}$$ if and only if $$\varphi(X)\in\mathscr{I}'$$. The prototypical example of a matroid is the vector matroid$$M[A]$$ of a matrix $$A$$: the ground set $$E$$ of $$M[A]$$ is taken to be the set of columns of $$A$$, and a subset $$I\subseteq E$$ is independent if and only if the corresponding columns are linearly independent. A matroid is representable over a field $$K$$ if it is isomorphic to the vector matroid of a matrix over $$K$$. It should be noted, however, that not every matroid is representable over some field; see [24, Proposition 6.1.10]. Let $$M$$ be a matroid on the ground set $$E$$. Let $$\mathscr{B}$$ be the collection of bases of $$M$$. Then $$\mathscr{B}^*=\{E-B : B\in\mathscr{B}\}$$ is also the collection of bases of a matroid $$M^*$$. We call this matroid the dual of $$M$$. For example, $$M[A]^*\cong M[A^*]$$ for any matrix $$A$$, where $$A^*$$ is a matrix whose row space is the orthogonal space of the row space of $$A$$. An element $$e\in E$$ is called a loop if $$\{e\}$$ is a circuit of $$M$$. We say that $$M$$ is loopless if it has no loops. A loop of $$M^*$$ is called a coloop of $$M$$. More generally, circuits of $$M^*$$ are called cocircuits of $$M$$. A series class$$S$$ of $$M$$ is a maximal subset of $$E$$ such that $$S$$ contains no coloops and if $$e,f$$ are distinct elements of $$S$$, then $$\{e,f\}$$ is a cocircuit of $$M$$. A series class is non-trivial if it contains at least two elements. Notice that if $$S$$ is a series class and $$C$$ is a circuit of $$M$$, then either $$C\cap S=\emptyset$$ or $$S\subseteq C$$. This follows from the well-known fact that a circuit and a cocircuit of $$M$$ cannot have just a single element in common; see [24, Proposition 2.1.11]. Let $$X$$ be a subset of $$E$$. The deletion of $$X$$ from $$M$$, denoted $$M-X$$, is the matroid on ground set $$E-X$$ whose independent sets are the independent sets of $$M$$ that are contained in $$E-X$$. The contraction of $$X$$ from $$M$$ is defined to be $$M/X=(M^*-X)^*$$. Note that the operations of deletion and contraction commute, that is, $$(M-X)/Y=M/Y-X$$ for disjoint subsets $$X$$ and $$Y$$ of $$E$$. A minor of $$M$$ is a matroid which can be obtained from $$M$$ by a sequence of deletions and contractions. A class of matroids $$\mathscr{M}$$ is said to be minor-closed if for every $$M\in\mathscr{M}$$, all minors of $$M$$ are also members of $$\mathscr{M}$$. Let $$M_1$$ and $$M_2$$ be matroids on disjoint ground sets $$E_1$$ and $$E_2$$. Their direct sum$$M_1\oplus M_2$$ is the matroid on ground set $$E_1\cup E_2$$ whose independent sets are all possible unions of an independent set of $$M_1$$ with an independent set of $$M_2$$. The direct sum of a finite collection of matroids is then defined by iterating the previous construction. A matroid is called connected if it is not the direct sum of two smaller matroids. Otherwise, it is called disconnected. An arbitrary matroid $$M$$ can be decomposed uniquely (up to ordering) as a direct sum $$M=M_1\oplus\cdots\oplus M_k$$, where $$M_1,\ldots,M_k$$ are connected matroids. In that case, the matroids $$M_1,\ldots,M_k$$ are called the connected components of $$M$$. Let $$M$$ be a connected matroid on $$E$$. Then $$M$$ is called minimally connected if $$M-e$$ is disconnected for every $$e\in E$$. On the other hand, a series class $$S$$ of $$M$$ is said to be removable if $$M-S$$ is connected. Evidently, every removable series class of a minimally connected matroid is non-trivial. For the existence of removable series classes we will need the following result. Lemma 2.2. Let $$M$$ be a connected matroid on the ground set $$E$$ with at least two elements. Then $$M$$ contains a removable series class. In particular, if $$M$$ is minimally connected, then it contains a non-trivial removable series class. □ Proof. If $$M$$ has exactly one series class, then $$E$$ forms a circuit and hence $$E$$ itself is a removable series class of $$M$$. When $$M$$ contains at least two series classes, the result follows from [35, Proposition 5.3]. ■ Let $$M_1$$ and $$M_2$$ be matroids on ground sets $$E_1$$ and $$E_2$$ with $$E_1\cap E_2=\{e\}$$. Assume that $$e$$ is neither a loop nor a coloop of $$M_1$$ or $$M_2$$. Let $$\mathscr{C}(M_i)$$ denote the collection of circuits of $$M_i$$. The parallel connection$$P(M_1,M_2)$$ of $$M_1$$ and $$M_2$$ with respect to $$e$$ is the matroid on $$E_1\cup E_2$$ whose collection of circuits is given by   C(P(M1,M2))=C(M1)∪C(M2)∪{C1∪C2−e:e∈Ci∈C(Mi)  for  i=1,2}. The deletion $$P(M_1,M_2)-e$$ is called the $$2$$-sum of $$M_1$$ and $$M_2$$, denoted by $$M_1\oplus_2M_2$$. Note that the circuits of $$M_1\oplus_2M_2$$ are the circuits of $$P(M_1,M_2)$$ not containing $$e$$; see [24, 3.1.14]. Thus   C(M1⊕2M2)=C(M1−e)∪C(M2−e)∪{C1∪C2−e:e∈Ci∈C(Mi) for i=1,2}. (2) The following simple observation will be useful in Section 3. For brevity’s sake we call a matroid an $$m$$-circuit if its ground set is an $$m$$-circuit. Lemma 2.3. Let $$S$$ be a series class of a matroid $$M$$ with $$|S|=m$$. Set $$\widetilde{M}=M/(S-e)$$ for some $$e\in S$$. Then $$M\cong \widetilde{M}\oplus_2 C$$, where $$C$$ is an $$(m+1)$$-circuit containing $$e$$. □ Proof. By a slight abuse of notation we identify $$C$$ with its ground set. Then we may write $$C=S'\cup e$$, where $$|S'|=|S|$$. Notice that the collection $$\mathscr{C}(\widetilde{M})$$ of circuits of $$\widetilde{M}$$ consists of the minimal nonempty members of $$\mathscr{D}:=\{D-(S-e): D\in \mathscr{C}(M)\}$$; see [24, Proposition 3.1.11]. Since $$S$$ is a series class, either $$D\cap S=\emptyset$$ or $$S\subseteq D$$ for every $$D\in \mathscr{C}(M)$$. Hence, all members of $$\mathscr{D}$$ are minimal and nonempty. This yields   C(M~)=D={D:D∈C(M),D∩S=∅}∪{D−(S−e):D∈C(M),S⊆D}. Now by (2),   C(M~⊕2C) =C(M~−e)∪C(C−e)∪{C∪D−S:D∈C(M),S⊆D} ={D:D∈C(M),D∩S=∅}∪{S′∪(D−S):D∈C(M),S⊆D}. It then follows readily that $$M\cong \widetilde{M}\oplus_2 C$$, as desired. ■ Example 2.4. Let $$M$$ be the cycle matroid of the complete bipartite graph $$K_{2,3}$$, with the edges labelled as in Figure 1(a). Then $$S=\{1,2\}$$ is a series class of $$M$$. The 2-sum of $$\widetilde{M}=M/\{1\}$$ and the 3-circuit $$C=\{2,1',2'\}$$, which is the cycle matroid of the graph depicted in Figure 1(d), is clearly isomorphic to $$M$$. □ Fig. 1. View largeDownload slide $$M \cong \widetilde{M}\oplus_2 C$$. Fig. 1. View largeDownload slide $$M \cong \widetilde{M}\oplus_2 C$$. By iterating, the operation of parallel connection can be defined for special families of more than two matroids. Let $$M_1,\ldots,M_n$$ be matroids on ground sets $$E_1,\ldots,E_n$$ such that $$E_{i+1}\cap(\bigcup_{j=1}^i E_j)=\{e_i\}$$ for $$i=1,\ldots,n-1$$. Here, $$e_1,\ldots,e_{n-1}$$ need not be distinct. Assume further that each $$e_i$$ is neither a loop nor a coloop of the matroids containing it. Then we can form $$P(M_1,M_2)$$, $$P(P(M_1,M_2),M_3)$$, and so on. The last matroid obtained in this way, denoted by $$P(M_1,\ldots,M_n)$$, is called the parallel connection of $$M_1,\ldots,M_n$$ with respect to $$e_1,\ldots,e_{n-1}$$. Assume $$M$$ is a connected matroid on $$E$$. Then $$M$$ is called parallel irreducible at $$e\in E$$ if either $$|E|=1$$ or $$M$$ is not a parallel connection of two smaller matroids with respect to $$e$$. We say that $$M$$ is parallel irreducible if it is parallel irreducible at every element of $$E$$. The following result, which was essentially proved by Brylawski [5, Propositions 5.8, 5.9] (see also [19, Lemma 2.1]), indicates that in certain matroid arguments the general result can be obtained by restricting attention to the parallel irreducible case. Lemma 2.5. Let $$M$$ be a connected matroid on the ground set $$E$$. Then the following statements hold: (i) If $$M=P(M_1,M_2)$$ with respect to $$e$$, then $$M/e$$ is disconnected:   M/e=M1/e⊕M2/e. Conversely, if $$M/e$$ is disconnected, then $$M$$ is a parallel connection of two smaller matroids with respect to $$e$$. Hence, $$M$$ is parallel irreducible if and only if $$M/e$$ is connected for every $$e\in E$$. (ii) $$M$$ admits a decomposition $$M=P(M_1,\ldots,M_n)$$, where each $$M_i$$ is connected and parallel irreducible. □ 2.2 Broken circuit complexes Let $$M$$ be a matroid, whose ground set $$E$$ is endowed with a linear order $$<$$. We further assume that $$M$$ is loopless, since otherwise $$BC(M)=\emptyset$$, which is not interesting for us here. Let $$r=r(M)$$. Then it is well-known that $$BC(M)$$ is an $$(r-1)$$-dimensional shellable simplicial complex; see [26] or [2, 7.4]. Let $$f(M)=(f_0(M),\ldots,f_r(M))$$ be the $$f$$-vector of $$BC(M)$$, where $$f_i(M)$$ is the number of faces of $$BC(M)$$ of cardinality $$i$$. Notice that $$f(M)$$ is independent of the chosen order $$<$$, as is easily seen from the Whitney–Rota formula (1). Define the $$h$$-vector$$h(M)=(h_0(M),\ldots,h_r(M))$$ and the $$h$$-polynomial (or shelling polynomial) $$h(M;t)=\sum_{i=0}^rh_i(M)t^{r-i}$$ of $$BC(M)$$ by the polynomial identity $$h(M;t)=(-1)^r\chi(M;1-t)$$. Thus, the $$f$$-vector and the $$h$$-vector of $$BC(M)$$ are correlated as follows   fi(M)=∑j=0i(r−ji−j)hj(M)  and  hi(M)=∑j=0i(−1)i−j(r−ji−j)fj(M),  i=0,…,r. In the sequel, for convenience, we make the convention that $$h_i(M)=0$$ for $$i<0$$ or $$i>r$$. Moreover, when it is clear from the context which matroid we are referring to, we will just write $$h_i$$ instead of $$h_i(M)$$. Both $$\chi(M;t)$$ and $$h(M;t)$$ are, up to sign, evaluations of the Tutte polynomial$$T(M;x,y)$$ of $$M$$, which is defined by   T(M;x,y)=∑X⊆E(x−1)r(E)−r(X)(y−1)|X|−r(X). Evidently, $$\chi(M;t)=(-1)^rT(M;1-t,0)$$. Hence, $$h(M;t)=T(M;t,0)$$. For later usage we collect here several basis properties of the $$h$$-polynomial of $$BC(M)$$. They follow easily from the corresponding properties of the Tutte polynomial of $$M$$; see [8, 6.2] and [7, p. 182]. Lemma 2.6. Let $$M$$ be a loopless matroid of rank $$r$$ on the ground set $$E$$. Let $$h(M;t)=\sum_{i=0}^rh_it^{r-i}$$ be the $$h$$-polynomial of $$BC(M)$$. Then the following statements hold: (i) $$h_i\geq 0$$ for $$i=0,\ldots,r$$. Moreover, if $$M$$ has $$c$$ connected components, then $$r-c$$ is the largest index $$i$$ such that $$h_{i}\ne0$$. (ii) (Deletion-contraction) Suppose $$|E|\geq 2$$ and $$e\in E$$. Then   h(M;t)={th(M−e;t) if e is a coloop of M,h(M−e;t)+h(M/e;t) otherwise.  Thus, in particular, if $$M$$ is connected, then either $$M-e$$ or $$M/e$$ is connected. (iii) If $$M$$ is an $$(r+1)$$-circuit, then $$h(M;t)=t^r+t^{r-1}+\cdots+t$$ . (iv) Assume that $$M$$ is either the direct sum or the parallel connection of two matroids $$M_1$$ and $$M_2$$. Then   h(M;t)={h(M1;t)h(M2;t) if M=M1⊕M2,t−1h(M1;t)h(M2;t) if M=P(M1,M2).  □ As an important step in the proof of Theorem 1.2, we will relate the $$h$$-vector of $$BC(M)$$ to the $$h$$-vectors of broken circuit complexes of certain minors of $$M$$ which are obtained from $$M$$ by deleting or contracting elements in a series class. For this, the following simple facts will be necessary. Lemma 2.7. Let $$S=\{e_1,\ldots,e_m\}$$ be a series class of a loopless matroid $$M$$. For $$0\le j\le m-1$$, set $$M_j=M/\{e_1,\ldots,e_j\}$$ and $$S_j=\{e_{j+1},\ldots,e_m\}$$. Then the following statements hold: (i) $$r(M-S)=r(M)-m+1$$. (ii) $$r(M_j)=r(M)-j$$, and if $$M$$ is connected, so is $$M_j$$. (iii) $$S_j$$ is a series class of $$M_j$$ and $$M_j-S_j= M-S$$. (iv) For every $$e\in S$$ and $$e'\in S_j$$ the $$h$$-vectors of the broken circuit complexes of the matroids $$M-e,\ M-S$$ and $$M_j-e'$$ coincide. □ Proof. (i) Since, by definition, $$e_1$$ is not a coloop of $$M$$, we have that $$r(M-e_1)=r(M)$$; see, for example, [24, 3.1.5]. Now as every element of $$S_1$$ is a coloop of $$M-e_1$$, it holds that   r(M−S)=r((M−e1)−S1)=r(M−e1)−|S1|=r(M)−m+1. (ii) As $$M_j=M_{j-1}/e_j$$ and $$e_j$$ is not a loop of $$M_{j-1}$$, we have $$r(M_j)=r(M_{j-1})-1$$; see for example, [24, 3.1.7]. In addition, $$M_{j-1}-e_j$$ is not connected since every element of $$S_j$$ is a coloop of this matroid. Hence, by Lemma 2.6(ii), $$M_j$$ is connected if $$M_{j-1}$$ is so. The assertion now follows by induction. (iii) By definition, it is easy to see that $$S_j$$ is a series class of $$M_j$$. Now since $$e_1,\ldots,e_j$$ are coloops of $$M-S_j$$, it follows from [24, Corollary 3.1.25] that   Mj−Sj=(M−Sj)/{e1,…,ej}=(M−Sj)−{e1,…,ej}=M−S. (iv) Since the elements of $$S-e$$ are coloops of $$M-e$$, Lemma 2.6(ii) yields $$h(M-e;t)=t^{m-1}h(M-S;t)$$. Similarly, $$h(M_j-e';t)=t^{m-j-1}h(M_j-S_j;t)$$. As $$M-S=M_j-S_j$$ by (iii), the assertion follows. ■ 3 Flawlessness of $$h$$-vectors of broken circuit complexes This section is devoted to the proof of Theorem 1.2 and its applications. We begin with the following lemma, which is essential for reducing the proof of Theorem 1.2 to the case of minimally connected matroids. Recall that a sequence $$(a_0,a_1,\ldots,a_n)$$ is symmetric if $$a_i=a_{n-i}$$ for $$0\le i\le n$$. Let us say that a polynomial $$a_0t^{n+u}+a_1t^{n+u-1}+\cdots +a_nt^u$$ with $$a_0, a_n\ne 0$$ and $$u\ge0$$ has a certain property (such as symmetric, unimodal or strongly flawless) if its coefficient sequence $$(a_0,a_1,\ldots,a_n)$$ has that property. Lemma 3.1. If $$\varphi(t)$$ and $$\psi(t)$$ are strongly flawless polynomials with nonnegative coefficients, then so is their product. □ Proof. By definition, a polynomial is strongly flawless if and only if its product with any power $$t^u$$ ($$u\ge0$$) is so. Hence without loss of generality we may assume that $$\varphi(t)$$ and $$\psi(t)$$ have the following form:   φ(t) =a0tn+a1tn−1+⋯+an−1t+an,ψ(t) =b0tm+b1tm−1+⋯+bm−1t+bm, where $$a_0,a_n,b_0,b_m> 0$$. We will argue by induction on   dφ,ψ:=|{0≤i≤⌊n/2⌋:ai<an−i}|+|{0≤j≤⌊m/2⌋:bj<bm−j}|. If $$d_{\varphi, \psi}=0$$, then $$\varphi(t)$$ and $$\psi(t)$$ are symmetric polynomials. Observe that for a symmetric polynomial, being strongly flawless is equivalent to being unimodal. So $$\varphi(t)$$ and $$\psi(t)$$ are symmetric and unimodal. It follows that their product $$\varphi(t)\psi(t)$$ is also symmetric and unimodal (see, for example, [33, Proposition 1]). Thus, $$\varphi(t)\psi(t)$$ is strongly flawless, and we are done in this case. Now consider the case $$d_{\varphi, \psi}>0$$. We may suppose that $$a_i<a_{n-i}$$ for some $$0\le i\le \lfloor n/2\rfloor$$. Set $$k:=\min\{0\le i\le \lfloor n/2\rfloor: a_i<a_{n-i}\}$$. Let $$\overline{\varphi}(t)$$ be the polynomial obtained from $$\varphi(t)$$ by replacing the term $$a_{n-k}t^k$$ of $$\varphi(t)$$ with $$a_{k}t^k$$, that is, $$\overline{\varphi}(t)=\varphi(t)+(a_k-a_{n-k})t^{k}.$$ Then it is readily seen that $$\overline{\varphi}(t)$$ is strongly flawless. Moreover, $$d_{\overline{\varphi}, \psi}=d_{\varphi, \psi}-1$$. Writing $$\varphi(t)\psi(t)=\sum_{i=0}^{m+n}c_it^{m+n-i}$$ and $$\overline{\varphi}(t)\psi(t)=\sum_{i=0}^{m+n}c'_it^{m+n-i}$$, we get   ci=∑u+v=iaubv={ci′if  i<n−k or  i>m+n−k,ci′+(an−k−ak)bi+k−notherwise.  (3) Since $$a_{n-k}>a_k$$ and the coefficients of $$\psi(t)$$ are nonnegative, it holds that $$c_i\ge c'_i$$ for all $$i$$. Now let $$0\le i\le j\le m+n-i$$. We have to show that $$c_i\le c_j$$. Note that $$c'_i\le c'_j$$ by induction. So, if $$i<n-k$$, then it follows from (3) that $$c_i=c'_i\le c'_j\le c_j$$. Now suppose $$i\ge n-k$$. Then $$i\ge k$$ since $$k\le \lfloor n/2\rfloor$$. Hence $$j\le m+n-i \le m+n-k$$. Again by (3) we have   cj−ci=cj′−ci′+(an−k−ak)(bj+k−n−bi+k−n)≥(an−k−ak)(bj+k−n−bi+k−n). Thus, the inequality $$c_i\le c_j$$ will be confirmed once we have shown that $$b_{i+k-n}\le b_{j+k-n}$$. But the last inequality holds since $$0\le i+k-n\le j+k-n\le m-(i+k-n)$$ (which follows easily from $$n-k\le i\le j\le m+n-i$$ and $$k\le \lfloor n/2\rfloor$$) and $$\psi(t)$$ is strongly flawless. This completes the proof. ■ In the sequel, for our purposes, it will be convenient to consider $$h$$-vectors with zero entries at the end removed. So, if we say that $$h(M)=(h_0(M),h_1(M),\ldots,h_s(M))$$ is the $$h$$-vector of $$BC(M)$$, then $$s$$ is the largest index $$i$$ with $$h_i(M)\ne 0$$. In this case, recall from Lemma 2.6(i) that $$s=r-c$$, where $$r=r(M)$$ and $$c$$ is the number of connected components of $$M$$. Now let $$M$$ be a loopless matroid and let $$h(M)=(h_0(M),h_1(M),\ldots,h_s(M))$$ be the $$h$$-vector of $$BC(M)$$. Define   h¯i(M):={hs−i(M)−hi(M)for  0≤i≤⌊s/2⌋,0otherwise.  Following Swartz [36], we call $$\bar h(M):=(\bar h_0(M),\bar h_1(M),\ldots,\bar h_{\lfloor s/2\rfloor}(M))$$ the complementary $$h$$-vector of $$BC(M)$$. For convenience we set $$\bar h(M)=(0)$$ if $$M$$ contains a loop. The next two lemmas present two different interpretations of the complementary $$h$$-vector of $$BC(M)$$ which involve the $$h$$-vector of $$BC(M/S)$$, where $$S$$ is a (removable) series class of $$M$$. Recall our convention that $$h_i(M/S)=0$$ for $$i<0$$ or $$i>r(M/S)$$. Lemma 3.2. Let $$M$$ be a connected matroid and $$S$$ a non-trivial removable series class of $$M$$ with $$|S|=m$$. Let $$h(M)=(h_0(M),h_1(M),\ldots,h_s(M))$$ be the $$h$$-vector of $$BC(M)$$. Then for every $$e\in S$$ and $$0\le i\le \lfloor s/2\rfloor$$,   h¯i(M)=h¯i(M/e)+h¯i−m+1(M−S)+(hi−m+1(M/S)−hi−m(M/S)). □ Proof. If $$M$$ is a $$2$$-circuit, then the statement is easily seen to be true. So assume that $$M$$ is not a $$2$$-circuit. Suppose $$S=\{e_1,\ldots,e_m\}$$ with $$e=e_1$$. Set $$M_j=M/\{e_1,\ldots,e_j\}$$ for $$j=1,\ldots,m$$. We will show via induction that   h¯i(M) =h¯i(M/e1)+h¯i−m+1(M−S)+(hi−j+1(Mj)−hi−j(Mj)) +(hi−m+1(M−S)−hi−j+1(M−S)) (4) for $$j=1,\ldots,m$$. The case $$j=m$$ then gives the desired assertion. Using the deletion-contraction formula (Lemma 2.6(ii)) and Lemma 2.7(iv), we have   h¯i(M) =hs−i(M)−hi(M) =(hs−i(M−e1)+hs−i−1(M/e1))−(hi(M−e1)+hi−1(M/e1)) =(hs−i−1(M/e1)−hi(M/e1))+(hs−i(M−S)−hi−m+1(M−S)) +(hi(M/e1)−hi−1(M/e1))+(hi−m+1(M−S)−hi(M−S)). (5) By Lemma 2.7(ii), $$M/e_1$$ is connected and $$r(M/e_1)=r(M)-1$$. Thus, in particular, $$M/e_1$$ is loopless since $$M$$ is not a 2-circuit. So from Lemma 2.6(i) it follows that $$h_{s-1}(M/e_1)\ne 0$$, and hence $$\bar h_i(M/e_1)=h_{s-i-1}(M/e_1)-h_{i}(M/e_1)$$. Similarly, as $$M-S$$ is connected and $$r(M-S)=r(M)-m+1$$ (see Lemma 2.7(i)), it holds that $$h_{s-m+1}(M-S)\ne 0$$ and $$\bar h_{i-m+1}(M-S)=h_{s-i}(M-S)-h_{i-m+1}(M-S)$$. Thus (5) implies that (4) is true for $$j=1$$. To complete the induction argument, it suffices to show that    (hi−j+1(Mj)−hi−j(Mj))+(hi−m+1(M−S)−hi−j+1(M−S)) =(hi−j(Mj+1)−hi−j−1(Mj+1))+(hi−m+1(M−S)−hi−j(M−S)), or equivalently,   hi−j+1(Mj)−hi−j(Mj)= (hi−j(Mj+1)+hi−j+1(M−S)) −(hi−j−1(Mj+1)+hi−j(M−S)). But the last equality follows from the deletion-contraction formula, since $$M_{j+1}=M_j/e_{j+1}$$ and $$h_k(M-S)=h_k(M_j-e_{j+1})$$ (by Lemma 2.7(iv)). This finishes the proof. ■ Lemma 3.3. Let $$M$$ be a connected matroid and $$S$$ a series class of $$M$$ with $$|S|=m$$. Set $$\widetilde{M}=M/(S-e)$$ for some $$e\in S$$. Let $$h(M)=(h_0(M),h_1(M),\ldots,h_s(M))$$ be the $$h$$-vector of $$BC(M)$$. Then   h¯i(M)={∑j=0min{i,s−m−i}h¯j(M~)+∑j=1i(hi−j(M/S)−hs−m+1−j(M/S)) if  0≤i≤min{m−1,s−m+1},∑j=i−m+1min{i,s−m−i}h¯j(M~)+∑j=1m−1(hi−j(M/S)−hs−i−j(M/S)) if  m−1≤s−m+1  and  m−1≤i≤⌊s/2⌋,0 if  s−m+1≤m−1  and  s−m+1≤i≤⌊s/2⌋.  □ Proof. Note that $$\widetilde{M}$$ is connected by Lemma 2.7(ii). So $$\widetilde{M}$$ contains a loop if and only if it is itself a loop, which means that $$M$$ is a circuit. Since the lemma is clearly true in this case, we may henceforth assume that $$\widetilde{M}$$ is loopless. By Lemma 2.3, $$M\cong P(\widetilde{M}, C)-e$$, where $$C$$ is an $$(m+1)$$-circuit containing $$e$$. Thus, the deletion-contraction formula, Lemma 2.5(i) and Lemma 2.6(iv) yield   h(M;t) =h(P(M~,C);t)−h(P(M~,C)/e;t) =h(P(M~,C);t)−h(M~/e⊕C/e;t) =h(M~;t)h(C;t)t−h(M/S;t)h(C/e;t). (6) Since $$r(\widetilde{M})=r(M)-m+1=s-m+2$$ (see Lemma 2.7(ii)) and $$r(M/S)=r(\widetilde{M})-1=s-m+1$$, we may write   h(M~;t) =h0(M~)ts−m+2+h1(M~)ts−m+1+⋯+hs−m+1(M~)t and h(M/S;t) =h0(M/S)ts−m+1+h1(M/S)ts−m+⋯+hs−m(M/S)t. Plugging these polynomials into (6) and using Lemma 2.6(iii) we get   h(M;t)=(∑j=0s−m+1hj(M~)ts−m+2−j)(∑k=0m−1tk)−(∑j=0s−mhj(M/S)ts−m+1−j)(∑k=1m−1tk). (7) From this formula we will derive formulas for the coefficients of $$h(M;t)$$, and thereby obtain the desired formula for the complementary $$h$$-vector. We distinguish two cases: $${\sf Case 1:}$$$$m-1\le s-m+1$$. Note that $$h_i(M)$$ is the coefficient of $$t^{s-i+1}$$ in $$h(M;t)$$. So from (7) we get   hi(M)={∑j=0ihj(M~)−∑j=0i−1hj(M/S) for  i≤m−1,∑j=i−m+1ihj(M~)−∑j=i−m+1i−1hj(M/S) for  m−1≤i≤s−m+1,∑j=i−m+1s−m+1hj(M~)−∑j=i−m+1s−mhj(M/S) for  s−m+1≤i≤s.  (8) As $$\widetilde{M}$$ is loopless and connected, it follows from Lemma 2.6(i) that $$h_{s-m+1}(\widetilde{M})\ne 0$$. Thus $$\bar h_{j}(\widetilde{M})=h_{s-m+1-j}(\widetilde{M})-h_{j}(\widetilde{M})$$ for $$0\le j\le \lfloor \frac{s-m+1}{2}\rfloor$$. Now it is readily seen from (8) that   h¯i(M)=hs−i(M)−hi(M)={∑j=0min{i,s−m−i}h¯j(M~)+∑j=1i(hi−j(M/S)−hs−m+1−j(M/S)) for  0≤i≤m−1,∑j=i−m+1min{i,s−m−i}h¯j(M~)+∑j=1m−1(hi−j(M/S)−hs−i−j(M/S)) for  m−1≤i≤⌊s/2⌋.  $${\sf Case 2:}$$$$s-m+1< m-1$$. In this case, (7) gives   hi(M)={∑j=0ihj(M~)−∑j=0i−1hj(M/S) for  i≤s−m+1,∑j=0s−m+1hj(M~)−∑j=0s−mhj(M/S) for  s−m+1≤i≤m−1,∑j=i−m+1s−m+1hj(M~)−∑j=i−m+1s−mhj(M/S) for  m−1≤i≤s.  Hence   h¯i(M)={∑j=0min{i,s−m−i}h¯j(M~)+∑j=1i(hi−j(M/S)−hs−m+1−j(M/S)) for  0≤i≤s−m+1,0 for  s−m+1≤i≤⌊s/2⌋.  The desired formula for $$\bar h_i(M)$$ is obtained by combining the two cases above. ■ Example 3.4. Let us revisit the cycle matroid $$M$$ of the complete bipartite graph $$K_{2,3}$$ discussed in Example 2.4. Notice that the series class $$S=\{1,2\}$$ of $$M$$ is removable. The graphs corresponding to the minors $$\widetilde{M}=M/\{1\},\ M-S,\ M/S$$ of $$M$$ are depicted in Figure 2. Using Lemma 2.6 one easily finds that $$h(M/S;t)=t^2,$$$$h(M-S;t)=t^3+t^2+t,$$$$h(\widetilde{M};t)=t^3+2t^2+t$$, and $$h(M;t)=t^4+2t^3+3t^2+t$$. Thus $$\bar h(M)=(1-1,3-2)=(0,1)$$. This agrees with the computation of $$\bar h(M)$$ using Lemma 3.2 or Lemma 3.3. For example, by Lemma 3.2,   h¯1(M)=h¯1(M/{1})+h¯0(M−S)+(h0(M/S)−h−1(M/S))=0+0+(1−0)=1. On the other hand, by Lemma 3.3,   h¯1(M)=h¯0(M~)+(h0(M/S)−h1(M/S))=0+(1−0)=1. □ Fig. 2. View largeDownload slide Minors of $$M$$ related to removable series class $$S=\{1,2\}$$. Fig. 2. View largeDownload slide Minors of $$M$$ related to removable series class $$S=\{1,2\}$$. We are now ready to prove our main result. Proof of Theorem 1.2. Let $$M\in\mathscr{M}$$ and let $$h(M)=(h_0(M),h_1(M),\ldots,h_s(M))$$ be the $$h$$-vector of $$BC(M)$$. Since $$h(M)$$ is unimodal by assumption, it suffices to prove that $$h(M)$$ is flawless, that is, the complementary $$h$$-vector of $$BC(M)$$ is nonnegative. We proceed by induction on the cardinality of the ground set $$E$$ of $$M$$. If $$|E|=1$$, then $$h(M)=(1)$$ and we have nothing to prove. So suppose $$|E|\ge2$$. We first show that we can reduce to the case where $$M$$ is minimally connected. By Lemmas 2.5(ii), 2.6(iv) and 3.1, we may assume that $$M$$ is connected, and furthermore, parallel irreducible. Thus, by Lemma 2.5(i), $$M/e$$ is connected for every $$e\in E$$. We will show that $$\bar h_i(M)\ge 0$$ for $$0\le i\le \lfloor s/2\rfloor$$ if there exists $$e\in E$$ with $$M-e$$ connected. Indeed, if $$s$$ is even and $$i=s/2$$, then $$\bar h_i(M)=0$$. Now assume that $$s$$ is odd or $$i<s/2$$. Then $$i\le \lfloor (s-1)/2\rfloor$$. Using the deletion-contraction formula we have   h¯i(M) =hs−i(M)−hi(M) =(hs−i(M−e)+hs−i−1(M/e))−(hi(M−e)+hi−1(M/e)) =(hs−i(M−e)−hi(M−e))+(hs−i−1(M/e)−hi(M/e)) +(hi(M/e)−hi−1(M/e)) =h¯i(M−e)+h¯i(M/e)+(hi(M/e)−hi−1(M/e)). (9) The last equality follows since $$M-e$$ and $$M/e$$ are connected. By the induction hypothesis, the $$h$$-vectors of $$BC(M-e)$$ and $$BC(M/e)$$ are strongly flawless, implying that each summand of $$\bar h_i(M)$$ in the last row of (9) is nonnegative. Therefore, $$\bar h_i(M)$$ is nonnegative as well. Henceforth we may assume that $$M$$ is minimally connected. Then $$M$$ contains a non-trivial removable series class by Lemma 2.2. Let $$S$$ be such a series class of $$M$$ with $$|S|=m$$. Given $$0\le i\le \lfloor s/2\rfloor$$, let us verify that $$\bar h_i(M)\ge 0$$. If $$i\le m-1$$, then $$\bar h_i(M)\ge 0$$ by Lemma 3.2 and the induction hypothesis. Now consider the case $$i>m-1$$. Since $$i\le \lfloor s/2\rfloor$$, we must have $$m-1<s-m+1$$. It then follows from Lemmas 3.2, 3.3 and the induction hypothesis that   h¯i(M)≥max{hi−m+1(M/S)−hi−m(M/S),∑j=1m−1(hi−j(M/S)−hs−i−j(M/S))}. Thus, if $$h_{i-m+1}(M/S)\ge h_{i-m}(M/S)$$, then $$\bar h_i(M)\ge 0$$. Suppose now that $$h_{i-m}(M/S)>h_{i-m+1}(M/S)$$. Then the unimodality of the $$h$$-vector of $$BC(M/S)$$ yields   hi−m+1(M/S)≥⋯≥hi−1(M/S)≥⋯≥hs−i−1(M/S). It follows that for $$1\le j \le m-1$$, we have $$h_{i-j}(M/S)\ge h_{s-i-j}(M/S)$$, because $$i-m+1\le i-j\le s-i-j\le s-i-1$$. Hence   ∑j=1m−1(hi−j(M/S)−hs−i−j(M/S))≥0, which also implies that $$\bar h_i(M)\ge 0$$. The proof is complete. ■ As a consequence of Theorem 1.2, we verify Conjecture 1.1 for matroids representable over a field of characteristic zero. Corollary 3.5. Let $$M$$ be a matroid representable over a field of characteristic zero. Then the $$h$$-vector of $$BC(M)$$ is strongly flawless. □ Proof. Let $$\mathscr{M}$$ be the class of matroids representable over a field of characteristic zero. Then it is well-known that $$\mathscr{M}$$ is minor-closed; see [24, Proposition 3.2.4]. Moreover, it follows from Huh’s log-concavity result [16, Theorem 3] that $$\mathscr{M}$$ is unimodal. So $$\mathscr{M}$$ is strongly flawless by Theorem 1.2. ■ Let us now derive an application of Corollary 3.5 to Orlik–Terao algebras. Recall that a (central) complex hyperplane arrangement$$\mathscr{A}=\{H_1,\ldots,H_n\}$$ is a collection of hyperplanes in $$\mathbb{C}^{r}$$, all of which contain the origin of $$\mathbb{C}^{r}$$. Suppose each hyperplane $$H_i$$ of $$\mathscr{A}$$ is given as the kernel of a linear form $$\alpha_i$$. Then the Orlik–Terao algebra of $$\mathscr{A}$$ is defined to be the $$\mathbb{C}$$-algebra generated by reciprocals of the $$\alpha_i$$’s:   C(A):=C[1/α1,…,1/αn]. This algebra was introduced by Orlik and Terao in [23]. Since then it has appeared in different contexts and received considerable attention; see for example, [4, 11, 18, 20–22, 25, 29–31, 37]. An interesting property of $$C(\mathscr{A})$$ is that it degenerates flatly to the Stanley–Reisner ring of the broken circuit complex of the underlying matroid$$M(\mathscr{A})$$ of $$\mathscr{A}$$ [25, Theorem 4]. Thus, in particular, $$C(\mathscr{A})$$ is a Cohen–Macaulay ring and its $$h$$-vector coincides with the $$h$$-vector of $$BC(M(\mathscr{A}))$$. Recall that the underlying matroid $$M(\mathscr{A})$$ is defined to be the matroid on ground set $$\mathscr{A}$$ such that a subset $$B = \{H_{ i_ 1},\ldots , H _{i_ p}\}$$ of $$\mathscr{A}$$ is independent if and only if the corresponding linear forms $$\alpha_{i_1}, \ldots,\alpha_{i_p}$$ are linearly independent. Evidently, $$M(\mathscr{A})$$ is representable over $$\mathbb{C}$$. So from Corollary 3.5 we immediately get the following: Corollary 3.6. Let $$\mathscr{A}$$ be a complex hyperplane arrangement. Then the $$h$$-vector of the Orlik–Terao algebra of $$\mathscr{A}$$ is strongly flawless. □ It should be noted here that $$C(\mathscr{A})$$ has a canonical linear system of parameters [25, Proposition 7] and that, similar to Swartz’s examples mentioned in the introduction, the corresponding Artinian reduction of $$C(\mathscr{A})$$ needs not have $$g$$-elements [25, Remark 8]. It would therefore be difficult to provide an algebraic proof of the above corollary. 4 Concluding remarks In view of our main result (Theorem 1.2), Conjecture 1.1 would follow from the first one of the following successively stronger conjectured assertions: Conjecture 4.1. Let $$h(M)=(h_0,h_1,\ldots,h_s)$$ be the $$h$$-vector of the broken circuit complex of a matroid $$M$$. Set $$h'_i=h_i/\binom{h_1+i-1}{i}$$ for $$i=0,1,\ldots, s$$. Then (i) $$h(M)$$ is unimodal. (ii) $$h(M)$$ is log-concave. (iii) $$h(M)$$ is strongly log-concave, that is the sequence $$(h'_0,h'_1,\ldots,h'_s)$$ is log-concave. □ This still wide open conjecture was proposed by Brylawski [7, p. 232]. Therein, he also showed that Conjecture 4.1(ii) is stronger than Rota–Heron’s conjecture [13, 28] and Welsh’s conjecture [38]. As we mentioned before, significant progress towards proving Conjecture 4.1(ii) was made by Huh [16], who verified it for matroids representable over a field of characteristic zero. Concerning Conjecture 1.1 it is also worth noting the following question: Question 4.2. Let $$M$$ be a matroid and let $$h(M)=(h_0,h_1,\ldots,h_s)$$ be the $$h$$-vector of $$BC(M)$$, where $$h_s\neq 0$$. Define $$g(M)=(1,h_1-h_0,\ldots,h_{\lfloor s/2\rfloor}-h_{\lfloor s/2\rfloor -1})$$ to be the $$g$$-vector of $$BC(M)$$. Is it always true that $$g(M)$$ is an $$O$$-sequence? □ This question together with Conjecture 1.1 was posed by Swartz in [34], where he gave an affirmative answer to the question in the case of independence complexes. We believe that this question should also have an affirmative answer for broken circuit complexes in general. However, we would like to remark that it is not clear whether the question can be reduced to the case of parallel irreducible matroids. For this, one would, in analogy with Lemma 3.1, need that the property of the $$g$$-vector being an $$O$$-sequence is preserved under taking products. Currently, in joint work with Uwe Nagel, the first author is investigating this problem. Funding The first author was supported by the German Research Council DFG-GRK, 1916. Acknowledgments We wish to thank Ed Swartz for pointing out that Conjecture 4.1 is due to Brylawski. We are also grateful to the referees, whose thoughtful comments and useful suggestions led to several improvements in the article. References [1] Adiprasito K. Huh J. and Katz. E. Hodge theory for combinatorial geometries , arXiv:1511.02888 (accessed on November 11, 2016). [2] Björner A. The homology and shellability of matroids and geometric lattices. In Matroid Applications , vol. 40, 226– 83. Encyclopedia of Mathematics and its Applications. Edited by White. N. Cambridge: Cambridge University Press, 1992. Google Scholar CrossRef Search ADS   [3] Boij M. Migliore J. Miró-Roig R. Nagel U. and Zanello. F. “On the shape of a pure O-sequence.” Memoirs of the American Mathematical Society  218, no. 1024. American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA, ( 2012). Google Scholar CrossRef Search ADS   [4] Braden T. and Proudfoot. N. “The hypertoric intersection cohomology ring.” Inventiones mathematicae  177, no. 2 ( 2009): 337– 79. Google Scholar CrossRef Search ADS   [5] Brylawski T. “A combinatorial model for series–parallel networks.” Transactions of the American Mathematical Society  154 ( 1971): 1– 22. Google Scholar CrossRef Search ADS   [6] Brylawski T. “The broken-circuit complex.” Transactions of the American Mathematical Society  234 ( 1977): 417– 33. Google Scholar CrossRef Search ADS   [7] Brylawski T. The Tutte polynomial. General. I. theory. In Matroid Theory and its Applications , edited by Barlotti A. 125– 275, Naples: Liguori, 1982. Google Scholar CrossRef Search ADS   [8] Brylawski T. and Oxley. J. The Tutte polynomial and its applications. In Matroid Applications , edited by White. N. vol. 40, 123– 225, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, 1992. Google Scholar CrossRef Search ADS   [9] Chari M. K. “Two decompositions in topological combinatorics with applications to matroid complexes.” Transactions of the American Mathematical Society  349 ( 1997): 3925– 943. Google Scholar CrossRef Search ADS   [10] Crapo H. “A higher invariant for matroids.” Journal of Combinatorial Theory  2 ( 1967): 406– 17. Google Scholar CrossRef Search ADS   [11] Denham G. Garrousian M. and Tohăneanu. S. “Modular decomposition of the Orlik–Terao algebra.” Annals of Combinatorics  18, no. 2 ( 2014): 289– 312. Google Scholar CrossRef Search ADS   [12] Eisenbud D. Popescu S. and Yuzvinsky. S. “Hyperplane arrangement cohomology and monomials in the exterior algebra.” Transactions of the American Mathematical Society  355 ( 2003): 4365– 383. Google Scholar CrossRef Search ADS   [13] Heron A. P. Matroid polynomials.  In Combinatorics (Proceedings of the Conference on Combinatorial Mathematics, Mathematical Institute, Oxford, 1972), edited by Welsh D. Woodall D. 164– 202, Southend-on-Sea: The Institute of Mathematics and its Applications, 1972. [14] Hibi T. “What can be said about pure O-sequences?” Journal of Combinatorial Theory, Series A  50 ( 1989): 319– 22. Google Scholar CrossRef Search ADS   [15] Hibi T. “Face number inequalities for matroid complexes and Cohen–Macaulay types of Stanley–Reisner rings of distributive lattices.” Pacific Journal of Mathematics  154 ( 1992): 253– 64. Google Scholar CrossRef Search ADS   [16] Huh J. “h-vectors of matroids and logarithmic concavity.” Advances in Mathematics  270 ( 2015): 49– 59. Google Scholar CrossRef Search ADS   [17] Kämpf G. and Römer. T. “Homological properties of Orlik–Solomon algebras.” Manuscripta Mathematica  129 ( 2009): 181– 210. Google Scholar CrossRef Search ADS   [18] Le D. V. “On the Gorensteinness of broken circuit complexes and Orlik–Terao ideals.” Journal of Combinatorial Theory, Series A  123, no. 1 ( 2014): 169– 85. Google Scholar CrossRef Search ADS   [19] Le D. V. “Broken circuit complexes of series–parallel networks.” European Journal of Combinatorics  51 ( 2016): 12– 36. Google Scholar CrossRef Search ADS   [20] Le D. V. and Mohammadi. F. “On the Orlik–Terao ideal and the relation space of a hyperplane arrangement.” Advances in Applied Mathematics  71 ( 2015): 34– 51. Google Scholar CrossRef Search ADS   [21] Le D. V. and Römer. T. “Broken circuit complexes and hyperplane arrangements.” Journal of Algebraic Combinatorics  38, no. 4 ( 2013): 989– 1016. Google Scholar CrossRef Search ADS   [22] McBreen M. and Proudfoot. N. “Intersection cohomology and quantum cohomology of conical symplectic resolutions.” Algebraic Geometry  2, no. 5 ( 2015): 623– 41. Google Scholar CrossRef Search ADS   [23] Orlik P. and Terao. H. “Commutative algebras for arrangements.” Nagoya Mathematical Journal  134 ( 1994): 65– 73. Google Scholar CrossRef Search ADS   [24] Oxley J. Matroid Theory , vol. 21, 2nd ed., Oxford Graduate Texts in Mathematics. Oxford: Oxford University Press, 2011. [25] Proudfoot N. and Speyer. D. “A broken circuit ring.” Beiträge zur Algebra und Geometrie  47, no. 1 ( 2006): 161– 6. [26] Provan J. S. Decompositions, Shellings, and Diameters of Simplicial Complexes and Convex Polyhedra . Thesis, Cornell University, 1977. [27] Rota G.-C. “On the foundations of combinatorial theory. I. Theory of Möbius functions.” Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete.  2 ( 1964): 340– 68. Google Scholar CrossRef Search ADS   [28] Rota G.-C. “Combinatorial theory, old and new.” In: Proceedings of the International Congress of Mathematicians  (Nice, 1970), 229– 233. Paris: Gauthier-Villars, 1971. [29] Sanyal R. Sturmfels B. and Vinzant. C. “The entropic discriminant.” Advances in Mathematics  244 ( 2013): 678– 707. Google Scholar CrossRef Search ADS   [30] Schenck H. “Resonance varieties via blowups of $${\mathbb P}^2$$ and scrolls.” International Mathematics Research Notices  20 ( 2011): 4756– 778. [31] Schenck H. and Tohăneanu. Ş. “The Orlik–Terao algebra and 2-formality.” Mathematical Research Letters  16 ( 2009): 171– 82. Google Scholar CrossRef Search ADS   [32] Stanley. R. P. “Cohen–Macaulay complexes. In Higher Combinatorics , edited by Aigner M. 51– 62. Dordrecht: Reidel, 1977. Google Scholar CrossRef Search ADS   [33] Stanley R. P. “Log-concave and unimodal sequences in algebra, combinatorics, and geometry.” In Graph Theory and Its Applications,  edited by Capobianco M. Guan M.G. Hsu D.F. and Tian F. 576, 500– 35, New York Academy of Sciences, New York, 1989. Google Scholar CrossRef Search ADS   [34] Swartz E. “g-elements of matroid complexes.” Journal of Combinatorial Theory, Series B  88, no. 2 ( 2003): 369– 75. Google Scholar CrossRef Search ADS   [35] Swartz E. “Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes.” SIAM Journal on Discrete Mathematics  18, no. 3 ( 2004/05): 647– 61. Google Scholar CrossRef Search ADS   [36] Swartz E. “g-Elements, finite buildings and higher Cohen–Macaulay connectivity.” Journal of Combinatorial Theory, Series A  113, no. 7 ( 2006): 1305– 20. Google Scholar CrossRef Search ADS   [37] Terao H. “Algebras generated by reciprocals of linear forms.” Journal of Algebra  250 ( 2002): 549– 58. Google Scholar CrossRef Search ADS   [38] Welsh D. Matroid Theory , London: Academic Press, 1976. [39] Whitney H. “A logical expansion in mathematics.” Bulletin of the American Mathematical Society  38 ( 1932): 572– 79. Google Scholar CrossRef Search ADS   [40] Whitney H. “On the abstract properties of linear dependence.” American Journal of Mathematics  57, no. 3 ( 1935): 509– 33. Google Scholar CrossRef Search ADS   [41] Zaslavsky T. The Möbius function and the characteristic polynomial.  In Combinatorial Geometries , vol. 29, 114– 38, Encyclopedia of Mathematics and its Applications, edited by White N. Cambridge, Cambridge University Press, 1987. © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]

Bertola, Marco;Dubrovin, Boris;Yang, Di

doi: 10.1093/imrn/rnw285pmid: N/A

Abstract For a simple Lie algebra $$\mathfrak{g}$$ we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of $$\mathfrak{g}$$-type. The dimension of the space of solutions regular at infinity is equal to the rank of the Lie algebra. For the simplest example $$\mathfrak{g}=sl_2(\mathbb C)$$ the regular solution can be expressed via products of Airy functions and their derivatives; this matrix-valued function was used in our previous work [4] for computing logarithmic derivatives of the Witten–Kontsevich tau-function. For an arbitrary simple Lie algebra we construct a basis in the space of regular solutions to the topological equation called generalized Airy resolvents. We also outline applications of the generalized Airy resolvents for computing the Witten and Fan–Jarvis–Ruan invariants of the Deligne–Mumford moduli spaces of stable algebraic curves. 1 Introduction In this paper to an arbitrary simple Lie algebra we will associate two families of special functions defined by certain systems of linear ODEs with polynomial coefficients along with appropriate asymptotic conditions. Let us begin with the first family. 1.1 Topological ODE and generalized Airy resolvents Let $$(\mathfrak{g},[\cdot,\cdot])$$ be a simple Lie algebra over $$\mathbb{C}$$ of rank $$n$$. We sometimes refer to elements of $$\mathfrak{g}$$ as matrices, even if the constructions will not depend on the choice of a matrix realization of the Lie algebra. Choose $$\mathfrak{h}$$ a Cartan subalgebra of $$\mathfrak{g}$$, $$\Pi=\{\alpha_1,\dots,\alpha_n\}\subset\mathfrak{h}^*$$ a set of simple roots, and $$\triangle\subset \mathfrak{h}^*$$ the root system. Then $$\mathfrak{g}$$ admits the root space decomposition   g=h⊕⨁α∈△gα, (1.1) where $$\mathfrak{g}_\alpha=\{x\in \mathfrak{g} \, | \,[h,x]=\alpha(h)\,x, \, \forall \, h\in \mathfrak{h}\}.$$ Let $$\theta$$ be the highest root w.r.t. $$\Pi$$. Denote by $$(\cdot|\cdot): \mathfrak{g} \times\mathfrak{g}\rightarrow \mathbb{C}$$ the normalized Cartan–Killing form [24] such that $$(\theta|\theta)=2.$$ For any root $$\alpha$$, let $$H_{\alpha}\in \mathfrak{h}$$ denote the root vector of $$\alpha$$ w.r.t. $$(\cdot|\cdot)$$. Choose a set of Weyl generators $$E_i\in\mathfrak{g}_{\alpha_i},\,F_i\in\mathfrak{g}_{-\alpha_i}$$, $$H_i={2H_{\alpha_i}}/{(\alpha_i|\alpha_i)}$$ satisfying   [Ei,Fi]=Hi,[Hi,Ej]=AijEj,[Hi,Fj]=−AijFj, (1.2) where $$(A_{ij})$$ denotes the Cartan matrix associated with $$\mathfrak{g}$$. In (1.2) and throughout this paper we will follow the convention that, free Latin indices take integer values from $$1$$ to $$n$$ unless otherwise indicated. Choose $$E_{-\theta}\in \mathfrak{g}_{-\theta},\, E_\theta \in\mathfrak{g}_\theta$$. They can be normalized by the conditions $$(E_\theta\,|\,E_{-\theta})=1$$ and $$\omega(E_{-\theta})=-E_\theta$$, where $$\omega:\mathfrak{g}\to\mathfrak{g}$$ is the Chevalley involution. Let $$I_+:=\sum_{i=1}^n E_i$$ be a principal nilpotent element of $$\mathfrak{g}$$. Define   Λ=I++λE−θ. (1.3) Consider the following differential equation for a $$\mathfrak{g}$$-valued function $$M=M(\lambda)$$ of an independent variable $$\lambda$$  M′=[M,Λ],′=ddλ. (1.4) Definition 1.1. We call eq. (1.4) the topological differential equation (in short: topological ODE) of $$\mathfrak{g}$$-type. Solutions of the differential equation (1.4) are entire functions of the complex variable $$\lambda$$. At $$\lambda=\infty$$ they may have a singularity. A solution $$M(\lambda)$$ is called regular if it grows at most polynomially at $$|\lambda|\to\infty$$ within a certain chosen sector of the complex $$\lambda$$-plane. Denote by $$\mathfrak{S}_\infty^{{\rm reg}}(\mathfrak{g})$$ the vector space of regular solutions to (1.4). Our goal is to describe the space $$\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})$$ for an arbitrary simple Lie algebra $$\mathfrak{g}$$. Actually, we will only deal with asymptotic expansions of regular solutions without entering into details about the region of their validity that can be described by means of standard techniques of asymptotic analysis of ODEs (see, e.g., [34]). As a way of introductory example we consider now the simplest instance $$\mathfrak{g}=sl_2(\mathbb C)$$. In this case $$\Lambda=\left({0 \atop \lambda} \quad {1 \atop 0}\right)$$. Write $$M(\lambda)=\left({a(\lambda) \atop c(\lambda)}\quad {b(\lambda) \atop -a(\lambda)}\right)$$. The equation (1.4) reads   (a′b′c′)=(0λ−1200−2λ00)(abc). (1.5) The type of behavior at infinity is essentially specified by the eigenvalues of the matrix of coefficients. In this case they are equal to 0 and $$\pm 2\sqrt{\lambda}$$. Solutions corresponding to the non-zero eigenvalues have an essential singularity $$\sim \lambda^{-\frac12}{\rm e}^{\pm \frac43 \lambda^{3/2}}$$ at infinity. The regular solution is unique up to a constant factor. It corresponds to the eigenvector with eigenvalue zero. The solution has the form   M(λ)=λ−122(−12∑g=1∞(6g−5)!!96g−1⋅(g−1)!λ−3g+22∑g=0∞(6g−1)!!96g⋅g!λ−3g−2∑g=0∞6g+16g−1(6g−1)!!96g⋅g!λ−3g+112∑g=1∞(6g−5)!!96g−1⋅(g−1)!λ−3g+2). (1.6) This matrix-valued function appeared in our paper [4]. It was used, in a slightly modified normalization, as key tool within an efficient algorithm for computation of intersection numbers of tautological classes on the Deligne–Mumford moduli spaces (see eq. (5.14)). The topological ODE for the Lie algebra $$sl_2(\mathbb C)$$ is closely related to the theory of Airy functions. Indeed, from (1.5) it readily follows that   a=12b′,c=−12b″+λb (1.7) while for $$b=b(\lambda)$$ one arrives at a third-order ODE   −12b‴+2λb′+b=0. (1.8) Solutions to this equation are products of solutions to the Airy equation   y″=λy. (1.9) That is, $$b(\lambda)$$ is the diagonal value of the resolvent of the Airy equation. The matrix $$M(\lambda)$$ can also be considered as resolvent of the matrix version of the Airy equation   [ddλ+(01λ0)]y→=0,y→=(y−y′). (1.10) Let us proceed now with the general case. The eigenvectors of the matrix of coefficients of the system (1.4) with zero eigenvalue correspond to matrices commuting with $$\Lambda$$. As observed in [27], the kernel of the linear operator   adΛ(λ0):g→g for any $$\lambda_0\in \mathbb{C}$$ has dimension $$n={\rm rk}\,\mathfrak{g}$$. This suggests that the dimension of the space of regular solutions to the topological ODE (1.4) is equal to $$n$$. We show in Theorem 1.2 that this is the case indeed. Moreover, we will construct a particular basis in the space $$\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})$$. To this end denote $$L(\mathfrak{g})=\mathfrak{g} \otimes \mathbb{C}[\lambda, \lambda^{-1}]$$ the loop algebra of $$\mathfrak{g}$$. Introduce the principal gradation on the Lie algebra $$L(\mathfrak{g})$$ in the following way:   deg⁡Ei=1, deg⁡Hi=0, deg⁡Fi=−1, i=1,…,n,deg⁡λ=h. (1.11) Here $$h$$ is the Coxeter number of $$\mathfrak{g}$$. According to this gradation we have $$\deg \Lambda=1.$$ Recall [23] that $$\mathrm{Ker} \, \mathrm {ad}_\Lambda\subset L(\mathfrak{g})$$ has the following decomposition   KeradΛ=⨁j∈ECΛj,deg⁡Λj=j∈E:=⨆i=1n(mi+hZ), (1.12) where the integers   1=m1<m2≤⋯≤mn−1<mn=h−1 (1.13) are the exponents of $$\mathfrak{g}$$. The matrices $$\Lambda_i$$ commute pairwise   [Λi,Λj]=0,∀i,j∈E. (1.14) They can be normalized in such a way [24, 25] that   Λma+kh=Λmaλk,k∈Z, (1.15)  (Λma|Λmb)=hλδa+b,n+1. (1.16) In particular, we choose $$\Lambda_1=\Lambda.$$ Theorem 1.2. (1) $$\dim_\mathbb{C}\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})={\rm rk}\, \mathfrak{g}.$$ (2) There exists a unique basis $$M_1(\lambda)$$, $$\dots$$, $$M_n(\lambda)$$ in the space $$\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})$$ of the form   Ma(λ)=λ−mah[Λma+∑k=1∞Ma,k(λ)],Ma,k(λ)∈L(g),deg⁡Ma,k(λ)=ma−(h+1)k,a=1,…,n. (1.17) □ The proof is given in Section 2. Definition 1.3. The solutions (1.17) to the topological ODE of $$\mathfrak{g}$$-type are called generalized Airy resolvents of $$\mathfrak{g}$$-type. □ The topological ODE (1.4) can be recast into the form   [ddλ+Λ,M]=0. (1.18) Using such a representation along with (1.14), (1.16) as well as the invariance property of the Cartan–Killing form one arrives at Proposition 1.4. The generalized Airy resolvents commute pairwise   [Ma(λ),Mb(λ)]=0,a,b=1,…,n. (1.19) Moreover, $$(M_a(\lambda)\,|\,M_b(\lambda))=h\,\delta_{a+b, n+1}.$$ □ Example 1.5. Consider the $$A_n$$ case, $$\mathfrak{g}=sl_{n+1}(\mathbb C)$$. The $$(n+1)\times (n+1)$$ matrix $$\Lambda$$ reads   Λ=(010⋯0001⋱0⋮⋱⋱⋱⋮00⋱⋱1λ00⋯0). The exponents are $$m_a=a$$, $$a=1, \dots, n$$, the Coxeter number is $$h=n+1$$. One can choose   Λi=Λi,i=1,…,n. Then   Ma(λ)=λ−an+1[Λa+1(n+1)λ∑ν=1nζνa−14sin2⁡πνn+1ΩνΛa−1+…], (1.20) where $$\zeta={\rm e}^{\frac{2\pi i}{n+1}}$$ is a root of unity, $$\Omega_\nu={\rm diag}\left(1, \zeta^\nu, \zeta^{2\nu}, \dots , \zeta^{n\, \nu}\right)\!.$$ The recursion procedure for computing subsequent terms will be explained below, see Example 2.4. In the particular case $$n=1$$ one obtains the solution (1.6). □ The recursive procedure for computing the series expansions of the generalized Airy resolvents outlined in the Example 1.5 can be represented in an alternative form. The basic idea can be already seen from the $$sl_2(\mathbb C)$$ example: the topological ODE for a matrix-valued function reduces to a scalar differential equation (1.8) for one of the entries of the matrix (called $$b(\lambda)$$ above); other matrix entries are expressed in terms of $$b(\lambda)$$ and its derivatives (see eq. (1.7)). Let us explain a general method for reducing the topological ODE for an arbitrary simple Lie algebra of rank $$n$$ to a system of ODEs with $$n$$ dependent variables. To this end we will use the Jordan decomposition of the operator $$\mathrm {ad}_{I_+}$$ described hereafter. Let $$\rho^\vee\in \mathfrak{h}$$ be the Weyl co-vector of $$\mathfrak{g}$$, whose defining equations are   αi(ρ∨)=1,i=1,…,n. (1.21) Write $$\rho^\vee=\sum_{i=1}^n x_i \,H_i,\,x_i\in\mathbb{C}$$ and define $$I_-=2\sum_{i=1}^n x_i \,F_i.$$ Then $$I_+,I_-,\rho^\vee$$ form an $$sl_2(\mathbb{C})$$ Lie algebra:   [ρ∨,I+]=I+,[ρ∨,I−]=−I−,[I+,I−]=2ρ∨. (1.22) According to [2, 27], there exist elements $$\gamma^1,\dots,\gamma^n\in \mathfrak{g}$$ such that   KeradI−=SpanC{γ1,…,γn},[ρ∨,γi]=−miγi. (1.23) Fix $$\{\gamma^1,\dots,\gamma^n\}$$, then the lowest weight decomposition of $$\mathfrak{g}$$ has the form   g=⨁i=1nLi,Li=SpanC{γi,adI+γi,…,adI+2miγi}. (1.24) Here each $$\mathcal L^i$$ is an $$sl_2(\mathbb{C})$$-module. Any $$\mathfrak{g}$$-valued function $$M(\lambda)$$ can be uniquely represented in the form   M(λ)=∑i=1nSi(λ)adI+2miγi+∑i=1n∑m=02mi−1Kim(λ)adI+mγi, (1.25) where $$S_{i}(\lambda)$$, $$K_{im}(\lambda)$$ are certain complex-valued functions. Note that $$\mathrm {ad}_{I_+}^{2m_i} \gamma^i$$ is the highest weight vector of $$\mathcal L^i$$, $$i=1,\dots, n$$. Theorem 1.6. Let $$M$$ be any solution of (1.4). The functions $$K_{im}$$ have the following expressions:   Kim=(−1)mSi(2mi−m)+∑u=1n∑v=02mi−1−mkimuv0Su(v)+λ∑u=1n∑v=0mi−1−mkimuv1Su(v), m=0,…,mi−1;Kim=(−1)mSi(2mi−m), m=mi,…,2mi−1 (1.26) with constant coefficients $$k_{imuv}^0,\,k_{imuv}^1$$ independent of the choice of the solution $$M.$$ Moreover, the topological ODE (1.4) is equivalent to a system of linear ODEs for $$S_1,\dots,S_n$$ of the form   Si(2mi+1)=∑u=1n∑v=02mikiuv0Su(v)+λ∑u=1n∑v=1mikiuv1Su(v),i=1,…,n. (1.27) Here $$k_{iuv}^0,\,k_{iuv}^1$$ are constants independent of the choice of the solution $$M.$$ □ The proof is given in Section 3. The coefficients $$k^0_{iuv}$$, $$k^1_{iuv}$$ coincide with $$k^0_{imuv}$$, $$k^1_{imuv}$$ at $$m=-1$$. See details in the proof. We call eqs. (1.27) the reduced topological ODEs of $$\mathfrak{g}$$-type. Denote by $$S_{a;1}(\lambda), \dots, S_{a;n}(\lambda)$$ the $$S$$-coefficients in the decomposition (1.25) of the generalized Airy resolvent $$M_a(\lambda)$$, $$a=1, \dots, n$$. Definition 1.7. The series $$S_{a; i}(\lambda)$$ are called essential series of $$\mathfrak{g}$$-type. □ Example 1.8 (The $$A_1$$ case). The essential series $$S_{1;1}$$ coincides with the $$(1,2)$$-entry of the Airy resolvent (1.6) (the function $$b(\lambda)$$ in (1.5)). The expression (1.26) is given explicitly by (1.7). □ Example 1.9. (The $$A_2$$ case). The decomposition (1.25) reads   M=(2K11+K22−2S1−3K236S22K10+K21−2K223K23−2S1K202K10−K21−2K11+K22). (1.28) The expression (1.26) is given explicitly by   K10=S1″+3λS2,K11=−S1′, (1.29)  K20=S2(4)−2λS1,  K21=−S2‴,  K22=S2″,  K23=−S2′. (1.30) The reduced equations (1.27) read   6S2+2S1(3)+9λS2′=0, (1.31)  2S1+2S1(4)+15S2′−S2(5)+λ(9S2″+6S1′)=0. (1.32) Explicit expressions of the essential series $$S_{a;j}$$ of the $$A_2$$-type can be found in Section 5. □ Proposition 1.10. (1) For any $$a\in\{1,2,\dots n\}$$, the essential series $$S_{a;i}$$ satisfy   (i)  Sa;1(λ),…,Sa;n(λ)∈λ−mah⋅C[[λ−1]];(ii)  the vectors (Sa;i)i=1…n,a=1,…,n are linearly independent;(iii)  Sa;a is non-zero. (2) Let $$\mathfrak{g}\neq D_{n=2k}$$ and $$M$$ be a regular solution of the topological ODE (1.4) such that the $$S_a$$-coefficient of $$M$$ coincides with $$S_{a;a}$$; then $$M=M_a.$$ If $$\mathfrak{g}$$ is of type $$D_{n}$$ with $$n$$ even, the statement remains valid for $$a\not\in\{n/2, n/2+1\}$$, and it should be modified for $$a=n/2$$ and $$a=n/2+1$$ in the following way: $$M=M_a$$ if the $$S_{n/2},\,S_{n/2+1}$$-coefficients of $$M$$ coincide with $$S_{a;n/2}$$, $$S_{a;n/2+1}$$, respectively. □ The proof will be given in Section 3. We call $$S_{a;a},\,a=1,\dots,n$$ the fundamental series of $$\mathfrak{g}$$-type. 1.2 Dual topological ODE and its normal form Solutions to (1.4) can be expressed by a suitable integral transform via another class of functions satisfying somewhat simpler differential equations. These functions will be constructed as solutions to another system of linear ODEs for a $$\mathfrak{g}$$-valued function $$G=G(x)$$, $$x\in\mathbb C$$. In the notations of the previous section it reads   [G′,E−θ]+[G,I+]+xG=0,′=ddx. (1.33) Definition 1.11. We call equation (1.33) the dual topological differential equation (dual topological ODE for short). The space of solutions of (1.33) is denoted by $$\mathcal{G}(\mathfrak{g}).$$ □ Solutions to the dual topological ODE are related with the generalized Airy resolvents by the following Laplace–Borel transform:   M(λ)=∫CG(x)e−λxdx, (1.34) where $$C$$ is a suitable contour on the complex $$x$$-plane, which can depend on the choice of a solution $$M(\lambda)\in\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})$$. Example 1.12. For $$\mathfrak{g}=sl_2(\mathbb C)$$ the dual topological equation for the matrix-valued function $$G=\left({\alpha(x) \atop \gamma(x)}\quad{\beta(x) \atop -\alpha(x)}\right)$$ reduces to   β′=(x24−12x)β. (1.35) This yields the following integral representation for solutions to the differential equation (1.8)   b(λ)=1π∫0∞ex312−λxdxx. (1.36) Here and below the Laplace-type integrals are understood in a formal sense by applying a term-by-term integration   1π∫0∞ex312−λxdxx=1π∑k=0∞∫0∞1k!(x312)ke−λxdxx=1π∑k=0∞Γ(3k+12)k!12kλ−3k−12. In order to arrive at an integral representation of a solution with the same asymptotic expansion within a certain sector near $$\lambda=\infty$$ one can integrate over a ray of the form $$x=r\,{\rm e}^{\frac{(2k+1)\pi\,i}3}$$, $$0<r<\infty$$ for $$k=0, \, 1,$$ or $$2$$. □ For an arbitrary simple Lie algebra we will now describe the reduction procedure for the dual topological ODE similar to the one used in the previous subsection. Write   G(x)=∑i=1nϕi(x)adI+2miγi+∑i=1n∑m=02mi−1K~im(x)adI+mγi (1.37) (cf. (1.25)). Theorem 1.13. (i) Let $$G$$ be any solution of (1.33). Then $$\widetilde K_{im}$$ have the following expressions:   K~im=∑u=1n∑v=02mi−m(−x)vkimuv0ϕu+∑u=1n∑v=0mi−1−m(−1)vkimuv1(xvϕu′+vxv−1ϕu), m=0,…,mi−1,K~im=x2mi−mϕi, m=mi,…,2mi, (1.38) where $$k_{imuv}^0,\,k_{imuv}^1$$ are the same constants as in Theorem 1.6. Moreover, the dual topological ODE (1.33) is equivalent to a system of first-order ODEs   x2mi+1ϕi+∑u=1n∑v=02mi(−1)vkiuv0xvϕu+∑u=1n∑v=1mi(−1)vkiuv1(vxv−1ϕu+xvϕu′)=0. (1.39) Here, $$i=1, \dots, n,$$ and $$k^0_{iuv},\,k^1_{iuv}$$ are constants (the same as in Theorem 1.6). (ii) The point $$x=0$$ is a Fuchsian singularity of the ODE system (1.39). More precisely, equations (1.39) have the following normal form  ϕ′=V−1xϕ+∑k=02h−2xkVkϕ,ϕ=(ϕ1,…,ϕn)T, (1.40)  V−1=diag(−mn+1−a/h)a=1,…,n,Vk∈Mat(n,C),k≥0. (1.41) (iii) $$\dim_\mathbb{C} \mathcal{G}(\mathfrak{g})={\rm rk}\,\mathfrak{g}.$$ □ Remark 1.14. Choosing the contour $$C$$ in several ways in (1.34) one obtains a complete basis of solutions of (1.4). □ Example 1.15. For $$\mathfrak{g}=sl_2(\mathbb{C})$$ one reads from (1.35) that $$V_{-1}=-\frac12,\,V_0=V_1=0,\,V_2\,{=}\,\frac14.$$ □ Example 1.16. For $$\mathfrak{g}=sl_3(\mathbb{C})$$ the normal form (1.40) of the dual topological equation reads   ϕ′=[−13x(2001)−x29(0020)+x46(0100)]ϕ. (1.42) □ Let $$G_a$$ denote the Laplace–Borel transform of $$M_a$$, and $$\phi_{a;1}, \dots, \phi_{a;n}$$ the $$\phi$$-coefficients of $$G_a$$, $$a=1, \dots, n$$. Then the matrix $$\Phi$$ defined by $$\Phi_{ia}:=\phi_{a;i}$$ is a fundamental solution matrix of the normal form (1.40). Definition 1.17. We call $$\phi_{a;i}$$ the dual essential series, and $$\phi_{a;a}$$ the dual fundamental series. □ For low ranks, the dual fundamental series $$\phi_{a;a}$$ can be expressed via elementary functions or combinations of elementary and Bessel functions, see Table 1. However, already for the case of the Lie algebra $$sl_5(\mathbb C)$$ we were not able to identify the dual fundamental series with a classical special function. Table 1. Special functions arising from the dual topological ODE of $$\mathfrak{g}$$-type Type of $$\mathfrak{g}$$  Special functions arising in dual fundamental series  $$A_1$$  Exponential function  Rank $$2$$  Bessel functions  Rank $$3$$  Bessel functions, exponential functions  $$A_4$$  Solutions to scalar ODEs of order $$4$$  $$D_4$$  Bessel functions, exponential functions  Type of $$\mathfrak{g}$$  Special functions arising in dual fundamental series  $$A_1$$  Exponential function  Rank $$2$$  Bessel functions  Rank $$3$$  Bessel functions, exponential functions  $$A_4$$  Solutions to scalar ODEs of order $$4$$  $$D_4$$  Bessel functions, exponential functions  Last but not least: why do we call eq. (1.4) the topological differential equation? For the case of $$\mathfrak{g}=sl_2(\mathbb C)$$ the Airy resolvent (1.6) appears in the expression of [4] for generating series of intersection numbers of $$\psi$$-classes on the Deligne–Mumford moduli spaces $$\overline{\mathcal M}_{g,N}$$; see also the formula (5.14). It turns out that for a simply laced simple Lie algebra $$\mathfrak{g}$$ a representation similar to that of [4] can be used for computing the E. Witten and H. Fan–T. Jarvis–Y. Ruan (FJRW) intersection numbers. We give the precise expressions in the last section of the present paper, see Theorems 5.1 and 5.5. Connection between FJRW invariants and tau-functions of the Drinfeld–Sokolov hierarchy of $$A\,D\,E$$-type conjectured in [37] and proven in [17, 18] plays an important role in derivation of these expressions. The non-simply laced analogue of the FJRW intersection numbers, according to the recent paper by Liu $$\textit{et al.}$$ [30], is related to tau-functions of the Drinfeld–Sokolov hierarchies of $$B\, C\, F\, G$$-type. 1.3 Organization of the paper In Section 2 we prove Theorem 1.2. In Section 3 we prove Theorem 1.6, Proposition 1.10, and Theorem 1.13. In Section 4, we calculate the dual essential series, in particular the dual fundamental series for several special cases. Concluding remarks are given in Section 5. We give definition and examples of generalized Airy functions in the Appendix. 2 Proof of Theorem 1.2 Before proceeding with the proof, we introduce the gradation operator  gr=hλddλ+adρ∨. (2.1) Recall that $$\rho^\vee\in \mathfrak{h}$$ is the Weyl co-vector. The following statement [25] will be useful in the proof. Lemma 2.1. An element $$a\in L(\mathfrak{g})$$ is homogeneous of principal degree $$k$$if and only if  gra=ka. (2.2) □ The action of the gradation operator can be obviously extended to expressions containing fractional powers of $$\lambda$$. The following lemma will also be useful. Lemma 2.2. The following formulæ hold true:   L(g)=KeradΛ⊕ImadΛ,ImadΛ=(KeradΛ)⊥, (2.3) where the orthogonality is with respect to the Cartan–Killing form. □ The proof can be found, for example, in [11]. Proof of Theorem 1.2. Let us fix an exponent $$m_a$$ for some $$a\in \{1,\dots,n\}.$$ We first prove existence and uniqueness of formal solutions of (1.4) of the following form:   M(λ)=λ−mah∑k=0∞Mk(λ),M0(λ)=Λma, (2.4)  Mk(λ)∈L(g),deg⁡Mk(λ)=ma−(h+1)k. (2.5) Substituting (2.4) in (1.4) and equating the terms of equal principal degrees we obtain    [Λ,M0]=0, (2.6)   [Λ,Mk+1]=−Mk′+mah1λMk,k≥0. (2.7) The first condition follows due to the choice $$M_0=\Lambda_{m_a}$$ (see eq. (1.14)). In order to resolve eq. (2.7) for $$k=0$$ one has to check that   hλΛma′−maΛma∈ImadΛ. Indeed, since $$\deg \Lambda_{m_a}=a$$ we have   grΛma≡hλΛma′+[ρ∨,Λma]=maΛma. So, the equation (2.7) for $$k=0$$ becomes   [Λ,M1]=1hλ[ρ∨,Λma]. Due to Lemma 2.2 it suffices to check that the r.h.s. is orthogonal to $${\rm Ker}\, \mathrm {ad}_\Lambda$$. Indeed,   ([ρ∨,Λma]|Λj)=(ρ∨|[Λma,Λj])=0. This completes the first step of the recursive procedure. Assume, by induction, that the $$k$$th term $$M_k(\lambda)$$ exists and, moreover, it satisfies   −Mk′+mah1λMk∈ImadΛ. Then there exists a matrix $$M_{k+1}$$ satisfying eq. (2.7). It is determined uniquely modulo an element in $${\rm Ker}\, \mathrm {ad}_\Lambda$$. Write   Mk+1=A+B,A∈ImadΛ,B∈KeradΛ, (2.8)  degA=degB=ma−(h+1)(k+1). (2.9) Since the map   adΛ:ImadΛ→ImadΛ is an isomorphism it remains to prove that the matrix $$B=B(\lambda)$$ is uniquely determined by the condition   −Mk+1′+mah1λMk+1∈ImadΛ. Like above, using the degree condition (2.9) along with the gradation operator (2.1) recast the last condition into the form   [ρ∨,A+B]+(h+1)(k+1)(A+B)⊥KeradΛ. (2.10) By assumption $$A\perp {\rm Ker}\, \mathrm {ad}_\Lambda$$. Also $$\left[ \rho^\vee, B\right]\perp {\rm Ker}\, \mathrm {ad}_\Lambda$$ due to commutativity of $${\rm Ker}\, \mathrm {ad}_\Lambda$$. Thus eq. (2.10) reduces to the system   (h+1)(k+1)(B|Λj)=−([ρ∨,A]|Λj),j∈E. (2.11) Decompose now $$B$$ as follows:   B=∑b=1ncb(λ)Λmb(λ). Choosing $$j=m_1, \dots, m_n$$ and using the normalization (1.16) one finally obtains   B=−1λh(h+1)(k+1)∑b=1n([ρ∨,A]|Λmn+1−b)Λmb. This concludes the inductive step and thus completes the proof of existence and uniqueness of solutions (1.17) to the topological ODE (1.4). The series $$M_1(\lambda),\dots,M_n(\lambda)$$ are linearly independent since their leading terms are. It is known from [34] that there exist analytic solutions to the topological ODE whose asymptotic expansions in certain sector coincide with $$M_i$$. These solutions we denote again by $$M_i$$. We conclude that   SpanC{M1,…,Mn}⊂S∞reg(g). (2.12) In particular $$\dim_\mathbb{C}\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})\geq n.$$ On another hand, according to Kostant [27] for any $$\lambda_0\in\mathbb{C}$$ the kernel of $${\rm ad}_{\Lambda(\lambda_0)}: \mathfrak{g} \to \mathfrak{g}$$ is of dimension $$n$$. This implies that   dimC⁡S∞reg≤n. (2.13) Hence $$\dim_\mathbb{C}\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})=n$$ and $$\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})={\rm Span}_\mathbb{C} \{M_1,\dots,M_n\}.$$ Theorem 1.2 is proved. □ The above proof gives an algorithm for computing $$M_a,\,a=1,\dots,n$$ recursively. Remark 2.3. Theorem 1.2 and Proposition 1.4 tell that $$\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})$$ forms a one-parameter family of Abelian subalgebras of $$\mathfrak{g}.$$ □ Example 2.4. (The $$A_n$$ case). Take the same matrix realization of $$\mathfrak{g}$$ as in Example 1.5, so that the normalized Cartan–Killing form coincides with the matrix trace. Recall that $$h=n+1,\,m_a=a$$. We have   Λj=Λj,ρ∨=diag(n2,n−22,…,−n2), (2.14)  Hj=Ej,j−Ej+1,j+1,h=SpanC{Hj}j=1n. (2.15) Fix an exponent $$m_a$$. Consider $$[L,M]=0,\,L=\partial_{\lambda}+\Lambda$$ and write $$M=\lambda^{-\frac{a}h}\sum_{k=0}^\infty M_k,$$ then   Mk=Ak+Bk,Ak∈ImadΛ,Bk∈KeradΛ, k≥0. (2.16) Here $$A_k$$ and $$B_k$$ have the principal degree $$\deg A_k=\deg B_k=a-(h+1)k,$$ and   Bk=−1λkh(h+1)∑b=1n(adρ∨Ak|Λn+1−b)Λb,k≥1. (2.17) It can be easily seen that   ImadΛ=SpanC[λ,λ−1]{HiΛj|i=1,…,n, j=0,…,n}, (2.18)  Tr(XΛj[ρ∨,Λℓ])={−hλxjδn+1,j+ℓj=1,…,n,0j=0,.  for X=∑i=1nxiHi. (2.19) Thus if $$A_k\,(k\geq 1)$$ is of form $$A_k = \sum_{j=0}^n X^{(j)}_k(\lambda) \, \Lambda^j(\lambda),~ X^{(j)}_k= \sum_{i=1}^n x_{k,i}^{(j)} \, H_i ~(0\leq j \leq n),$$ then   Bk(λ)=1(h+1)k∑j=1nxk,j(j)(λ)Λj(λ),k≥1. (2.20) Denote by $$\zeta=\exp({2\pi \sqrt{-1}}/{h})$$ a root of unity. The map $$\mathrm {ad}_\Lambda$$ acts on $${\rm Im}\,\mathrm {ad}_\Lambda$$ as follows:   adΛ(XΛj)=X~Λj+1,j=0,…,n, X,X~∈h, (2.21)  (x~1x~2⋮x~n)=−(2−10…011−1…0101−1…0⋮⋱⋱10…1−110…01)(x1x2⋮xn). (2.22) Define $$\Omega_\nu = {\rm diag} (\zeta^{\nu(j-1)} )_{j=1\dots n+1}, ~ \nu = 1,\dots n,$$ then we have   adΛ(ΩνΛj)=(ζν−1)ΩνΛj+1,j=0,…,n. (2.23) A direct computation gives   ρ∨=∑ν=1nζνζν−1Ων,adρ∨Λ0=0,adρ∨Λℓ=(∑ν=1nζν−ζν(ℓ+1)ζν−1Ων)Λℓ,  ℓ≥1; (2.24)  adΛ−1adρ∨Λℓ=(∑ν=1nζν−ζν(ℓ+1)(ζν−1)2Ων)Λℓ−1,  ℓ≥1. (2.25) Using these formulæ and consider the leading term of $$M$$ given by $$M_0 = \Lambda_a,$$ one immediately obtains the formula of the first-order approximation (1.20). □ 3 Essential Series and Dual Essential Series In this section we apply a reduction approach and the technique of Laplace–Borel transform to prove Theorem 1.6, Proposition 1.10, and Theorem 1.13. First of all, we introduce the lowest weight structure constants. Definition 3.1. Fix a choice of $$\gamma^1,\dots,\gamma^n$$ and $$I_+$$ as above. The coefficients $$c^{(p,q),(m,i)}_{(j,s)}$$ uniquely determined by   [adI+pγq,adI+mγi]=∑j=1n∑s=02mjc(j,s)(p,q),(m,i)adI+sγj. (3.1) are called the lowest weight structure constants of $$\mathfrak{g}$$. □ In this paper we will only use part of these structure constants, namely $$c^m_{ijs}:=c^{(0,n),(m,i)}_{(j,s)}$$. The above definition (3.1) becomes   [γn,adI+mγi]=∑j=1n∑s=02mjcijsmadI+sγj. (3.2) Lemma 3.2. The constants $$c^m_{i j s}$$ are zero unless   s−mj=m−mi−(h−1). (3.3) □ Proof By using the fact that   degadI+sγj=s−mj,∀s∈{0,…,2mj} and by comparing the principal degrees of both sides of (3.2). ■ Lemma 3.3. If $$i,j\in\{1,\dots,n\}$$ satisfy $$m_i+m_j=h-1$$ then $$c_{ij0}^{2m_i}=0.$$ □ Proof It suffices to consider the case $$i\leq n-1$$. By Lemma 3.2 we have   [γn,adI+2miγi]=∑0≤mi+mj−(h−1)≤2mjcij,mi+mj−(h−1)2miadI+mi+mj−(h−1)γj. (3.4) We will use the following orthogonality conditions [27]   (γi1|adI+2mi2γi2)=0,if i1≠i2. (3.5) It follows that   (ρ∨|[γn,adI+2miγi])=([ρ∨,γn]|adI+2miγi)=−(h−1)(γn|adI+2miγi)=0. (3.6) On another hand, for $$m_i+m_j-(h-1)\geq 2$$ we have   (ρ∨|adI+mi+mj−(h−1)γj)=(I+|adI+mi+mj−(h−1)−1γj)=0. (3.7) For $$m_i+m_j-(h-1)=1$$ we have   (ρ∨|adI+γj)=(I+|γj)=0. (3.8) The last equality is due to $$i\neq n$$ and so $$j$$ cannot be $$1$$, and $$I_+=const \cdot \mathrm {ad}_{I_+}^2\,\gamma^1$$. Hence $$c_{ij0}^{2m_i}=0.$$ The lemma is proved. ■ Proof of Theorem 1.6. Let $$M$$ be any solution of (1.4). It can be uniquely decomposed in the form   M=∑i=1n∑m=02miKimadI+mγi (3.9) for some functions $$K_{im}=K_{im}(\lambda)$$. Substituting this expression into eq. (1.4) we have   ∑i=1n∑m=02miKim′adI+mγi+∑i=1n∑m=12miKi,m−1adI+mγi+λ[γn,∑i=1n∑m=02miKimadI+mγi]=0. (3.10) Here $$'$$ means the derivative w.r.t. $$\lambda.$$ Substituting (3.2) into (3.10) we find   ∑j=1n∑s=02mj(Kjs′+Kj,s−1)adI+sγj+λ∑j=1n∑s=02mj∑i=1n∑m=02miKimcijsadI+sγj=0, (3.11) where $$K_{j,-1}:=0.$$ Now using Lemma 3.2 we find   Kj,s−1=−Kjs′−λ∑mi≥s−mj+(h−1)cijss+mi−mj+(h−1)Ki,s+mi−mj+(h−1), (3.12) where $$0\leq s\leq 2m_j.$$ Due to the inequality $$m_i-m_j+h-1\geq m_i > 0$$ the system (3.12) is of triangular form. So we can solve it for the vector $$(K_{j,s-1})_{j=1}^{n}$$ as long as the vectors $$(K_{j s'})_{j=1}^n,\ s'\geq s+1$$, are known. By definition $$S_{j}=K_{j,\,2m_{j}}$$. Thus $$K_{im},\,m=0,\dots, 2m_{i}-1$$ are determined as linear combinations of $$S_{1},\dots,S_{n}$$ and their derivatives. Further noticing that   s+mi−mj+(h−1)≥s+mi,∀i,j (3.13) we obtain the more subtle expressions (1.26). Indeed, it is easy to see that   Kim=(−1)mSi(2mi−m),mi≤m≤2mi−1. (3.14) This observation together with (3.13), (3.12) yields (1.26). The coefficients $$k^0_{imuv}$$ and $$k^1_{imuv}$$ in (1.26) are uniquely specified by the lowest weight structure constants of $$\mathfrak{g}.$$ Taking $$s=0$$ in eqs. (3.12) we obtain a system of linear ODEs (1.27) of the form   Si(2mi+1)=∑u=1n∑v=02mikiuv0Su(v)+λ∑u=1n∑v=0mikiuv1Su(v),i=1,…,n, (3.15) where the constant coefficients $$k^0_{iuv}$$ and $$k^1_{iuv}$$ are determined by the lowest weight structure constants of $$\mathfrak{g}$$. It is now clear that there is a one-to-one correspondence between solutions of the topological ODE (1.4) and solutions of the system (3.15). Thus these two systems of ODEs are equivalent. It remains to show that $$k^1_{iu0}=0,\,\forall \,i,u\,\in\{1,\dots,n\}.$$ The $$(s=0)$$-equations in (3.12) read as follows:   Kj0′+λ∑mi+mj≥h−1cij0mi−mj+(h−1)Ki,mi−mj+(h−1)=0. (3.16) Taking the $$\lambda$$-derivative in the $$(m=0,\,i=j)$$-equations of (1.26) we obtain   Kj0′=Sj(2mj+1)+∑u=1n∑v=02mj−1kj0uv0Su(v+1)+∑u=1n∑v=0mj−1kj0uv1Su(v)+λ∑u=1n∑v=0mj−1kj0uv1Su(v+1). (3.17) From equations (3.17), (3.16), (3.15) we know that $$K_{j0}'$$ does not contribute to $$k^1_{iu0}$$. The only possible terms that can contribute to $$k^1_{iu0}$$ are   ∑mi+mj≥h−1cij0mi−mj+(h−1)Ki,mi−mj+(h−1). (3.18) Noticing again the inequalities $$m_i-m_j+(h-1)\geq m_i$$ as well as (3.14), we find that (3.18) does not contribute to $$k^1_{iu0}$$ unless   mi−mj+(h−1)=2mi. (3.19) However, by Lemma 3.3 we know that $$c_{ij0}^{2m_i}=0$$ if $$m_i+m_j=h-1.$$ Hence $$k^1_{iu0}\equiv0.$$ The theorem is proved. □ Equations (1.27) can be written in the following form (which will be used later):   ∑v=02h−1pv0S(v)+λ∑v=1hpv1S(v)=0,S=(S1,…,Sn)T, (3.20) where $$p^0_{v}$$ and $$p^1_{ v}$$ are constant $$n\times n$$ matrices, $$(p^0_v)_{iu}=k^0_{iuv},\,(p^1_v)_{iu}=k^1_{iuv}.$$ These matrices satisfy $$(p^0_{2m_i+1})_{\,i, \,2m_i+1}=-1$$ and the following vanishing conditions:    (pv0)iu=0ifv>2mi+1, (pv1)iu=0ifv>mi. (3.21) Before proving Proposition 1.10 and Theorem 1.13 we do some preparations. Recall that the essential series $$S_{a;i}$$ of $$\mathfrak{g}$$ are defined as the $$S$$-coefficients of the generalized Airy resolvents $$M_a.$$ Lemma 3.4. For any $$a\in \{1,\dots,n\}$$, the essential series $$S_{a;i},\,i=1,\dots,n$$ have the form   Sa;i=λ−saih∑m=0∞ci,maλ−m(h+1),ci,ma∈C, (3.22) where $$s_{ai}$$ is a positive integer satisfying   sai≡ma  modh,sai≡mi  mod(h+1). (3.23) In particular, the essential series $$S_{a;a}$$ is non-zero and has the form   Sa;a=λ−mah∑m=0∞cmaλ−m(h+1),c0a≠0, cma∈C. (3.24) □ Proof Fix an integer $$a\in\{1,\dots,n\}.$$ By (1.17) we know that $$S_{a;i}\in \, \lambda^{-\frac{m_a}h} \cdot \mathbb{C}((\lambda^{-1})), \quad i=1,\dots,n,$$ where $$\mathbb{C}((\lambda^{-1}))$$ denotes the set of formal Laurent series at $$\lambda=\infty.$$ If $$S_{a;i}$$ is zero then it has the form (3.22) with $$c_{i,m}^a=0,\,m\geq 0.$$ Let $$S_{a;i}$$ be non-zero. Recall the expansion (1.17)   Ma(λ)=λ−mah[Λma+∑k=1∞Ma,k(λ)],Ma,k(λ)∈L(g),deg⁡Ma,k(λ)=ma−(h+1)k,a=1,…,n. Let $$S_{a;i,k}(\lambda)$$ denote the $$S$$-coefficients of $$M_{a,k}(\lambda)$$, we have   Sa;i=λ−mah∑k=0∞Sa;i,k(λ). (3.25) Noticing that $$\deg \, \mathrm {ad}_{I_+}^{2m_i}\gamma^i=m_i$$ we obtain   degSa;i,k(λ)=ma−mi−(h+1)k,k≥0. (3.26) Since $$\deg\,\lambda=h$$ we conclude that   ma−mi−(h+1)k≡0 (modh). (3.27) Hence $$S_{a;i,k}=0$$ unless there exists an integer $$u$$ such that $$k=m_a-m_i+u \, h,$$ and in this case   Sa;i,k=const⋅λ−(ma−mi)−u(h+1), (3.28) which yields (3.23). Here, note that $$k\geq 0$$ implies $$u\geq 0$$, hence $$s_{ai}>0.$$ To prove (3.24) it suffices to show that $$c^a_{0}\neq 0$$. It has been proved by Kostant that the elements $$\Lambda_{m_a},\,a=1,\dots, n$$ have the following decompositions (cf. p. 1014 in [27])   Λma=λβ+vadI+2maγa,deg⁡β=−(h−ma), (3.29) where $$v$$ are some non-zero constant and $$\beta\in\mathfrak{g}$$. (We would like to thank Yassir Dinar for bringing this useful result of [27] to our attention.) Here, we have also applied the facts   KeradI+=Span{adI+2m1γ1,…,adI+2mnγn},degadI+2miγi=mi. (3.30) The lemma is proved. ■ Let us now prove Proposition 1.10. Proof The statement (i) follows from (3.22). The statement (ii) follows from (1.26) because $$M_a$$ are linearly independent. The statement (iii) is contained in Lemma 3.4. Let $$M$$ be a regular solution to (1.4) such that the $$S_a$$-coefficient of $$M$$ coincides with $$S_{a;a}$$. If the exponents of $$\mathfrak{g}$$ are pairwise distinct then $$M$$ must be equal to $$M_a$$ since $$S_{a;a}$$ is non-zero. Slight modifications can be done for $$D_n$$ with $$n$$ even as described in the content of the proposition. The proposition is proved. ■ For any $$a\in\{1,\dots,n\}$$, Theorem 1.6 reduces $$M_a$$ consisting of $$n(h+1)$$ scalar series to $$n$$ scalar series $$S_{a;j},\,j=1,\dots,n.$$ Proposition 1.10 further reduces $$\{S_{a;j}\}_{j=1}^n$$ to one scalar series $$S_{a;a}$$. Finally let us consider the Laplace–Borel transform and prove Theorem 1.13. Proof Equations (1.38), (1.39) follow directly from equations (1.26), (1.27). So (i) is proved. To show (ii), we will construct a basis of formal solutions to (1.39). Indeed, we consider the Laplace–Borel transform of the essential series $$S_{a;i}(\lambda)$$  Sa;i(λ)=∫Caϕa;i(x)e−λxdx (3.31) for suitable contours $$C_a.$$ By Lemma 3.4 we have   ϕa;i=xsaih−1∑m=0∞ci,maΓ(saih+m(h+1))xm(h+1), (3.32) where the coefficients $$c^a_{i,m}$$ are defined by eq. (3.22). For $$i\neq a$$, equations (3.23) imply that the integers $$s_{ai}$$ satisfy   sai≥ma+h and sai≥mi+h+1 (3.33) if all exponents are pairwise distinct; for the case $$\mathfrak{g}$$ is of the $$D_{n=2k}$$ type, the above estimates are also true after possibly a linear change   ϕn/2↦a1ϕn/2+a2ϕn/2+1, (3.34)  ϕn/2+1↦b1ϕn/2+b2ϕn/2+1 (3.35) of two solutions $$\phi_{n/2},\,\phi_{n/2+1}$$. In any case, for $$i=a$$ we have   ϕa;a=xmah−1∑m=0∞cmaΓ(mah+m(h+1))xm(h+1),c0a≠0. (3.36) Hence we have obtained a set of independent formal solutions of (1.39)   ϕ1;j,…,ϕn;j, and we collect them to a matrix solution $$\Phi$$ by defining $$\Phi_{ia}=\phi_{a;i}$$, then we have   Φ=(I+O(x))(xm1h−10⋯00xm2h−1⋱⋮⋮⋱⋱000⋯xmnh−1),x→0. (3.37) Equations (1.39) can be written as   ∑v=02h−1(−x)vpv0ϕ+∑v=1h(−1)vpv1(xvϕ′+vxv−1ϕ)=0,ϕ=(ϕ1,…,ϕn)T, (3.38) where $$p^0_v$$ and $$p^1_v$$ are $$n\times n$$ matrices defined in (3.20). So we have   B(x)Φ′(x)=C(x)Φ(x), (3.39) where $$B,\,C$$ are $$n\times n$$ matrices whose entries are polynomials in $$x$$ with coefficients determined uniquely by the lowest weight structure constants. More explicitly, the expressions of $$B,C$$ are given by   B(x)=∑v=1h(−1)vxvpv1,C(x)=∑v=02h−1(−x)vpv0+∑v=1h(−1)vvxv−1pv1. We are going to show that the matrix $$B(x)$$ is of the following form:   B=(0⋯0b1⋅x⋮⋱b2⋅x∗0⋱⋱$⋮bn⋅x∗⋯∗), (3.40) where the asterisks denote polynomials in $$x$$ possessing at least a double zero at $$x=0.$$ Recall that $$p^1_v$$ are defined as the matrix of coefficients of $$\lambda\times S_j^{(v)},\,j=1,\dots,n$$ in the following equations:   Kj0′+λ∑mi+mj≥h−1cij0mi−mj+(h−1)(−1)mi+mj+h−1Si(mi+mj−(h−1))=0,′=ddλ. (3.41) The summand with $$m_i+m_j=h-1$$ in (3.41) does not contribute to the matrix $$B(x)$$, since we have already proved in Theorem 1.6 that $$p^1_0=0.$$ Let us consider the summands in (3.41) with $$m_i+m_j\geq h$$. Consider first the case $$\mathfrak{g}\neq D_{n}$$ with $$n$$ even where all exponents are distinct. In this case the $$(m_i+m_j=h)$$-summand of the second term of l.h.s. of (3.41) contributes to the anti-diagonal entries of $$B(x)$$. For $$m_i+m_j=h+1,h+2,\dots, 2h-2$$ the corresponding summands contribute to the entries of $$B(x)$$ marked with an asterisk. The first term $$K_{j0}'$$ also contributes to $$B(x)$$ through the last term of r.h.s. of (3.17), namely,   λ∑u=1n∑v=0mj−1kj0uv1Su(v+1). To understand this term we look at the $$(s=1)$$-equations of (3.12):   Kj0=−Kj1′−λ∑mi+mj≥hcij1mi−mj+h(−1)mi+mj−hSi(mi+mj−h). (3.42) The contribution is of the form (3.40). We proceed in a similar way with analyzing $$K_{j1},K_{j2},\dots,$$ in order to arrive at the matrix $$B(x)$$ in a finite number of steps. In the case of $$D_n$$ with $$n$$ even, the above arguments remain valid with suitable choices of $$\gamma^{n/2},\,\gamma^{n/2+1}.$$ We have $$\det \, B=\pm \,b_1\cdots b_n \,x^n.$$ Since the dimension of the space of solutions of (1.4) is finite and since we have already obtained $$n$$ linearly independent solutions of (3.39), we conclude that each $$b_i$$ is not zero. Using (3.21) and noting that the $$(m_i+m_j=k,\,k\geq h+1)$$-off diagonal entries of $$B(x)$$ (which are the terms marked with asterisk) have at least a $$(k-(h-1))$$-tuple zero, we conclude that the equations   ϕ′=B−1Cϕ,ϕ=(ϕ1,…,ϕn)T (3.43) have the precise normal form (1.40), where the expression $$V_{-1}$$ is due to (3.37). The part (ii) of Theorem 1.13 is proved. Finally, (iii) follows from the normal form (1.40) automatically. The theorem is proved. ■ The series $$\phi_{a;i}$$ in the above proof are called the dual essential series of $$\mathfrak{g}$$. The final remark is about uniqueness of essential series. We fix a principal nilpotent element $$I_+$$. Then, for a given choice of the basis $$\{\gamma^i\}_{i=1}^n$$ and of $$\{\Lambda_{m_j}\}_{j=1}^n$$, the series $$\{S_{i;j}\}_{i,j=1}^n$$ are determined uniquely. With a particular choice of only $$\{\Lambda_{m_j}\}_{j=1}^n$$, the element $$\gamma^j$$ for any $$j\in\{1,\dots,n\}$$ and hence each of the series $$\{S_{i;j}\}_{i=1}^n$$ is unique up to a constant factor except for the $$D_n$$ with even $$n$$ case. Because of this for presenting the expression of an essential or a dual essential series, we allow a free constant factor. For the exceptional case $$D_n$$ with $$n$$ even, $$\gamma^{n/2},\,\gamma^{n/2+1}$$ and hence $$\{S_{i;n/2},\,S_{i;n/2+1}\}_{i=1}^n$$ are unique up to invertible linear combinations with constant coefficients. 4 Examples In this section we study examples of low ranks. We take a faithful matrix realization $$\pi$$ of $$\mathfrak{g},$$ and derive the normal form (1.40) and dual essential series. Denote by $$\chi$$ the unique constant such that   (a|b)=χTr(π(a)π(b)),∀a,b∈g. (4.1) Below, we will often write $$\pi(a),\,a\in\mathfrak{g}$$ just as $$a,$$ for simplicity. 4.1 Simply laced cases Example 4.1 (The cases $$A_1,A_2$$). These two cases have already been studied in detail in Introduction (see (1.5–1.8), Examples 1.9, 1.15, 1.16). Here we will make a short summary. The essential series for $$A_1$$ has the following explicit expression:   S=∑g=0∞(6g−1)!!96g⋅g!1λ6g+12, which was derived in [4] by an alternative method. The dual fundamental series has the expression   ϕ=x−12ex312. The essential series for $$A_2$$ are given below in equations (5.15–5.18). The dual fundamental series are given explicitly via confluent hypergeometric limit functions   ϕ1;1=x−23⋅0F1(;13;−x81728),ϕ2;2=x−13⋅0F1(;23;−x81728). They satisfy the following ODEs, respectively:   27x2ϕ1;1″−81xϕ1;1′+(x8−84)ϕ1;1=0,27x2ϕ2;2″−27xϕ2;2′+(x8−21)ϕ2;2=0. □ Recall that the confluent hypergeometric limit function is defined by the convergent series    0F1(;a;z)=∑k=0∞1(a)kzkk!, where   (a)k=a(a+1)…(a+k−1) (4.2) is the Pochhammer symbol. It is also closely related to the Bessel functions   Jν(x)=(z2)νΓ(ν+1)0F1(;ν+1;−z24). Example 4.2 (The $$A_3$$ case). In this case, $$\mathfrak{g}=sl_4(\mathbb{C}),\,h=4.$$ The normal form of the dual topological ODE reads as follows:   ϕ′=(−34000−12000−14)ϕx+(−316x40−158x+18x60−14x4018x200)ϕ. The dual fundamental series satisfy the following ODEs:   64(x5−15)x2ϕ1;1″−4(125−3x5)x6ϕ1;1′−(x15−9x10+1134x5−1260)ϕ1;1=0,4xϕ2;2′+(x5+2)ϕ2;2=0,64x2ϕ3;3″−(64x−12x6)ϕ3;3′−(x10−18x5+36)ϕ3;3=0, and they are given explicitly by   ϕ1;1=x−34e−3x5160[ 0F1(;14;x104096)−3x5800F1(;54;x104096)]ϕ2;2=x−12e−x520,ϕ3;3=x−14e−3x5160⋅0F1(;34;x104096). □ Example 4.3 (The $$A_4$$ case). In this case $$h=5$$ and $$\mathfrak{g}=sl_5(\mathbb{C}).$$ The normal form of the dual topological ODE of $$A_4$$-type reads as follows:   ϕ′=(−450000−350000−250000−15)ϕx+(0−6625x6−7925x110x811210x40−125x6−1112x07300x400−235x2000)ϕ. (4.3) Let $$\theta=x\,\partial_x;$$ then $$\phi_{4;4}$$ satisfies the following ODE:   [3125(x12+155)θ4−12500(7x12+620)θ3+625(x24+1277x12+54870)θ2.−1250(x24+321x12+31124)θ+(x36−1495x24+510995x12−9215525)]ϕ4;4=0. (4.4) Noting that $$x=0$$ is a regular singularity of this ODE and that the indicial equation at $$x=0$$ reads   (k+15)(k−115)(k−235)(k−475)=0, (4.5) one then recognizes that the dual fundamental series $$\phi_{4;4}$$ is the (unique up to a constant factor) solution corresponding to the root $$k=-\frac15$$,   ϕ4;4=x−1/5(1−161210⋅35x12+26605753223⋅312⋅52x24−…). In a similar way one can treat the other three dual fundamental series. For any solution $$\phi=(\phi_1,\dots,\phi_4)^T$$ of the normal form (4.3), $$\phi_4$$ satisfies (4.4), and the series $$\phi_{a;4},\,a=1,\dots,4$$ are all non-zero. Hence, similarly to the proof of Proposition 1.10, for $$A_4$$, one can use one ODE to determine all solutions of (4.3). In particular, the interesting numbers $$s_{a;4},\,a=1,\dots,4$$ (see eq. (3.22)) can be read off immediately from (4.5):   s1;4=16, s2;4=52, s3;4=28, s4;4=4. (4.6) We expect that this phenomenon always occurs when $$h$$ is an odd number; this is motivated by the fact (deduced from the so-called dimension counting for certain FJRW invariants; see in Section 5 for some remarks on these invariants) that if $$h$$ is odd then $$\phi_{a;n},\, a=1,\dots,n$$ are all non-zero. □ Example 4.4 (The case $$D_4$$). We have $$n=4$$, $$h=6$$ and $$m_1=1,\,m_2=3,\,m_3=3',\,m_4=5$$ and   g={B∈Mat(8,C)|B+SηBTηS=0}, (4.7) where $$\eta_{ij}=\delta_{i+j,9},\,1\leq i,j\leq 8$$ and $$S={\rm diag}\,(1,-1,1,-1,-1,1,-1,1).$$ In this case, $$\chi=\frac12$$ and   Λ=(01000000001000000001120000000012000000010000000010λ200000010λ2000000). (4.8) The matrices $$\Lambda_1,\,\Lambda_3,\,\Lambda_{3'},\,\Lambda_5$$ can be chosen as follows:   Λ1=Λ,Λ5=2Λ5,Λ3=Λ3,Λ3′=2(00001000000001000000001000000001λ00000000λ00000000λ00000000λ0000). (4.9) The normal form of the dual topological ODEs of the $$D_4$$-type reads as follows:   ϕ′=(−560000−120000−120000−16)ϕx+(2627x600919x3+16x100x60000x6029x2000)ϕ. (4.10) The dual fundamental series are given by   ϕ1;1=x−56exp⁡(13x7189)⋅[0F1(;16;x1436)+1363x70F1(;76;x1436)],ϕ2;2=ϕ3;3=x−12exp⁡(x77),ϕ4;4=x−16exp⁡(13x7189)⋅0F1(;56;x1436). They are solutions to the following ODEs, respectively:   108(3x7+182)x2ϕ1;1″−4(78x14+5461x7+9828)xϕ1;1′ −(12x21+260x14+107029x7+62790)ϕ1;1=0,ϕ2;2′=−ϕ2;22x+x6ϕ2;2,ϕ3;3′=−ϕ3;32x+x6ϕ3;3,108x2ϕ4;4″−(104x8+108x)ϕ4;4′−(4x14+260x7+39)ϕ4;4=0. □ 4.2 Non-simply laced cases Example 4.5 (The case $$B_2$$). We use the realization given in [11]: $$\mathfrak{g}= \{B\in {\rm Mat} (5,\mathbb{C})\,|\, B+ S\eta B \eta S =0\}, \chi=\frac12$$ where $$\eta_{ij}=\delta_{i+j,6}\,(1\leq i,j\leq 5),$$$$S={\rm diag}\,(1,-1,1,-1,1).$$ We have   Λ=(010000010000010λ200010λ2000). Recall that in this case $$h=4,\,m_1=1,\,m_2=3$$, and we can choose $$\Lambda_{m_i}$$ as follows:   Λ1=Λ,Λ3=2Λ3. (4.11) The normal form of the dual topological ODE is given by   ϕ′=(−3400−14)ϕx+(34x41516x+14x6x20)ϕ. (4.12) Solving this ODE system we obtain the dual fundamental series   ϕ1;1=x−34e3x540[0F1(;14;x1028)+3x5200F1(;54;x1028)],ϕ2;2=x−14e3x5400F1(;34;x1028). They are solutions to the following ODEs   (64x5+240)x2ϕ1;1″−(48x5+500)x6ϕ1;1′−(16x15+36x10+1134x5+315)ϕ1;1=0,16x2ϕ2;2″−4(3x5+4)xϕ2;2′−(4x10+18x5+9)ϕ2;2=0. □ Example 4.6 (The $$B_3$$ case). We have $$h=6,\,\mathfrak{g}= \{B\in {\rm Mat} (7,\mathbb{C})\,|\, B+ S\eta B \eta S =0\},\,\chi=\frac12$$ where $$\eta_{ij}=\delta_{i+j,8},\,1\leq i,j\leq 7$$ and $$S={\rm diag}\,(1,-1,1,-1,1,-1,1).$$ The cyclic element is   Λ=(01000000010000000100000001000000010λ20000010λ200000). We choose $$\Lambda_{m_i}$$ as follows: $$\Lambda_1=\Lambda,\quad \Lambda_5=2\,\Lambda^5,\quad \Lambda_3=\sqrt{2}\,\Lambda^3.$$ The normal form of the dual topological ODEs read   ϕ′=(−56000−12000−16)ϕx+(2627x60919x3+16x100x6029x200)ϕ, which is obviously equivalent to the normal form of $$D_4$$ restricted to $$\phi_3\equiv0.$$ Hence   ϕ1;1=x−56exp⁡(13x7189)⋅[0F1(;16;x1436)+1363x70F1(;76;x1436)],ϕ2;2=x−12exp⁡(x77),ϕ3;3=x−16exp⁡(13x7189)⋅0F1(;56;x1436). □ Example 4.7 (The $$G_2$$ case). We have $$h=6,\,m_1=1,\,m_2=5.$$ Using the matrix realization given in [2], we obtain the normal form of the dual topological ODE of $$G_2$$-type   ϕ′=(−5600−16)ϕx+(1354x618209x3+56x101360x20)ϕ. (4.13) The dual fundamental series have the following explicit expressions:   ϕ1;1=x−56e13x7756[ 0F1(;16;x1424⋅36)+13x72520F1(;76;x1424⋅36)],ϕ2;2=x−16e13x7756⋅0F1(;56;x1424⋅36). We remark that the normal form of $$G_2$$-type coincides with the normal form of $$B_3$$ restricted to $$\phi_2\equiv0$$ after a rescaling of $$\Lambda$$ by $$\Lambda\mapsto 2^{-\frac13}\,\Lambda.$$ □ 5 Concluding Remarks 5.1 On the analytic properties of solutions to the topological ODEs We have proved the dimension of the space of solutions regular at $$\lambda=\infty$$ of the topological equation of $$\mathfrak{g}$$-type   [L,M]=0,L=∂λ+Λ (5.1) is equal to the rank of $$\mathfrak{g}.$$ We have also proved Theorem 1.6 on reduction of this equation through the lowest weight gauge. One can also consider the vector space of all solutions to the topological ODE; denote it by $$\mathfrak{S}(\mathfrak{g})$$. As $$\lambda=0$$ is a regular point for the system (1.4), every solution $$M(\lambda)\in \mathfrak{S}(\mathfrak{g})$$ is uniquely determined by the initial data $$M(0)$$. Thus $$\dim_\mathbb{C} \, \mathfrak{S}(\mathfrak{g})=\dim_\mathbb{C} \, \mathfrak{g}=n(h+1).$$ Denote $$M^{ak}$$ the unique analytic solution of (1.4) determined by the following initial data:   Mak(0)=adI+kγa,a=1,…,n,k=0,…,2ma. (5.2) Clearly $$M^{ak}(\lambda)$$ form a basis of $$\mathfrak{S}(\mathfrak{g}).$$ With a particular choice of basis $$\{M_1,\dots,M_n\}$$ of $$\mathfrak{S}_\infty^{\rm reg}(\mathfrak{g})$$, there exist unique partial connection numbers$$C_{abk}$$ such that   Ma=∑b=1n∑k=02mbCabkMbk, (5.3) where $$C_{abk}$$ are constants. (Such a choice depends on a choice of an appropriate sector in the complex plane near $$\lambda=\infty$$.) We will study these connection numbers for the topological ODEs as well as the monodromy data for the dual topological ODEs in a subsequent publication. 5.2 Application of topological ODEs to computation of intersection numbers on the moduli spaces of $$r$$-spin structures First, it should be noted that all main results of the previous sections remain valid after a change of normalization of the topological equation. It will be convenient to introduce a normalization factor $$\kappa$$ in the following way:   [L,M]=0,L=∂λ+κΛ. (5.4) This parameter will be useful for applications of solutions to (5.4) to computation of certain topological invariants of Deligne–Mumford moduli spaces. In particular, here we explain two applications of the topological ODEs (5.4). The proofs follow the scheme used in [4] for computing the coefficients of Taylor expansion of the logarithm of the Witten–Kontsevich tau-function of the Korteweg–de Vries hierarchy. Details of the proofs can be found in an upcoming publication [5]. Let $$\overline{\mathcal{M}}_{g,N}$$ denote the Deligne–Mumford moduli space of stable curves $$C$$ of genus $$g$$ with $$N$$ marked points $$x_1$$, $$\dots$$, $$x_N$$, $$\mathcal{L}_k$$ the $$k$$th-tautological line bundle over $$\overline{\mathcal{M}}_{g,N}$$, $$\psi_k=c_1\left(\mathcal{L}_k \right)\in H^2\left( \overline{\mathcal{M}}_{g,N}\right)$$, $$k=1,\dots,N.$$ Let $$r\geq 2$$ be an integer. Fix a set of integers $$\{a_k\}_{k=1}^N \subset \{0,\dots,r-1\}$$ such that $$2g-2-\sum_{k=1}^N a_k$$ is divisible by $$r$$. Then the degree of the line bundle $$K_C-\sum_{i=1}^N a_i x_i$$ is divisible by $$r$$. Here $$K_C$$ is the canonical class of the curve $$C$$. Hence there exists a line bundle $${\mathcal T}$$ such that   T⊗r≃KC−∑i=1Naixi. (5.5) Such a line bundle is not unique; there are $$r^{2g}$$ choices of $${\mathcal T}$$. A choice of such a line bundle $${\mathcal T}$$ is called an $$r$$-spin structure [17, 37] on the curve $$(C, x_1, \dots, x_N)\in\overline{\mathcal M}_{g,N}$$. The space of all $$r$$-spin structures admits a natural compactification $$\overline{\mathcal M}^{1/r}_{g; a_1, \dots, a_N}$$ along with a forgetful map   p:M¯g;a1,…,aN1/r→M¯g,N. There is a natural complex vector bundle $${\mathcal V}\to \overline{\mathcal M}^{1/r}_{g; a_1, \dots, a_N}$$ such that the fiber over the generic point $$(C, x_1, \dots, x_N, {\mathcal T})\in \overline{\mathcal M}^{1/r}_{g; a_1, \dots, a_N} $$ coincides with $$H^1(C, {\mathcal T})$$. Here the genericity assumption means that $$H^0(C,{\mathcal T})=0$$. Under this assumption, using the Riemann–Roch theorem one concludes that $${\rm rank}\, {\mathcal V}=s+g-1$$ where   s=a1+⋯+aN−(2g−2)r. The construction of the vector bundle $${\mathcal V}$$ over non-generic points in $$\overline{\mathcal M}^{1/r}_{g; a_1, \dots, a_N}$$ is more subtle; for details see [17] and references therein. The Witten class$$c_W(a_1,\dots,a_N)\in H^{2(s+g-1)}\left( \overline{\mathcal{M}}_{g,N}\right)$$ is defined via the push-forward of the Euler class of the dual vector bundle $${\mathcal V}^\vee$$  cW(a1,…,aN)=1rgp∗(euler(V∨)). (5.6) See more details on properties of $$c_W(a_1,\dots,a_N)$$ in [17, 22, 32, 36, 37]. Witten’s $$r$$-spin intersection numbers are nonnegative rational numbers defined by   ⟨τi1,k1⋯τiN,kN⟩r−spin:=∫M¯g,Nψ1k1∧⋯∧ψNkN∧cW(i1−1,…,iN−1). (5.7) Here $$i_\ell \in \{1,\dots, r-1\}, \,k_\ell\geq 0,~ \ell=1,\dots, N.$$ We have assumed that the genus in l.h.s. of (5.7) is reconstructed by the dimension counting:   i1−1r+⋯+iN−1r+r−2r(g−1)+k1+⋯+kN=3g−3+N. (5.8) See [3, 6, 7, 10, 22, 28, 29, 38] for some explicit calculations of Witten’s $$r$$-spin intersection numbers. According to Witten’s $$r$$-spin conjecture [37] the intersection numbers (5.7) are coefficients of Taylor expansion of logarithm of tau-function of a particular solution to the Drinfeld–Sokolov integrable hierarchy of the $$A_n$$ type, $$n=r-1$$. This conjecture was proved by Faber $$\textit{et al.}$$ [17]. We will now use the constructed above generalized Airy resolvents for a simple algorithm for computing the $$r$$-spin intersection numbers in all genera. For a given $$N\geq 1$$ and a given collection of indices $$i_1$$, $$\dots$$, $$i_N$$ satisfying $$1\leq i_\ell\leq r-1$$ define the following generating function of the $$N$$-point Witten’s $$r$$-spin intersection numbers by   Fi1,…,iNr−spin(λ1,…,λN)=(−κ−r)N∑k1,…,kN≥0(−1)k1+⋯+kN∏ℓ=1N(iℓr)kℓ+1(κλℓ)iℓr+kℓ+1⟨τi1,k1…τiN,kN⟩r−spin. (5.9) Here $$\lambda_1$$, $$\dots$$, $$\lambda_N$$ are indeterminates, $$\kappa=(\sqrt{-r})^{-r}$$. We use the standard notation (4.2) for the Pochhammer symbol. Theorem 5.1 ([5]). Let $$n=r-1,\, \mathfrak{g}= sl_{n+1} (\mathbb{C})$$, $$\Lambda=\sum_{i=1}^n E_{i,i+1}+\lambda\, E_{n+1,1}.$$ Let $$M_i=M_i(\lambda)$$ be the basis of generalized Airy resolvents of $$\mathfrak{g}$$-type, uniquely determined by the topological ODE   M′=κ[M,Λ],κ=(−r)−r (5.10) normalized by   Mi=−λ−ihΛi+lower degree terms w.r.t. deg,Λi:=Λi. (5.11) Denote $$\eta_{ij}=\delta_{i+j,n+1}.$$ Then the generating functions (5.9) of $$N$$-point correlators of $$r$$-spin intersection numbers have the following expressions:   dFir−spindλ(λ) = −κ(Mi)1,n+1(λ)−κλ−r−1rδi,n, (5.12)  Fi1,…,iNr−spin(λ1,…,λN) = −1N∑s∈SNTr(Mis1(λs1)…MisN(λsN))∏j=1N(λsj−λsj+1) −δN2ηi1i2λ1−i1hλ2−i2h(i1λ1+i2λ2)(λ1−λ2)2,N≥2. (5.13) □ Up to a constant factor the entry $$(M_i)_{1,n+1}$$ is nothing but the essential series $$S_{i;n}.$$ Then, from (5.12) we expect that all one-point $$r$$-spin intersection numbers can be read off from coefficients of solutions to one linear ODE (for $$S_{;n}$$) expanded near $$\lambda=\infty$$. We also remark that alternative closed expressions for one-point $$r$$-spin intersection numbers have been obtained in [28] by using the Gelfand–Dickey pseudo-differential operators. Example 5.2 ($$r=2$$). Witten’s two-spin invariants coincide with intersection numbers of $$\psi$$-classes over $$\overline{\mathcal{M}}_{g,N}$$ [17, 26, 35]. So Theorem 5.1 in the choice $$r=2$$ recovers the result of [4, 39]: for $$N\geq 2,$$   ∑g=0∞∑p1,…,pN≥0(2p1+1)!!⋯(2pN+1)!!∫M¯g,Nψ1p1⋯ψNpNλ1−2p1+32⋯λN−2pN+32 =−1N∑r∈SNTr(M(λr1)⋯M(λrN))∏j=1N(λrj−λrj+1)−δN2λ1−12λ2−12(λ1+λ2)(λ1−λ2)2, (5.14) where $$M=-\frac{\lambda^{-\frac12}}{2}\left({-\frac{1}{2\kappa} \sum_{g=1}^\infty \frac{(6g-5)!!}{(96\,\kappa^2)^{g-1}\cdot (g-1)!} \lambda^{-3g+2} \atop -2 \sum_{g=0}^\infty\frac{6g+1}{6g-1} \frac{(6g-1)!!}{(96\,\kappa^2)^g\cdot g!} \lambda^{-3g+1}}\quad{2 \sum_{g=0}^\infty \frac{(6g-1)!!}{(96\,\kappa^2)^g\cdot g!} \lambda^{-3g} \atop \frac1{2\kappa} \sum_{g=1}^\infty \frac{(6g-5)!!}{(96\,\kappa^2)^{g-1}\cdot (g-1)!} \lambda^{-3g+2}} \right)\!,~\kappa=(-2)^{-1}.$$ The formula (5.14) is closely related to a new formula recently proved by Zhou [40] (see Theorem 6.1 therein). For $$N=1$$, it follows easily from (5.12) the well-known formula   ⟨τ3g−2⟩g=124g⋅g!forg≥1. □ Example 5.3 ($$r=3$$). Define $$M=\left(\begin{array}{ccc} 2 K_{11} +K_{22} & - 2 S_1 -3 K_{23} & 6 S_2\\ 2 K_{10}+K_{21} & -2 K_{22} & 3 K_{23}-2 S_1\\ K_{20} & 2 K_{10} -K_{21} & -2 K_{11} +K_{22}\end{array}\right)\!,$$ where   K10=S1″κ2+3λS2,K11=−S1′κ,K20=S2(4)κ4−2λS1,K21=−S2‴κ3,K22=S2″κ2,K23=−S2′κ. The generalized Airy resolvents $$M_1(\lambda)$$ and $$M_2(\lambda)$$ are obtained by replacing the functions $$S_1$$ and $$S_2$$ in the above expressions by   S1;1 =12233κ2∑g=0∞(−1)g⋅37gΓ(8g+13)24g⋅g!⋅(54κ2)3g−1Γ(g+13)λ−24g+13, (5.15)  S1;2 = −12332κ2∑g=0∞(−1)g⋅37gΓ(8g+103)24g⋅g!⋅(54κ2)3gΓ(g+43)λ−24g+103 (5.16) and   S2;1 = −3423∑g=0∞(−1)g⋅37gΓ(8g+173)24g⋅g!⋅(54κ2)3g+2Γ(g+53)λ−24g+173, (5.17)  S2;2 = −16∑g=0∞(−1)g⋅37gΓ(8g+23)24g⋅g!⋅(54κ2)3g⋅Γ(g+23)λ−24g+23, (5.18) respectively. Setting $$\kappa=(\sqrt{-r})^{-r}$$ we easily obtain from Theorem 5.1 that   ⟨τ1,8g−7⟩g=129646656g⋅(g−1)!⋅(13)g(g≥1),⟨τ2,8g−2⟩g=146656g⋅g!⋅(23)g(g≥1),Fi1,…,iN3−spin(λ1,…,λN)=−1N∑s∈SNTr(Mis1(λs1)…MisN(λsN))∏j=1N(λsj−λsj+1)−δN2ηi1i2λ1−i1hλ2−i2h(i1λ1+i2λ2)(λ1−λ2)2,N≥2. The above formulae of one-point three-spin intersection numbers agree with known formulae in [7, 28, 29, 33]; for example, the first several of these numbers are given by   ⟨τ1,1⟩1=112,⟨τ2,6⟩1=131104,⟨τ1,9⟩2=1746496,⟨τ2,14⟩2=14837294080. □ 5.3 Topological ODEs, Drinfeld–Sokolov hierarchies, and FJRW “quantum singularity theory” According to [18, 30], the partition function of FJRW invariants for an $$A\,D\,E,\,D^T_n$$ singularity with the maximal diagonal symmetry group is a tau-function of its mirror Drinfeld–Sokolov hierarchy; the partition function of FJRW invariants for a $$(D_{2k},\langle J\rangle)$$ singularity is a particular tau-function of the $$D_{2k}$$-Drinfeld–Sokolov hierarchy; the partition function of non-simply laced analogue of the FJRW intersection numbers is defined as a tau-function of the Drinfeld–Sokolov hierarchies of $$B\, C\, F\, G$$-type [30]. Definition 5.4. The Drinfeld–Sokolov partition function $$Z$$ is a tau-function of the Drinfeld–Sokolov hierarchy of $$\mathfrak{g}$$-type [11] uniquely specified by the following string equation:   ∑i=1n∑k=0∞tk+1i∂Z∂tki+12∑i,j=1nηijt0it0jZ=∂Z∂t01. (5.19) Here, $$t^i_k$$ are time variables of the Drinfeld–Sokolov hierarchy and $$\eta_{ij}=\delta_{i+j,n+1}.$$ □ It should be noted that the terminology “the Drinfeld–Sokolov hierarchy of $$\mathfrak{g}$$-type” that we use refers to the Drinfeld–Sokolov hierarchy (under the choice of a principal nilpotent element) associated with the non-twisted affine Lie algebra $${{\widehat{\mathfrak{g}}}^{(1)}}.$$ Since the construction of this integrable hierarchy does not depend on the central extension of the loop algebra, it is essentially associated with the simple Lie algebra. Define the following generating functions of $$N$$-point correlators of $$Z$$ by   Fi1,…,iN(λ1,…,λN)=(−κ−h)N∑k1,…,kN≥0∞(−1)k1+⋯+kN∏ℓ=1N(miℓh)kℓ+1(κλℓ)miℓh+kℓ+1∂Nlog⁡Z∂tk1i1…∂tkNiN(0). (5.20) We will now express these generating functions in terms of the generalized Airy resolvents of $$\mathfrak{g}$$-type. To this end we will need to use the following multilinear forms on the Lie algebra:   B(a1,…,aN)=tr(ada1∘⋯∘adaN),∀a1,…,aN∈g. (5.21) Theorem 5.5. ([5]). Let $$\mathfrak{g}$$ be a simple Lie algebra of rank $$n$$. Let $$M_i=M_i(\lambda)$$, $$i=1, \dots, n$$ be the generalized Airy resolvents of $$\mathfrak{g}$$-type, which are the unique solutions to   M′=κ[M,Λ],κ=(−h)−h (5.22) subjected to   Mi=−λ−mihΛmi+lower degree terms w.r.t. deg. (5.23) Then the generating functions (5.9) for the $$N$$-point correlators of the Drinfeld–Sokolov partition function associated with $$\mathfrak{g}$$ are given by the following expressions:   dFidλ(λ) = −κ2h∨B(E−θ,Mi)−κλ−h−1hδi,n, (5.24)  Fi1,…,iN(λ1,…,λN) = −12Nh∨∑s∈SNB(Mis1(λs1),…,MisN(λsN))∏j=1N(λsj−λsj+1) −δN2ηi1i2λ1−mi1hλ2−mi2h(mi1λ1+mi2λ2)(λ1−λ2)2,N≥2. (5.25) Here, $$h^\vee$$ is the dual Coxeter number. □ Similarly as before, we expect that all one-point correlators of the Drinfeld–Sokolov partition function can be read off from coefficients of solutions to one linear ODE. The proof will be given in [5]. It is interesting to mention that for the $$ADE$$ cases, the correction term appearing in (5.25) for $$N=2$$ coincides with the propagators derived in [31] in the vertex algebra approach [1] to FJRW invariants; in these cases, the Drinfeld–Sokolov partition function coincides with the total descendant potential [15, 16, 20] of the corresponding Frobenius manifolds [12, 13]. It would also be interesting to investigate relations between the dual fundamental series and the hypergeometric functions considered in [21]. In subsequent publications we will continue the study of topological ODEs as well as their applications to computation of invariants of the Deligne–Mumford moduli spaces considering generalized Drinfeld–Sokolov hierarchies [8]. Funding The work was partially supported by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of Italian Ministry of Universities and Researches; and the research of M. B. was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN/261229–2011 and by the Fonds de recherche Nature et téchnologie du Québec (FQRNT) grant “Matrices Aléatoires, Processus Stochastiques et Systèmes Intégrables” (2013–PR–166790). Acknowledgments We would like to thank Philip Candelas, Yassir Dinar, Si-Qi Liu, Youjin Zhang, and Jian Zhou for helpful discussions. D. Y. is grateful to Youjin Zhang for his advising. Appendix: Simple Lie Algebras and Generalized Airy Functions Let $$\mathfrak{g}$$ be an arbitrary simple Lie algebra of rank $$n$$. We consider in this appendix a $$\mathfrak{g}$$-generalization of equation (1.10). Let $$E_{-\theta}$$ and $$I_+$$ be defined as in the Introduction. Let $$\Lambda:=I_++\lambda\,E_{-\theta}$$. Throughout this appendix we will fix a matrix realization $$\pi:\,\mathfrak{g}\rightarrow {\rm Mat}\,(m,\mathbb{C})$$. Again for any $$A\in\mathfrak{g}$$ we simply write $$\pi(A)$$ as $$A.$$ Definition A.1. The system of $$m$$ linear ODEs   dy→dλ+Λy→=0,y→=(y1,…,ym)T∈Cm,λ∈C (A.1) is called the generalized Airy system of $$\mathfrak{g}$$-type. □ The dimension of the space of solutions to (A.1) is $$m$$. Let $$Y:\mathbb{C}\rightarrow {\rm Mat}\,(m,\mathbb{C})$$ be a fundamental solution matrix of (A.1), that is,   Y′+ΛY=0,det(Y)=const≠0. (A.2) Proposition A.2. Let $$A$$ be any constant matrix in $$\pi(\mathfrak{g})$$. The matrix-valued function $$M:=YAY^{-1}$$ is a solution of the topological ODE, that is, we have   M′=[M,Λ]. (A.3) □ Proof By straightforward calculations. ■ Noting that $$\lambda=0$$ is a regular point of the generalized Airy system (A.1), we define a particular fundamental solution matrix $$Y$$ of (A.1) by using the following initial data:   Y(0)=Im, (A.4) where $$I_m$$ denote the $$m\times m$$ identity matrix. Then we have Proposition A.3. Let $$\{M^{ak}\}_{a=1,\dots,n,\,k=0,\dots,2m_a}$$ denote the basis of $$\mathfrak{S}(\mathfrak{g})$$ defined by (5.2). Then   π(Mak)=Yπ(adI+kγa)Y−1. (A.5) □ Proof By using Proposition A.2 and by using the standard uniqueness theorem for linear ODEs. ■ Corollary A.4. Let $$\{M_a\}_{a=1,\dots,n}$$ be a chosen (analytic) basis of $$\mathfrak{S}_\infty^{{\rm reg}}(\mathfrak{g})$$. Then we have   π(Ma)=∑b=1n∑k=02mbCabkYπ(adI+kγa)Y−1, (A.6) where $$C_{abk}$$ are the corresponding partial connection numbers (cf. (5.3)). □ Example A.5 ($$A_n,\,n\geq 1$$). Take $$\mathfrak{g}=sl_{n+1}(\mathbb{C}).$$ We have $$m=n+1.$$ Any solution $$\vec{y}= (y_1=y,\,y_2,\dots,\,y_m)^T$$ of (A.1) satisfies that   (−1)n+1y(n+1)=λy, (A.7)  yk=(−1)k−1y(k−1), k=2,…,m. (A.8) Solutions to (A.7) can be represented by a Pearcey-type integral   y(λ)=∫Cexp⁡(xn+2n+2−λx)dx, (A.9) where $$C$$ is a suitable contour on the complex $$x$$-plane. □ Example A.6. ($$D_n,\,n\geq 4$$). Take the matrix realization of $$\mathfrak{g}$$ as in [11]. We have $$m=2n.$$ The cyclic element $$\Lambda$$ takes the form   Λ=(01⋱112⋱012⋱1⋱λ210λ20). For any solution $$\vec{y}=(y_1=y,\,y_2,\dots,\,y_m)^T$$ to (A.1) we have   y(2n−1)=λy′+12y,, (A.10)  yk=(−1)k−1y(k−1),k=2,…,n−1, (A.11)  yk=(−1)k−2y(k−2),k=n+2,…,2n−1, (A.12)  2yn′=yn+1′=(−1)n−1y(n),y2n=y(2n−2)−12λy. (A.13) Solutions to (A.10) can be represented by a Pearcey-type integral   y(λ)=∫Cexp⁡(−x4n−22n−1−λx2)dx, (A.14) where $$C$$ is a suitable contour in the complex $$x$$-plane. □ Example A.7. ($$B_n,\,n\geq 2$$). Take the matrix realization of $$\mathfrak{g}$$ as in [11]. We have $$m=2n+1,$$ and $$\mathfrak{g}= \{B\in {\rm Mat} (m,\mathbb{C})\,|\, B+ S\eta B \eta S =0\}$$ where $$\eta_{ij}=\delta_{i+j,2n+2}\,(1\leq i,j\leq 2n+1),$$$$S={\rm diag}\,(1,-1,1,-1,\dots,1,-1).$$ We have   Λ=(010⋯0⋮⋱⋱⋱⋮00⋱⋱0λ20⋱⋱10λ20⋯0). Then any solution $$\vec{y}=(y_1=y,\,y_2,\dots,\,y_m)^T$$ of (A.1) satisfies   y(2n+1)=λy′+12y, (A.15)  yk=(−1)k−1y(k−1), k=2,…,m−1, (A.16)  ym=(−1)m−1y(m−1)−λ2y. (A.17) Solutions to (A.15) can be represented by the following Pearcey-type integral   y(λ)=∫Cexp⁡(−x4n+22n+1−λx2)dx, (A.18) where $$C$$ is a suitable contour in the complex $$x$$-plane. □ Example A.8 ($$E_6$$). We take the matrix realization the same as in [14], where $$m=27.$$ The matrices $$I_+$$ and $$E_{-\theta}$$ can also be read off from [14]. The corresponding generalized Airy system reduces to two linear ODEs for $$y_1,\,y_6.$$ Denote by $$u(x),\,v(x)$$ the Laplace–Borel transforms of $$y_1,\,y_6$$, respectively:   y1(λ)=∫Cu(x)e−λxdx,y6(λ)=∫Cv(x)e−λxdx. (A.19) Then $$u,v$$ satisfy the following system of ODEs:   (u′v′)=1x(−1300−23)(uv)+(5x12−263x8−269x165x12)(uv). (A.20) It follows that   −27x2u″(x)+(270x14+189x)u′+(x26+675x13+75)u=0, (A.21)  −27x2v″(x)+(270x14+405x)v′+(x26−405x13+300)v=0. (A.22) Solving (A.21) we have   u=c1x−13e5x13130F1(;23;−x2627)+c2x253e5x13130F1(;43;−x2627), (A.23) where $$c_1,c_2$$ are arbitrary constants. □ Applying the saddle point technique to the above integral representations one can derive asymptotic expansions for the generalized Airy functions. In principle, Proposition A.2 can be used for computing the asymptotic expansions of solutions to the topological ODE. However, the methods explained in the main part of the present paper seem to be more efficient for obtaining such expansions for regular solutions. Remark A.9. For the original construction [26] of the Witten–Kontsevich tau-function it was introduced a matrix analogue of the Airy integral. More recently in [19] the Kontsevich’s matrix Airy function was generalized for an arbitrary compact Lie group. For $$\mathfrak{g}=sl_{n+1}(\mathbb C)$$ the generalized matrix Airy function appears in [9] in representation of the tau-function of the Drinfeld–Sokolov hierarchy associated with the $$r$$-spin intersection numbers on $$\overline{\mathcal M}_{g,N}$$, $$n=r-1$$. It is not clear whether the generalized matrix Airy functions of [19] for other simple Lie algebras can be used for computing the FJRW intersection numbers. □ References [1] Bakalov B. and Milanov. T. “$$\mathcal{W}$$-constraints for the total descendant potential of a simple singularity.” Compositio Mathematica  149, no. 5 (2013): 840– 88. Google Scholar CrossRef Search ADS   [2] Balog J. Fehér L. O’Raifeartaigh L. Forgacs P. and Wipf A. “Toda theory and W-algebra from a gauged WZNW point of view.” Annals of Physics  203, no. 1 (1990): 76– 136. Google Scholar CrossRef Search ADS   [3] Balogh F. Yang D. and Zhou. J. “Explicit formula for Witten’s $$r$$-spin partition function” (in press). [4] Bertola M. Dubrovin B. and Yang. D. “Correlation functions of the KdV hierarchy and applications to intersection numbers over $$\overline{\mathcal M}_{g,n}$$.” Physica D. Nonlinear Phenomena  327 (2016): 30– 57. Google Scholar CrossRef Search ADS   [5] Bertola M. Dubrovin B. and Yang. D. “Simple Lie algebras, Drinfeld–Sokolov hierarchies, and multipoint correlation functions.” (2016): preprint arXiv:1610.07534. [6] Brézin E. and Hikami. S. “Intersection numbers of Riemann surfaces from Gaussian matrix models.” Journal of High Energy Physics 2007 , no. 10 (2007): 96. [7] Brézin E. and Hikami. S. “Computing topological invariants with one and two-matrix models.” Journal of High Energy Physics 2009,  no. 4 (2009): 110. [8] De Groot M. F. Hollowood T. J. and Miramontes. J. L. “Generalized Drinfel’d-Sokolov hierarchies.” Communications in Mathematical Physics  145, no. 1 (1992): 57– 84. Google Scholar CrossRef Search ADS   [9] Dijkgraaf R. “Intersection theory, integrable hierarchies and topological field theory.” In New Symmetry Principles in Quantum Field Theory , edited by Fröhlich J. et al.  , NATO ASI Series 295, 95– 158, Plenum Press, New York, 1992. Google Scholar CrossRef Search ADS   [10] Ding X.-M. Li Y. and Meng. L. “From $$r$$-Spin Intersection Numbers to Hodge Integrals.” Journal of High Energy Physics  2016, no. 1 (2016): 15. Google Scholar CrossRef Search ADS   [11] Drinfeld V. G. and Sokolov. V. V. “Lie algebras and equations of Korteweg–de Vries type.” Journal of Mathematical Sciences  30, no. 2 (1985): 1975– 2036. ( Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya) 24 (1984): 81–180.) [12] Dubrovin B. “Geometry of 2D topological field theories.” In Integrable Systems and Quantum Groups (Montecatini Terme, 1993) , edited by Francaviglia M. and Greco. S. Springer Lecture Notes in Mathematics 1620, 120– 348. 1996. Springer-Verlag, Berlin, Heidelberg. Google Scholar CrossRef Search ADS   [13] Dubrovin B. “Painlevé transcendents in two-dimensional topological field theory.” In The Painlevé Property , edited by Conte R. 287– 412. New York: Springer, 1999. Google Scholar CrossRef Search ADS   [14] Dubrovin B. Liu S.-Q. and Zhang. Y. “Frobenius manifolds and central invariants for the Drinfeld–Sokolov biHamiltonian structures.” Advances in Mathematics  219, no. 3 (2008): 780– 837. Google Scholar CrossRef Search ADS   [15] Dubrovin B. and Zhang. Y. “Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation.” Communications in Mathematical Physics  198, no. 2 (1998): 311– 61. Google Scholar CrossRef Search ADS   [16] Dubrovin B. and Zhang. Y. “Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants.” (2001): Preprint arXiv:math/0108160. [17] Faber C. Shadrin S. and Zvonkine. D. “Tautological relations and the $$r$$-spin Witten conjecture.” In Annales Scientifiques de l’Ecole Normale Supérieure  43, no. 4 (2010): 621– 58. Google Scholar CrossRef Search ADS   [18] Fan H. Jarvis T. and Ruan. Y. “The Witten equation, mirror symmetry, and quantum singularity theory.” Annals of Mathematics  178, no. 1 (2013): 1– 106. Google Scholar CrossRef Search ADS   [19] Fernandez R. N. and Varadarajan. V. S. “Airy functions for compact Lie groups.” International Journal of Mathematics  20 (2009): 945– 77. Google Scholar CrossRef Search ADS   [20] Givental A. “Gromov–Witten invariants and quantization of quadratic Hamiltonians.” Moscow Mathematical Journal  1, no. 4 (2001): 551– 68. [21] Heckman G. J. and Opdam. E. M. “Root systems and hypergeometric functions. I.” Compositio Mathematica  64, no. 3 (1987): 329– 52. [22] Jarvis T. J. Kimura T. and Vaintrob. A. “Spin Gromov–Witten invariants.” Communications in Mathematical Physics  259, no. 3 (2005): 511– 43. Google Scholar CrossRef Search ADS   [23] Kac V. G. “Infinite-dimensional algebras, Dedekind’s $$\eta$$-function, classical Möbius function and the very strange formula.” Advances in Mathematics  30, no. 2 (1978): 85– 136. Google Scholar CrossRef Search ADS   [24] Kac V. G. Infinite-Dimensional Lie Algebras , vol. 44. Cambridge, Cambridge University Press, 1994. [25] Kac V. G. and Wakimoto. M. “Exceptional Hierarchies of Soliton Equations.” In Proceedings of Symposia in Pure Mathematics , edited by Ehrenpreis L. et al.  . (Theta Functions Bowdoin 1987), vol. 49, 191– 237, Providence, RI, American Mathematical Society, 1989. Google Scholar CrossRef Search ADS   [26] Kontsevich M. “Intersection theory on the moduli space of curves and the matrix Airy function.” Communications in Mathematical Physics  147, no. 1 (1992): 1– 23. Google Scholar CrossRef Search ADS   [27] Kostant B. “The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group.” American Journal of Mathematics  81 (1959): 973– 1032. Google Scholar CrossRef Search ADS   [28] Liu K. Vakil R. and Xu. H. “Formal pseudodifferential operators and Witten’s $$r$$-spin numbers.” Journal für die reine und angewandte Mathematik (Crelles Journal)  (2011). DOI: https://doi.org/10.1515/crelle-2014-0102 (accessed on December 23, 2014). [29] Liu K. and Xu. H. “Descendent integrals and tautological rings of moduli spaces of curves.” In Geometry and Analysis , vol. 2. Advanced Lecturers in Mathematics, edited by Ji L. 18, 137– 72, Somerville, MA, USA, International Press of Boston, 2010. [30] Liu S.-Q. Ruan Y. and Zhang. Y. “BCFG Drinfeld–Sokolov hierarchies and FJRW-theory.” Inventiones Mathematicae  201, no. 2 (2015): 711– 72. Google Scholar CrossRef Search ADS   [31] Liu S.-Q. Yang D. and Zhang. Y. “Uniqueness theorem of $$\mathcal{W}$$-constraints for simple singularities.” Letters in Mathematical Physics  103, no. 12 (2013): 1329– 45. Google Scholar CrossRef Search ADS   [32] Polishchuk A. and Vaintrob. A. “Algebraic construction of Witten’s Top Chern Class.” In Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000) , edited by Previato E. 229– 50. AMS, Providence, RI, Special Session on Enumerative Geometry in Physics. Contemporary Mathematics 276, 2001. Google Scholar CrossRef Search ADS   [33] Shadrin S. V. “Geometry of meromorphic functions and intersections onmoduli spaces of curves.” International Mathematics Research Notices  2003, no. 38 (2003): 2051– 94. Google Scholar CrossRef Search ADS   [34] Wasow W. Asymptotic Expansions for Ordinary Differential Equations . New York/London/Sydney: Interscience Publishers, A Division of John Wiley and Sons Inc., 1965. [35] Witten E. “Two-dimensional gravity and intersection theory on moduli space.” Surveys in Differential Geometry  1 (1991): 243– 310. Google Scholar CrossRef Search ADS   [36] Witten E. “The $$N$$ matrix model and gauged WZW models.” Nuclear Physics B 371  (1992): 191– 245. [37] Witten E. “Algebraic geometry associated with matrix models of two-dimensional gravity.” Topological Methods in Modern Mathematics (Stony Brook, NY, 1991) , 235– 69. Houston, TX: Publish or Perish, 1993. [38] Zhou J. “Solution of W-constraints for R-spin intersection numbers.” (2013): arXiv:1305.6991. [39] Zhou J. “On absolute N-point function associated with Gelfand–Dickey polynomials.” (2015): preprint. [40] Zhou J. “Emergent geometry and mirror symmetry of a point.” (2015): preprint arXiv:1507.01679. © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]

Behrstock, Jason;Falgas-Ravry, Victor;Hagen, Mark F;Susse, Tim

doi: 10.1093/imrn/rnw287pmid: N/A

Abstract We study two global structural properties of a graph $$\Gamma$$, denoted $$\mathcal{AS}$$ and $$\mathcal{CFS}$$, which arise in a natural way from geometric group theory. We study these properties in the Erdős–Rényi random graph model $${\mathcal G}(n,p)$$, proving the existence of a sharp threshold for a random graph to have the $$\mathcal{AS}$$ property asymptotically almost surely, and giving fairly tight bounds for the corresponding threshold for the $$\mathcal{CFS}$$ property. As an application of our results, we show that for any constant $$p$$ and any $$\Gamma\in{\mathcal G}(n,p)$$, the right-angled Coxeter group $$W_\Gamma$$ asymptotically almost surely has quadratic divergence and thickness of order $$1$$, generalizing and strengthening a result of Behrstock–Hagen–Sisto [8]. Indeed, we show that at a large range of densities a random right-angled Coxeter group has quadratic divergence. 1 Introduction In this article, we consider two properties of graphs motivated by geometric group theory. We show that these properties are typically present in random graphs. We repay the debt to geometric group theory by applying our (purely graph-theoretic) results to the large-scale geometry of Coxeter groups. Random graphs Let $${\mathcal G}(n,p)$$ be the random graph model on $$n$$ vertices obtained by including each edge independently at random with probability $$p=p(n)$$. The parameter $$p$$ is often referred to as the density of $${\mathcal G}(n,p)$$. The model $${\mathcal G}(n,p)$$ was introduced by Gilbert [23], and the resulting random graphs are usually referred to as the “Erdős–Rényi random graphs” in honor of Erdős and Rényi’s seminal contributions to the field, and we follow this convention. We say that a property $$\mathcal{P}$$ holds asymptotically almost surely (a.a.s.) in $${\mathcal G}(n,p)$$ if for $$\Gamma\in{\mathcal G}(n,p),$$ we have $$\mathbb{P}(\Gamma \in \mathcal{P})\rightarrow 1$$ as $$n\rightarrow \infty$$. In this article, we will be interested in proving that certain global properties hold a.a.s. in $${\mathcal G}(n,p)$$ both for a wide range of probabilities $$p=p(n)$$. A graph property is (monotone) increasing if it is closed under the addition of edges. A paradigm in the theory of random graphs is that global increasing graph properties exhibit sharp thresholds in $${\mathcal G}(n,p)$$: for many global increasing properties $$\mathcal{P}$$, there is a critical density$$p_c=p_c(n)$$ such that for any fixed $$\epsilon>0$$ if $$p<(1-\epsilon)p_c$$ then a.a.s. $$\mathcal{P}$$ does not hold in $${\mathcal G}(n,p)$$, while if $$p>(1+\epsilon)p_c$$ then a.a.s. $$\mathcal{P}$$ holds in $${\mathcal G}(n,p)$$. A quintessential example is the following classical result of Erdős and Rényi which provides a sharp threshold for connectedness: Theorem (Erdős–Rényi; [21]). There is a sharp threshold for connectivity of a random graph with critical density $$p_c(n)=\frac{\log(n)}{n}$$. □ The local structure of the Erdős–Rényi random graph is well understood, largely due to the assumption of independence between the edges. For example, Erdős–Rényi and others have obtained threshold densities for the existence of certain subgraphs in a random graph (see e.g., [21, Theorem 1, Corollaries 1–5]). In earlier applications of random graphs to geometric group theory, this feature of the model was successfully exploited in order to analyze the geometry of right-angled Artin and Coxeter groups presented by random graphs; this is notable, for example, in the work of Charney and Farber [14]. In particular, the presence of an induced square implies non-hyperbolicity of the associated right-angled Coxeter group [14, 21, 31]. In this article, we take a more global approach. Earlier work established a correspondence between some fundamental geometric properties of right-angled Coxeter groups and large-scale structural properties of the presentation graph, rather than local properties such as the presence or absence of certain specified subgraphs. The simplest of these properties is the property of being the join of two subgraphs that are not cliques. One large scale graph property relevant in the present context is a property studied in [8] which, roughly, says that the graph is constructed in a particular organized, inductive way from joins. In this article, we discuss a refined version of this property, $$\mathcal{CFS}$$, which is a slightly-modified version of a property introduced by Dani–Thomas [17]. We also study a stronger property, $$\mathcal{AS}$$, and show it is generic in random graphs for a large range of $$p(n)$$, up $$1-\omega(n^{-2})$$. $$\boldsymbol{\mathcal{AS}}$$ graphs The first class of graphs we study is the class of augmented suspensions, which we denote $$\mathcal{AS}$$. A graph is an augmented suspension if it contains an induced subgraph which is a suspension (see Section 2 for a precise definition of this term), and any vertex which is not in that suspension is connected by edges to at least two nonadjacent vertices of the suspension. Theorem 4.4 and 4.5 (Sharp Threshold for $$\boldsymbol{\mathcal{AS}}$$).Let $$\epsilon>0$$ be fixed. If $$p=p(n)$$ satisfies $$p\ge(1+\epsilon)\left(\displaystyle\frac{\log n}{n}\right)^{\frac{1}{3}}$$ and $$(1-p)n^2\to\infty$$, then $$\Gamma\in{\mathcal G}(n,p(n))$$ is a.a.s. in $$\mathcal{AS}$$. On the other hand, if $$p\le(1-\epsilon)\left(\displaystyle\frac{\log{n}}{n}\right)^{\frac{1}{3}}$$, then $$\Gamma\in{\mathcal G}(n,p)$$ a.a.s. does not lie in $$\mathcal{AS}$$. Intriguingly, Kahle proved that a function similar to the critical density in Theorem 4.4 is the threshold for a random simplicial complex to have vanishing second rational cohomology [28]. Remark (Behavior near $$p=1$$). Note that property $$\mathcal{AS}$$ is not monotone increasing, since it requires the presence of a number of non-edges. In particular, complete graphs are not in $$\mathcal{AS}$$. Thus unlike the global properties typically studied in the theory of random graphs, $$\mathcal{AS}$$ will cease to hold a.a.s. when the density $$p$$ is very close to $$1$$. In fact, [8, Theorem 3.9] shows that if $$p(n)=1-\Omega\left(\frac{1}{n^2}\right)$$, then a.a.s. $$\Gamma$$ is either a clique or a clique minus a fixed number of edges whose endpoints are all disjoint. Thus, with positive probability, $$\Gamma\in\mathcal{AS}$$. However, [14, Theorem 1] shows that if $$(1-p)n^2\to 0$$ then $$\Gamma$$ is asymptotically almost surely a clique, and hence not in $$\mathcal{AS}$$. □ $$\boldsymbol{\mathcal{CFS}}$$ graphs The second family of graphs, which we call $$\mathcal{CFS}$$ graphs (“Constructed From Squares”), arise naturally in geometric group theory in the context of the large–scale geometry of right–angled Coxeter groups, as we explain below and in Section 3. A special case of these graphs was introduced by Dani–Thomas to study divergence in triangle-free right-angled Coxeter groups [17]. The graphs we study are intimately related to a property called thickness, a feature of many key examples in geometric group theory and low dimensional topology that is, closely related to divergence, relative hyperbolicity, and a number of other topics. This property is, in essence, a connectivity property because it relies on a space being “connected” through sequences of “large” subspaces. Roughly speaking, a graph is $$\mathcal{CFS}$$ if it can be built inductively by chaining (induced) squares together in such a way that each square overlaps with one of the previous squares along a diagonal (see Section 2 for a precise definition). We explain in the next section how this class of graphs generalizes $$\mathcal{AS}$$. Our next result about genericity of $$\mathcal{CFS}$$ combines with Proposition 3.1 below to significantly strengthen [8, Theorem VI]. This result is an immediate consequence of Theorems 5.1 and 5.7, which, in fact, establish slightly more precise, but less concise, bounds. Theorem 5.1 and 5.7 Suppose $$(1-p)n^2\to\infty$$ and let $$\epsilon>0$$. Then $$\Gamma\in{\mathcal G}(n,p)$$ is a.a.s. in $$\mathcal{CFS}$$ whenever $$p(n)>n^{-\frac{1}{2}+\epsilon}$$. Conversely, $$\Gamma\in{\mathcal G}(n,p)$$ is a.a.s. not in $$\mathcal{CFS}$$, whenever $$p(n)<n^{-\frac{1}{2} -\epsilon}.$$ We actually show, in Theorem 5.1, that at densities above $$5\sqrt{\frac{\log n}{n}}$$, with $$(1-p)n^2\to\infty$$, the random graph is a.a.s. in $$\mathcal{CFS}$$, while in Theorem 5.7 we show a random graph a.a.s. not in $$\mathcal{CFS}$$ at densities below $$\frac{1}{\sqrt{n}\log{n}}$$. Theorem 5.1 applies to graphs in a range strictly larger than that in which Theorem 4.4 holds (though our proof of Theorem 5.1 relies on Theorem 4.4 to deal with the large $$p$$ case). Theorem 5.1 combines with Theorem 4.5 to show that, for densities between $$\left(\log n/n\right)^{\frac{1}{2}}$$ and $$\left(\log n/n\right)^{\frac{1}{3}}$$, a random graph is asymptotically almost surely in $$\mathcal{CFS}$$ but not in $$\mathcal{AS}$$. We also note that Babson–Hoffman–Kahle [3] proved that a function of order $$n^{-\frac{1}{2}}$$ appears as the threshold for simple-connectivity in the Linial–Meshulam model for random 2–complexes [29]. It would be interesting to understand whether there is a connection between genericity of the $$\mathcal{CFS}$$ property and the topology of random 2-complexes. Unlike our results for the $$\mathcal{AS}$$ property, we do not establish a sharp threshold for the $$\mathcal{CFS}$$ property. In fact, we believe that neither the upper nor lower bounds, given in Theorem 5.1 and Theorem 5.7, for the critical density around which $$\mathcal{CFS}$$ goes from a.a.s. not holding to a.a.s. holding are sharp. Indeed, we believe that there is a sharp threshold for the $$\mathcal{CFS}$$ property located at $$p_c(n)=\theta(n^{-\frac{1}{2}})$$. This conjecture is linked to the emergence of a giant component in the “square graph” of $$\Gamma$$ (see the next section for a definition of the square graph and the heuristic discussion after the proof of Theorem 5.7). Applications to geometric group theory Our interest in the structure of random graphs was sparked largely by questions about the large-scale geometry of right-angled Coxeter groups. Coxeter groups were first introduced in [15] as a generalization of reflection groups, that is, discrete groups generated by a specified set of reflections in Euclidean space. A reflection group is right-angled if the reflection loci intersect at right angles. An abstract right-angled Coxeter group generalizes this situation: it is defined by a group presentation in which the generators are involutions and the relations are obtained by declaring some pairs of generators to commute. Right-angled Coxeter groups (and more general Coxeter groups) play an important role in geometric group theory and are closely-related to some of that field’s most fundamental objects, for example, CAT(0) cube complexes [18, 27, 33] and (right-angled) Bruhat-Tits buildings (see e.g., [18]). A right-angled Coxeter group is determined by a unique finite simplicial presentation graph: the vertices correspond to the involutions generating the group, and the edges encode the pairs of generators that commute. In fact, the presentation graph uniquely determines the right-angled Coxeter group [32]. In this article, as an application of our results on random graphs, we continue the project of understanding large-scale geometric features of right-angled Coxeter groups in terms of the combinatorics of the presentation graph, begun in [8, 14, 17]. Specifically, we study right-angled Coxeter groups defined by random presentation graphs, focusing on the prevalence of two important geometric properties: relative hyperbolicity and thickness. Relative hyperbolicity, in the sense introduced by Gromov and equivalently formulated by many others [10, 22, 24, 35], when it holds, is a powerful tool for studying groups. On the other hand, thickness of a finitely-generated group (more generally, a metric space) is a property introduced by Behrstock–Druţu–Mosher in [6] as a geometric obstruction to relative hyperbolicity and has a number of powerful geometric applications. For example, thickness gives bounds on divergence (an important quasi-isometry invariant of a metric space) in many different groups and spaces [5, 7, 11, 17, 37]. Thickness is an inductive property: in the present context, a finitely generated group $$G$$ is thick of order $$0$$ if and only if it decomposes as the direct product of two infinite subgroups. The group $$G$$ is thick of order $$n$$ if there exists a finite collection $$\mathcal H$$ of undistorted subgroups of $$G$$, each thick of order $$n-1$$, whose union generates a finite-index subgroup of $$G$$ and which has the following “chaining” property: for each $$g,g'\in G$$, one can construct a sequence $$g\in g_1H_1,g_2H_2,\ldots,g_kH_k\ni g'$$ of cosets, with each $$H_i\in\mathcal H$$, so that consecutive cosets have infinite coarse intersection. Many of the best-known groups studied by geometric group theorists are thick, and indeed thick of order $$1$$: one-ended right-angled Artin groups, mapping class groups of surfaces, outer automorphism groups of free groups, fundamental groups of three-dimensional graph manifolds, etc. [6]. The class of Coxeter groups contains many examples of hyperbolic and relatively hyperbolic groups. There is a criterion for hyperbolicity purely in terms of the presentation graph due to Moussong [31] and an algebraic criterion for relative hyperbolicity due to Caprace [13]. The class of Coxeter groups includes examples which are non-relatively hyperbolic, for instance, those constructed by Davis–Januszkiewicz [19] and, also, ones studied by Dani–Thomas [17]. In fact, in [8], this is taken further: it is shown that every Coxeter group is actually either thick, or hyperbolic relative to a canonical collection of thick Coxeter subgroups. Further, there is a simple, structural condition on the presentation graph, checkable in polynomial time, which characterizes thickness. This result is needed to deduce the applications below from our graph theoretic results. Charney and Farber initiated the study of random graph products (including right-angled Artin and Coxeter groups) using the Erdős-Rényi model of random graphs [14]. The structure of the group cohomology of random graph products was obtained in [20]. In [8], various results are proved about which random graphs have the thickness property discussed above, leading to the conclusion that, at certain low densities, random right-angled Coxeter groups are relatively hyperbolic (and thus not thick), while at higher densities, random right-angled Coxeter groups are thick. In this article, we improve significantly on one of the latter results, and also prove something considerably more refined: we isolate not just thickness of random right-angled Coxeter groups, but thickness of a specified order, namely $$1$$: Corollary 3.2 (Random Coxeter groups are thick of order 1.)There exists a constant $$C>0$$ such that if $$p\colon{\mathbb{N}}\rightarrow(0,1)$$ satisfies $$\left(\displaystyle\frac{C\log{n}}{n}\right)^{\frac{1}{2}}\leq p(n)\leq 1-\displaystyle\frac{(1+\epsilon)\log{n}}{n}$$ for some $$\epsilon>0$$, then the random right-angled Coxeter group $$W_{G_{n,p}}$$ is asymptotically almost surely thick of order exactly $$1$$, and in particular has quadratic divergence. Corollary 3.2 significantly improves on Theorem 3.10 of [8], as discussed in Section 3. This theorem follows from Theorems 5.1 and 4.4, the latter being needed to treat the case of large $$p(n)$$, including the interesting special case in which $$p$$ is constant. Remark 1.1. We note that characterizations of thickness of right-angled Coxeter groups in terms of the structure of the presentation graph appear to generalize readily to graph products of arbitrary finite groups and, probably, via the action on a cube complex constructed by Ruane and Witzel in [36], to arbitrary graph products of finitely generated abelian groups, using appropriate modifications of the results in [8]. □ Organization of the article In Section 2, we give the formal definitions of $$\mathcal{AS}$$ and $$\mathcal{CFS}$$ and introduce various other graph-theoretic notions we will need. In Section 3, we discuss the applications of our random graph results to geometric group theory, in particular, to right-angled Coxeter groups and more general graph products. In Section 4, we obtain a sharp threshold result for $$\mathcal{AS}$$ graphs. Section 5 is devoted to $$\mathcal{CFS}$$ graphs. Finally, Section 6 contains some simulations of random graphs with density near the threshold for $$\mathcal{AS}$$ and $$\mathcal{CFS}$$. 2 Definitions Convention 2.1 A graph is a pair of finite sets $$\Gamma=(V,E)$$, where $$V=V(\Gamma)$$ is a set of vertices, and $$E=E(\Gamma)$$ is a collection of pairs of distinct elements of $$V$$, which constitute the set of edges of $$G$$. A subgraph of $$\Gamma$$ is a graph $$\Gamma'$$ with $$V(\Gamma')\subseteq V(\Gamma)$$ and $$E(\Gamma')\subseteq E(\Gamma)$$; $$\Gamma'$$ is said to be an induced subgraph of $$\Gamma$$ if $$E(\Gamma')$$ consists exactly of those edges from $$E(\Gamma)$$ whose vertices lie in $$V(\Gamma')$$. In this article, we focus on induced subgraphs, and we generally write “subgraph” to mean “induced subgraph”. In particular, we often identify a subgraph with the set of vertices inducing it, and we write $$\vert \Gamma\vert$$ for the order of $$\Gamma$$, that is, the number of vertices it contains. A clique of size $$t$$ is a complete graph on $$t\geq0$$ vertices. This includes the degenerate case of the empty graph on $$t= 0$$ vertices. □ Fig. 1. View largeDownload slide A graph in $$\mathcal{AS}$$. A block exhibiting inclusion in $$\mathcal{AS}$$ is shown in bold; the two (left-centrally located) ends of the bold block are highlighted. (A colour version of this figure is available from the journal’s website.) Fig. 1. View largeDownload slide A graph in $$\mathcal{AS}$$. A block exhibiting inclusion in $$\mathcal{AS}$$ is shown in bold; the two (left-centrally located) ends of the bold block are highlighted. (A colour version of this figure is available from the journal’s website.) Definition 2.2 (Link, join). Given a graph $$\Gamma$$, the link of a vertex $$v\in\Gamma$$, denoted $${\mathrm Lk}_\Gamma(v)$$, is the subgraph spanned by the set of vertices adjacent to $$v$$. Given graphs $$A,B$$, the join$$A\star B$$ is the graph formed from $$A\sqcup B$$ by joining each vertex of $$A$$ to each vertex of $$B$$ by an edge. A suspension is a join where one of the factors $$A,B$$ is the graph consisting of two vertices and no edges. □ We now describe a family of graphs, denoted $$\mathcal{CFS}$$, which satisfy the global structural property that they are “constructed from squares.” Definition 2.3 ($$\boldsymbol{\mathcal{CFS}}$$). Given a graph $$\Gamma$$, let $$\square(\Gamma)$$ be the auxiliary graph whose vertices are the induced $$4$$–cycles from $$\Gamma$$, with two distinct $$4$$–cycles joined by an edge in $$\square(\Gamma)$$ if and only if they intersect in a pair of non-adjacent vertices of $$\Gamma$$ (i.e., in a diagonal). We refer to $$\square(\Gamma)$$ as the square-graph of $$\Gamma$$. A graph $$\Gamma$$ belongs to $$\mathcal{CFS}$$ if $$\Gamma=\Gamma'\star K$$, where $$K$$ is a (possibly empty) clique and $$\Gamma'$$ is a non-empty subgraph such that $$\square(\Gamma')$$ has a connected component $$C$$ such that the union of the $$4$$–cycles from $$C$$ covers all of $$V(\Gamma')$$. Given a vertex $$F\in\square(\Gamma)$$, we refer to the vertices in the $$4$$–cycle in $$\Gamma$$ associated to $$F$$ as the support of $$F$$. □ Remark 2.4. Dani–Thomas introduced component with full support graphs in [17], a subclass of the class of triangle-free graphs. We note that each component with full support graph is constructed from squares, but the converse is not true. Indeed, since we do not require our graphs to be triangle-free, our definition necessarily only counts induced 4–cycles and allows them to intersect in more ways than in [17]. This distinction is relevant to the application to Coxeter groups, which we discuss in Section 3. □ Definition 2.5 (Augmented suspension). The graph $$\Gamma$$ is an augmented suspension if it contains an induced subgraph $$B=\{w,w'\}\star \Gamma'$$, where $$w,w'$$ are nonadjacent and $$\Gamma'$$ is not a clique, satisfying the additional property that if $$v\in\Gamma-B$$, then $$\mathrm{Lk}_\Gamma(v)\cap \Gamma'$$ is not a clique. Let $$\mathcal{AS}$$ denote the class of augmented suspensions. Figure 1 shows a graph in $$\mathcal{AS}$$. □ Remark 2.6. Neither the $$\mathcal{CFS}$$ nor the $$\mathcal{AS}$$ properties introduced above are monotone with respect to the addition of edges. This stands in contrast to the most commonly studied global properties of random graphs. □ Definition 2.7 (Block, core, ends). A block in $$\Gamma$$ is a subgraph of the form $$B(w,w')=\{w,w'\}\star \Gamma'$$ where $$\{w,w'\}$$ is a pair of non-adjacent vertices and $$\Gamma'\subset \Gamma$$ is a subgraph of $$\Gamma$$ induced by a set of vertices adjacent to both $$w$$ and $$w'$$. A block is maximal if $$V(\Gamma')=\mathrm{Lk}_{\Gamma}(w)\cap \mathrm{Lk}_{\Gamma}(w')$$. Given a block $$B=B(w,w')$$, we refer to the non-adjacent vertices $$w,w'$$ as the ends of $$B$$, denoted $$\mathrm{end}(B)$$, and the vertices of $$\Gamma'$$ as the core of $$B$$, denoted $$\mathrm{core}(B)$$. □ Note that $$\mathcal{AS}\subsetneq\mathcal{CFS}$$, indeed Theorems 5.1 and 4.5 show that there must exist graphs in $$\mathcal{CFS}$$ that are not in $$\mathcal{AS}$$. Here we explain how any graph in $$\mathcal{AS}$$ is in $$\mathcal{CFS}$$. Lemma 2.8. Let $$\Gamma$$ be a graph in $$\mathcal{AS}$$. Then $$\Gamma \in \mathcal{CFS}$$. □ Proof Let $$B(w,w')=\{w,w'\}\star \Gamma'$$ be a maximal block in $$\Gamma$$ witnessing $$\Gamma \in \mathcal{AS}$$. Write $$\Gamma'=A\star D$$, where $$D$$ is the collection of all vertices of $$\Gamma'$$ which are adjacent to every other vertex of $$\Gamma'$$. Note that $$D$$ induces a clique in $$\Gamma$$. By definition of the $$\mathcal{AS}$$ property $$\Gamma'$$ is not a clique, whence, $$A$$ contains at least one pair of non-adjacent vertices. Furthermore, by the definition of $$D$$, for every vertex $$a\in A$$ there exists $$a' \in A$$ with $$\{a,a'\}$$ non-adjacent. The $$4$$–cycles induced by $$\{w,w'a,a'\}$$ for non-adjacent pairs $$a, a'$$ from $$A$$ are connected in $$\square(\Gamma)$$. Denote the component of $$\square(\Gamma)$$ containing them by $$C$$. Consider now a vertex $$v\in \Gamma - B(w,w')$$. Since $$B(w,w')$$ is maximal, we have that at least one of $$w,w'$$ is not adjacent to $$v$$ — without loss of generality, let us assume that it is $$w$$. By the $$\mathcal{AS}$$ property, $$v$$ must be adjacent to a pair $$a,a'$$ of non-adjacent vertices from $$A$$. Then $$\{w, a,a', v\}$$ induces a $$4$$–cycle, which is adjacent to $$\{w,w',a,a'\}\in\square(\Gamma)$$ and hence lies in $$C$$. Finally, consider a vertex $$d \in D$$. If $$v$$ is adjacent to all vertices of $$\Gamma$$, then $$\Gamma$$ is the join of a graph with a clique containing $$d$$, and we can ignore $$d$$ with respect to establishing the $$\mathcal{CFS}$$ property. Otherwise $$d$$ is not adjacent to some $$v\in \Gamma-B(w,w')$$. By the $$\mathcal{AS}$$ property, $$v$$ is connected by edges to a pair of non-adjacent vertices $$\{a,a'\}$$ from $$A$$. Thus $$\{d,a,a',v\}$$ induces a $$4$$–cycle. Since (as established above) there is some $$4$$–cycle in $$C$$ containing $$\{a,a',v\}$$, we have that $$\{d,a,a',v\} \in C$$ as well. Thus $$\Gamma= \Gamma'' \star K$$, where $$K$$ is a clique and $$V(\Gamma'')$$ is covered by the union of the $$4$$–cycles in $$C$$, so that $$\Gamma \in \mathcal{CFS}$$ as claimed. ■ 3 Geometry of right-angled Coxeter groups If $$\Gamma$$ is a finite simplicial graph, the right-angled Coxeter group$$W_\Gamma$$presented by$$\Gamma$$ is the group defined by the presentation   ⟨Vert(Γ)∣{w2,uvu−1v−1:u,v,w∈Vert(Γ),{u,v}∈Edge(Γ)⟩. A result of Mühlherr [32] shows that the correspondence $$\Gamma\leftrightarrow W_\Gamma$$ is bijective. We can thus speak of “the random right-angled Coxeter group” — it is the right-angled Coxeter group presented by the random graph. (We emphasize that the above presentation provides the definition of a right-angled Coxeter group: this definition abstracts the notion of a reflection group – a subgroup of a linear group generated by reflections — but infinite Coxeter groups need not admit representations as reflection groups.) Recent articles have discussed the geometry of Coxeter groups, especially relative hyperbolicity and closely-related quasi-isometry invariants like divergence and thickness, cf. [8, 13, 17]. In particular, Dani–Thomas introduced a property they call having a component of full support for triangle-free graphs (which is exactly the triangle-free version of $$\mathcal{CFS}$$) and they prove that under the assumption $$\Gamma$$ is triangle-free, $$W_\Gamma$$ is thick of order at most $$1$$ if and only if it has quadratic divergence if and only if $$\Gamma$$ is in $$\mathcal{CFS}$$, see [17, Theorem 1.1 and Remark 4.8]. Since the densities where random graphs are triangle-free are also square-free (and thus not $$\mathcal{CFS}$$ — in fact, they are disconnected!), we need the following slight generalization of the result of Dani–Thomas: Proposition 3.1. Let $$\Gamma$$ be a finite simplicial graph. If $$\Gamma$$ is in $$\mathcal{CFS}$$ and $$\Gamma$$ does not decompose as a nontrivial join, then $$W_\Gamma$$ is thick of order exactly $$1$$. □ Proof Theorem II of [8] shows immediately that, if $$\Gamma\in\mathcal{CFS}$$, then $$W_\Gamma$$ is thick, being formed by a series of thick unions of $$4$$–cycles; since each $$4$$–cycle is a join, it follows that $$\Gamma$$ is thick of order at most $$1$$. On the other hand, [8, Proposition 2.11] shows that $$W_\Gamma$$ is thick of order at least $$1$$ provided $$\Gamma$$ is not a join. ■ Our results about random graphs yield: Corollary 3.2. There exists $$k>0$$ so that if $$p\colon{\mathbb{N}}\rightarrow(0,1)$$ and $$\epsilon>0$$ are such that $$\sqrt{\frac{k\log n}{n}}\leq p(n)\leq 1-\displaystyle\frac{(1-\epsilon)\log{n}}{n}$$ for all sufficiently large $$n$$, then for $$\Gamma\in{\mathcal G}(n,p)$$ the group $$W_\Gamma$$ is asymptotically almost surely thick of order exactly $$1$$ and hence has quadratic divergence. □ Proof Theorem 5.1 shows that any such $$\Gamma$$ is asymptotically almost surely in $$\mathcal{CFS}$$, whence $$W_\Gamma$$ is thick of order at most $$1$$. We emphasize that to apply this result for sufficiently large functions $$p(n)$$ the proof of Theorem 5.1 requires an application of Theorem 4.4 to establish that $$\Gamma$$ is a.a.s. in $$\mathcal{AS}$$ and hence in $$\mathcal{CFS}$$ by Lemma 2.8. By Proposition 3.1, to show that the order of thickness is exactly one, it remains to rule out the possibility that $$\Gamma$$ decomposes as a nontrivial join. However, this occurs if and only if the complement graph is disconnected, which asymptotically almost surely does not occur whenever $$p(n)\le 1-\frac{(1-\epsilon)\log{n}}{n}$$, by the sharp threshold for connectivity of $${\mathcal G}(n,1-p)$$ established by Erdős and Rényi in [21]. Since this holds for $$p(n)$$ by assumption, we conclude that asymptotically almost surely, $$W_\Gamma$$ is thick of order at least $$1$$. Since $$W_\Gamma$$ is CAT(0) and thick of order exactly $$1$$, the consequence about divergence now follows from [5]. ■ This corollary significantly generalizes Theorem 3.10 of [8], which established that, if $$\Gamma\in{\mathcal G}(n,\frac{1}{2})$$, then $$W_\Gamma$$ is asymptotically almost surely thick. Theorem 3.10 of [8] does not provide effective bounds on the order of thickness and its proof is significantly more complicated than the proof of Corollary 3.2 given above — indeed, it required several days of computation (using 2013 hardware) to establish the base case of an inductive argument. Remark 3.3 (Higher-order thickness). A lower bound of $$p(n)=n^{-\frac{5}{6}}$$ for membership in a larger class of graphs whose corresponding Coxeter groups are thick can be found in [8, Theorem 3.4]. In fact, this argument can be adapted to give a simple proof that a.a.s. thickness does not occur at densities below $$n^{-\frac{3}{4}}$$. The correct threshold for a.a.s. thickness is, however, unknown. □ Remark 3.4 (Random graph products versus random presentations). Corollary 3.2 and Remark 3.3 show that the random graph model for producing random right-angled Coxeter groups generates groups with radically different geometric properties. This is in direct contrast to other methods of producing random groups, most notably Gromov’s random presentation model [25, 26] where, depending on the density of relators, groups are either almost surely hyperbolic or finite (with order at most $$2$$). This contrast speaks to the merits of considering a random right-angled Coxeter group as a natural place to study random groups. For instance, Calegari–Wilton recently showed that in the Gromov model a random group contains many subgroups which are isomorphic to the fundamental group of a compact hyperbolic 3–manifold [12]; does the random right-angled Coxeter group also contain such subgroups? Right-angled Coxeter groups, and indeed thick ones, are closely related to Gromov’s random groups in another way. When the parameter for a Gromov random groups is $$<\frac{1}{6}$$ such a group is word-hyperbolic [25] and acts properly and cocompactly on a CAT(0) cube complex [34]. Hence the Gromov random group virtually embeds in a right-angled Artin group [4]. Moreover, at such parameters such a random group is one-ended [16], whence the associated right-angled Artin group is as well. By [4] this right-angled Artin group is thick of order 1. Since any right-angled Artin group is commensurable with a right-angled Coxeter group [19], one obtains a thick of order $$1$$ right-angled Coxeter group containing the randomly presented group. □ 4 Genericity of $$\boldsymbol{\mathcal{AS}}$$ We will use the following standard Chernoff bounds (see e.g., [2, Theorems A.1.11 and A.1.13]): Lemma 4.1 (Chernoff bounds). Let $$X_1,\ldots,X_n$$ be independent identically distributed random variables taking values in $$\{0,1\}$$, let $$X$$ be their sum, and let $$\mu=\mathbb E[X]$$. Then for any $$\delta\in(0,2/3)$$  P(|X−μ|≥δμ)≤2e−δ2μ3. □ Corollary 4.2. Let $$\varepsilon, \delta>0$$ be fixed. (i) If $$p(n)\ge \left(\frac{(6+\varepsilon)\log n}{\delta^2n}\right)^{1/2}$$, then a.a.s. for all pairs of distinct vertices $$\{x,y\}$$ in $$\Gamma \in {\mathcal G}(n,p)$$ we have $$\left\vert \vert \mathrm{Lk}_{\Gamma}(x)\cap \mathrm{Lk}_{\Gamma}(y)\vert-p^2(n-2)\right\vert < \delta p^2(n-2)$$. (ii) If $$p(n)\ge \left(\frac{(9+\varepsilon)\log n}{\delta^2n}\right)^{1/3}$$, then a.a.s. for all triples of distinct vertices $$\{x,y, z\}$$ in $$\Gamma \in {\mathcal G}(n,p)$$ we have $$\left\vert \vert \mathrm{Lk}_{\Gamma}(x)\cap \mathrm{Lk}_{\Gamma}(y)\cap \mathrm{Lk}_{\Gamma}(z)\vert-p^3(n-3)\right\vert < \delta p^3(n-3)$$. □ Proof For (i), let $$\{x,y\}$$ be any pair of distinct vertices. For each vertex $$v \in \Gamma-\{x,y\}$$, set $$X_v$$ to be the indicator function of the event that $$v\in \mathrm{Lk}_{\Gamma}(x)\cap \mathrm{Lk}_{\Gamma}(y)$$, and set $$X=\sum_v X_v$$ to be the size of $$\mathrm{Lk}_{\Gamma}(x)\cap \mathrm{Lk}_{\Gamma}(y)$$. We have $$\mathbb{E}X=p^2(n-2)$$ and so by the Chernoff bounds above, $$\Pr\left(\vert X-p^2(n-2)\vert\geq \delta p^2(n-2)\right) \leq 2e^{-\frac{\delta^2p^2(n-2)}{3}}$$. Applying Markov’s inequality, the probability that there exists some “bad pair” $$\{x,y\}$$ in $$\Gamma$$ for which $$\vert\mathrm{Lk}_{\Gamma}(x)\cap \mathrm{Lk}_{\Gamma}(y)\vert$$ deviates from its expected value by more than $$\delta p^2(n-2)$$ is at most   (n2)2e−δ2p2(n−2)3=o(1), provided $$\delta^2 p^2n\ge (6+\varepsilon) \log n$$ and $$\varepsilon, \delta>0$$ are fixed. Thus for this range of $$p=p(n)$$, a.a.s. no such bad pair exists. The proof of (ii) is nearly identical. ■ Lemma 4.3. (i) Suppose $$1-p\geq \frac{\log n}{2n}$$. Then asymptotically almost surely, the order of a largest clique in $$\Gamma\in{\mathcal G}(n,p)$$ is $$o(n)$$. (ii) Let $$\eta$$ be fixed with $$0<\eta<1$$. Suppose $$1-p\geq \eta$$. Then asymptotically almost surely, the order of a largest clique in $$\Gamma \in {\mathcal G}(n,p)$$ is $$O(\log n)$$. □ Proof For (i), set $$r=\alpha n$$, for some $$\alpha$$ bounded away from $$0$$. Write $$H(\alpha)=\alpha\log\frac{1}{\alpha}+(1-\alpha)\log\frac{1}{1-\alpha}$$. Using the standard entropy bound $$\binom{n}{\alpha n}\leq e^{H(\alpha)n}$$ and our assumption for $$(1-p)$$, we see that the expected number of $$r$$-cliques in $$\Gamma$$ is   (nr)p(r2) ≤eH(α)nelog⁡(1−(1−p))(α2n22+O(n))≤exp⁡(−α22nlog⁡n+O(n))=o(1). Thus by Markov’s inequality, a.a.s. $$\Gamma$$ does not contain a clique of size $$r$$, and the order of a largest clique in $$\Gamma$$ is $$o(n)$$. The proof of (ii) is similar: suppose $$1-p>\eta$$. Then for any $$r\leq n$$,   (nr)p(r2) <nr(1−η)r(r−1)/2=exp⁡(r(log⁡n−r−12log⁡11−η)), which for $$\eta>0$$ fixed and $$r-1>\frac{2}{\log (1/(1-\eta))}(1+ \log n )$$ is as most $$n^{-\frac{2}{\log (1/(1-\eta))}}=o(1)$$. We may thus conclude as above that a.a.s. a largest clique in $$\Gamma$$ has order $$O(\log n)$$. ■ Theorem 4.4 (Genericity of $$\mathcal{AS}$$). Suppose $$p(n)\ge(1+\epsilon)\left(\displaystyle\frac{\log{n}}{n}\right)^{\frac{1}{3}}$$ for some $$\epsilon>0$$ and $$(1-p)n^2\to\infty$$. Then, a.a.s. $$\Gamma\in{\mathcal G}(n,p)$$ is in $$\mathcal{AS}$$. □ Proof Let $$\delta>0$$ be a small constant to be specified later (the choice of $$\delta$$ will depend on $$\epsilon$$). By Corollary 4.2 (i) for $$p(n)$$ in the range we are considering, a.a.s. all joint links have size at least $$(1-\delta)p^2(n-2)$$. Denote this event by $$\mathcal{E}_1$$. We henceforth condition on $$\mathcal{E}_1$$ occurring (not this only affects the values of probabilities by an additive factor of $$\mathbb{P}(\mathcal{E}_1^c)=O(n^{-\varepsilon})=o(1)$$). With probability $$1-p^{\binom{n}{2}}=1-o(1)$$, $$\Gamma$$ is not a clique, whence, there there exist non-adjacent vertices in $$\Gamma$$. We henceforth assume $$\Gamma\neq K_n$$, and choose $$v_1, v_2\in \Gamma$$ which are not adjacent. Let $$B$$ be the maximal block associated with the pair $$(v_1,v_2)$$. We separate the range of $$p$$ into three. Case 1: $$\boldsymbol{p}$$ is “far” from both the threshold and $$1$$. Let $$\alpha>0$$ be fixed, and suppose $$\alpha n^{-1/4}\leq p \leq 1- \frac{\log n}{2n}$$. Let $$\mathcal{E}_2$$ be the event that for every vertex $$v\in\Gamma-B$$ the set $$\mathrm{Lk}_{\Gamma}(v)\cap B$$ has size at least $$\frac{1}{2}p^3(n-3)$$. By Corollary 4.2, a.a.s. event $$\mathcal{E}_2$$ occurs, that is, all vertices in $$\Gamma-B$$ have this property. We claim that a.a.s. there is no clique of order at least $$\frac{1}{2}p^3(n-3)$$ in $$\Gamma$$. Indeed, if $$p<1-\eta$$ for some fixed $$\eta>0$$, then by Lemma 4.3 part (ii), a largest clique in $$\Gamma$$ has order $$O(\log n)=o(p^3n)$$. On the other hand, if $$1-\eta <p\leq 1- \frac{\log n}{2n}$$, then by Lemma 4.3 part (i), a largest clique in $$\Gamma$$ has order $$o(n)=o(p^3n)$$. Thus in either case a.a.s. for every$$v\in \Gamma-B$$, $$\mathrm{Lk}_{\Gamma}(v)\cap B$$ is not a clique and hence $$v \in \overline{B}$$, so that a.a.s. $$\overline{B}=\Gamma$$, and $$\Gamma \in \mathcal{AS}$$ as required. Case 2: $$\boldsymbol{p}$$ is “close” to the threshold. Suppose that $$(1+\epsilon)\left(\displaystyle\frac{\log{n}}{n}\right)^{\frac{1}{3}}\le p(n)$$ and $$np^4\to 0$$. Let $$\vert B\vert=m+2$$. By our conditioning, we have $$(1-\delta)(n-2)p^2\leq m \leq (1+\delta)(n-2)p^2$$. The probability that a given vertex $$v\in \Gamma$$ is not in $$\overline{B}$$ is given by:   P(v∉B¯|{|B|=m})=(1−p)m+mp(1−p)m+1+∑r=2m(mr)pr(1−p)m−rp(r2). (1) In this equation, the first two terms come from the case where $$v$$ is connected to $$0$$ and $$1$$ vertex in $$B\setminus\{v_1,v_2\}$$ respectively, while the third term comes from the case where the link of $$v$$ in $$B\setminus\{v_1,v_2\}$$ is a clique on $$r\geq 2$$ vertices. As we shall see, in the case $$np^4\to 0$$ which we are considering, the contribution from the first two terms dominates. Let us estimate their order:   (1−p)m+mp(1−p)m−1=(1+mp1−p)(1−p)m ≤(1+mp1−p)e−mp ≤(1+(1−δ)(1+ϵ)3log⁡n1−p)n−(1−δ)(1+ϵ)3. Taking $$\delta<1-\frac{1}{(1+\epsilon)^3}$$ this expression is $$o(n^{-1})$$. We now treat the sum making up the remaining terms in Equation 1. To do so, we will analyze the quotient of successive terms in the sum. Fixing $$2\le r\le m-1$$ we see:   (mr+1)pr+1(1−p)m−r−1p(r+12)(mr)pr(1−p)m−rp(r2)=m−r−1r+1⋅pr+11−p≤mpr+1≤mp3. Since $$np^4\to 0$$ (by assumption), this also tends to zero as $$n\to\infty$$. The quotients of successive terms in the sum thus tend to zero uniformly as $$n\to \infty$$, and we may bound the sum by a geometric series:   ∑r=2m(mr)pr(1−p)m−rp(r2)≤(m2)p3(1−p)m−2∑i=0m−2(mp3)i≤(12+o(1))m2p3(1−p)m−2. Now, $$m^2p^3(1-p)^{m-2}=\frac{mp^2}{1-p}\cdot mp(1-p)^{m-1}$$. The second factor in this expression was already shown to be $$o(n^{-1})$$, while $$mp^2\le(1+\delta)np^4 \to 0$$ by assumption, so the total contribution of the sum is $$o(n^{-1})$$. Thus for any value of $$m$$ between $$(1-\delta)p^2(n-2)$$ and $$(1+\delta)p^2(n-2)$$, the right hand side of Equation (1) is $$o(n^{-1})$$, and we conclude:   P(v∉B¯|E1)≤o(n−1). Thus, by Markov’s inequality,   P(B¯=Γ)≥P(E1)(1−∑vP(v∉B¯|E1))=1−o(1), establishing that a.a.s. $$\Gamma\in\mathcal{AS}$$, as claimed. Case 3: $$p$$ is “close” to $$1$$. Suppose $$n^{-2}\ll (1-p)\leq \frac{\log n}{2n}$$. Consider the complement of $$\Gamma$$, $$\Gamma^c\in {\mathcal G}(n, 1-p)$$. In the range of the parameter $$\Gamma^c$$ a.a.s. has at least two connected components that contain at least two vertices. In particular, taking complements, we see that $$\Gamma$$ is a.a.s. a join of two subgraphs, neither of which is a clique. It is a simple exercise to see that such as graph is in $$\mathcal{AS}$$, thus a.a.s. $$\Gamma\in\mathcal{AS}$$. ■ As we now show, the bound obtained in the above theorem is actually a sharp threshold. Analogous to the classical proof of the connectivity threshold [21], we consider vertices which are “isolated” from a block to prove that graphs below the threshold strongly fail to be in $$\mathcal{AS}$$. Theorem 4.5. If $$p\le\left(1-\epsilon\right)\left(\displaystyle\frac{\log{n}}{n}\right)^{\frac{1}{3}}$$ for some $$\epsilon>0$$, then $$\Gamma\in{\mathcal G}(n,p)$$ is asymptotically almost surely not in $$\mathcal{AS}$$. □ Proof We will show that, for $$p$$ as hypothesized, every block has a vertex “isolated” from it. Explicitly, let $$\Gamma\in{\mathcal G}(n,p)$$ and consider $$B=B_{v,w}=\mathrm{Lk}(v)\cap \mathrm{Lk}(w)\cup\{v,w\}$$. Let $$X(v,w)$$ be the event that every vertex of $$\Gamma-B$$ is connected by an edge to some vertex of $$B$$. Clearly $$\Gamma\in \mathcal{AS}$$ only if the event $$X(v,w)$$ occurs for some pair of non-adjacent vertices $$\{v,w\}$$. Set $$X=\bigcup_{\{v,w\}} X(v,w)$$. Note that $$X$$ is a monotone event, closed under the addition of edges, so that the probability it occurs in $$\Gamma\in {\mathcal G}(n,p)$$ is a non-decreasing function of $$p$$. We now show that when $$p=\left(1-\epsilon\right)\left(\log{n}/ n\right)^{\frac{1}{3}}$$, a.a.s. $$X$$ does not occur, completing the proof. Consider a pair of vertices $$\{v,w\}$$, and set $$k=\vert B_{v,w}\vert$$. Conditional on $$B_{v,w}$$ having this size and using the standard inequality $$(1-x)\le e^{-x}$$, we have that   P(X(v,w))=(1−(1−p)k)n−k≤e−(n−k)(1−p)k. Now, the value of $$k$$ is concentrated around its mean: by Corollary 4.2, for any fixed $$\delta>0$$ and all $$\{v,w\}$$, the order of $$B_{v,w}$$ is a.a.s. at most $$(1+\delta)np^2$$. Conditioning on this event $$\mathcal{E}$$, we have that for any pair of vertices $$v,w$$,   P(X(v,w)|E)≤maxk≤(1+δ)np2e−(n−k)(1−p)k=e−(n−(1+δ)np2)(1−p)(1+δ)np2. Now $$(1-p)^{(1+\delta)np^2} = e^{(1+\delta)np^2\log(1-p)} $$ and by Taylor’s theorem $$\log(1-p)=-p+O(p^2)$$, so that:   P(X(v,w)|E)≤e−n(1+O(p2))e−(1+δ)np3(1+O(p))=e−n1−(1+δ)(1−ϵ)3+o(1). Choosing $$\delta<\displaystyle\frac{1}{(1-\epsilon)^3}-1$$, the expression above is $$o(n^{-2})$$. Thus   P(X) ≤P(Ec)+∑{v,w}P(X(v,w)|E) =o(1)+(n2)o(n−2)=o(1). Thus a.a.s. the monotone event $$X$$ does not occur in $$\Gamma\in{\mathcal G}(n,p)$$ for $$p=\left(1-\epsilon\right)\left(\log{n}/ n\right)^{\frac{1}{3}}$$, and hence a.a.s. the property $$\mathcal{AS}$$ does not hold for $$\Gamma\in{\mathcal G}(n,p)$$ and $$p(n)\leq \left(1-\epsilon\right)\left(\log{n}/ n\right)^{\frac{1}{3}}$$. ■ 5 Genericity of $$\boldsymbol{\mathcal{CFS}}$$ The two main results in this section are upper and lower bounds for inclusion in $$\mathcal{CFS}$$. These results are established in Theorems 5.1 and 5.7. Theorem 5.1. If $$p\colon{\mathbb{N}}\rightarrow(0,1)$$ satisfies $$(1-p)n^2\to\infty$$ and $$p(n)\geq 5\sqrt{\frac{\log n}{n}}$$ for all sufficiently large $$n$$, then a.a.s. $$\Gamma \in {\mathcal G}(n,p)$$ lies in $$\mathcal{CFS}$$. □ The proof of Theorem 5.1 divides naturally into two ranges. First of all for large $$p$$, namely for $$p(n)\geq 2\left(\log{n}/n\right)^{\frac{1}{3}}$$, we appeal to Theorem 4.4 to show that a.a.s. a random graph $$\Gamma\in {\mathcal G}(n,p)$$ is in $$\mathcal{AS}$$ and hence, by Lemma 2.8, in $$\mathcal{CFS}$$. In light of our proof of Theorem 4.4, we may think of this as the case when we can “beam up” every vertex of the graph $$\Gamma$$ to a single block $$B_{x,y}$$ in an appropriate way, and thus obtain a connected component of $$\square(\Gamma)$$ whose support is all of $$V(\Gamma)$$ Secondly we have the case of “small $$p$$” where   5log⁡nn≤p(n)≤2(log⁡nn)13, which is the focus of the remainder of the proof. Here we construct a path of length of order $$n/\log n$$ in $$\square(\Gamma)$$ on to which every vertex $$v\in V(\Gamma)$$ can be “beamed up” by adding a $$4$$–cycle whose support contains $$v$$. This is done in the following manner: we start with an arbitrary pair of non-adjacent vertices contained in a block $$B_{0}$$. We then pick an arbitrary pair of non-adjacent vertices in the block $$B_0$$ and let $$B_1$$ denote the intersection of the block they define with $$V(\Gamma)\setminus B_0$$. We repeat this procedure, to obtain a chain of blocks $$B_0, B_1, B_2, \ldots, B_t$$, with $$t=O(n/\log n)$$, whose union contains a positive proportion of $$V(\Gamma)$$, and which all belong to the same connected component $$C$$ of $$\square(\Gamma)$$. This common component $$C$$ is then large enough that every remaining vertex of $$V(\Gamma)$$ can be attached to it. The main challenge is showing that our process of recording which vertices are included in the support of a component of the square graph does not die out or slow down too much, that is, that the block sizes $$\vert B_i\vert$$ remains relatively large at every stage of the process and that none of the $$B_i$$ form a clique. Having described our strategy, we now fill in the details, beginning with the following upper bound on the probability of $$\Gamma\in {\mathcal G}(n,p)$$ containing a copy of $$K_{10}$$, the complete graph on $$10$$ vertices. The following lemma is a variant of [21, Corollary 4]: Lemma 5.2. Let $$\Gamma\in{\mathcal G}(n,p)$$. If $$p=o(n^{-\frac{1}{4}})$$, then the probability that $$\Gamma\in {\mathcal G}(n,p)$$ contains a clique with at least $$10$$ vertices is at most $$o(n^{-\frac{5}{4}})$$. □ Proof The expected number of copies of $$K_{10}$$ in $$\Gamma$$ is   (n10)p(102)≤n10p45=o(n−5/4). The statement of the lemma then follows from Markov’s inequality. ■ Proof of Theorem 5.1. As remarked above, Theorem 4.4 proves Theorem 5.1 for “large” $$p$$, so we only need to deal with the case where   5log⁡nn≤p(n)≤2(log⁡nn)13. We iteratively build a chain of blocks, as follows. Let $$\{x_0,y_0\}$$ be a pair of non-adjacent vertices in $$\Gamma$$, if such a pair exists, and an arbitrary pair of vertices if not. Let $$B_0$$ be the block with ends $$\{x_0, y_0\}$$. Now assume we have already constructed the blocks $$B_0, \ldots, B_i$$, for $$i\geq 0$$. Let $$C_i=\bigcup_i B_i$$ (for convenience we let $$C_{-1}=\emptyset$$). We will terminate the process and set $$t=i$$ if any of the three following conditions occur: $$\vert \mathrm{core}(B_i)\vert\leq 6\log n $$ or $$i\geq n/6\log n$$ or $$\vert V(\Gamma)\setminus C_i\vert \leq n/2$$. Otherwise, we let $$\{x_{i+1}, y_{i+1}\}$$ be a pair of non-adjacent vertices in $$\mathrm{core}(B_i)$$, if such a pair exists, and an arbitrary pair of vertices from $$\mathrm{core}(B_i)$$ otherwise. Let $$B_{i+1}$$ denote the intersection of the block whose ends are $$\{x_{i+1}, y_{i+1}\}$$ and the set $$\left(V(\Gamma)\setminus(C_i)\right)\cup\{x_{i+1}, y_{i+1}\}$$. Repeat. Eventually this process must terminate, resulting in a chain of blocks $$B_0, B_1, \ldots, B_t$$. We claim that a.a.s. both of the following hold for every $$i$$ satisfying $$0 \leq i \leq t$$: (i) $$\vert\mathrm{core}(B_i)\vert > 6\log n$$; and (ii) $$\{x_i,y_i\}$$ is a non-edge in $$\Gamma$$. Part (i) follows from the Chernoff bound given in Lemma 4.1: for each $$i\geq -1$$ the set $$V(\Gamma)\setminus C_i$$ contains at least $$n/2$$ vertices by construction. For each vertex $$v\in V(\Gamma)\setminus C_i$$, let $$X_v$$ be the indicator function of the event that $$v$$ is adjacent to both of $$\{x_{i+1}, y_{i+1}\}$$. The random variables $$(X_v)$$ are independent identically distributed Bernoulli random variables with mean $$p$$. Their sum $$X=\sum_v X_v$$ is exactly the size of the core of $$B_{i+1}$$, and its expectation is at least $$p^2n/2$$. Applying Lemma 4.1, we get that   P(X<6log⁡n) ≤P(X≤12EX) ≤2e−(12)225log⁡n6n=2e−2524log⁡n. Thus the probability that $$\vert \mathrm{core}(B_i)\vert <6\log n$$ for some $$i$$ with $$0\leq i \leq t$$ is at most:   t2e−2524log⁡n≤4n5log⁡n2e−2524log⁡n=o(1). Part (ii) is a trivial consequence of part (i) and Lemma 5.2: a.a.s. $$\mathrm{core}(B_i)$$ has size at least $$6\log n$$ for every $$i$$ with $$0 \leq i \leq t$$, and a.a.s. $$\Gamma$$ contains no clique on $$10 < \log n$$ vertices, so that a.a.s. at each stage of the process we could choose an non-adjacent pair $$\{x_i,y_i\}$$. From now on we assume that both (i) and (ii) occur, and that $$\Gamma$$ contains no clique of size $$10$$. In addition, we assume that $$\vert \mathrm{core}(B_0)\vert < 8n^{\frac{1}{3}}(\log{n})^{\frac{2}{3}}$$, which occurs a.a.s. by the Chernoff bound. Since $$\mathrm{core}(B_i)\geq 6\log n$$ for every $$i$$, we must have that by time $$0<t\leq n/6\log n$$ the process will have terminated with $$C_t=\bigcup_{i=0}^t B_i$$ supported on at least half of the vertices of $$V(\Gamma)$$. Lemma 5.3. Either one of the assumptions above fails or there exists a connected component $$F$$ of $$\square(\Gamma)$$ such that: (i) for every $$i$$ with $$0 \leq i \leq t$$ and every pair of non-adjacent vertices $$\{v,v'\}\in B_i$$, there is a vertex in $$F$$ whose support in $$\Gamma$$ contains the pair $$\{v,v'\}$$; and (ii) the support in $$\Gamma$$ of the $$4$$–cycles corresponding to vertices of $$F$$ contains all of $$C_t$$ with the exception of at most $$9$$ vertices of $$\mathrm{core}(B_0)$$; moreover, these exceptional vertices are each adjacent to all the vertices of $$\mathrm{core}(B_0)$$. □ Proof By assumption the ends $$\{x_0, y_0\}$$ of $$B_0$$ are non-adjacent. Thus, every pair of non-adjacent vertices $$\{v,v'\}$$ in $$\mathrm{core}(B_0)$$ induces a $$4$$–cycle in $$\Gamma$$ when taken together with $$\{x_0,y_0\}$$, and all of these squares clearly lie in the same component $$F$$ of $$\square(\Gamma)$$. Repeating the argument with the non-adjacent pair $$\{x_1, y_1\}\in \mathrm{core}(B_0)$$ and the block $$B_1$$, and then the non-adjacent pair $$\{x_2, y_2\}\in \mathrm{core}(B_1)$$ and the block $$B_2$$, and so on, we see that there is a connected component $$F$$ in $$\square(\Gamma)$$ such that for every $$0\leq i \leq t$$, every pair of non adjacent vertices $$\{v,v'\} \in B_i$$ lies in a $$4$$–cycle corresponding to a vertex of $$F$$. This establishes (i). We now show that the support of $$F$$ contains all of $$C_t$$ except possibly some vertices in $$B_0$$. We already established that every pair $$\{x_i,y_i\}$$ is in the support of some vertex of $$F$$. Suppose $$v\in \mathrm{core}(B_i)$$ for some $$i>0$$. By construction, $$v$$ is not adjacent to at least one of $$\{x_{i-1}, y_{i-1}\}$$, say $$x_{i-1}$$. Thus, $$\{x_{i-1},x_i, y_i, v\}$$ induces a $$4$$–cycle which contains $$v$$ and is associated to a vertex of $$F$$. Finally, suppose $$v\in \mathrm{core}(B_0)$$. By (i), $$v$$ fails to be in the support of $$F$$ only if $$v$$ is adjacent to all other vertices of $$\mathrm{core}(B_0)$$. Since, by assumption, $$\Gamma$$ does not contain any clique of size $$10$$, there are at most $$9$$ vertices not in the support of $$F$$, proving (ii). ■ Lemma 5.3, shows that a.a.s. we have a “large” component $$F$$ in $$\square(\Gamma)$$ whose support contains “many” pairs of non-adjacent vertices. In the last part of the proof, we use these pairs to prove that the remaining vertices of $$V(\Gamma)$$ are also supported on our connected component. For each $$i$$ satisfying $$0 \leq i \leq t$$, consider a a maximal collection, $$M_{i}$$, of pairwise-disjoint pairs of vertices in $$\mathrm{core}(B_i)\setminus\{x_{i+1},y_{i+1}\}$$. Set $$M=\bigcup_i M_i$$, and let $$M'$$ be the subset of $$M$$ consisting of pairs, $$\{v,v'\}$$, for which $$v$$ and $$v'$$ are not adjacent in $$\Gamma$$. We have   |M|=∑i=1t(⌊12|core(Bi)|⌋−1)≥|Ct|2−2t≥n4(1−o(1)). The expected size of $$M'$$ is thus $$(1-p)n(\frac{1}{4}-o(1))=\frac{n}{4}(1-o(1))$$, and by the Chernoff bound from Lemma 4.1 we have   P(|M′|≤n5) ≤2e−(15+o(1))2(1−p)n12≤e−(1300+o(1))n, which is $$o(1)$$. Thus a.a.s. $$M'$$ contains at least $$n/5$$ pairs, and by Lemma 5.3 each of these lies in some $$4$$–cycle of $$F$$. We now show that we can “beam up” every vertex not yet supported on $$F$$ by a $$4$$–cycle using a pair from $$M'$$. By construction we have at most $$n/2$$ unsupported vertices from $$V(\Gamma)\setminus C_t$$ and at most $$9$$ unsupported vertices from $$\mathrm{core}(B_0)$$. Assume that $$\vert M' \vert \geq n/5$$. Fix a vertex $$w\in V(\Gamma)\setminus C_t$$. For each pair $$\{v,v'\}\in M'$$, let $$X_{v,v'}$$ be the event that $$w$$ is adjacent to both $$v$$ and $$v'$$. We now observe that if $$X_{v,v'}$$ occurs for some pair $$\{v,v'\}\in M'\cap \mathrm{core}(B_i)$$, then $$w$$ is supported on $$F$$. By construction, $$w$$ is not adjacent to at least one of $$\{x_i, y_i\}$$, let us say without loss of generality $$x_i$$. Hence, $$\{x_i,v,v',w\}$$ is an induced $$4$$–cycle in $$\Gamma$$ which contains $$w$$ and which corresponds to a vertex of $$F$$. The probability that $$X_{v,v'}$$ fails to happen for every pair $$\{v,v'\}\in M'$$ is exactly   (1−p2)|M′|≤(1−p2)n/5≤e−p2n5=e−5log⁡n. Thus the expected number of vertices $$w \in V(\Gamma)\setminus C_t$$ which fail to be in the support of $$F$$ is at most $$\frac{n}{2}e^{-5\log n}=o(1)$$, whence by Markov’s inequality a.a.s. no such bad vertex $$w$$ exists. Finally, we deal with the possible $$9$$ left-over vertices $$b_1, b_2, \ldots b_9$$ from $$\mathrm{core}(B_0)$$ we have not yet supported. We observe that since $$\mathrm{core}(B_0)$$ contains at most $$8n^{\frac{1}{3}}(\log{n})^{\frac{2}{3}}$$ vertices (as we are assuming and as occurs a.a.s. , see the discussion before Lemma 5.3 ), we do not stop the process with $$B_0$$, $$\mathrm{core}(B_1)$$ is non-empty and contains at least $$6\log n$$ vertices. As stated in Lemma 5.3, each unsupported vertex $$b_i$$ is adjacent to all other vertices in $$\mathrm{core}(B_0)$$, and in particular to both of $$\{x_1, y_1\}$$. If $$b_i$$ fails to be adjacent to some vertex $$v\in \mathrm{core}(B_1)$$, then the set $$\{b_i, x_1, y_1, v\}$$ induces a $$4$$–cycle corresponding to a vertex of $$F$$ and whose support contains $$b_i$$. The probability that there is some $$b_i$$ not supported in this way is at most   9P(bi adjacent to all of core(B1)) =9p6log⁡n=o(1). Thus a.a.s. we can “beam up” each of the vertices $$b_1, \ldots b_9$$ to $$F$$ using a vertex $$v\in \mathrm{core}(B_1)$$, and the support of the component $$F$$ in the square graph $$\square(\Gamma)$$ contains all vertices of $$\Gamma$$. This shows that a.a.s. $$\Gamma \in \mathcal{CFS}$$, and concludes the proof of the theorem. ■ Remark 5.4. The constant $$5$$ in Theorem 5.1 is not optimal, and indeed it is not hard to improve on it slightly, albeit at the expense of some tedious calculations. We do not try to obtain a better constant, as we believe that the order of the upper bound we have obtained is not sharp. We conjecture that the actual threshold for $$\mathcal{CFS}$$ occurs when $$p(n)$$ is of order $$n^{-1/2}$$ (see the discussion below Theorem 5.7), but a proof of this is likely to require significantly more involved and sophisticated arguments than the present article. □ A simple lower bound for the emergence of the $$\mathcal{CFS}$$ property can be obtained from the fact that if $$\Gamma\in\mathcal{CFS}$$, then $$\Gamma$$ must contain at least $$n-3$$ squares; if $$p(n)\ll n^{-\frac{3}{4}}$$, then by Markov’s inequality a.a.s. a graph in $${\mathcal G}(n,p)$$ contains fewer than $$o(n)$$ squares and thus cannot be in $$\mathcal{CFS}$$. Below, in Theorem 5.7, we prove a better lower bound, showing that the order of the upper bound we proved in Theorem 5.1 is not off by a factor of more than $$(\log n)^{3/2}$$. Lemma 5.5. Let $$\Gamma$$ be a graph and let $$C$$ be the subgraph of $$\Gamma$$ supported on a given connected component of $$\square(\Gamma)$$. Then there exists an ordering $$v_1<v_2< \cdots <v_{\vert C\vert}$$ of the vertices of $$C$$ such that for all $$i\ge3$$, $$v_i$$ is adjacent in $$\Gamma$$ to at least two vertices preceding it in the order. □ Proof As $$C$$ is a component of $$\square(\Gamma)$$, it contains at least one induced $$4$$–cycle. Let $$v_1, v_2$$ be a pair of non-adjacent vertices from such an induced $$4$$–cycle. Then the two other vertices $$\{v_3,v_4\}$$ of the $$4$$–cycle are both adjacent in $$\Gamma$$ to both of $$v_{1}$$ and $$v_{2}$$. If this is all of $$C$$, then we are done. Otherwise, we know that each $$4$$–cycle in $$C$$ is “connected” to the cycle $$F=\{v_{1},v_{2},v_{3},v_{4}\}$$ via a sequence of induced $$4$$–cycles pairwise intersecting in pairwise non-adjacent vertices. In particular, there is some such $$4$$–cycle whose intersection with $$F$$ is either a pair of non-adjacent vertices in $$F$$ or three vertices of $$F$$; either way, we may add the new vertex next in the order. Continuing in this way and using the fact that the number of vertices not yet reached is a monotonically decreasing set of positive integers, the lemma follows. ■ Proposition 5.6. Let $$\delta>0$$. Suppose $$p\leq\frac{1}{\sqrt{n}\log n}$$. Then a.a.s. for $$\Gamma\in{\mathcal G}(n,p)$$, no component of $$\square(\Gamma)$$ has support containing more than $$4\log n$$ vertices of $$\Gamma$$. □ Proof Let $$\delta>0$$. Let $$m=\left\lceil \min\left(4\log n, 4\log \left(\frac{1}{p}\right) \right)\right\rceil$$, with $$p\leq 1/\left(\sqrt{n}\log n\right)$$. We shall show that a.a.s. there is no ordered $$m$$–tuple of vertices $$v_1<v_2<\cdots <v_m$$ from $$\Gamma$$ such that for every $$i\ge 2$$ each vertex $$v_i$$ is adjacent to at least two vertices from $$\{v_j: 1\leq j <i\}$$. By Lemma 5.5, this is enough to establish our claim. Let $$v_1<v_2<\cdots < v_m$$ be an arbitrary ordered $$m$$–tuple of vertices from $$V(\Gamma)$$. For $$i\geq 2$$, let $$A_i$$ be the event that $$v_i$$ is adjacent to at least two vertices in the set $$\{v_j: 1\leq j <i\}$$. We have:   Pr(Ai)=∑j=2i−1(i−1j)pj(1−p)i−j−1. (2) As in the proof of Theorem 4.5 we consider the quotients of successive terms in the sum to show that its order is given by the term $$j=2$$. To see this, observe:   (i−1j+1)pj+1(1−p)i−j−2(i−1j)pj(1−p)i−j−1≤i−j−1j+1⋅p1−p<mp where the final inequality holds for $$n$$ sufficiently large and $$p=p(n)$$ satisfying our assumption. Since $$m=O\left(\log n\right)$$ and $$p=o(n^{-1/2})$$ we have, again for $$n$$ large enough, that $$mp=o(1)$$, and we may bound the sum in equation (2) by a geometric series to obtain the bound:   Pr(Ai) =(i−12)p2(1−p)i−3(1+O(mp)) ≤(i−1)22p2(1+O(mp)). Now let $$A=\bigcap_{i=1}^m A_i$$. Note that the events $$A_i$$ are mutually independent, since they are determined by disjoint edge-sets. Thus we have:   Pr(A)=∏i=3mPr(Ai) ≤∏i=3m((i−1)22p2(1+O(mp))) =((m−1)!)2p2m−42m−2(1+O(m2p)), where in the last line we used the equality $$(1+O(mp))^{m-2} =1+O(m^2p)$$ to bound the error term. Thus we have that the expected number $$X$$ of ordered $$m$$–tuples of vertices from $$\Gamma$$ for which $$A$$ holds is at most:   E(X)=n!(n−m)!P(A) ≤nm4e2(m2p2−4me22)m(1+O(m2p)) =4e2(nm2p2−4/m2e2)m(1+O(m2p)), where in the first line we used the inequality $$(m-1)!\leq e (m/e)^{m}$$. We now consider the quantity   f(n,m,p)=nm2p2−4/m2e2 which is raised to the $$m^{\textrm{th}}$$ power in the inequality above. We claim that $$f(n,m,p)\leq e^{-1+\log 2+o(1)}$$. We have two cases to consider: Case 1: $$m=\lceil 4\log n\rceil $$. Since $$4\log n \leq 4\log(1/p)$$, we deduce that $$p\leq n^{-1}$$. Then   f(n,m,p) =n(4log⁡n)2p2−o(1)2e2≤n−1+o(1)≤e−1+log⁡2+o(1). Case 2: $$m=\lceil 4\log(1/p)\rceil $$. First, note that $$p^{-4/m}=\exp\left(\frac{4\log(1/p)}{\lceil 4 \log 1/p\rceil}\right)\leq e$$. Also, for $$p$$ in the range $$[0, n^{-1/2}(\log n)^{-1}]$$ and $$n$$ large enough, $$p^2(\log(1/p))^2$$ is an increasing function of $$p$$ and is thus at most:   1n(log⁡n)2(12log⁡n)2(1+O(log⁡log⁡nlog⁡n))=14n−1(1+o(1)). Plugging this into the expression for $$f(n,m,p)$$, we obtain:   f(n,m,p) =(1+o(1))16n(log⁡(1/p))2p2−4/m2e2 ≤(1+o(1))2e=e−1+log⁡2+o(1). Thus, in both cases (1) and (2) we have $$f(n,m,p)\leq e^{-1 +\log 2+o(1)}$$, as claimed, whence   E(X) ≤4e2(f(n,m,p))m(1+O(m2p))≤4e2e−(1−log⁡2)m+o(m)(1+O(m2p))=o(1). It follows from Markov’s inequality that the non-negative, integer-valued random variable $$X$$ is a.a.s. equal to $$0$$. In other words, a.a.s. there is no ordered $$m$$–tuple of vertices in $$\Gamma$$ for which the event $$A$$ holds and, hence by Lemma 5.5, no component in $$\square(\Gamma)$$ covering more than $$m\leq 4\log n$$ vertices of $$\Gamma$$. ■ Theorem 5.7. Suppose $$p\leq\frac{1}{\sqrt{n} \log n}$$. Then a.a.s. $$\Gamma\in{\mathcal G}(n,p)$$ is not in $$\mathcal{CFS}$$. □ Proof To show that $$\Gamma\not\in\mathcal{CFS}$$, we first show that, for $$p\leq \frac{1}{\sqrt{n} \log n}$$, a.a.s. there is no non-empty clique $$K$$ such that $$\Gamma=\Gamma' \star K$$. Indeed the standard Chernoff bound guarantees that we have a.a.s. no vertex in $$\Gamma$$ with degree greater than $$\sqrt{n}$$. Thus to prove the theorem, it is enough to show that a.a.s. there is no connected component $$C$$ in $$\square(\Gamma)$$ containing all the vertices in $$\Gamma$$. Theorem 5.6 does this by establishing the stronger bound that a.a.s. there is no connected component $$C$$ covering more than $$4\log n$$ vertices. ■ While Theorem 5.7 improves on the trivial lower bound of $$n^{-3/4}$$, it is still off from the upper bound for the emergence of the $$\mathcal{CFS}$$ property established in Theorem 5.1. It is a natural question to ask where the correct threshold is located. Remark 5.8. We strongly believe that there is a sharp threshold for the $$\mathcal{CFS}$$ property analogous to the one we established for the $$\mathcal{AS}$$ property. What is more, we believe this threshold should essentially coincide with the threshold for the emergence of a giant component in the auxiliary square graph $$\square(\Gamma)$$. Indeed, our arguments in Proposition 5.6 and Theorems 5.1 both focus on bounding the growth of a component in $$\square(\Gamma)$$. Heuristically, we would expect a giant component to emerge in $$\square(\Gamma)$$ to emerge for $$p(n)=cn^{-1/2}$$, for some constant $$c>0$$, when the expected number of common neighbors of a pair of non-adjacent vertices in $$\Gamma$$ is $$c^2$$, and thus the expected number of distinct vertices in $$4$$–cycles which meet a fixed $$4$$–cycle in a non-edge is $$2c^2$$. What the precise value of $$c$$ should be is not entirely clear (a branching process heuristics suggests $$\sqrt{\displaystyle\frac{\sqrt{17}-3}{2}}$$ as a possible value, see Remark 6.2), however, and the dependencies in the square graph make its determination a delicate matter. □ 6 Experiments Theorems 4.4 and 4.5 show that $$\left(\log n/n\right)^{\frac{1}{3}}$$ is a sharp threshold for the family $$\mathcal{AS}$$ and Theorem 5.1 shows that $$n^{-\frac{1}{2}}$$ is the right order of magnitude of the threshold for $$\mathcal{CFS}$$. Below we provide some empirical results on the behavior of random graphs near the threshold for $$\mathcal{AS}$$ and the conjectured threshold for $$\mathcal{CFS}$$. We also compare our experimental data with analogous data at the connectivity threshold. Our experiments are based on various algorithms that we implemented in $$\texttt{C++}$$; the source code is available from the authors. At www.wescac.net/research.html or math.columbia.edu/~jason, all source code and data can be downloaded. We begin with the observation that computer simulations of $$\mathcal{AS}$$ and $$\mathcal{CFS}$$ are tractable. Indeed, it is easily seen that there are polynomial-time algorithms for deciding whether a given graph is in $$\mathcal{AS}$$ and/or $$\mathcal{CFS}$$. Testing for $$\mathcal{AS}$$ by examining each block and determining whether it witnesses $$\mathcal{AS}$$ takes $$O(n^5)$$ steps, where $$n$$ is the number of vertices. The $$\mathcal{CFS}$$ property is harder to detect, but essentially reduces to determining the component structure of the square graph. The square graph can be produced in polynomial time and, in polynomial time, one can find its connected components and check the support of these components in the original graph. Using our software, we tested random graphs in $${\mathcal G}(n,p)$$ for membership in $$\mathcal{AS}$$ for $$n\in\{300+100k\mid0\leq k\leq12\}$$ and $$\{p(n)=\alpha\left(\log n/n\right)^{\frac{1}{3}}\mid \alpha=0.80+0.1k,\,0\leq k\leq 9\}$$. For each pair $$(n,p)$$ of this type, we generated $$400$$ random graphs and tested each for membership in $$\mathcal{AS}$$. (This number of tests ensures that, with probability approximately $$95\%$$, the measured proportion of $$\mathcal{AS}$$ graphs is within at most $$0.05$$ of the actual proportion.) The results are summarized in Figure 2. The data suggest that, fixing $$n$$, the probability that a random graph is in $$\mathcal{AS}$$ increases monotonically in the range of $$p$$ we are considering, rising sharply from almost zero to almost one. Fig. 2. View largeDownload slide Experimental prevalence of $$\mathcal{AS}$$ at density $$\alpha\left(\frac{\log n}{n}\right)^{\frac{1}{3}}$$. (A colour version of this figure is available from the journal’s website.) Fig. 2. View largeDownload slide Experimental prevalence of $$\mathcal{AS}$$ at density $$\alpha\left(\frac{\log n}{n}\right)^{\frac{1}{3}}$$. (A colour version of this figure is available from the journal’s website.) In Figure 3, we display the results of testing random graphs for membership in $$\mathcal{CFS}$$ for $$n\in\{100+100k\mid 0\leq k\leq 15\}$$ and $$\{p(n)=\alpha n^{-\frac{1}{2}} \mid \alpha=0.700+0.025k,\,0\leq k\leq 8\}$$. For each pair $$(n,p)$$ of this type, we generated $$400$$ random graphs and tested each for membership in $$\mathcal{CFS}$$. (This number of tests ensures that, with probability approximately $$95\%$$, the measured proportion of $$\mathcal{AS}$$ graphs is within at most $$0.05$$ of the actual proportion.) The data suggest that, fixing $$n$$, the probability that a random graph is in $$\mathcal{CFS}$$ increases monotonically in considered range of $$p$$: rising sharply from almost zero to almost one inside a narrow window. Fig. 3. View largeDownload slide Experimental prevalence of $$\mathcal{CFS}$$ membership at density $$\alpha n^{-\frac{1}{2}}$$. (A colour version of this figure is available from the journal’s website.) Fig. 3. View largeDownload slide Experimental prevalence of $$\mathcal{CFS}$$ membership at density $$\alpha n^{-\frac{1}{2}}$$. (A colour version of this figure is available from the journal’s website.) Remark 6.1 (Block and core sizes). For each graph $$\Gamma$$ tested, the $$\mathcal{AS}$$ software also keeps track of how many nonadjacent pairs $$\{x,y\}$$ — that is, how many blocks — were tested before finding one sufficient to verify membership in $$\mathcal{AS}$$; if no such block is found, then all non-adjacent pairs have been tested and the graph is not in $$\mathcal{AS}$$. ({A set of such data comes with the source code, and more is available upon request.}) At densities near the threshold, this number of blocks is generally very large compared to the number of blocks tested at densities above the threshold. For example, in one instance with $$(n,\alpha)=(1000,0.89)$$, verifying that the graph was in $$\mathcal{AS}$$ was accomplished after testing just $$86$$ blocks, while at $$(1000,0.80)$$, a typical test checked all $$422961$$ blocks (expected number: $$423397$$) before concluding that the graph is not $$\mathcal{AS}$$. At the same $$(n,\alpha)$$, another test found that the graph was in $$\mathcal{AS}$$, but only after $$281332$$ tests. These data are consonant with the spirit of our proofs of Theorems 4.5 and 4.4: in the former case, we showed that no “good” block exists, while in the latter we show that every block is good. We believe that right at the threshold we should have some intermediate behavior, with the expected number of “good” blocks increasing continuously from $$0$$ to $$(1-p)\binom{n}{2}(1-o(1))$$. What is more, we expect that the more precise threshold for the $$\mathcal{AS}$$ property, coinciding with the appearance of a single “good” block, should be located “closer” to our lower bound than to our upper one, that is, at $$p(n)= \left((1-\epsilon)\log n/n\right)^{1/3}$$, where $$\epsilon(n)$$ is a sequence of strictly positive real numbers tending to $$0$$ as $$n\rightarrow \infty$$ (most likely decaying at a rate just faster than $$(\log n)^{-2}$$, see below). Our experimental data, which exhibit a steep rise in $$\Pr(\mathcal{AS})$$ strictly before the value $$\alpha$$ hits one, gives some support to this guess. Finally, our observations on the number of blocks suggests a natural way to understand the influence of higher-order terms on the emergence of the $$\mathcal{AS}$$ property: at exactly the threshold for $$\mathcal{AS}$$, the event $$E(v,w)$$ that a pair of non-adjacent vertices $$\{v,w\}$$ gives rise to a “good” block is rare and, despite some mild dependencies, the number $$N$$ of pairs $$\{v,w\}$$ for which $$E(v,w)$$ occurs is very likely to be distributed approximatively like a Poisson random variable. The probability $$\Pr(N\geq 1)$$ would then be a very good approximation for $$\Pr(\mathcal{AS})$$. “Good” blocks would thus play a role for the emergence of the $$\mathcal{AS}$$ property in random graphs analogous to that of isolated vertices for connectivity in random graphs. When $$p=\left((1-\epsilon) \log n/n\right)^{\frac{1}{3}}$$, the expectation of $$N$$ is roughly $$ne^{-n^{\epsilon}(1-\epsilon)\log n}$$. This expectation is $$o(1)$$ when $$0<\epsilon(n)=\Omega(1/n)$$ and is $$1/2$$ when $$\epsilon(n)=(1+o(1)) \log 2/(\log n)^2$$. This suggest that the emergence of $$\mathcal{AS}$$ should occur when $$\epsilon(n)$$ decays just a little faster than $$(\log n)^2$$. □ Remark 6.2. Our data suggest that the prevalence of $$\mathcal{CFS}$$ is closely related to the emergence of a giant component in the square graph. Indeed, below the experimentally observed threshold for $$\mathcal{CFS}$$, not only is the support of the largest component in the square graph not all of $$\Gamma \in {\mathcal G}(n,p)$$, but in fact the size of the support of the largest component is an extremely small proportion of the vertices (see Figure 4). In the Erdős–Rényi random graph, a giant component emerges when $$p$$ is around $$1/n$$, that is, when vertices begin to expect at least one neighbor; this corresponds to a paradigmatic condition of expecting at least one child for survival of a Galton–Watson process (see [9] for a modern treatment of the topic). The heuristic observation that when $$p=cn^{-1/2}$$ the vertices of a diagonal $$e$$ in a fixed $$4$$-cycle $$F$$ expect to be adjacent to $$cn^2$$ vertices outside the $$4$$-cycle, giving rise to an expected $$\binom{cn^2+2}{2}-1$$ new $$4$$-cycles connected to $$F$$ through $$e$$ in $$\square(\Gamma)$$ suggests that $$c=\sqrt{\displaystyle\frac{\sqrt{17}-3}{2}}\approx 0.7494$$ could be a reasonable guess for the location of the threshold for the $$\mathcal{CFS}$$ property. 2 Our data, although not definitive, appears somewhat supportive of this guess: see Figure 4 which is based on the same underlying data set as Figure 3. We note that unlike an Erdős–Rényi random graph, the square graph $$\square(\Gamma)$$ exhibits some strong local dependencies, which may make the determination of the exact location of its phase transition a much more delicate affair. □ Fig. 4. View largeDownload slide Fraction of vertices supporting the largest $$\mathcal{CFS}$$–subgraph at density $$\alpha n^{-\frac{1}{2}}$$. (A colour version of this figure is available from the journal’s website.) Fig. 4. View largeDownload slide Fraction of vertices supporting the largest $$\mathcal{CFS}$$–subgraph at density $$\alpha n^{-\frac{1}{2}}$$. (A colour version of this figure is available from the journal’s website.) Remark 6.3. For comparison with the threshold data for $$\mathcal{AS}$$ and $$\mathcal{CFS}$$, we include below a similar figure of experimental data for connectivity for $$\alpha$$ from $$0.8$$ to $$1.4$$, where $$p=\frac{\alpha\log n}{n}$$. Given, what we know about the thresholds for connectivity and the $$\mathcal{AS}$$ property, this last set of data together with Figure 2 should serve as a warning not to draw overly strong conclusions: the graphs we tested are sufficiently large for the broader picture to emerge, but probably not large enough to allow us to pinpoint the exact location of the threshold for $$\mathcal{CFS}$$. □ Fig. 5. View largeDownload slide Experimental prevalence of connectedness at density $$\alpha\frac{\log n}{n}$$. (A colour version of this figure is available from the journal’s website.) Fig. 5. View largeDownload slide Experimental prevalence of connectedness at density $$\alpha\frac{\log n}{n}$$. (A colour version of this figure is available from the journal’s website.) Funding This work was supported by National Science Foundation [DMS-0739392 to J.B, 1045119 to M.F.H.] and also supported by a Simons Fellowship to J.B. Acknowledgments J.B. thanks the Barnard/Columbia Mathematics Department for their hospitality during the writing of this article. J.B. thanks Noah Zaitlen for introducing him to the joy of cluster computing and for his generous time spent answering remedial questions about programming. Also, thanks to Elchanan Mossel for several interesting conversations about random graphs. M.F.H. and T.S. thank the organizers of the 2015 Geometric Groups on the Gulf Coast conference, at which some of this work was completed. This work benefited from several pieces of software, the results of one of which is discussed in Section 6. Some of the software written by the authors incorporates components previously written by J.B. and M.F.H. jointly with Alessandro Sisto. Another related useful program was written by Robbie Lyman under the supervision of J.B. and T.S. during an REU program supported by NSF Grant DMS-0739392, see [30]. This research was supported, in part, by a grant of computer time from the City University of New York High Performance Computing Center under NSF Grants CNS-0855217, CNS-0958379, and ACI-1126113. We thank the anonymous referee for their helpful comments. References [1] Agol I. Groves D. and Manning. J. “The virtual Haken conjecture.” Documenta Mathematica  18 (2013): 1045– 87. [2] Alon N. and Spencer. J. H. The probabilistic method , 3rd ed. Wiley-Interscience Series in Discrete Mathematics and Optimization. Hoboken, NJ: John Wiley & Sons, Inc., 2008. [3] Babson E. Hoffman C. and Kahle. M. “The fundamental group of random 2-complexes.” Journal of the American Mathematical Society  24, no. 1 (2011): 1– 28. Google Scholar CrossRef Search ADS   [4] Behrstock J. and Charney. R. “Divergence and quasimorphisms of right-angled Artin groups.” Mathematische Annalen  352 (2012): 339– 56. Google Scholar CrossRef Search ADS   [5] Behrstock J. and Druţu. C. “Divergence, thick groups, and short conjugators.” Illinois Journal of Mathematics  58, no. 4 (2014): 939– 80. [6] Behrstock J. Druţu C. and Mosher. L. “Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity.” Mathematische Annalen  344, no. 3 (2009): 543– 95. Google Scholar CrossRef Search ADS   [7] Behrstock J. and Hagen. M. “Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries.” Geometry, Groups, and Dynamics  10, no. 2 (2016): 649– 707. Google Scholar CrossRef Search ADS   [8] Behrstock J. M. Hagen F. and Sisto. A. “Thickness, relative hyperbolicity, and randomness in Coxeter groups.” Algebraic and Geometric Topology, to appear with an appendix written jointly with Pierre-Emmanuel Caprace . [9] Bollobás B. and Riordan. O. “A simple branching process approach to the phase transition in $$G_{n, p}$$.” The Electronic Journal of Combinatorics  19, no. 4 (2012): P21. [10] Bowditch B. H. “Relatively hyperbolic groups.” International Journal of Algebra and Computation  22, no. 3 (2012): 1250016, 66. Google Scholar CrossRef Search ADS   [11] Brock J. and Masur. H. “Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity.” Geometry & Topology  12 (2008): 2453– 95. Google Scholar CrossRef Search ADS   [12] Calegari D. and Wilton H. 3–manifolds everywhere. ArXiv:math.GR/1404.7043  (accessed on April 29, 2014). [13] Caprace P.-E. “Buildings with isolated subspaces and relatively hyperbolic Coxeter groups.” Innovations in Incidence Geometry , 10 (2009): 15– 31. In proceedings of Buildings & Groups, Ghent. [14] Charney R. and Farber M. Random groups arising as graph products. Algebraic and Geometric Topology  12 (2012): 979– 96. Google Scholar CrossRef Search ADS   [15] Coxeter H. S. “Discrete groups generated by reflections.” Annals of Mathematics  (1934): 588– 621. [16] Dahmani F. Guirardel V. and Przytycki. P. “Random groups do not split.” Mathematische Annalen  349, no. 3 (2011): 657– 73. Google Scholar CrossRef Search ADS   [17] Dani P. and Thomas. A. “Divergence in right-angled Coxeter groups.” Transactions of the American Mathematical Society  367, no. 5 (2015): 3549– 77. Google Scholar CrossRef Search ADS   [18] Davis M. W. The geometry and topology of Coxeter groups, vol. 32 of London Mathematical Society Monographs Series . Princeton, NJ, Princeton University Press, 2008. [19] Davis M. W. and Januszkiewicz. T. “Right-angled Artin groups are commensurable with right-angled Coxeter groups.” Journal of Pure and Applied Algebra  153, no. 3 (2000): 229– 35. Google Scholar CrossRef Search ADS   [20] Davis M. W. and Kahle. M. “Random graph products of finite groups are rational duality groups.” Journal of Topology  7, no. 2 (2014): 589– 606. Google Scholar CrossRef Search ADS   [21] Erdős P. and Rényi. A. “On the evolution of random graphs. II.” Bulletin de l’Institut International de Statistique  38 (1961): 343– 47. [22] Farb B. “Relatively hyperbolic groups.” Geometric and Functional Analysis (GAFA)  8, no. 5 (1998): 810– 40. Google Scholar CrossRef Search ADS   [23] Gilbert E. “Random graphs.” Annals of Mathematical Statistics  4 (1959): 1141– 44. Google Scholar CrossRef Search ADS   [24] Gromov M. Hyperbolic groups. In Essays in group theory, edited by Gersten S. vol. 8 of MSRI Publications . New York, NY, Springer, 1987. [25] Gromov M. Asymptotic invariants of infinite groups, Geometric Group Theory, edited by Niblo G. A. and Roller M. A. vol. 2. London Mathematical Society Lecture Notes  182 (1993): 1– 295. [26] Gromov M. “Random walk in random groups.” Geometric and Functional Analysis (GAFA)  13, no. 1 (2003): 73– 146. Google Scholar CrossRef Search ADS   [27] Haglund F. and Wise. D. T. “Coxeter groups are virtually special.” Advances in Mathematics  224, no. 5 (2010): 1890– 903. Google Scholar CrossRef Search ADS   [28] Kahle M. “Sharp vanishing thresholds for cohomology of random flag complexes.” Annals of Mathematics (2)  179, no. 3 (2014): 1085– 107. Google Scholar CrossRef Search ADS   [29] Linial N. and Meshulam. R. “Homological connectivity of random 2-complexes.” Combinatorica  26, no. 4 (2006): 475– 87. Google Scholar CrossRef Search ADS   [30] Lyman R. Algorithmic Computation of Thickness in Right-Angled Coxeter Groups . Undergraduate honors thesis, Columbia University, 2015. [31] Moussong G. Hyperbolic Coxeter groups . PhD thesis, Ohio State University, 1988. Available at https://people.math.osu.edu/davis.12/papers/Moussongthesis.pdf (accessed on November 21, 2016). [32] Mühlherr B. “Automorphisms of graph-universal coxeter groups.” Journal of Algebra  200, no. 2 (1998): 629– 49. Google Scholar CrossRef Search ADS   [33] Niblo G. A. and Reeves. L. D. “Coxeter groups act on CAT (0) cube complexes.” Journal of Group Theory  6, no. 3 (2003): 399. Google Scholar CrossRef Search ADS   [34] Ollivier Y. and Wise. D. “Cubulating random groups at density less than 1/6.” Transactions of the American Mathematical Society  363, no. 9 (2011): 4701– 733. Google Scholar CrossRef Search ADS   [35] Osin D. V. “Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems.” Memoirs of the American Mathematical Society  179, no. 843 (2006): vi+ 100. Google Scholar CrossRef Search ADS   [36] Ruane K. and Witzel. S. “CAT(0) cubical complexes for graph products of finitely generated abelian groups.” arXiv preprint arXiv:1310.8646  (2013), to appear in New York Journal of Mathematics. [37] Sultan H. “The asymptotic cones of Teichmüller space and thickness.” Algebraic and Geometric Topology  15, no. 5 (2015): 3071– 106. Google Scholar CrossRef Search ADS   © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. 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Westerholt-Raum, Martin

doi: 10.1093/imrn/rnw288pmid: N/A

Abstract Given a non-holomorphic Saito-Kurokawa lift we construct a preimage under the vector-valued lowering operator. In analogy with the case of harmonic weak elliptic Maaß forms, this preimage allows for a natural decomposition into a meromorphic and a non-holomorphic part. In this way every harmonic weak Siegel–Maaß form gives rise to a Siegel mock modular form. More than 10 years ago Zwegers [41] and Bruinier and Funke [8] independently found definitions of harmonic weak Maaß forms. Zwegers focused on mock theta functions whose modular properties had been a long standing miracle since Ramanujan came up with them in 1920 in his death bed letter. Bruinier and Funke were inspired by the Kudla program and they used harmonic weak Maaß forms to describe dualities between theta lifts. Both approaches to harmonic weak Maaß forms together produced a rich and novel research area, leading to the resolution of vital conjectures in combinatorics, topology, and many other fields—see, for example, [29] for an overview and [3, 6, 9, 11] for some of the applications. In this paper, we suggest an extension of the concept of harmonic weak Maaß forms to the case of Siegel modular forms of genus $$2$$, by building up on the ideas of [8]. In order to state some defining formulas, we let $$\tau = x + {\rm i} y \in \mathbb{H}$$ be a variable in the Poincaré upper half plane and $$0 \le k$$ an even integer. Analytic aspects of the theory of harmonic weak Maaß forms have been dominated by two facts. First, the weight-$$k$$ hyperbolic Laplace operator $$\Delta_k$$ can be written as a composition $$-\xi_{2-k} \circ \xi_{k}$$ of two $$\xi$$-operators defined by $$\xi_k(\,f) = 2i y^k \overline{\partial_{\overline\tau}\,f}$$. Second, the $$\xi$$-operator gives rise to a short exact sequence of the space $${}^!{\mathrm{M}}_k$$ of weakly holomorphic modular forms, the space $${\mathrm{S}}_{2-k}$$ of cusp forms, and the space $$\mathbb{S}_k$$ of harmonic weak Maaß forms whose image under $$\xi_k$$ is a cusp form. This sequence puts the operator $$\xi_k$$ into the center of the theory. One can define harmonic weak Maaß forms as the modular preimages of weakly holomorphic modular forms of weight $$2-k$$ under $$\xi_k$$. The kernel of $$\xi_k$$ consists of holomorphic (or meromorphic) modular forms, and by the factorization of the weight-$$k$$ Laplace operator $$\Delta_k = - \xi_{2-k} \circ \xi_k$$, the given definition is indeed equivalent to being harmonic. The above sequence also gives rise to the connection of harmonic weak Maaß forms and mock modular forms, which are their “holomorphic parts.” Finally, note that the $$\xi$$-operator is essentially the Maaß lowering operator $${\mathrm{L}}$$. Concretely, we have $$\xi_k(\,f) = 2 i y^{k-2} \overline{\mathrm{L} f}$$. The Laplace operator for $${\mathrm{SL}_{2}}({\mathbb{R}})$$ generates the center of the algebra of invariant differential operators on the Poincaré upper half plane $$\mathbb{H}$$. It is customary in the classical theory of modular forms to normalize it in such a way that holomorphic functions are harmonic. One goal of this paper is to show existence of real-analytic Siegel modular forms (1) that are eigenforms with respect to the center of the algebra of invariant differential operators on the Siegel upper half plane $$\mathbb{H}^{(2)}$$, and (2) that have a holomorphic part, thus producing Siegel mock modular forms. The notion of holomorphic parts is not rigorous so far, and deserves further study. For the purpose of this paper, we content ourselves with the fact that harmonic weak Siegel–Maaß forms that are constructed in this paper generate a Harish-Chandra module with a holomorphic composition factor. By the theory of $$(\mathfrak{g}, \mathrm{K})$$-modules, the latter implies that eigenvalues under invariant differential operators of every composition factor are the same as those of holomorphic forms. Uniqueness of Bessel models and the theory of Eisenstein series show that harmonic weak Siegel–Maaß forms already have the same eigenvalues as holomorphic Siegel modular forms. In this sense, it is justified to call them harmonic. Our Section 4 will link them to harmonic weak Maaß forms for $${\mathrm{SL}_{2}}({\mathbb{R}})$$ via the Kohnen limit process initially studied in [7]. It also explains how to invoke uniqueness of Bessel models. Recall that every holomorphic representation of $${\mathrm{GL}_{2}}(\mathbb{C})$$ can be viewed as a weight for genus-$$2$$ Siegel modular forms. The classical case of weight $$k$$ modular forms corresponds to the representation $$\det^k :\, {\mathrm{GL}_{2}}(\mathbb{C}) {\rightarrow} \mathbb{C},\, g {\mapsto} \det(g)^k$$. In our case of genus-$$2$$ Siegel modular forms, every holomorphic, irreducible representation of $${\mathrm{GL}_{2}}(\mathbb{C})$$ can be written as a (tensor) product $$\det^k{\mathrm{sym}}^l$$ where $${\mathrm{sym}}^l$$ is the $$l$$th symmetric power representation of $${\mathrm{GL}_{2}}(\mathbb{C})$$. It is necessary to introduce vector-valued Siegel modular forms, because the above target space $${\mathrm{S}}_{2-k}$$ of $$\xi_k$$ will be replaced by    SKS(2)(det−k/2symk) :space of non-holomorphic Saito-Kurokawa lifts of holomorphicweight k cusp forms. We revisit the non-holomorphic Saito-Kurokawa lift in Section 1. Non-holomorphic Saito-Kurokawa lifts do not occur for scalar weights $$\det^k$$ for any $$k$$. The space of harmonic weak Siegel–Maaß forms is defined as the space of real-analytic functions with possible meromorphic singularities that transform like Siegel modular forms and that are mapped to non-holomorphic Saito-Kurokawa lifts under the vector-valued lowering operators $${\mathrm{L}}$$. This space contains the space of meromorphic Siegel modular forms $${}^!{\mathrm{M}}^{(2)}\big( \det^{2 - k / 2} {\mathrm{sym}}^{k-2} \big)$$, which are annihilated by the lowering operator. In analogy with the above short exact sequence for harmonic weak elliptic modular forms, we show that every non-holomorphic Saito-Kurokawa lift allows for a modular preimage. Theorem I There is an exact sequence of weak Siegel–Maaß forms Elements of $$^\mathrm{sk}\mathbb S^{(2)}\big( \det^{2 - k / 2}\mathrm{sym}^{k-2} \big)$$ will henceforth be called harmonic weak Siegel–Maaß forms. □ Remark The vector-valued lowering operator defined in [20], which we will use in this paper takes weight-$$\det^k{\mathrm{sym}}^l$$ forms to forms of weight $$\det^{k-2}{\mathrm{sym}}^2{\mathrm{sym}}^l$$, containing the weight $$\det^{k-2}{\mathrm{sym}}^{2+l}$$ as a subrepresentation. The above sequence should be read in such a way that the image of harmonic weak Siegel–Maaß forms under $${\mathrm{L}}$$ is a priori modular of weight $$\det^{-k / 2}{\mathrm{sym}}^2{\mathrm{sym}}^{k-2}$$ and that this image vanishes outside of $$\det^{-k / 2}{\mathrm{sym}}^{k} \subset \det^{-k / 2} {\mathrm{sym}}^2{\mathrm{sym}}^{k-2}$$. This in particular means that the kernel of the third map vanishes under the actual lowering operator $${\mathrm{L}}$$, which characterizes meromorphic Siegel modular forms. □ Remark The above kind of harmonic weak Siegel–Maaß forms is by no means the only one that we expect to exist. It is so far unclear how to construct harmonic weak Siegel–Maaß forms whose image under the raising operator is a Siegel holomorphic modular form. This space would naturally contain Siegel modular forms with singularities whose analytic behavior is analogous with the one of non-holomorphic Saito-Kurokawa lifts. Even such Siegel modular forms have not yet been explored. □ Remark The following is a heuristic argument on holomorphic parts and why appearance of Saito-Kurokawa lifts is rather natural: our requirement that harmonic weak Siegel–Maaß forms have a holomorphic part and meromorphic singularities includes, at least, that one composition factor is holomorphic. Restricting to indecomposable Harish-Chandra modules of length $$2$$. We may further require that this holomorphic representation is a quotient, since if it appeared as a subrepresentation, then we would deal with usual meromorphic Siegel modular forms. There are at least three families of holomorphic representations for $${\mathrm{Sp}_{2}}({\mathbb{R}})$$. The holomorphic discrete series are tempered. In addition, there are non-tempered holomorphic representations in some principal series for both the Siegel parabolic and the Klingen parabolic. Let $$f$$ be a “harmonic weak Siegel–Maaß form” in the above vague sense, and assume that it has a holomorphic part. Write $$\varpi(\,f)$$ for the Harish-Chandra module that it generates (see Section 2.2). We have already argued that $$\varpi(\,f) \twoheadrightarrow \varpi^{\mathrm{hol}}$$ for an irreducible holomorphic representation $$\varpi^{\mathrm{hol}}$$. Denote by $$\varpi' \hookrightarrow \varpi(\,f)$$ the irreducible subrepresentation. In analogy with conditions on harmonic weak Maaß forms that were imposed by Bruinier and Funke in [8], we assume that $$\varpi'$$ appears as a component at infinity of some cuspidal automorphic representation. We have $$f \not\in \varpi'$$, since otherwise $$f$$ would only generate $$\varpi' \subsetneq \varpi(\,f)$$. On the other hand, we have $$\frak{m} f \cap \varpi' \neq \emptyset$$, where $$\frak{g}$$ is the Lie algebra of $$G = {\mathrm{Sp}_{2}}({\mathbb{R}})$$ and $$\frak{g} = \frak{k} \oplus \frak{m}$$ is its Cartan decomposition. In classical language, the action of $$\frak{m}$$ corresponds to the Maaß lowering and raising operators (again, see Section 2.2). We reformulate this as a condition on the $$K$$-type supports of $$\varpi'$$ and $$\varpi^{\mathrm{hol}}$$. The lowest $$K$$-type of $$\varpi^{\mathrm{hol}}$$ should neighbor the $$K$$-type support of $$\varpi'$$. For the representations in Section 2.5 this is satisfied. It is so far not clear to the author in any of the remaining cases whether extensions with this property exist. For example holomorphic and large discrete series of compatible Harish-Chandra parameter in general do not satisfy this condition. As for Harish-Chandra modules attached to the non-holomorphic Saito-Kurokawa lift, after fixing the Harish-Chandra module that appears in the principal series for the Siegel parabolic, we observe that its only automorphic realization is the non-holomorphic Saito-Kurokawa lift itself—cf. p. 55 of [30]. □ The occurrence of vector-valued Siegel modular forms cannot be avoided, since non-holomorphic Saito-Kurokawa lifts do not allow for a scalar-valued realization. However, harmonic weak Siegel–Maaß forms can be forced into scalar weights by applying the vector-valued raising operator sufficiently often. Denote the space of weight $$k / 2$$ almost meromorphic Siegel modular forms of depth $$k / 2 - 1$$ by $${}^!{\mathrm{M}}^{(2)}\big( (\det^{k / 2})^{[k / 2 - 1]} \big)$$. A definition and study of almost meromorphic Siegel modular forms can be found in [20]. We call a singularity $$f$$ almost meromorphic if locally it can be written as $$f = f_{\mathrm{num}} / f_{\mathrm{den}} + f_{\mathrm{reg}}$$ for a regular function $$f_{\mathrm{reg}}$$, a holomorphic function $$f_{\mathrm{den}}$$, and an almost holomorphic function $$f_{\mathrm{num}}$$. An almost meromorphic Siegel modular form is a function on the upper half space with almost meromorphic singularities that transforms like a Siegel modular form. We define almost harmonic weak Siegel–Maaß forms of weight $$\det^{k / 2}$$ as real-analytic modular forms with almost meromorphic singularities whose image under $${\mathrm{L}}^{k / 2}$$ is a non-holomorphic Saito-Kurokawa lift. We denote the space of such forms by $${}^\mathrm{SK}\mathbb {S}^{(2)}\big( (\det^{k / 2})^{[k / 2 - 1]} \big)$$. Corollary II There is an exact sequence of weak Siegel–Maaß forms □ Remark The power of the lowering operator in the previous statement is a tensor power whose target weight is $$\det^{-k / 2} {\mathrm{sym}}^k \subset \det^{-k / 2} ({\mathrm{sym}}^2)^{k / 2}$$. The exact sequence has to be interpreted as the one in Theorem I. □ One source of interest in harmonic weak Maaß forms for $${\mathrm{SL}_{2}}({\mathbb{R}})$$ is their Fourier series expansion and mock modular forms associated with them. Writing $$e(x) = \exp(2 \pi {\rm i}\, x)$$ for $$x \in {\mathbb{C}}$$, any harmonic weak Maaß form $$f$$ can be naturally decomposed as   f(τ)=∑n≫−∞c+(f;n)e(nτ)+∑n≪∞c−(f;n)a−ak(1)⁡(n,y)e(nτ), where, if $$k \ne 1$$, then $$\sideset{^-}{^{(1)}_k}{\mathop{a}}$$ is essentially the $$W$$-Whittaker function if $$n < 0$$, the $$M$$-Whittaker function if $$n > 0$$, and a power of $$y = \Im\frak{m}\tau$$ if $$n = 0$$. The first sum is called the holomorphic part of $$f$$ and the second one is called its non-holomorphic part. The holomorphic part of $$f$$ is a mock modular form and conversely by definition every mock modular form has a modular completion of the above form. Harmonic weak Maaß forms with $$c^-(f;\, n) = 0$$ for $$n \ge 0$$ are best understood. Their image under $$\xi_k$$ is a cusp form that essentially has Fourier coefficients $$c^-(f;\,n)$$. Their Fourier coefficients $$c^+(f;\,n)$$ for $$n \le 0$$ are not completely understood, but they were successfully related to derivatives of twisted $$L$$-values in [5]. To formulate the genus-$$2$$ analogue of the above decomposition, we set $$e(x) = \exp(2 \pi {\rm i}\, \mathrm{trace}(x))$$ for any complex square matrix $$x$$. We put the superscript $$\pm$$ to the left of the analytic part $$a(t, y)$$ of the Fourier coefficient, reserving the right superscript for annotation with regard to the genus (which in our case is $$2$$). Theorem III Every $$f \in ^\mathrm{SK}\mathbb S^{(2)}\big( \det^{2 - k / 2}\mathrm{sym}^{k-2} \big)$$ has a Fourier series expansion of the form   f(τ)=∑tc+(f;t)e(tτ)+∑t indefinitec−(f;t)a−ak(2)⁡(t,y)e(tx), where   ∑t indefinitec−(f;t)aSKak(2)⁡(t,y)e(tx)∈SKS(2)(det−k2symk) for real-analytic functions $$\sideset{^-}{^{(2)}_k}{\mathop{a}}$$ and $$\sideset{^\mathrm{SK}}{^{(2)}_k}{\mathop{a}}$$ defined in Section 4. □ Given the analytic characterization of harmonic weak Siegel–Maaß forms, it seems justified to call the first summand in Theorem III a Siegel mock modular form. We give a precise definition in Section 4.3. We plan to investigate the Fourier coefficients of Siegel mock modular forms in a sequel to this paper. We discuss the proof of Theorem I. The proof generalizes ideas in [8], but it requires additional input from the theory of $$(\frak{g},\mathrm{K})$$-modules. The following description of $$(\frak{g},\mathrm{K})$$-modules associated with some harmonic weak Maaß forms was independently found by Ralf Schmidt in an unpublished note. The author was notified that also Schulze-Pillot found similar results in [37]. Recall that in the case of nonpositive, even weight $$k$$, there are two differential operators that map harmonic weak Maaß forms to weakly holomorphic ones. One of them, the $$\xi_k$$-operators, has already been mentioned, and another one arises from Bol’s identity. First note that from the perspective of $$(\frak{g},\mathrm{K})$$-modules it is not very natural to include complex conjugation into the above diagram. Rather, one sacrifices holomorphicity and employs the lowering operators $${\mathrm{L}}$$ instead of $$\xi_k$$. This yields a map   Mk⟶y2−kM!M2−k¯, where the right-hand side consists of functions transforming like modular forms of weight $$k-2$$. In other words, the $$(\frak{g},\mathrm{K})$$-module associated with a non-holomorphic $$f \in \mathbb M_k$$ contains the direct sum of holomorphic and antiholomorphic (limits of) discrete series. The quotient is a finite dimensional and irreducible $$(\frak{g},\mathrm{K})$$-module. It corresponds to the images $$\mathrm{R}^j\,f$$ with $$0 \le j \le -k$$ of $$f \in \mathbb M_k$$ under powers of the raising operator. One can phrase the above observation as follows: the $$(\frak{g},\mathrm{K})$$-module associated with a proper harmonic weak Maaß form $$f$$ is isomorphic to a suitable degenerate principal series. This observation guides our proof of existence of harmonic weak Siegel–Maaß forms. The Harish-Chandra module attached to non-holomorphic Saito-Kurokawa lifts can be realized as a Langlands quotient. Lee [25] gave a very explicit description of the corresponding principal series. The Saito-Kurokawa lift corresponds to the unique irreducible quotient. The second composition factor is the direct sum of a holomorphic and antiholomorphic representation, whose minimal $$K$$-type corresponds to a vector-valued Siegel modular form. The precise description of the Harish-Chandra module delivered by Lee enables us to extend the argument in [8] to Siegel modular forms of genus-$$2$$. Bruinier and Funke employ sheaf cohomology. Since they dealt with complex curves, the Dolbeault resolution of the structure sheaf is surjective on to $$(0,1)$$-forms. This can be interpreted in such a way that locally for every $$(0,1)$$-form there is a preimage in $$0$$-forms. Translated into classical language, every weakly holomorphic modular form is locally the image of some harmonic $$0$$-form. This is no longer true in the case of complex three-folds, which we deal with in this paper. Vanishing under the Dolbeault derivative $$\overline\partial$$ is necessary for a $$(0,1)$$-form to locally admit a preimage under $$\overline\partial$$. We reformulate this vanishing condition in terms of $$(\frak{g},\mathrm{K})$$-modules, and establish it for non-holomorphic Saito-Kurokawa lifts. The main step in the proof of Bruinier and Funke is the second step in ours. To guarantee global existence of preimages of harmonic weak Maaß forms they prove vanishing of a corresponding obstruction space, constituted by the first cohomology of a certain holomorphic line bundle. Using Serre duality they reduce it to a classical vanishing result for elliptic modular forms. In our case, the obstruction space is the first cohomology of a vector bundle. We employ a vanishing theorem by Grauert–Riemenschneider [13] and Igusa’s study [18] of Siegel modular forms to deduce its vanishing. We start the paper with preliminaries in Section 1. Section 2 contains the theory of Harish-Chandra modules and degenerate principal series that we need in the paper. We proceed to the proof of existence of harmonic weak Siegel modular forms in Section 3. That section contains most of the complex geometry that is important to the paper. The last Section 4 focuses on Siegel mock modular forms, which arise from a natural splitting of the Fourier expansion of harmonic Siegel modular forms. 1 Preliminaries 1.1 The symplectic group of genus 2 Throughout, we focus on the group $$G = {\mathrm{Sp}_{2}}({\mathbb{R}})$$ with complex Lie algebra $$\frak{g} = \frak{sp}_2$$, whose definitions are   G=Sp2(R) ={g=(abcd)∈Mat4(R):tgJ2g=J2}withJ2=(0−I2I20),I2=(1001),andg0=sp2(R) ={(abc−ta):a∈Mat2(R),b,c∈Mat2t(R)},g=g0⊗RC. A Cartan involution $$\theta$$ of $$\frak{g}_{\mathbb{R}}$$ is given by the map sending $$\left( {a \atop c} \quad {b \atop -\,{}^\mathrm{t}\! a} \right)$$ to $$\left( {-\,{}^\mathrm{t}\! a \atop -b} \quad {-c \atop a} \right)$$. The associated Cartan decomposition is $$\frak{g} = \frak{k} \oplus \frak{m}$$ with   k={(ab−ba):a∈Mat2(C),b∈Mat2t(C),a=−ta}andm={(abb−a):a,b∈Mat2t(C)}. The corresponding compact subgroup $$\mathrm{U}_{2}({\mathbb{R}}) \cong K({\mathbb{R}}) \subset G({\mathbb{R}})$$ consists of matrices $$\left({a \atop -b} \quad {b \atop a} \right) \in {\mathrm{Sp}_{2}}({\mathbb{R}})$$ with $$a - {\rm i}b \in \mathrm{U}_{2}({\mathbb{R}})$$. We write $$\frak{g}_0$$ and $$\frak{k}_0$$ for the real Lie algebras of $$G({\mathbb{R}})$$ and $$K({\mathbb{R}})$$. We now fix a bases of $$\frak{k}$$ and of $$\frak{m}$$. Define $$\mathfrak{h}_\mathfrak{c} = -i \left({ \atop I_2} \quad {-I_2 \atop 0}\right)$$ and Names are chosen in such a way that the action of $$\frak{k}$$ is the customary one: $$\big[ \frak{e}_\frak{k}, \frak{e}_\frak{m}^+ \big] = 0$$, $$\big[ \frak{e}_\frak{k}, \frak{h}_\frak{m}^+ \big] = -2 \frak{e}_\frak{m}^+$$, etc. The element $$\frak{h}_\frak{c}$$ acts on $$\frak{m}^\pm$$ by multiplication with $$\pm 2$$. Compact and non-compact roots of $$\frak{g}$$ are   ±2hk∨;±2hc∨,±2(hc∨+hk∨),±2(hc∨−hk∨). Note that $$i \frak{h}_\frak{c}$$ and $$i \frak{h}_\frak{k}$$ span a Cartan subalgebra of both $$\frak{k}_0$$ and $$\frak{g}_0$$. Thus we can, as is customary, identify weights of $$K$$ and $$G$$. 1.2 The upper half space The genus-$$2$$ Siegel upper half space $$\mathbb{H}^{(2)} = \{ \tau = x + {\rm i}y \in \mathrm{Mat}^\mathrm{t}_{2}({\mathbb{C}}) \,:\, y > 0 \}$$ consists of symmetric complex matrices $$\tau$$ with positive definite imaginary part. It carries an action of $${\mathrm{Sp}_{2}}({\mathbb{R}})$$ by means of   Sp2(R)↻H(2):(abcd)τ=(aτ+b)(cτ+d)−1. The entries of $$\tau$$, $$x$$, and $$y$$ are given by   τ=(τ1zzτ2),y=(y1vvy2),andx=(x1uux2). We associate with every $$\tau \in \mathbb{H}^{(2)}$$ the element   gτ=(yxy−1y−1)∈Sp2(R), where $$\sqrt{y}$$ is the unique positive definite symmetric square root of $$y$$. 1.3 Weights, types, and slash actions We call a finite-dimensional, holomorphic representation $$\sigma$$ of $${\mathrm{GL}_{2}}(\mathbb{C})$$ a weight for genus-$$2$$ Siegel modular forms. In genus-$$2$$, any weight is isomorphic to a direct sum of representations $$\det^k{\mathrm{sym}}^l$$ for some $$k \in \mathbb{Z}$$ and some $$0 \le l \in \mathbb{Z}$$. The representation space of $$\sigma$$ will be denoted by $$V(\sigma)$$. A finite-dimensional, complex representation of $$\Gamma^{(2)} = {\mathrm{Sp}_{2}}(\mathbb{Z})$$ is called a type. We focus throughout and without further mentioning it on types whose kernel has finite index in $$\Gamma^{(2)}$$. The representation space of $$\rho$$, in analogy with weights, is denoted by $$V(\rho)$$. By slight abuse of notation we denote the trivial weight and trivial type by the same letter $$\mathbb{1}$$. Fix the $${\mathrm{GL}_{2}}(\mathbb{C})$$-valued cocycle given by $$j(g, \tau) = c \tau + d$$ for $$g \in {\mathrm{Sp}_{2}}({\mathbb{R}})$$ and $$\tau \in \mathbb{H}^{(2)}$$. For a weight $$\sigma$$ and a type $$\rho$$, we define the slash action of weight $$\sigma$$ and type $$\rho$$ on functions $$f :\, \mathbb{H}^{(2)} {\rightarrow} V(\sigma) \otimes V(\rho)$$ by   f|σ,ργ=σ(cτ+d)−1⊗ρ(γ)−1(f∘γ). If the type is trivial, we suppress it in the notation, writing $$\big|_{\sigma}$$ instead of $$\big|_{\sigma,\rho}$$. 1.4 Lowering and raising operators Order-$$1$$ lowering and raising operators for Siegel modular forms are necessarily vector-valued. Existence and basic properties follow from general work by Helgason [15, 16]. Concrete expressions in classical terms were computed in [20]. As usual, we define the matrix differentials   ∂τ=(∂τ112∂z12∂z∂τ2)and∂τ¯=(∂τ1¯12∂z¯12∂z¯∂τ2¯). The lowering operator can be written independently of the weight. Given a differentiable function $$f :\, \mathbb{H}^{(2)} {\rightarrow} V(\sigma) \otimes V(\rho)$$, we set   Lf=Lσf=yt(y∂τ¯)⊗f. The tensor notation above makes use of the inclusion   C∞(H→V(sym∨2))⊗C∞(H→V(σ))↪C∞(H→V(sym∨2σ)), where $${\mathrm{sym}}^{\vee\,2}$$ denotes the dual representation of $${\mathrm{sym}}^2$$. Throughout, we will use this notation to denote functions with values in tensor products of representation spaces. The raising operator depends on the weight, and we give a formula specific to weights $$\det^k{\mathrm{sym}}^l$$. We define a symmetrization map that was devised in [20]. Observe that elements of $$V({\mathrm{sym}}^l)$$ can be written in terms of (tensor) products of $$l$$ elements in the standard representation $$\mathrm{std}$$ of $${\mathrm{GL}_{2}}(\mathbb{C})$$, which satisfies $$V(\mathrm{std}) \cong {\mathbb{C}}^2$$. For $$0 \le l \in \mathbb{Z}$$, we define   tdetksyml :sym2⊗detksyml ⟶sym2⊗detksyml,v1v2⊗∏j=1lv2+j ⟼12l∑j′=1l(v2+lv2⊗v1∏j=1j≠j′lv2+j+v1v2+l⊗v2∏j=1j≠j′lv2+j). The raising operator on a differentiable function $$f :\, \mathbb{H}^{(2)} {\rightarrow} V(\det^k{\mathrm{sym}}^l) \otimes V(\rho)$$ is given by   Rf=Rdetksymlf=∂τ⊗f−ik2y−1⊗f−il2tdetksyml(y−1⊗f). We will suppress the weight from notation, if it becomes clear from the context. The covariance properties of $${\mathrm{L}}$$ and $$\mathrm{R}$$ define them uniquely up to scalar multiples by proposition 6.9 of [20]. For every $$g \in {\mathrm{Sp}_{2}}({\mathbb{R}})$$, we have   (Lf)|sym∨2σg=L(f|σg)and(Rf)|sym2σg=R(f|σg). (1.1) To simplify stating the next definition, we set $${\mathrm{L}} \sigma = {\mathrm{sym}}^{\vee\,2}\sigma = \det^{-2}{\mathrm{sym}}^2 \sigma$$ and $$\mathrm{R} \sigma = {\mathrm{sym}}^2 \sigma$$. By viewing $${\mathrm{GL}_{2}}(\mathbb{C})$$-representations as $$\mathrm{U}_{2}({\mathbb{R}})$$-representations, we define the following orthogonal projections for $$\sigma = \det^k {\mathrm{sym}}^l$$ if $$l \ge 2$$.   πL,+ :Lσ →detk−2syml+2,πL,0 :Lσ →detk−1syml,πL,− :Lσ →detksyml−2;πR,+ :Rσ →detksyml+2,πR,0 :Rσ →detk+1syml,πR,− :Rσ →detk+2syml−2. It is clear that $$\pi_{{\mathrm{L}},+} \circ {\mathrm{L}}$$ and all analogous differential operators satisfy covariance properties derived from (1.1). 1.5 The non-holomorphic Saito-Kurokawa lift Existence of the non-holomorphic Saito-Kurokawa lift was first indicated in work of Howe and Piatetski-Shapiro [17]. Piatetski-Shapiro [31] used it to determine cuspidal automorphic representations for $$\mathrm{PGSp}_{2}$$ whose $$L$$-function has a pole. A special case of the Saito-Kurokawa lift that is of particular interest from a geometrical perspective was investigated by Bruinier; see Example 5.13 of [4]. Since the (non-holomorphic) Saito-Kurokawa lift can be expressed as a theta lift (or Weil lifting as Piatetski-Shapiro calls it), it appears in Li’s study of theta lifts [26]. In the present paper, we will focus on the formulation by Schmidt [36], who established functoriality, and Miyazaki [27], who computed Fourier expansions of non-holomorphic Saito-Kurokawa lifts. Given a real-analytic Siegel modular eigenform $$f$$ of weight $${\det^{-k / 2}\mathrm{sym}^k}$$ with $$k \in 2\mathbb{Z}$$ for some finite index subgroup $$\Gamma' \subseteq \Gamma^{(2)}$$, we say that $$f$$ is a non-holomorphic Saito-Kurokawa lift if its spin $$L$$-function has a pole. The linear span of such Saito-Kurokawa lifts is denoted by $${{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k},\, \Gamma' \big)$$. Theorem 1.1 (Piatetski-Shapiro). Given an elliptic modular form $$f$$ of level $$1$$ and weight $$k$$ with $$4 \, | \, k$$, there is a cuspidal, real-analytic Siegel modular form $$g$$ of level $$1$$ and weight $${\det^{-k / 2}\mathrm{sym}^k}$$ whose spin $$L$$-function has a pole and which contains the $$L$$-function of $$f$$ as a factor. In particular, for $$k \ge 12$$ divisible by $$4$$, the space $${{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k},\, \Gamma^{(2)} \big)$$ is not empty. □ Since we work with vector-valued Siegel modular forms for $$\Gamma^{(2)}$$, we now explain how to pass from Saito-Kurokawa lifts in the classical sense to vector-valued ones. The reader who prefers to work in a slightly more classical setting can skip this and assume $$4 \, | \, k$$ and $$\rho = \mathbb{1}$$ in the rest of the paper. None of the steps of our construction involves the type in a crucial way. For a finite index subgroup $$\Gamma' \subseteq \Gamma^{(2)}$$, we denoted the induction of the trivial representation from $$\Gamma'$$ to $$\Gamma^{(2)}$$ by $$\rho_{\Gamma'}$$. Recall the induction map from modular forms for a subgroup $$\Gamma' \subseteq \Gamma^{(2)}$$ to modular forms of type $$\rho_{\Gamma'}$$, which, for example, was spelled out in [40]. With slightly different meaning, induction of vector-valued modular forms also appears in Scheithauer’s work [34]. The image of $${{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( \det^{-k / 2 }{\mathrm{sym}}^{k},\, \Gamma' \big)$$ under the induction map is denoted by $${{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( \det^{-k / 2}{\mathrm{sym}}^{k} \otimes \rho_{\Gamma'} \big)$$. The decomposition $$\bigoplus_j \rho_j$$ of $$\rho_{\Gamma'}$$ into irreducible components yields a natural decomposition   SKS(2)(det−k/2symk⊗ρΓ′)=⨁jSKS(2)(det−k/2symk⊗ρj), which we employ to define the spaces on the right-hand side. Consider Theorem 5.3 of [27]. It tells us that there is a polynomial $$p_k(t,\, \sqrt{y})$$ that takes values in $$V({\det^{-k / 2}\mathrm{sym}^k})$$, and that depends on the entries of indefinite $$t \in \mathrm{Mat}_{2}({\mathbb{R}})$$ and the positive definite square root $$\sqrt{y}$$ of $$y$$ such that any level-$$1$$ Saito-Kurokawa lift has Fourier expansion   ∑tindefinitec~(f;t)SKa~det−k/2symk(2)(t;y)e(tx) (1.2) with coefficients $$\widetilde{c}(f;\,t) \in {\mathbb{C}}$$ and    SKa~det−k/2symk(2)(t,y)=det(y)k+24pk(t,y)Kk/2(trace(ty)2−det(t)det(y))(trace(ty)2−det(t)det(y))k/4, (1.3) where $$K_{k / 2}$$ is the $$K$$-Bessel function. The precise shape of the polynomials $$p_k$$ is irrelevant to use, since later in Section 4 we replace it by a more suitable choice. The coefficients $$\widetilde{c}(f;\,t)$$ grow polynomially in the entries of $$t$$. From the construction of Saito-Kurokawa lifts as theta lifts it becomes clear that for Saito-Kurokawa lifts of nontrivial type $$f \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$ we have an analogous Fourier expansion with $$\widetilde{c}(f;\,t) \in V(\rho)$$. 2 Harish-Chandra Modules This section contains most of the theory of $$(\frak{g},\mathrm{K})$$-modules that we apply in the present paper. We outline some of the basic concepts in Section 2.1, but generally the reader is referred to [21, 39] for details and precise definitions. In Section 2.2 we describe how to pass from functions on the upper half space to $$(\frak{g},\mathrm{K})$$-modules and how the behavior under differential operators is connected to $$K$$-types. Section 2.3 contains a description of some Vogan–Zuckerman modules. In Section 2.4, we identify the non-holomorphic Saito-Kurokawa lift at the infinite place as a suitable module $$A_\frak{q}(\lambda)$$. In Section 2.5, we consider the composition series of some degenerate generalized principal series. In particular, we recognize the Saito-Kurokawa $$(\frak{g},K)$$-module as a submodule in a specific principle series. 2.1 Preliminaries on Harish-Chandra modules Given a Lie group $$G$$ with complex Lie algebra $$\frak{g}$$ and a maximal compact subgroup $$K \subseteq G$$, a $$(\frak{g},\mathrm{K})$$-module is a $$\frak{g}$$-module which carries a compatible action of $$K$$. The definition is spelled out in Section 3.3.1 of [39]. A $$(\frak{g},\mathrm{K})$$-module is called admissible if it is a unitary $$K$$-module and the $$K$$-isotypical components are finite dimensional. An admissible $$(\frak{g},\mathrm{K})$$-module is called a Harish-Chandra module. It is a theorem by Harish-Chandra (cf. Theorem 3.4.10 in [39]) that every irreducible unitary representation $$\pi$$ of $$G$$ gives rise to an admissible $$(\frak{g},\mathrm{K})$$-module by taking smooth vectors. It is called the Harish-Chandra module associated with $$\pi$$. The $$K$$-types that occur in a $$(\frak{g},\mathrm{K})$$-module and transitions between them are the only feature that we will make use of. It is not necessary but convenient to draw them. As throughout the whole paper, we focus on the case $$G = {\mathrm{Sp}_{2}}({\mathbb{R}})$$. Then irreducible representations of $$K$$ can be parametrized by dominant weights of $$K$$, which can be expressed in terms of roots of $$\frak{g}$$. We denote the weight $$(a+b) \frak{h}_\frak{c}^\vee + (a-b)\frak{h}_\frak{k}^\vee$$ by the pair $$(a,b)$$. As a $${\mathrm{GL}_{2}}(\mathbb{C})$$ representation it corresponds to $$\det^b {\mathrm{sym}}^{a-b}$$. Possible $$K$$-types range in $$\Lambda = \{ (a,b) \in \mathbb{Z}^2 \,:\, a \ge b \}$$. We depict a $$(\frak{g},\mathrm{K})$$-module by an arrangement of circles that correspond to all possible $$(a,b)$$. Filled circles indicate $$K$$-types that occur with multiplicity at least one, and the remaining ones indicate those that do not occur. The two diagrams below are typical such diagrams of $$K$$-types. We read the diagram as if it would continue infinitely to the right and to the bottom without indicating this. In the left diagram we have encircled the $$K$$-isotrivial component, which corresponds to the pair $$(0,0)$$. In general we will assume that our diagram is symmetric and therefore refrain from marking $$(0,0)$$. The action of $$\frak{m} \subset \frak{g}$$ gives rise to transitions of $$K$$-types, whose possible targets can be computed by the Clebsch–Gordan rules. For $$a - b \ge 2$$ the image of a $$K$$-type $$(a,b)$$ under $$\frak{m}^\pm$$ is a direct sum of $$K$$-types $$(a\pm 2,b)$$, $$(a\pm 1,b\pm 1)$$, and $$(a,b\pm 2)$$. The image of $$(a+1,a)$$ under $$\frak{m}^+$$ and $$\frak{m}^-$$ is a direct sum of $$(a+3,a)$$ and $$(a+2,a+1)$$, and $$(a+1,a-2)$$ and $$(a,a-1)$$, respectively. The image of $$(a,a)$$ under $$\frak{m}^+$$ and $$\frak{m}^-$$ is $$(a+2,b)$$ and $$(a,a-2)$$. For a given $$(\frak{g},\mathrm{K})$$-module some of those transition maps might be nonzero. This is obviously the case if they hit an unfilled circle, and no further indication in our diagrams is needed. A second phenomenon, which we call (transition) walls, can occur. A wall is a linear relation $$w_a a + w_b b = w_c$$ for $$(a,b)$$ and a sign $$w_\sigma \in \{\pm 1\}$$ such that every transition from $$(a,b)$$ to $$(a',b')$$ with $$w_a a + w_b b = w_c$$ is zero, if $$w_\sigma \cdot \big(w_a a' + w_b b' - w_c \big) < 0$$. We mark transition walls by a line determined by the linear relation, and a decorating arrow attached to the line that indicates the associated sign. For example, in the right-hand diagram below, the vertical line corresponds to the relation $$a = 2$$ with sign $$+1$$. No $$K$$-type to the left of that line occurs. The horizontal line corresponds to the relation $$b = -2$$ with sign $$-1$$. Presence of $$K$$-types below this line indicates that the depicted $$(\frak{g},\mathrm{K})$$-module is reducible.   2.2 Functions on the upper half space We now explain how to pass back and forth between $$(\frak{g},\mathrm{K})$$-modules and functions $$\mathbb{H}^{(2)} {\rightarrow} V(\sigma) \otimes V(\rho)$$. Since the group $$\Gamma^{(2)}$$ and its representation $$\rho$$ do not enter the construction, we can safely assume that $$\rho = \mathbb{1}$$. As for the weight, note that any representation $$\sigma$$ of $${\mathrm{GL}_{2}}(\mathbb{C})$$ defines a representation of $$K$$ by restricting along the map $$K {\rightarrow} \mathrm{U}_{2}({\mathbb{R}}) \subset {\mathrm{GL}_{2}}(\mathbb{C}),\, \left({a \atop -b} \quad {b \atop a}\right) {\mapsto} a - i b$$. When referring to a weight as a $$K$$-representation we throughout mean this restriction. Given $$f :\, \mathbb{H}^{(2)} {\rightarrow} V(\sigma)$$ we consider the function $$\mathrm{A}_{{\mathbb{R}}}(\,f) :\, G {\rightarrow} V(\sigma)$$ defined by   AR(f)(g)=AR,σ(f)(g)=σ−1(j(g,iI2))f(gi). Diverging from the common approach, we do not assume any invariance property of $$f$$. Only later it will be crucial to choose a suitable $$\sigma$$ to obtain from slash invariance of $$f$$ a translation invariance of $$\mathrm{A}_{\mathbb{R}}(\,f)(g)$$. By contraction of $$\mathrm{A}_{\mathbb{R}}(\,f)$$ and $$V(\sigma)^\vee$$ we obtain a space $$\overline{\mathrm{A}}_{\mathbb{R}}(\,f)$$ of functions on $$G$$, which under right translation yields a $$K$$-representation that is isomorphic to $$\sigma^\vee$$. More precisely, for any $$v^\vee \in V(\sigma)^\vee$$, we have   v∨(AR(f)(gk))=v∨(σ−1(j(gk,iI2))f(gi))=v∨(σ−1(j(k,iI2))σ−1(j(g,iI2))f(gi)) =(σ(j(k,iI2))v∨)(σ−1(j(g,iI2))f(gi))=(σ(j(k,iI2))v∨)(AR(f)(g)). We refer to the $$(\frak{g},\mathrm{K})$$-modules generated by the components of $$\mathrm{A}_{\mathbb{R}}(\,f)$$ as the $$(\frak{g},\mathrm{K})$$-module associated with $$f$$. By the $$G$$-representation associated with $$f$$ we mean the representation which is generated by $$\mathrm{A}_{\mathbb{R}}(\,f)$$ under right translation. Its Harish-Chandra module is the $$(\frak{g},\mathrm{K})$$-module that we have associated with $$f$$. Vice versa it is possible to extract a function on $$\mathbb{H}^{(2)}$$ from any vector in a $$K$$-type of a $$(\frak{g},\mathrm{K})$$-module that is realized by smooth functions on $$G$$. For any such $$\tilde f$$ in a representation isomorphic to $$\sigma^\vee$$, we set   AR−1(f~)(τ)=σ(j(gτ,iI2))f~(gτi). Obviously, we have $$\mathrm{A}_{\mathbb{R}}^{-1} \mathrm{A}_{\mathbb{R}} (\,f) = f$$. We next describe a connection between the differential operators in [20] and transition of $$K$$-types in a $$(\frak{g},\mathrm{K})$$-module associated with $$f :\, \mathbb{H}^{(2)} {\rightarrow} V(\sigma)$$. Covariance of the lowering and raising operator and compatibility of $$\frak{g}$$\nbd\ and $$K$$-actions allows us to focus on values of $$f$$ at $$i I_2 \in \mathbb{H}^{(2)}$$ and values of $$\mathrm{A}_{\mathbb{R}}(\,f)$$ at the identity element $$e \in G$$. Helgason’s formalism [15, 16] implies that the action of $$\mathrm{R}$$ at $$i I_2$$ is given by   H0(k,m+⊗sym∨2)⊗1⊆H0(k,m+⊗sym∨2)⊗H0(k,σ⊗σ∨)⊆H0(k,m+⊗σ⊗(sym2σ)∨), where $$\mathbb{1}$$ refers to the canonical copy of the trivial representation in $$\sigma \otimes \sigma^\vee$$. Analogously, the lowering operator $${\mathrm{L}}$$ is given by $$\frak{k}$$-invariants in $$\frak{m}^- \otimes (\det^{-2}{\mathrm{sym}}^2)^\vee$$. Since the action of $$\frak{m}$$ defines transition of $$K$$-types, we therefore conclude that the following diagrams commute up to nonzero scalar multiples for all $$f \in \mathrm{C}^\infty \big( \mathbb{H}^{(2)} {\rightarrow} V(\sigma) \big)$$.   The map $$\mathrm{A}_{\mathbb{R}}^{-1}$$ allows us to deduce vanishing statements with respect to lowering and raising operators from the structure of $$(\frak{g},\mathrm{K})$$-modules. 2.3 Vogan-Zuckerman modules We recall from [38] the $$(\frak{g},\mathrm{K})$$-modules $$A_\frak{q}(\lambda)$$ that are attached to $$\theta$$-stable parabolic subalgebras of $$\frak{g}$$ and central characters $$\lambda \in \frak{h}_{\mathbb{C}}^\vee$$ satisfying specific conditions. According to Theorem 5.4 in [38] such representations are uniquely determined by their $$K$$-types and the central character. The notion of $$A_\frak{q}(\lambda)$$ builds up on $$\theta$$-stable parabolic subalgebras $$\frak{q} \subset \frak{g}$$, whose definition is revisited on page 56 of [38]. As pointed out there, the name is chosen in an unfortunate way, since not every parabolic subalgebra that is stable under the Cartan involution is necessary $$\theta$$-stable in the sense of the following definition. Given $$x$$ in the Cartan subalgebra of $$i \frak{k}_0$$, we attach a parabolic subalgebra to $$x$$ by means of   q=qx=l⊕u,l=ker⁡(ad(x)),u =sum of positive eigenspaces of ad(x). We give details for the parabolic subalgebras corresponding to holomorphic and antiholomorpic representations, and to the non-holomorphic Saito-Kurokawa lift. The following choices are suitable elements of $$i \frak{k}_0$$.   xhol=hc,xhol¯=−hc,xSK=hk. For the purpose of studying transitions of $$K$$-types in $$A_\frak{q}(\lambda)$$ it suffices to determine the intersection of $$\frak{u}$$ and $$\frak{m}$$, which is   uhol∩m=span⁡{hm+,em+,fm+},uhol¯∩m=span⁡{hm−,em−,fm−},uSK∩m=span⁡{em+,em−}. (2.1) In particular, the half-sums of weights in $$\frak{u} \cap \frak{m}$$ are $$\rho_{\mathrm{hol}} = 3 \frak{h}_\frak{c}^\vee$$, $$\rho_{\overline{\mathrm{hol}}} = -3 \frak{h}_\frak{c}^\vee$$, and $$\rho_{\mathrm{SK}} = 2 \frak{h}_\frak{k}^\vee$$. 2.4 The Saito-Kurokawa lift From Theorem 2.5 of [38], we infer that the minimal $$K$$-type of $$A_\frak{q}(\lambda)$$ has highest weight $$2\rho_\frak{q} + \lambda$$, where $$\rho_\frak{q}$$ is the Weyl vector associated with $$\frak{q}$$. In conjunction with Proposition 7.7 of [27], we find that the Harish-Chandra modules associated with $$f \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$ are isomorphic to $$A_{\frak{q}_\mathrm{SK}}\big( (k-2) \frak{h}_\frak{k}^\vee \big) \otimes V(\rho)$$. Note that $$(k-2)\frak{h}_\frak{k}^\vee$$ can and will be written as the pair $$(k / 2 - 1, 1-k / 2)$$. When comparing the Harish-Chandra parameter of $$A_\frak{q}(\lambda)$$ to, for example, what Schmidt found on p. 225 of [36], keep in mind that we normalize weights in a way that is convenient from a classical point of view. Lemma 2.1 Given $$f \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \big)$$, the following vanishing results hold:   πL,0Lf=0,πL,−Lf=0,πR,0Rf=0,πR,−Rf=0. □ Proof The $$K$$-type $$\overline{\mathrm{A}}_{\mathbb{R}}(\,f)$$ is the minimal one in the Harish-Chandra module associated with $$f$$. The lowering and raising operators correspond to transitions of $$K$$-types, as explained in Section 2.2. To prove the lemma it therefore suffices to check that the given lowering and raising operators correspond to transitions by $$\frak{h}_\frak{m}^-$$, $$\frak{f}_\frak{m}^-$$, $$\frak{h}_\frak{m}^+$$, and $$\frak{f}_\frak{m}^+$$. None of them occurs in $$\frak{u}_\mathrm{SK} \cap \frak{m}$$, and therefore the $$K$$-types corresponding to target weights of the covariant differential operators in question do not occur in $$A_{\frak{q}_\mathrm{SK}}\big( (k-2) \frak{h}_k^\vee \big)$$. ■ 2.5 Degenerate generalized principle series We determine the composition series of some generalized principal series related to the Eisenstein series that made appearance in [7]. The result that we use is due to Lee [25]. Kudla and Rallis [22] previously studied the same principal series, but they did not arrive at the exact conclusion that we need for our purposes. Note also that we could use Muić’s study of principal series [28], but then again his approach does not yield exactly what we need. The next proposition describes induction from the parabolic subgroup   P={(ab0d)∈G}with|(ab0d)|=det(a), employing unitary normalization. Proposition 2.2 (Kudla–Rallis,Lee). Fix a positive, even $$k \in \mathbb{Z}$$. The induced $$(\frak{g},K)$$-module $$\mathrm{Ind}_P^G |\,\cdot\,|^{k - 1 / 2}$$ has socle series   AqSK(λ),Aqhol(λ)⊕Aqhol¯(λ),V(λ), where $$\lambda = (2k - 2) \frak{h}_\frak{k}^\vee$$ and $$V(\lambda)$$ is the finite-dimensional $$G$$-representation with infinitesimal character $$\lambda$$. □ Proof Theorem 5.2 of [25] applies for even $$k$$. We recall some of the notation of Section 5 in [25]. In Lee’s notation we have $$n=2$$, $$m=1$$, and therefore $$r=1$$, and $$\alpha = -1-k$$. He defines subquotients $$L_{p,q}$$ of $$\mathrm{Ind}_P^G |\,\cdot\,|^{k-\frac{1}{2}}$$ with $$1 \le p,q \le 2$$ by   X11 ={(a,b)∈2Λ:a<k},X21 ={(a,b)∈2Λ:a≥k},Y11 ={(a,b)∈2Λ:b≤−k},Y21 ={(a,b)∈2Λ:b>−k}. The definition of $$L_{p,q}$$ comprises four cases   L1,1=X11∩Y11,L1,2=X11∩Y21,L2,1=X21∩Y11,andL2,2=X21∩Y21. Theorem 5.2 of [25] tells us that the socle series of $$\mathrm{Ind}_P^G |\,\cdot\,|^{k - \frac{1}{2}}$$ is $$L_{2,1},\, L_{2,2} \oplus L_{1,1},\ L_{1,2}$$. All $$K$$-types in $$\mathrm{Ind}_P^G |\,\cdot\,|^{k - \frac{1}{2}}$$ occur with multiplicity at most one by what is explained in Section 2 of [25]. More precisely, we have   (IndPG|⋅|k−12)K=⨁a,b∈2ΛVK,(a,b), where $$V_{K, (a,b)}$$ is the finite-dimensional $$K$$-representation with highest weight $$(a,b)$$. With this at hand, consider the following diagram $$\mathrm{Ind}_P^G |\,\cdot\,|^{k - \frac{1}{2}}$$. The minimal $$K$$-type of $$L_{2,1}$$ is encircled; It has weight $$2 k \frak{h}_\frak{k}^\vee$$.   To determine the infinitesimal character of $$\mathrm{Ind}_P^G |\,\cdot\,|^{k - \frac{1}{2}}$$, we can use the finite-dimensional representation $$L_{1,2}$$. Its maximal $$K$$-type has weight $$(2k - 4) \frak{h}_\frak{k}^\vee$$. The sum of positive roots in $$\frak{g}$$ is $$2 \delta = 2 ( \frak{h}_\frak{c}^\vee + \frak{h}_\frak{k}^\vee)$$, which is equivalent to $$4 \frak{h}_\frak{k}^\vee$$ under the action of the Weyl group. The infinitesimal character of $$L_{1,2}$$ therefore equals   (2k−4)hk∨+δ=(2k−2)hk∨. Notice that $$L_{1,2}$$ is finite dimensional, and therefore determined by its infinitesimal character. To verify that the modules $$L_{1,1}$$, $$L_{2,2}$$, and $$L_{2,1}$$ are the ones that occur in [38], we use Theorem 5.3 in [38]. Condition (5.1) from that paper holds for $$\lambda = (2k - 2) \frak{h}_\frak{k}^\vee$$, and Assumption (b) of Theorem 5.3 in [38] is also satisfied. To ease verification of Assumptions (a) and (c), consider the above drawing of $$K$$-types. Inspection of minimal $$K$$-types and possible transitions of $$K$$-types verifies assumptions (a) and (c) in each of the infinite-dimensional cases. ■ 3 Existence of Harmonic Weak Siegel–Maaß Forms The goal of this section is to prove the first main theorem. It is a consequence of the following one. Theorem 3.1. Fix $$g \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$. Then there is a function $$f :\, \mathbb{H}^{(2)} {\rightarrow} V(\det^{2 - k / 2}\mathrm{sym}^{k-2}) \otimes V(\rho)$$ with possible singularities along a suitable divisor $$D$$ such that: (i) We have $$\pi_{{\mathrm{L}},+} {\mathrm{L}}\, f = g$$, $$\pi_{{\mathrm{L}},0} {\mathrm{L}}\, f = 0$$, and $$\pi_{{\mathrm{L}},-} {\mathrm{L}}\, f = 0$$. (ii) For all $$\gamma \in \Gamma^{(2)}$$ we have $$f \big|_{\det^{2 - k / 2}\mathrm{sym}^{k-2},\rho}\,\gamma = f$$. (iii) Singularities of $$f$$ are meromorphic. That is, for every $$\tau \in \mathbb{H}^{(2)}$$ there is a neighborhood $$U$$ of $$\tau$$ and a meromorphic function $$f_U$$ on $$U$$ such that $$f - f_U$$ extends to $$U$$. □ A function that satisfies the properties in the previous theorem will be called a harmonic weak Siegel–Maaß form. The space of such forms will be denoted by $$^\mathrm{SK}\mathbb S^{(2)}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$. Remark 3.2. Harmonic weak Siegel–Maaß forms defined above are not the only generalizations of harmonic weak Maaß forms for $${\mathrm{SL}_{2}}({\mathbb{R}})$$ that we expect to exist. Natural candidates arise by searching for preimages under $$\mathrm{R}$$ of holomorphic Siegel modular forms. The methods in this paper, however, do not allow us to construct such preimages, since the correct characterization by means of differential operators would be as follows: given a holomorphic Siegel modular form $$g$$ of weight $$\det^k$$, then a harmonic weak Siegel–Maaß forms $$f$$ in a suitable sense should exist which satisfies   πR,−Rf=g,πR,0Rf=0,πL,0Lf=0. Such a description by differential operators does not fit into the Dolbeault resolution that we employ in this section. □ 3.1 Toroidal compactifications   Throughout this section, we let $$Y_\Gamma = \Gamma \backslash \mathbb{H}^{(2)}$$ for congruence subgroups $$\Gamma \subset {\mathrm{Sp}_{2}}(\mathbb{Z})$$. If $$\Gamma$$ is neat, then $$Y_\Gamma$$ is a smooth quasi-projective variety. To simplify notation we will abbreviate $$Y_\Gamma$$ by $$Y$$ for one fixed choice of neat $$\Gamma$$. Only these Sections 3.1 and 3.2 contain references to $$Y_\Gamma$$ different from $$Y$$. Recall from [1] that there is a smooth toroidal compactification $$X_\Gamma$$ of $$Y_\Gamma$$ if $$\Gamma$$ is neat. The corresponding compactification of $$Y$$ is denoted by $$X$$. By [33] the fundamental group of $$X_\Gamma$$ coincides with the one of $$Y_\Gamma$$. In the next subsection, we will need to pass between various $$X_\Gamma$$. Therefore, it is important to note that we may choose compactifications in such a way that for $$\Gamma' \subseteq \Gamma$$ there is the following commutative diagram of smooth maps between compactifications and coverings. This follows from the dominance statement in the Main Theorem II of [1] on page 287. 3.2 Weights and types as vector bundles Every representation $$\sigma$$ of $${\mathrm{GL}_{2}}(\mathbb{C})$$ and every representation $$\rho$$ of $${\mathrm{Sp}_{2}}(\mathbb{Z})$$ together define a vector bundle over $$\mathbb{H}^{(2)}$$, denoted by $$\mathbb V_{\sigma,\rho}$$. If the image of $$(-I_2,-I_4)$$ under $$\sigma \otimes \rho$$ is the identity, then it descends to the quotient $$Y_{\Gamma^{(2)}} = {\mathrm{Sp}_{2}}(\mathbb{Z}) \backslash \mathbb{H}^{(2)}$$. We denote the resulting vector bundle and its pullbacks to $$\Gamma \backslash \mathbb{H}^{(2)}$$ for any $$\Gamma \subseteq \Gamma^{(2)}$$ by the same symbol $$\mathbb V_{\sigma,\rho}$$. The holomorphic vector bundles $$\mathbb V_{\sigma,\rho}$$ extend uniquely to any toroidal compactification of $$Y_\Gamma$$. To see this for $$\mathbb V_{\sigma,\mathbb{1}}$$, one employs the Hodge bundle and Schur functors, as is explained, for example, on p. 209 of [2]. As for $$\mathbb V_{\mathbb{1},\rho}$$ it suffices to pass to $$Y_{\ker\,\rho}$$ and pull back a trivial vector bundle to $$X_\Gamma$$ along the morphisms in Section 3.1. Tensoring $$\mathbb V_{\sigma,\mathbb{1}}$$ and $$\mathbb V_{\mathbb{1},\rho}$$ then yields extensions to $$X_\Gamma$$ of all $$\mathbb V_{\sigma,\rho}$$. The dual $$\mathbb V_{\sigma,\rho}^\vee$$ of $$\mathbb V_{\sigma,\rho}$$ is the bundle that corresponds to $$\sigma^\vee$$ and $$\overline\rho$$. In the case of weights, we can make this more concrete: the dual of $$\det^k {\mathrm{sym}}^l$$ is $$\det^{-k-l} {\mathrm{sym}}^l$$. 3.3 Holomorphic $$\boldsymbol{n}$$-forms We determine the isomorphism classes of vector bundles associated with the sheafs $$\mathbb E^{0,1}$$ and $$\mathbb E^{0,2}$$ of differential $$(0,1)$$- and $$(0,2)$$-forms, and of the sheaf $$\Omega^3$$ of holomorphic three-forms. At this occasion we also fix notation $$\cal{O}_X$$ for the structure sheaf of $$X$$, and $$\cal{O}_X(D)$$ for the sheaf of meromorphic functions with divisor bounded from below by $$-D$$. It is quickest to first consider one-forms, which have local basis $$d\!\tau_1$$, $$d\!z$$, and $$d\!\tau_2$$. We use matrix notation for covariant differentials and obtain   dτ=(dτ1dzdzdτ2);d(τ−1)=−τ−1(dτ)τ−1,ℑ(τ−1)−1d(τ¯−1)ℑ(τ−1)−1=−τy−1(dτ¯)y−1τ. This implies that the sheaf of holomorphic one-forms is isomorphic to sections of $$\mathbb V_{{\mathrm{sym}}^2, \mathbb{1}}$$. Differential $$(0,1)$$-forms correspond to the vector bundle $$\mathbb V_{\det^{-2}{\mathrm{sym}}^2}$$. Taking wedge products we get an isomorphism of the sheaf of differential $$(0,2)$$-forms with $$\mathbb E^{0,0} \otimes \mathbb V_{\det^{-3}{\mathrm{sym}}^2}$$. Holomophic three-forms correspond to $$\mathbb V_{\det^{3}}$$. As a consequence of the above computation, we find that   E0,1⊗Vdet2−k/2symk−2,ρ=E0,0⊗(Vdet−k/2symk,ρ⊕Vdet1−k/2symk−1,ρ⊕Vdet2−k/2symk−2,ρ). This implies immediately the next lemma. Lemma 3.3 Given $$f \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \big)$$, there is a unique up to scalar multiples global section of the sheaf $$\mathbb E_X^{0,1}\otimes \mathbb V_{\det^{2 - k / 2}\mathrm{sym}^{k-2},\rho}$$ that corresponds to $$f$$ when pulled back to a section over $$\mathbb{H}^{(2)}$$. □ 3.3.1 Differential operators We study the differential $$\overline\partial$$ connecting $$\mathbb E^{0,0} \otimes \mathbb V_{\sigma,\rho}$$, $$\mathbb E^{0,1} \otimes \mathbb V_{\sigma,\rho}$$, and $$\mathbb E^{0,2} \otimes \mathbb V_{\sigma,\rho}$$. Sections of these vector bundles correspond to smooth functions on $$\mathbb{H}^{(2)}$$ that extend to the boundary of $$X$$ and which transform like Siegel modular forms of type $$\rho$$, and weight $$\sigma$$ in the first case, weight $$\det^{-2}{\mathrm{sym}}^2\sigma$$ in the second one, and weight $$\det^{-3}{\mathrm{sym}}^2\sigma$$ in the third one. Lemma 3.4. For fixed $$\sigma$$, there are automorphisms $$\phi_1$$ of $$\det^{-2}{\mathrm{sym}}^2\sigma$$ and $$\phi_2$$ of $$\det^{-3}{\mathrm{sym}}^2\sigma$$ such that the following diagram is commutative.   □ Proof Observe that $$\overline\partial$$ corresponds for functions on $$\mathbb{H}^{(2)}$$ to a differential operator of order $$1$$ that annihilates holomorphic functions. It is covariant with respect to $$\Gamma^{(2)}$$, and by approximation it is therefore covariant with respect to $$G$$. Further, we check that its image separates points by applying it to germs of polynomials. From the uniqueness statement in Proposition 6.9 of [20] we therefore deduce that is a nonzero multiple of constituents of the lowering operator $${\mathrm{L}}$$. ■ Corollary 3.5. The following composition of maps is zero.   □ Proof This follows when combining the previous lemma with Lemma 2.1. ■ 3.4 A vanishing statement Before we can proceed to the proof of Theorem 3.1, we need to establish the following vanishing statement. Lemma 3.6. Let $$D$$ be the divisor on $$X$$ of Igusa’s Siegel modular form $$\chi_{10}$$ (cf. [18]). Fix a weight $$\sigma$$ and a type $$\rho$$. Then for sufficiently large $$n$$, we have   H1(X,Vσ,ρ⊗OX(nD))=0. □ Proof Serre Duality relates the left-hand side to   H2(X,Ω3⊗Vσ∨,ρ¯⊗OX(−nD))=H2(X,Vdet3−10nσ∨,ρ¯). To show that this vanishes, we employ Corollary 5.6 (b) in [12] (formulated originally in [13]). We have to verify that sections of $$\mathbb V_{\det^{10n-3} \sigma,\rho}$$ yield an embedding of $$X$$ into some projective space. By Igusa’s work, the vector bundle $$\mathbb V_{\det^k}$$ is very ample for sufficiently large $$k$$. For fixed $$\sigma$$ and $$\rho$$, a result of Serre (cf. Theorem 5.17 of [14]) implies that for sufficiently large $$k$$ global sections of $$\mathbb V_{\det^k\sigma,\rho}$$ generate the corresponding sheaf. This establishes the lemma. ■ 3.5 A proof of the existence theorem This subsection will be fully occupied by our proof of Theorem 3.1. We start with a sequence of sheafs similar to the one that appears in the proof of Theorem 3.7 of [8]. As opposed to [8] we have to establish local lifting as an intermediate step. Let $$D$$ be the divisor of Igusa’s $$\chi_{10}$$ as in Lemma 3.6, and let $$n$$ be large enough so that the Lemma’s conclusion holds. We start by considering the following exact sheaf sequence over $$X$$, which can be obtained by tensoring the Dolbeault resolution of the structure sheaf $$\cal{O}_X$$ with $$\mathbb V_{\sigma,\rho} \otimes \cal{O}_X(n D)$$. Details on the Dolbeault complex can be found in Section I.3.C of [10].   Given $$f \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$ we obtain from it a global section in   H0(X,EX0,1⊗Vdet2−k/2symk−2,ρ⊗OX(nD))⊆H0(X,EX0,1⊗Vdet2−k/2symk−2,ρ). Lemma 3.5 says that this section vanishes under $$\overline\partial$$, mapping it to $$(0,2)$$-forms. In conjunction with exactness, this implies that it lies, locally, in the image of $$\overline\partial$$ mapping from $$0$$-forms to $$(0,1)$$-forms. We consider the following image sheaf under $$\overline\partial$$:   We next consider the following short exact sequence.   Since $$\mathbb E_X^{0,0}$$ is a fine sheaf, its higher cohomology vanishes. We therefore obtain the long exact sequence   Lemma 3.6 implies that the obstruction space $$\mathrm{H}^1\big( X,\, \mathbb V_{\sigma,\rho} \otimes \cal{O}_X(n D) \big)$$ vanishes. In particular, there is a global section of $$\mathbb E_X^{0,0} \otimes \mathbb V_{\sigma,\rho} \otimes \cal{O}_X(nD)$$ that maps under $$\overline\partial$$ to the section corresponding to $$g$$. Passing back to functions on $$\mathbb{H}^{(2)}$$ we finish the proof. 3.6 Scalar-valued almost harmonic weak Siegel–Maaß forms From the conditions on $${\mathrm{L}}\, f$$ we see that the Harish-Chandra module attached to any harmonic weak Siegel–Maaß form is the extension of $$A_{\frak{q}_{\mathrm{hol}}}(\lambda)$$ by $$A_{\frak{q}_{\mathrm{SK}}}(\lambda)$$ with $$\lambda = (k - 2) \frak{h}_\frak{k}^\vee$$, which is a quotient of $$\mathrm{Ind}^G_P |\,\cdot\,|^{(k-1)/ 2}$$ studied in Section 2.5. This implies that there is a nonzero constant $$c_k$$ that depends only on $$k$$ such that   (πL,+L)k/2−1(πR,−R)k/2−1f=ckffor allf∈SKS(2)(det2−k/2symk−2⊗ρ). From this and the characterization of almost meromorphic Siegel modular forms in terms of the lowering operator, we infer Corollary II. 4 Meromorphic and Non-Holomorphic Parts Given the proof of existence of harmonic weak Siegel–Maaß forms, which we achieved in the previous section, we now study Fourier expansions. In particular, we find a canonical lift of Fourier coefficients of non-holomorphic Saito-Kurokawa lifts. This allows us to define Siegel mock modular forms as the remaining terms. The canonical lift of Fourier coefficients stems from Eisenstein series that were investigated in [7]. 4.1 Fourier coefficients of some Eisenstein series Recall from Equation (1.2) that non-holomorphic Saito-Kurokawa lifts have Fourier expansions supported on indefinite coefficients. To phrase (1.2), we employed an ad hoc definition for the analytic part of Fourier coefficients. We are going to replace it by another type of Fourier coefficient that we have better control of. Lemma 4.3 is key in this context. For $$k > 2$$, we consider the classical Eisenstein series   Ek,12(2)=∑γ∈Γ∞(2)∖Γ(2)det(y)12|detkγ. (4.1) Proposition 4.1. For $$k > 2$$ the Eisenstein series $$E^{(2)}_{k,1 / 2}$$ converges. (1) It has Fourier expansion   ∑t∈Mat2(Q)cE(k;t)aEak,12(2)⁡(ty)e(tx), where $$c_E(k;\, t) \in {\mathbb{C}}$$. We have $$c_E(k;\, t) = 0$$, if $$t$$ is negative definite. The analytic part of the Fourier coefficients $$\sideset{^E}{^{(2)}_{k,1 / 2}}{\mathop{a}}(ty)$$ is a function that decays exponentially with respect to the sum of the absolute values of the eigenvalues of $$t y$$. (2) The Harish-Chandra module associated with $$E^{(2)}_{k,1 / 2}$$ is an extension of $$A_{\frak{q}_{\mathrm{hol}}}(\lambda)$$ by $$A_{\frak{q}_\mathrm{SK}}(\lambda)$$ with Harish-Chandra parameter $$\lambda = (2k - 2) \frak{h}_\frak{k}^\vee$$. □ Proof Observe that $$E^{(2)}_{k,1 / 2}$$ can be related to the Eisenstein series studied in [7] as follows:   Ek,12(2)=det(y)12∑γ∈Γ∞(2)∖Γ(2)1|k+12,12γ=det(y)12(∑γ∈Γ∞(2)∖Γ(2)1|12,(k+1)−12γ)¯, where the group $$\Gamma^{(2)}_\infty$$ and the slash action on the right-hand side are defined by   Γ∞(2)={(ab0d)∈Γ(2)}and(f|k,k′γ)(τ)=det(cτ+d)−kdet(cτ¯+d)−k′f(gτ). The slash action on the right-hand side of the above equality is the weight $$k+1$$ skew slash action that made appearance in [7]. For the purpose of reference, we set   EskEk+1(2)=∑γ∈Γ∞(2)∖Γ(2)1|12,(k+1)−12γ=∑t∈Mat2(Q)cE,sk(k+1;t)askak+1(2)⁡(ty)e(tx). The Fourier expansion of $${}^\mathrm{sk} E^{(2)}_{k+1}$$ establishes the one of $$E^{(2)}_{k,1 / 2}$$. From Theorem 4 in [7], we see that $$c_{E,\mathrm{sk}}(k+1;\,t) = 0$$, if $$t$$ is positive definite. The relation with $$E_{k,1 / 2}$$ implies that $$c_E(k;\, t) = 0$$ if $$t$$ is negative definite. Exponential decay of the analytic part of the Fourier coefficients follows from Shimura’s study of confluent hypergeometric functions [35]. To obtain the Harish-Chandra module associated with $$E^{(2)}_{k,1 / 2}$$ note that this Eisenstein series yields an intertwining operator from $$I_k = \mathrm{Ind}_P^G\,|\,\cdot\,|^{k-1 / 2}$$ into $$L^2(\Gamma^{(2)} \backslash G)$$. The standard reference for Eisenstein series is [23], but [24] is a survey article which classical readers might find more accessible. The induced representation $$I_k$$ contains all $$K$$-types with multiplicity at most one, and therefore $$\mathrm{A}_{{\mathbb{R}},\det^k}\big( E^{(2)}_{k,1 / 2} \big)$$ generates the $$K$$-type $$\det^k$$ of the image of $$I_k$$ in $$L^2(\Gamma^{(2)} \backslash G)$$. We conclude the proof by applying Proposition 2.2. ■ Remark 4.2. The previous proposition determines the Harish-Chandra module associated with $${}^{\mathrm{sk}}E^{(2)}_{k+1}$$ defined in the course of its proof. In particular, we see that Harish-Chandra modules associated with index $$t$$ Fourier coefficients of $${}^{\mathrm{sk}}E^{(2)}_{k+1}$$ for negative definite $$t$$ are antiholomorphic representations. In other words, for negative definite $$t$$ Fourier coefficients of $${}^{\mathrm{sk}}E^{(2)}_{k+1}$$ can be expressed as the product of $$e(t \overline\tau)$$ and a polynomial of degree $$k / 2$$ in the entries of $$y^{-1}$$. This implies that the Kohnen Limit process studied in [7] can also be defined for Fourier–Jacobi coefficients of negative index. □ 4.2 An analogue to non-holomorphic Eichler integrals We are now ready to define the analytic parts of Fourier coefficients of Saito-Kurokawa lifts. For positive weight $$k$$ with $$4 \, | \, k$$ we set    SKadet2−k/2symk−2(2)(t;y)e(tx) =(πL,+L)k/2−1 (aEak2,12(2)⁡(ty)e(tx)), SKadet2−k/2symk−2(2)(t;y)e(tx) =(πL,+L)k/2 (aEak2,12(2)⁡(ty)e(tx)). As a consequence of Proposition 4.1 and the connection of raising and lowering operators with $$(\frak{g},\mathrm{K})$$-modules discussed in Section 2.2, we find that   πL,0L(SKadet2−k/2symk−2(2)(t;y)e(tx))=0,πL,−L(SKadet2−k/2symk−2(2)(t;y)e(tx))=0,andπL,+L(SKadet2−k/2symk−2(2)(t;y)e(tx))=SKadet2−k/2symk−2(2)(t;y)e(tx). (4.2) Lemma 4.3. We have    SKadet2−k/2symk−2(2)(t;y)=ct,kSKa~det−k/2symk(2)(t,y) for nonzero normalizing factors $$c_{t,k} \in {\mathbb{C}}$$. □ Proof We determine the Harish-Chandra modules attached to both Fourier terms. The $$(\frak{g},\mathrm{K})$$-module associated with a Fourier term is a subquotient of the Harish-Chandra module arising from a Siegel–Maaß form. Consider   ϖk:=A¯R,det−k/2symk(SKa~det−k/2symk(2)(t;y)e(tx))andϖ~:=A¯R,det−k/2symk(SKadet2−k/2symk−2(2)(t;y)e(tx)). Lemma 2.1 implies that $$\varpi_k$$ is isomorphic to $$A_{\frak{q}_\mathrm{SK}}\big( (k-2)\frak{h}_\frak{k}^\vee \big)$$. From (2) of Proposition 4.1 and by comparing $$K$$-types we also infer that $$\widetilde\varpi_k$$ is isomorphic to $$A_{\frak{q}_\mathrm{SK}}\big( (k-2)\frak{h}_\frak{k}^\vee \big)$$. Both $$\varpi_k$$ and $$\widetilde\varpi_k$$ yield Bessel models over $${\mathbb{R}}$$ of the irreducible $${\mathrm{Sp}_{2}}({\mathbb{R}})$$ representation that corresponds to $$A_{\frak{q}_\mathrm{SK}}\big( (k-2)\frak{h}_\frak{k}^\vee \big)$$. By what is explained in Section 2.6.4 of [32], we can twist $$\varpi_k$$ and $$\widetilde\varpi_k$$ to obtain Bessel models for the same character (denoted by $$\Lambda$$ there). Section 2.6.4 of loc. cit. in particular allows us to assume that the central character of $$\varpi_k$$ and $$\widetilde\varpi_k$$ is trivial on $$\mathbb R_{> 0}$$, that is, is a sign character or trivial. From the realization of $$A_{\frak{q}_\mathrm{SK}}\big( (k-2) \frak{h}_\frak{k}^\vee \big)$$ as a quotient of $$\mathrm{Ind}_P^G |\,\cdot\,|^{k - \frac{1}{2}}$$ in Section 2.5, we see that the central character is trivial after an appropriate twist. In particular, we obtain a Bessel model for $$\mathrm{PGSp}_{2}({\mathbb{R}})$$ instead of $${\mathrm{Sp}_{2}}({\mathbb{R}})$$. We can now invoke uniqueness of local Bessel models for $$\mathrm{PGSp}_{2}({\mathbb{R}})$$, established in [19]. This implies the above equality for suitable $$c_{t,k}$$. ■ Proposition 4.4. Consider $$f \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$ with Fourier expansion   f=∑tindefinitec(f;t)SKadet2−k/2symk−2(2)(t;y)e(tx). Then the series   f∗=∑tindefinitec(f;t)SKadet2−k/2symk−2(2)(t;y)e(tx) converges locally absolutely and satisfies   πL,+Lf∗=f,πL,0Lf∗=0,andπL,−Lf∗=0. □ Proof Using exponential decay of $$\sideset{^E}{^{(2)}_{k / 2,1 / 2}}{\mathop{a}}(t y)$$ stated in Proposition 4.1, we find that   ∑tindefinitec(f;t)aEak2,12(2)⁡(ty)e(tx) converges locally absolutely. We can therefore interchange summation and application of lowering operators, to obtain $$f^\ast$$ as the image under the $$(k / 2 - 1)$$\thdash\ power of $$\pi_{{\mathrm{L}},+} {\mathrm{L}}$$. The behavior of $$f^\ast$$ under lowering operators follows from 4.2. ■ 4.3 Meromorphic parts of harmonic weak Siegel–Maaß forms The analogue of non-holomorphic Eichler integrals allows us to define Siegel mock modular forms. We call a meromorphic function $$f :\, \mathbb{H}^{(2)} {\rightarrow} V( \det^{2 - k / 2}\mathrm{sym}^{k-2} ) \otimes V(\rho)$$ a Siegel mock modular form of weight $$\det^{2 - k / 2}\mathrm{sym}^{k-2}$$ and type $$\rho$$, if there is $$g \in {{}^\mathrm{SK}\mathrm{S}^{(2)}}\big( {\det^{-k / 2}\mathrm{sym}^k} \otimes \rho \big)$$ such that   ∀γ∈Γ(2):(f+g∗)|det2−k/2symk−2,ργ=f+g∗. In this case $$g$$ is called the shadow of $$f$$. Conversely, Proposition 4.4 says that any harmonic weak Siegel–Maaß form $$f \in \mathrm{SK}\mathbb S^{(2)}\big( \det^{2 - k / 2}\mathrm{sym}^{k-2} \otimes \rho \big)$$ allows for a decomposition into a meromorphic and a non-holomorphic part. More precisely, we have   f=f++f−withf−=(πL,+Lf)∗andf+=f−f−. (4.3) Funding This work was partially supported by Vetenskapsrådet Grant 2015-04139 to M.W.-R. Acknowledgments The author is grateful to Özlem Imamoğlu and Olav Richter for many helpful discussions about harmonic weak Maaß forms and their generalizations. He thanks Rainer Weissauer for discussions about principal series and endoscopy. He appreciates very much comments on an early version of this paper by Kathrin Bringmann, Jan H. Bruinier, Olav Richter, and Ralf Schmidt. He also thanks one of the referees for his or her many remarks that helped improving the presentation of some statements, and for pinpointing a sign mistake in an earlier version. Communicated by Prof. Valentin Blomer References [1] Ash A. Mumford D. Rapoport M. and Tai. Y. Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications , vol. IV, pp. iv+ 335, Brookline, MA, Math. Sci. Press, 1975. Google Scholar CrossRef Search ADS   [2] Bruinier J. H. Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors . vol. 1780. viii+ 152 pp. Lecture Notes in Mathematics. Berlin: Springer, 2002. Google Scholar CrossRef Search ADS   [3] Bruinier J. H. and Funke. J. “On two geometric theta lifts.” Duke Mathematical Journal  125, no. 1 ( 2004): 45– 90. Google Scholar CrossRef Search ADS   [4] Bruinier J. H. and Ono. K. “Heegner divisors, $$L$$-functions and harmonic weakMaass forms.” Annals of Mathematics  172 ( 2010): 2135– 81. Google Scholar CrossRef Search ADS   [5] Bruinier J. H. van der Geer G. Harder G. and Zagier. D. B. The 1-2-3 of Modular Forms , pp. x+ 266, edited by Ranestad K. Universitext. Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004. Berlin: Springer, 2008. Google Scholar CrossRef Search ADS   [6] Bringmann K. and Ono. K. “The $$f(q)$$ mock theta function conjecture and partition ranks.” Inventiones Mathematicae  165, no. 2 ( 2006): 243– 66. Google Scholar CrossRef Search ADS   [7] Bringmann K. and Ono. K. “Dyson’s ranks and Maass forms.” Annals of Mathematics (2)  171, no. 1 ( 2010): 419– 49. Google Scholar CrossRef Search ADS   [8] Bringmann K. Raum M. and Richter. O. K. “Kohnen’s limit process for real-analytic Siegel modular forms.” Advances in Mathematics  231, no. 2 ( 2012): 1100– 18. Google Scholar CrossRef Search ADS   [9] Dabholkar A. Murthy S. and Zagier. D. B. Quantum Black Holes, Wall Crossing, and Mock Modular Forms.  arXiv: 1208.4074. 2012, pp. 571. [10] Demailly J.-P. Complex Analytic and Differential Geometry.  Demailly’s webpage, 2012. [11] Duke W. Imamoğlu Ö. and Tóth. Ä. “Cycle integrals of the $$j$$-function and mock modular forms.” Annals of Mathematics (2)  173, no. 2 ( 2011): 947– 81. Google Scholar CrossRef Search ADS   [12] Esnault H. and Viehweg. E. Lectures on vanishing Theorems.  vol. 20, vi+ 164, DMV Seminar. Basel: Birkhäuser, 1992. Google Scholar CrossRef Search ADS   [13] Grauert H. and Riemenschneider. O. “Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen.” Inventiones Mathematicae  11 ( 1970): 263– 92. Google Scholar CrossRef Search ADS   [14] Hartshorne R. Algebraic geometry . Graduate Texts in Mathematics, 52. New York, Heidelberg: Springer-Verlag, 1977, pages xvi+ 496. Google Scholar CrossRef Search ADS   [15] Helgason S. “Differential operators on homogenous spaces.” Acta Mathematica  102 ( 1959): 239– 99. Google Scholar CrossRef Search ADS   [16] Helgason S. Differential Geometry and Symmetric Spaces.  Pure and Applied Mathematics, vol. XII. New York, London: Academic Press, 1962, pages xiv+ 486. [17] Howe R. and Piatetski-Shapiro. I. I. “Some examples of automorphic forms on Sp4.” Duke Mathematical Journal  50, no. 1 ( 1983): 55– 106. Google Scholar CrossRef Search ADS   [18] Igusa J. “On Siegel modular forms of genus two.” American Journal of Mathematics  84 ( 1962): 175– 200. Google Scholar CrossRef Search ADS   [19] Jiang D. Sun B. and Zhu. C.-B. “Uniqueness of Besselmodels: the Archimedean case.” Geometric and Functional Analysis  20, no. 3 ( 2010): 690– 709. Google Scholar CrossRef Search ADS   [20] Klemm A. Poretschkin M. Schimannek T. and Westerholt-Raum. M. Direct Integration for Genus Two Mirror Curves , to appear in Communications in Number Theory and Physics, arXiv:1502.00557. 2015. [21] Knapp A. W. Representation Theory of Semisimple Groups . Princeton Landmarks in Mathematics. An overview based on examples, Reprint of the 1986 original. Princeton, NJ: Princeton University Press, 2001, pages xx+773. [22] Kudla S. S. and Rallis. S. “Degenerate principal series and invariant distributions.” Israel Journal of Mathematics  69, no. 1 ( 1990): 25– 45. Google Scholar CrossRef Search ADS   [23] Langlands R. P. On the Functional Equations Satisfied by Eisenstein Series.  Lecture Notes in Mathematics, vol. 544, pp. v+337, Berlin, New York: Springer, 1976. [24] Langlands R. P. “Eisenstein series, the trace formula, and the modern theory of Automorphic Forms.” Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987) . Boston, MA: Academic Press, 1989. Google Scholar CrossRef Search ADS   [25] Lee S. T. “Degenerate principal series representations of Sp(2n, R).” Compositio Mathematica  103, no. 2 ( 1996): 123– 51. [26] Li J.-S. “Theta lifting for unitary representations with nonzero cohomology.” Duke Mathematical Journal  61, no. 3 ( 1990): 913– 37. Google Scholar CrossRef Search ADS   [27] Miyazaki T. “On Saito-Kurokawa lifting to cohomological Siegelmodular forms.” Manuscripta Mathematica  114, no. 2 ( 2004): 139– 63. Google Scholar CrossRef Search ADS   [28] Muić G. “Intertwining operators and composition series of generalized and degenerate principal series for Sp(4, R).” Glasnik Matematički. Serija III  44, no. 2 ( 2009): 349– 99. Google Scholar CrossRef Search ADS   [29] Ono K. “Unearthing the visions of a master: harmonicMaass forms and number theory.” Current Developments in Mathematics , 2008. Somerville, MA: Int. Press, 2009. Google Scholar CrossRef Search ADS   [30] Petersen D. “Cohomology of local systems on the moduli of principally polarized abelian surfaces.” Pacific Journal of Mathematics  275, no. 1 ( 2015): 39– 61. Google Scholar CrossRef Search ADS   [31] Pitale A. and Schmidt. R. “Bessel models for lowest weight representations of GSp(4, R).” International Mathematics Research Notices  7 ( 2009): 1159– 212. [32] Piatetski-Shapiro I. I. “On the Saito-Kurokawa lifting.” Inventiones Mathematicae  71, no. 2 ( 1983): 309– 38. Google Scholar CrossRef Search ADS   [33] Sankaran G. K. “Fundamental group of locally symmetric varieties.” Manuscripta Mathematica  90, no. 1 ( 1996): 39– 48. Google Scholar CrossRef Search ADS   [34] Scheithauer N. “Generalized Kac-Moody algebras, automorphic forms and Conway’s group.” Advances in Mathematics  183, no. 2 ( 2004): 240– 70. Google Scholar CrossRef Search ADS   [35] Schulze-Pillot R. “WeakMaaßforms and $$(\mathfrak{g},K)$$-modules.” Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan  26, no. 3 ( 2011): 437– 45. [36] Schmidt R. “The Saito-Kurokawa lifting and functoriality.” American Journal of Mathematics  127, no. 1 ( 2005): 209– 40. Google Scholar CrossRef Search ADS   [37] Shimura G. “Confluent hypergeometric functions on tube domains.” Mathematische Annalen  260, no. 3 ( 1982): 269– 302. Google Scholar CrossRef Search ADS   [38] Vogan D. A., Jr., and Zuckerman. G. J. “Unitary representations with nonzero cohomology.” Compositio Mathematica  53, no. 1 ( 1984): 51– 90. [39] Wallach N. Real Reductive Groups. I. , vol. 132, xx+412 pp. Pure and Applied Mathematics. Boston, MA: Academic Press Inc., 1988. [40] Westerholt-Raum M. Products of Vector Valued Eisenstein Series . arXiv:1411.3877. 2014. [41] Zwegers S. “Mock theta functions.” PhD thesis, Universiteit Utrecht, 2002. © The Author(s) 2016. Published by Oxford University Press. All rights reserved. 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Kappeler, Thomas;Montalto, Riccardo

doi: 10.1093/imrn/rnw233pmid: N/A

Abstract In a case study for integrable PDEs, we construct real analytic, canonical coordinates for the defocusing NLS equation on the circle, specifically tailored to the needs in perturbation theory. They are defined in neighbourhoods of families of finite-dimensional invariant tori and are shown to satisfy together with their derivatives tame estimates. When expressed in these coordinates, the defocusing NLS Hamiltonian is in normal form up to order three. 1 Introduction In form of a case study for integrable PDEs (iPDEs), the goal of this paper is to construct canonical coordinates for the defocusing NLS (dNLS) equation, specifically tailored to the needs in perturbation theory. We consider the dNLS equation in one space dimension   i∂tu=−∂x2u+2|u|2u,x∈T:=R/Z (1.1) on the Sobolev space $$H^s_\mathbb C \equiv H^s({\mathbb T}, {\mathbb C})$$ of complex-valued functions on $${\mathbb T}$$, whose distributional derivatives up to order $$s \in {\mathbb Z}_{ \geq 0}$$ are in $$L^2(\mathbb T, \mathbb C)$$. Equation (1.1) can be viewed as a Hamiltonian PDE, obtained by restricting the Hamiltonian system on the phase space $$H^s_c := H^s_\mathbb C \times H^s_\mathbb C$$ with Poisson bracket and Hamiltonian given by   {F,G}(u,v)=−i∫01(∂uF∂vG−∂vF∂uG)dx,  Hnls(u,v)=∫01(∂xu∂xv+u2v2)dx (1.2) to the real subspace $$H^s_r$$ of $$H^s_c$$ consisting of elements $$(u, v)$$ with $$v = \overline u$$. Here $$\mathcal F, \mathcal G$$ are $${\cal C}^1$$-smooth complex-valued functionals on $$H^s_c$$ with sufficiently regular $$L^2$$-gradients. Equation (1.1) can then be rewritten as $$\partial _tu = - {\rm i} \partial _{v}{\mathcal H}^{\rm nls} \mid_{v = \overline u}$$. The dNLS equation is an iPDE and according to [10], admits global Birkhoff coordinates on $$H^s_\mathbb C$$ with $$s \in {\mathbb Z}_{\geq 0}$$. To state the main results of this paper we first need to describe these coordinates in more detail: for any $$s \in \mathbb Z_{\geq 0}$$, let   hCs≡hs(Z,C):={x=(xn)n∈Z⊆C:‖x‖s<+∞}, ‖x‖s:=∑n∈Z⟨n⟩2s|xn|2, ⟨n⟩:=max{1,|n|},hs≡hs(Z,R):={(xn)n∈Z∈hCs:xn∈R∀n∈Z} and   hcs:=hCs×hCs,hrs:=hs×hs. The Sobolev space $$H^s_\mathbb C$$ can then be described by   HCs={u=∑n∈Zune2πinx:(un)n∈Z∈hCs},‖u‖s:=‖(un)n∈Z‖s. Furthermore let   ℓ1,2≡ℓ1,2(Z,R):={x=(xn)n∈Z⊂R:‖x‖1,2:=∑n∈Z⟨n⟩2|xn|<+∞},ℓ+1,2:={(xn)n∈Z∈ℓ1,2:xn≥0,∀n∈Z} and define the following version $$F_{\rm nls}$$ of the Fourier transform, introduced in [10],   Fnls:Hc0→hc0,(u,v)↦(−12(u−n+vn),−i2(u−n−vn)), (1.3) where $$u_n$$ denotes the $$n$$th Fourier coefficient of $$u$$, $$u_n := \int_0^1 u(x) \,{\rm e}^{- 2 \pi {\rm i} n x}\, {\rm d} x$$. Note that for $$v = \overline u$$, one has $$v_n = \overline u_{- n}$$ for any $$n \in \mathbb Z$$, implying that   Fnls(u,u¯)=(−2Re(u−n),2Im(u−n)). The inverse of $$F_{\rm nls}$$ is then given by   Fnls−1:hc0→Hc0,((xn)n∈Z,(yn)n∈Z)↦(−12∑n∈Z(x−n−iy−n)e2πinx,−12∑n∈Z(xn+iyn)e2πinx). Finally we recall that a possibly nonlinear map $$F : U \to Y$$ of a subset $$U$$ of a Banach space $$X$$ into another Banach space $$Y$$ is said to be bounded if $$F(V)$$ is bounded for any bounded subset $$V$$ in $$U$$. Theorem 1.1. ([10, 14]). There exists a neighbourhood $${\cal W}$$ of $$H^0_r$$ in $$H^0_c$$ and an analytic map   Φnls:W→hc0,(u,v)↦((xn)n∈Z,(yn)n∈Z) with $$\Phi^{\rm nls}(0) = 0$$ such that the following holds: (B1) For any $$s \in \mathbb Z_{\geq 0}$$, $$\Phi^{\rm nls}(H^s_r) \subseteq h^s_r$$ and $$\Phi^{\rm nls}: H^s_r \to h^s_r$$ is a real analytic diffeomorphism. (B2) The map $$\Phi ^{\rm nls}$$ is canonical, meaning that on $${\cal W}$$, $$\{ x_n, y_n \} = - 1$$ and all the other brackets between coordinate functions vanish. (B3) The Hamiltonian $$H^{\rm nls} := {\cal H}^{\rm nls} \circ (\Phi^{\rm nls})^{- 1}\!$$, defined on $$h_r^1$$, is a function of the actions $$I_n := (x_n^2 + y_n^2)/2$$, $$n \in \mathbb Z$$, only and $$H^{\rm nls} : \ell^{1, 2}_+ \to \mathbb R, I \mapsto H^{\rm nls}(I)$$ is real analytic. (B4) The differential $$d_0\Phi^{\rm nls}$$ of $$\Phi^{\rm nls}$$ at $$0$$ is the Fourier transform $$F_{\rm nls}$$ defined in (1.3). (B5) The nonlinear parts $$A^{\rm nls} := \Phi^{\rm nls} - F_{\rm nls}$$ of $$\Phi^{\rm nls}$$ and $$B^{\rm nls} := \Psi^{\rm nls} - F_{\rm nls}^{- 1}$$ of $$\Psi^{\rm nls}:= (\Phi^{\rm nls})^{- 1}$$ are one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$,   Anls:Hrs→hrs+1andBnls:hrs→Hrs+1 are real analytic and bounded. The maps $$\Phi^{\rm nls}, \Psi^{\rm nls}$$ are referred to as Birkhoff maps and the coordinates $$((x_n, y_n))_{n \in \mathbb Z}$$ as Birkhoff coordinates for the dNLS equation. □ Birkhoff coordinates are a tool to study perturbations of the dNLS equation far away from the equilibrium. In particular, in [2] they were used to show the existence of finite-dimensional invariant tori of large size for Hamiltonian perturbations of this equation, involving no derivatives of $$u$$. So far, no such results have been obtained for perturbations involving $$\partial_x u$$ (and possibly $$\partial_x^2 u$$) — see [2, 4–7, 9, 12, 17] for results on perturbations of the dNLS equation on the circle obtained so far. In view of the recent results in [1] concerning the existence of small quasi-periodic solutions of quasi-linear Hamiltonian perturbations of the Karteweg de Vries (KdV) equation and our results in [2] described above, we expect that Hamiltonian perturbations of the dNLS equation, involving $$\partial_x u$$ (and possibly $$\partial_x^2 u$$), also admit large quasi-periodic solutions, also referred to as multi-solitons. For this purpose, the scheme developed in [2] has to be considerably refined. In particular, canonical coordinates are needed which together with their derivatives satisfy tame estimates. In [19], such estimates were derived for $$\Phi^{\rm nls}:H^0_r \to h^0_r$$ on the real subspaces $$H^s_r$$ and for its inverse $$\Psi^{\rm nls}: h^0_r \to H^0_r$$ on the real subspaces $$h^s_r$$ where $$s \in \mathbb Z_{s \ge 2}$$. But so far they are not available for their derivatives. In this paper, we prove how to use the Birkhoff coordinates to construct near bounded, integrable, finite-dimensional subsystems (iSS) of the dNLS equation, local canonical coordinates so that they satisfy tame estimates and the dNLS Hamiltonian, when expressed in these coordinates, is in normal form up to order three–see Theorem 1.2 for a precise statement. In future work, we will use these coordinates as a starting point for applying a KAM scheme to reduce certain linear operators with tame estimates, which come up in the Nash Moser iteration, to operators with constant coefficients. Recently, such schemes have been further developed in significant ways. In the context of the dNLS equation, results of this type in [3] will be particularly relevant. To state our main result, we need to introduce some more notation. For any $$S \subseteq \mathbb Z$$ with $$|S| < + \infty$$, let $$S^\bot := \mathbb Z \setminus S$$. By a slight abuse of notation, we identify $$h^s_c$$ with $$\mathbb C^S \times \mathbb C^S \times h^s_{ \bot c}$$ and $$h^s_r$$ with $$\mathbb R^S \times \mathbb R^S \times h^s_{\bot r}$$ where   h⊥cs:=hs(S⊥,C)×hs(S⊥,C),h⊥rs:=hs(S⊥,R)×hs(S⊥,R). Accordingly, an element $$z \in h^0_c$$ is written as   z=(zS,z⊥),zS=((xj)j∈S,(yj)j∈S),z⊥=((xj)j∈S⊥,(yj)j∈S⊥), and as norm we choose $$\| z \|_s := \| z_S\| + \| z_\bot\|_s$$ where   ‖zS‖≡‖zS‖0:=(∑j∈S|xj|2+|yj|2)12,‖z⊥‖s:=(∑j∈S⊥⟨j⟩2s(|xj|2+|yj|2))12. Furthermore, we introduce the bilinear form   (z⊥,z⊥′)r:=∑j∈S⊥xjxj′+yjyj′,z⊥=(x⊥,y⊥),z⊥′=(x⊥′,y⊥′)∈h⊥c (1.4) and write the sequence of actions $$I = (I_k)_{k \in \mathbb Z}$$ as $$(I_S, I_\bot)$$ where   IS:=(Ik)k∈S,I⊥:=(Ik)k∈S⊥,Ik≡Ik(z)=|zk|22=xk2+yk22,∀k∈Z. Finally, we introduce the dNLS frequencies   ωknls(I):=∂IkHnls(I),k∈Z. (1.5) They satisfy asymptotics of the form $$\omega_k(I) = 4 k^2 \pi^2 + O(1)$$ as $$k \to \pm \infty$$. More precisely, the map   ℓ+1,2→ℓ∞, (Ik)k∈Z↦(ωnnls(I)−4π2n2)n∈Z is real analytic and bounded—see Proposition 5.3 in Section 5.2. The main result of this paper is the following one. Theorem 1.2. Let $$S \subseteq \mathbb Z$$ be finite. For any compact subset $${\cal K} \subseteq \mathbb R^S \times \mathbb R^S$$, there exists an open, bounded, complex neighbourhood $${\cal V} \subseteq h^0_c$$ of $${\cal K} \times \{ 0 \}$$ and a bounded analytic map   Ψ:V→Hc0,(zn)n∈Z↦w so that the following holds: (C1) For any $$s \in \mathbb Z_{\geq 0}$$, $$\Psi({\cal V} \cap h^s_r) \subseteq H^s_r$$ and $$\Psi : {\cal V} \cap h^s_r \to H^s_r$$ is a real analytic diffeomorphism onto its image. (C2) $$\Psi$$ is canonical, meaning that on $$\Psi({\cal V} \cap h^0_r)$$, $$\{ x_n, y_n \} = - 1$$ for any $$n \in \mathbb Z$$, whereas all the other brackets between coordinate functions vanish. (C3) The transformation $$\Psi$$ is related to $$\Psi^{\rm nls} = (\Phi^{\rm nls})^{- 1}$$ by   Ψ∣K×{0}=Ψnls∣K×{0},dΨ(z)=dΨnls(z),∀z∈K×{0}. (C4) The Hamiltonian $${\cal H} := {\cal H}^{\rm nls} \circ \Psi$$, defined on $${\cal V} \cap h^1_r$$, is in normal form up to order three. More precisely,   H(z)=Hnls(IS,0)+∑n∈S⊥ωnnls(IS,0)In(z)+P3(z), (1.6) where the Hamiltonian $${\cal P}_3 : {\cal V} \cap h^0_r \to \mathbb R$$ is real analytic. Furthermore, $${\cal P}_3$$ satisfies the following tame estimates: for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V} \cap h^s_r,$$$$\widehat z \in h^s_c$$,    ‖∇P3(z)‖s≲s‖z⊥‖s‖z⊥‖0,‖d∇P3(z)[z^]‖s≲s‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s (1.7) and for any $$k \in \mathbb Z_{\geq 2}$$, $$ \widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P3(z)[z^1,…,z^k]‖s≲s∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. Here, the meaning of $$\lesssim_s$$ is the standard one. So, for example, $$\| \nabla {\cal P}_3(z)\|_s \lesssim_s \| z_\bot\|_s \| z_\bot\|_0$$ says that there exists a constant $$C \equiv C(s) > 0$$ so that   ‖∇P3(z)‖s≤C‖z⊥‖s‖z⊥‖0,∀z∈V∩hrs. (C5) The nonlinear maps $$B:= \Psi - F_{\rm nls}^{- 1} : {\cal V} \cap h^0_r \to H^0_r$$ and $$A := \Psi^{- 1} - F_{\rm nls} : \Psi({\cal V}) \cap H^0_r \to h^0_r$$ are real analytic maps and so is   A:V∩hr0→L(Hc0,hc0),z↦A(z):=dΨ(z)−1−Fnls. On $${\cal V} \cap h^0_r$$, the maps $$B$$ and $${\cal A }$$ satisfy the following estimates: for any $$z \in {\cal V} \cap h^0_r$$, $$k \in \mathbb Z_{\geq 1}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^0_c$$, and $$\widehat w \in H^0_c$$,   ‖B(z)‖0≲1,‖dkB(z)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0,‖A(z)[w^]‖0≲‖w^‖0,‖dk(A(z)[w^])[z^1,…,z^k]‖0≲k‖w^‖0∏j=1k‖z^j‖0. Furthermore, $$B$$ is one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, $$B: {\cal V} \cap h^s_r \to H^{s+1}_r$$ is real analytic, and satisfies the following tame estimates: for any $$k \in \mathbb Z_{\geq 1}$$, $$ z \in {\cal V} \cap h^s_r$$, and $$ \widehat z_1, \ldots , \widehat z_k \in h^s_c$$,    ‖B(z)‖s+1≲s1+‖z⊥‖s, ‖dkB(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. Similarly, the maps $$A$$ and $${\cal A }$$ are one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, $$A: \Psi( {\cal V} \cap h^s_r) \to h^{s+1}_r$$ and $${\cal A } : {\cal V} \cap h^s_r \to {\cal L}(H^s_c, h^{s + 1}_c)$$ are real analytic. Moreover, $${\cal A }$$ satisfies the following tame estimates: for any $$z \in {\cal V} \cap h^s_r$$, $$\widehat w \in H^s_c$$,   ‖A(z)[w^]‖s+1≲s‖z⊥‖s‖w^‖0+‖w^‖s and for any $$k \in \mathbb Z_{\geq 1},$$$$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk(A(z)[w^])[z^1,…,z^k]‖s+1 ≲s,k(‖z⊥‖s‖w^‖0+‖w^‖s)∏j=1k‖z^j‖0 +‖w^‖0∑j=1k‖z^j‖s∏i≠j‖z^i‖0. □ Remark 1.1. In Theorem 1.2, apart from being compact, no further assumptions on $${\cal K}$$ are being made. In particular, $${\cal K}$$ may contain the equilibrium point $$0$$ in which case $${\cal K}$$ does not admit action-angle coordinates. In subsequent work, the estimates for $${\cal A}(z) ={\rm d} \Psi(z)^{- 1} - F_{\rm nls}$$ will be used to study perturbations of the dNLS equation. Since such estimates are not needed for $$A(\Psi(z))$$, we have not included them in Theorem 1.2. □ Remark 1.2. In Section 7 we present additional results about the map $$\Psi.$$ In particular we study the restrictions of $$\Psi$$ to $${\cal V} \cap h^0_{r, 1}$$ and $${\cal V} \cap h^0_{r, 2}$$ where $$h^0_{r, 1}$$ and $$h^0_{r, 2}$$ are the dNLS invariant subspaces, corresponding via the Birkhoff map $$\Phi^{\rm nls}$$ to potentials $$\varphi \in H^0_r$$ which are even and, respectively, odd. □ Outline of the construction of$$\Psi$$: Let $${\cal V}$$ be of the form $${\cal V} = {\cal V}_S \times {\cal V}_\bot \subset h^0_c$$ where $${\cal V}_S$$ is a bounded, open neighbourhood of $${\cal K}$$ in $$\mathbb C^S \times \mathbb C^S$$ and $${\cal V}_\bot$$ an open ball in $$h^0_{\bot c}$$, centred at $$\{ 0 \}$$. By Theorem 1.1, $${\cal V}_S$$ and $${\cal V}_\bot$$ can be chosen so that the Birkhoff map $$\Psi^{\rm nls}$$ is defined on $${\cal V}$$ and all the estimates of $$\Psi^{\rm nls}$$ and its derivatives used in the sequel are uniform on $${\cal V}$$. The canonical map $$\Psi$$ is then defined to be the composition $$\Psi := \Psi_L \circ \Psi_C$$ where $$\Psi_L$$ is the Taylor expansion of $$\Psi^{\rm nls}$$ of order one in the normal directions $$z_\bot$$ around $$(z_s, 0)$$,   ΨL(zS,z⊥):=Ψnls(zS,0)+dΨnls(zS,0)[0,z⊥], (1.8) and $$\Psi_C$$, referred to as symplectic corrector, is chosen so that $$\Psi_L \circ \Psi_C$$ becomes symplectic and satisfies the claimed tame estimates. In his pioneering work [16], Kuksin presents a general scheme for proving KAM-type theorems for semilinear Hamiltonian perturbations of iPDEs in one space dimension, such as the KdV or the sine Gordon (sG) equations, which possess a Lax pair formulation and admit finite-dimensional integrable subsystems, foliated by invariant tori. One of the key elements of his work is a normal form theory for such PDEs. Expanding on work of Krichever [15], Kuksin considers bounded iSS of such an iPDE which admit action-angle coordinates. In the case of the KdV and the sG equations, the angle variables are given by the celebrated Its Matveev formulas. These action-angle coordinates are complemented by infinitely many coordinates whose construction is based on a set of time periodic solutions, referred to as Floquet solutions, of the PDE obtained by linearizing iPDE along solutions in iSS. The resulting coordinate transformation, denoted in [16] by $$\Phi$$, is typically not symplectic and to obtain canonical coordinates, an additional coordinate transformation needs to be applied. In [16], Kuksin constructs such a transformation, which he denotes by $$\phi$$, using arguments of Moser and Weinstein in the given infinite dimensional setup—see [16, Lemma 1.4 and Section 7.1]. To construct the map $$\Psi_C$$ we follow the same scheme of proof. Actually, the following result holds. Theorem 1.3. Assume that in addition to the assumptions made in Theorem 1.2, the set $${\cal K}$$ is contained in $$(\mathbb R \setminus 0)^S \times (\mathbb R \setminus 0)^S.$$ Then, up to normalizations and natural identifications, $$\Psi_L$$ coincides with the map $$\Phi$$, obtained by applying the scheme of construction in [16] to the dNLS equation. As a consequence, so does $$\Psi = \Psi_L \circ \Psi_C$$ with $$\Phi \circ \phi$$. □ Since Birkhoff coordinates provide a concise, self-contained, and efficient framework for proving Theorem 1.2—in particular the claimed tame estimates, the main goal of our study—Theorem 1.3 also provides in the case of the dNLS equation a valuable alternative for proving the normal form result for this equation, obtained by applying the scheme of proof in [16]. Note also that the assumptions on $${\cal K}$$ in Theorem 1.2 are slightly weaker than the ones made in the setup of [16]. Organization: The maps $$\Psi_L$$ and $$\Psi_C$$ are introduced and studied in Sections 3 and 4, respectively, after a short Section 2, describing the Hamiltonian setup. In Section 5, we prove Theorem 1.2: in Section 5.1, we show that the composition $$\Psi = \Psi_L \circ \Psi_C$$ satisfies the analytic properties, stated in Theorem 1.2, and in the subsequent Section 5.2, the expansion of the dNLS Hamiltonian in the new coordinates is computed up to order three. In Section 5.3 we summarize the proof of Theorem 1.2. Finally, in Section 6 we prove Theorem 1.3 and in Section 7 results, concerning the restriction of $$\Psi$$ to subsets, satisfying symmetry conditions. In Appendix 1, we recall an infinite-dimensional version of the Poincaré Lemma, needed in Section 4 (cf. from [16], [18]). Notation: For any $$C^1$$ map $$F : h^0_c \to X$$ with $$X$$ being a Banach space, we denote by $$d_\bot F(z)$$ the differential of $$F$$ at $$z$$ with respect to the variable $$z_\bot $$,   d⊥F(z)[z^⊥]=∑j∈S⊥x^j∂xjF(z)+y^j∂yjF(z),z^⊥:=((x^j)j∈S⊥,(y^j)j∈S⊥)∈h⊥c0, where for any $$j \in S^\bot$$, $$\partial_{x_j} F,\, \partial_{y_j} F \in X$$ denote the partial derivatives of $$F$$ with respect to the variables $$x_j$$ respectively $$y_j$$. Similarly, we define the gradient with respect to the variable $$z_\bot$$ as   ∇⊥F:=((∂xjF)j∈S⊥,(∂yjF)j∈S⊥). The gradient of $$F$$ with respect to $$z_S$$ is denoted by   ∇SF:=((∂xjF)j∈S,(∂yjF)j∈S) and the differential of $$F$$ at $$z$$ with respect to $$z_S$$ by $$d_SF(z)$$,   dSF(z)[z^S]=∑j∈Sx^j∂xjF(z)+y^j∂yjF(z),z^S:=((x^j)j∈S,(y^j)j∈S)∈CS×CS. For the partial derivatives of $$F$$ with respect to $$z_j$$, $$j \in S$$, we use the multi-index notation and write for any $$\alpha, \beta \in \mathbb Z^S_{\geq 0}$$  ∂Sα,βF:=(∏j∈S∂xjαj∂yjβj)F. If not stated otherwise, $${\cal K}$$ denotes a compact subset of $$\mathbb R^S \times \mathbb R^S$$ and $${\cal V}$$ an open, bounded neighbourhood of $${\cal K}\times \{0\}$$ in $$h_c^0$$ of the form $${\cal V}_S \times {\cal V}_\bot$$ where $${\cal V}_\bot$$ is a ball in $$h_{\bot c}$$, centred at $$0$$. We write $${\cal V}_\bot (\delta)$$ to indicate that the radius of the ball $${\cal V}_\bot$$ is $$\delta > 0.$$ Finally, we frequently will use the symbols $$ \lesssim$$, $$\lesssim_{s}$$,... to express that a quantity is bounded by another one up to a constant which is “universal,” respectively, depends only on the Sobolev index $$s$$. For example, given two real valued functionals $$A, B$$ on $${\cal V}$$ we write $$A \lesssim_{s} B$$ if there is a constant $$C\equiv C(s)$$ so that $$A(z) \le C B(z)$$ for any $$z \in {\cal V} \cap h_r^s$$. 2 Hamiltonian Setup In this preliminary section we discuss the Hamiltonian setup, introduced in Section 1 in more detail and introduce some additional notations. The Hamiltonian vector field associated with a sufficiently smooth functional $${\cal F} : H^0_c \to \mathbb C$$ and the Poisson bracket (1.2) on $$H^0_c$$ is denoted by   XF=iJ∇F,∇F:=(∇uF,∇vF), where $$\nabla_u {\cal F}$$, $$\nabla_v {\cal F}$$ denote the $$L^2$$ gradients with respect to $$u$$ and $$v$$, namely   dF[(u^,0)]=∫T∇uFu^dx,dF[(0,v^)]=∫T∇vFv^dx,J:=(0−IdId0):Hc0→Hc0, (2.1) and $${\rm Id} : H^0_\mathbb C \to H^0_\mathbb C$$ is the identity operator. Furthermore we introduce the non-degenerate bilinear form   ⟨⋅,⋅⟩r:Hc0×Hc0→C defined for any $$w= (u, v)$$, $$w' = (u', v') \in H^0_c$$ by   ⟨w,w′⟩r:=∫Tu(x)u′(x)dx+∫Tv(x)v′(x)dx. (2.2) The subscript $$r$$ indicates that in the latter integrals, no complex conjugation appears. The Poisson bracket (1.2) then reads   {F,G}=⟨∇F,iJ∇G⟩r and the symplectic form, associated with it, is the two form   (2.3) For any sufficiently smooth functionals $${\cal F}, {\cal G} : H^0_c \to \mathbb C$$, one has   Λ(XF,XG)={F,G}. In terms of the Fourier coefficients of $$\widehat w$$ and $$\widehat w'$$, $$\Lambda[\widehat w, \widehat w']$$ can be expressed as   Λ[w^,w^′]=i∑k∈Z(u^kv^−k′−v^−ku^k′) and hence $$\Lambda$$ can be conveniently written as   Λ=i∑k∈Zduk∧dv−k, where   (duk∧dv−k)[(u^,v^),(u^′,v^′)]=u^kv^−k′−v^−ku^k′,∀k∈Z. In addition, we define the one form $$\lambda$$ on $$H^0_c$$ as   λ≡λ(w)=i∑k∈Zukdv−k. Its action on a function $$\widehat w = (\widehat u, \widehat v) \in H^0_c$$ is given by   λ[w^]=i∫Tu(x)v^(x)dx=i∑k∈Zukv^−k. The exterior differential of $$\lambda$$, defined by $$d \lambda = {\rm i} \sum_{k \in \mathbb Z} {\rm d} u_k \wedge {\rm d} v_{- k}$$, thus satisfies $$d \lambda = \Lambda$$. The Poisson bracket on the model space $$h^0_c$$ is determined by defining it for the coordinate functions,   {xn,ym}M=−δnm,{yn,xm}M=δnm,{xn,xm}M=0,{yn,ym}M=0,∀n,m∈Z. By a slight abuse of terminology in connection with the definition (1.4), we also denote by $$\big(\cdot, \cdot \big)_r$$ the non-degenerate bilinear form $$\big(\cdot, \cdot \big)_r : h^0_c \times h^0_c \to \mathbb C$$  (z,z′)r:=x⋅x′+y⋅y′,∀z=(x,y),z′=(x′,y′)∈hc0 (2.4) where $$x \cdot x' := \sum_{k \in \mathbb Z} x_k x_k'$$. Given two sufficiently smooth functionals $$F, G : h^0_c \to \mathbb C$$, one has   {F,G}M=−∑k(∂xkF∂ykG−∂ykF∂xkG)=(∇F,J∇G)r where   J:=(0−IdId0):hc0→hc0, (2.5)$${\rm Id} : h^0_c \to h^0_c$$ is the identity operator and   ∇F=(∇xF,∇yF),∇xF=(∂xkF)k∈Z,∇yF=(∂ykF)k∈Z. The Hamiltonian vector field $$X_F$$ of $$F : h^0_c \to \mathbb C$$, corresponding to the Poisson bracket $$\{ \cdot, \cdot \}_M$$, is then given by   XF=J∇F (2.6) and the symplectic form $$\Lambda_M$$, associated with it, by   ΛM[z^,z^′]:=(J−1z^,z^′)r=y^⋅x^′−x^⋅y^′,∀z^=(x^,y^),z^′=(x^′,y^′)∈hc0. (2.7) Note that   ΛM=−∑k∈Zdxk∧dyk where as above, for any $$k \in \mathbb Z$$, the two form $${\rm d} x_k \wedge {\rm d} y_k$$ is defined as   (dxk∧dyk)[(x^,y^),(x^′,y^′)]=x^ky^k′−y^kx^k′. Then   ΛM(XF,XG)=(∇F,J∇G)r={F,G}M. The one form associated with $$\Lambda_M$$ is defined as   λM≡λM(z):=∑k∈Zykdxk. (2.8) Its action on a vector $$\widehat z = (\widehat x, \widehat y) \in h_c^0$$ is given by   λM[z^]=∑k∈Zykx^k. The exterior differential of $$\lambda_M$$ then satisfies $${\rm d} \lambda_M = \Lambda_M$$. 3 The Map $$\Psi_L$$ In this section, we study the map $$\Psi_L$$ introduced in (1.8). In particular, we prove tame estimates and one smoothing properties for $$\Psi_L$$. First we introduce some more notations. Denote by $$\Pi_S$$ and $$\Pi_\bot$$ the standard projections   ΠS:(CS×CS)×h⊥c0→(CS×CS)×{0},z=(zS,z⊥)↦(zS,0) (3.1)  Π⊥:(CS×CS)×h⊥c0→{0}×h⊥c0,z=(zS,z⊥)↦(0,z⊥). (3.2) The formula (1.8) for $$\Psi_L(z)$$ with $$z = (z_S, z_\bot)$$ then reads   ΨL(z)=Ψnls(ΠSz)+d⊥Ψnls(ΠSz)[z⊥]. (3.3) For a quite explicit formula for $$d_\bot \Psi^{\rm nls}(\Pi_S z) [ z_\bot], $$ we refer to Appendix 2. The map $$\Psi_L$$ is defined on   Vmax:=VSmax×h⊥cs,VSmax:=ΠSΦnls(W), where $${\cal W} \subseteq H^0_c$$ is the domain of definition of the Birkhoff map $$\Phi^{\rm nls}$$ of Theorem 1.1. Note that   RS×RS⊆VSmax⊆CS×CS,hr0⊂Vmax⊂hc0,ΨL(0)=0. Furthermore, the differential $$d \Psi_L(z)$$ of $$\Psi_L$$ at $$z = (z_S, z_\bot) \in {\cal V}^{\max}$$ applied to a vector $$\widehat z = (\widehat z_S, \widehat z_\bot) \in h^0_c$$ is given by   dΨL(z)[z^S,z^⊥] =dSΨnls(ΠSz)[z^S]+d⊥Ψnls(ΠSz)[z^⊥]+dS(d⊥Ψnls(ΠSz)[z⊥])[z^S] (3.4)   =dΨnls(ΠSz)[z^]+d2Ψnls(ΠSz)[ΠSz^,Π⊥z]. (3.5) The latter expression is independent of $$\Pi_\bot \widehat z$$ and that by Theorem 1.1, $$d \Psi_L(0) = {\rm d} \Psi^{\rm nls}(0) = F_{\rm nls}^{- 1}$$. First we establish the following auxiliary results. Lemma 3.1. (i) The map $$\Psi_L : {\cal V}^{\max} \to H^0_c$$ is analytic and for any $$s \in \mathbb Z_{\geq 0}$$, the restriction $$\Psi_L \mid_{h^s_r} : h^s_r \to H^s_r$$ is real analytic. Furthermore, for any $$z_S \in \mathbb R^S \times \mathbb R^S$$ and any $$s \in \mathbb Z_{\geq 0}$$, $$d \Psi_L(z_S, 0): h^s_c \to H^s_c$$ is a linear isomorphism. (ii) For any compact subset $${\cal K} \subseteq \mathbb R^S \times \mathbb R^S$$, there exists a ball $${\cal V}_\bot$$ in $$h^0_{\bot r}$$, centred at $$0$$, so that the restriction $$\Psi_L : {\cal K} \times {\cal V}_\bot \to H^0_r$$ is one to one. Furthermore, after shrinking the radius of the ball $${\cal V}_\bot$$, if necessary, the map $$\Psi_L : {\cal K} \times {\cal V}_\bot \to H^0_r$$ is a local diffeomorphism. □ Proof $$(i)$$ The claimed analyticity follows from the definition of $$\Psi_L$$ and the corresponding properties of $$\Psi^{\rm nls}$$, stated in Theorem 1.1. Concerning the statement on the differential $$d \Psi_L(z_S, 0)$$, note that by (3.5), $$d \Psi_L(z_S, 0) = {\rm d} \Psi^{\rm nls}(z_S, 0)$$ and hence by Theorem 1.1, $$d \Psi_L(z_S, 0) : h^s_c \to H^s_c$$ is a linear isomorphism for any $$s \in \mathbb Z_{\geq 0}$$. $$(ii)$$ Let $${\cal K} \subseteq \mathbb R^S \times \mathbb R^S$$ be a given compact subset. Assume that there exists no ball $${\cal V}_\bot$$ in $$h^0_{\bot r}$$, centred at $$0$$, so that $$\Psi_L\mid_{ {\cal K} \times {\cal V}_\bot}$$ is one to one. Then there exist two sequences $$z^{(j)} = (z^{(j)}_S, z_\bot^{(j)})$$, $${j \geq 1}$$, and $$\tilde z^{(j)} = (\tilde z^{(j)}_S, \tilde z_\bot^{(j)})$$, $${j \geq 1}$$, in $${\cal K} \times h^0_{\bot r}$$ such that for any $$j \geq 1$$  z(j)≠z~(j),ΨL(z(j))=ΨL(z~(j)),limj→∞z⊥(j)=limj→∞z~⊥(j)=0. Since by assumption $${\cal K}$$ is compact, there exist subsequences of $$(z^{(j)})_{j \geq 1}, (\tilde z^{(j)})_{j \geq 1}$$, denoted for simplicity in the same way, such that $$(z^{(j)}_S)_{j \geq 1}, (\tilde z^{(j)}_S)_{j \geq 1}$$ converge. Denote their limits by $$z_S^{(\infty)}$$ and $$\tilde z_S^{(\infty)}$$, respectively. Then   limj→∞z(j)=(zS(∞),0),limj→∞z~(j)=(z~S(∞),0) are elements in $${\cal K} \times \{ 0 \}$$. By the continuity of $$\Psi_L$$, one has $$\Psi_L(z_S^{(\infty)}, 0) = \Psi_L(\tilde z_S^{(\infty)}, 0)$$ and since $$\Psi_L$$ and $$\Psi^{\rm nls}$$ coincide on $${\cal V}_S^{\max} \times \{ 0 \}$$ it then follows from Theorem 1.1 that $$z_S^{(\infty)} = \tilde z_S^{(\infty)}$$. By item (i) and the local inversion theorem one then concludes that in contradiction to our assumption, $$z^{(j)} = \tilde z^{(j)}$$ for $$j$$ sufficiently large. This proves the first part of item (ii). Since according to item (i), for any given $$z_S \in {\cal K}$$, $$d \Psi_L(z_S, 0) : h^0_c \to H^0_c$$ is a linear isomorphism, $$d \Psi_L(z)$$ is such an operator for $$z$$ in a whole neighbourhood of $$(z_S, 0)$$. Using that $${\cal K}$$ is compact it then follows that after shrinking the radius of the ball $${\cal V}_\bot$$, if necessary, $$\Psi_L : {\cal K} \times {\cal V}_\bot \to H^0_r$$ is a local diffeomorphism. ■ Proposition 3.1. For any compact subset $${\cal K} \subseteq \mathbb R^S \times \mathbb R^S$$ there exists an open complex neighbourhood $${\cal V}$$ of $${\cal K} \times \{ 0 \}$$ in $$h^0_c$$ of the form $${\cal V}_S \times {\cal V}_\bot$$ where $$\overline{\cal V}_S$$ is compact with $$\overline{\cal V}_S \subseteq {\cal V}^{max}_S$$ and $${\cal V}_\bot \subset h^0_{\bot c}$$ is an open ball, centred at $$0$$, so that the restriction of $$\Psi_L$$ to $${\cal V}$$ has the following properties: $$(L1)$$$$\Psi_L$$ is analytic on $${\cal V}$$ and   ΨL∣VS×{0}=Ψnls∣VS×{0},dΨL(zS,0)=dΨnls(zS,0),∀zS∈VS. (3.6) Furthermore, $$\Psi_L : {\cal V} \cap h^0_r \to H^0_r$$ is a real analytic diffeomorphism onto its image. $$(L2)$$ The map $$B_L := \Psi_L - F_{\rm nls}^{- 1} : {\cal V} \to H^0_c$$ is analytic and one smoothing. More precisely, the analytic map $$B_L$$ is given by   BL(z)=Bnls(ΠSz)+d⊥Bnls(ΠSz)[z⊥] (3.7) with $$B^{\rm nls}$$ being the map introduced in Theorem 1.1, and for any $$s \in \mathbb Z_{\geq 1}$$, $$B_L : {\cal V} \cap h^s_r \to H^{s + 1}_r$$ is real analytic. Furthermore   d⊥BL(z)=d⊥Bnls(ΠSz),d⊥2BL(z)=0,∀z∈V and for any $$z \in {\cal V} \cap h^0_r$$, $$\alpha, \beta \in \mathbb Z_{\geq 0}^S$$,   ‖∂Sα,βBL(z)‖0≲α,β1,‖∂Sα,βd⊥BL(z)[z^⊥]‖0≲α,β‖z^⊥‖0,∀z^⊥∈h⊥c0 (3.8) and for any $$s \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V} \cap h^s_{r}$$,   ‖∂Sα,βBL(z)‖s+1≲s,α,β1+‖z⊥‖s,‖∂Sα,βd⊥BL(z)[z^⊥]‖s+1≲s,α,β‖z^⊥‖s,∀z^⊥∈h⊥cs. (3.9) $$(L3)$$ For any $$s \in \mathbb Z_{\geq 1}$$, the restriction $$\Psi_L \mid_{{\cal V} \cap h^s_r}$$ is a map $${\cal V} \cap h^s_r \to H^s_r$$ which is a real analytic diffeomorphism onto its image. $$(L4)$$ The map $$A_L := \Psi_L^{- 1} - F_{\rm nls}: \Psi_L({\cal V}) \to h^0_c$$ is analytic and one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, $$A_L : \Psi_L ({\cal V}) \cap H^s_r \to h^{s + 1}_r$$ is real analytic. □ Remark 3.1. For convenience, in the sequel, we always choose $${\cal V}_\bot$$ to be a ball of radius smaller than one. □ Proof Choose $${\cal V}_S$$ to be an open bounded neighbourhood of $${\cal K}$$ in $$\mathbb C^S \times \mathbb C^S$$ so that $$\overline{\cal V}_S \subseteq {\cal V}_S^{\max}$$ and let $${\cal V}_\bot$$ be an open ball in $$h^0_{\bot c}$$, centred at $$0$$, so that item (ii) of Lemma 3.1 applies to $${\cal V} := {\cal V}_S \times {\cal V}_\bot$$, implying that $$\Psi_L : {\cal V} \cap h^0_r \to H^0_r$$ is one to one and a local diffeomorphism. The identities (3.6) hold by the definition of $$\Psi_L$$ and the analyticity of $$\Psi_L$$, stated in $$(L1)$$, follows by Lemma 3.1(i). One then concludes that   ΨL:V∩hr0→Hr0 is a real analytic diffeomorphism on to its image. $$(L2)$$ follows from the definition of $$\Psi_L$$, Theorem 1.1, the compactness of $$\overline{\cal V}_S$$, and standard estimates in Sobolev spaces. Concerning $$(L3)$$, first note that by Theorem 1.1, for any $$s \in \mathbb Z_{\ge 1}$$, the restriction $$\Psi_L \mid_{{\cal V} \cap h^s_r}$$ is a map with values in $$ H^s_r$$ and as such real analytic. By item (L1), $$\Psi_L \mid_{{\cal V} \cap h^s_r}$$ is one to one and so is its differential $$d \Psi_L (z) : h^s_c \to H^s_c$$ at any point $$z \in {\cal V} \cap h^s_r$$. Since by $$(L2)$$ the map $$B_L$$ is one smoothing, $$d \Psi_L(z) : h^s_c \to H^s_c$$ is Fredholm and hence a linear isomorphism, implying that $$\Psi_L : {\cal V} \cap h^s_r \to H^s_r$$ is a real analytic diffeomorphism on to its image. Finally, item $$(L4)$$ follows from $$(L3)$$ and Theorem 1.1. ■ Whereas the tame estimates (3.9) for $$B_L$$ are an immediate consequence of the definition of $$\Psi_L$$, Theorem 1.1 and the compactness of $$\overline{\cal V}_S$$, this is not so for $$A_L$$. Actually, for the applications in perturbation theory considered in subsequent work, we only need to derive tame estimates for   AL:V∩hr0→L(Hc0,hc0),z↦AL(z):=dAL(ΨL(z))=dΨL(z)−1−Fnls (3.10) with $${\cal V}$$ denoting the neighbourhood of $${\cal K} \times \{ 0 \}$$ of Proposition 3.1. By formula (3.5), for any $$z \in {\cal V} \cap h^0_r,$$ the operator $$d \Psi_L(z) \in {\cal L}(h^0_c, H^{0}_c)$$ can be written as   dΨL(z)=T(z)+R(z),T(z):=dΨnls(ΠSz) (3.11) with $${\cal R}(z) \in {\cal L}(h^0_c , H^{0}_c)$$ given by   R(z):hc0→Hc0,z^↦R(z)[z^]:=d2Ψnls(ΠSz)[ΠSz^,Π⊥z]=d2Bnls(ΠSz)[ΠSz^,Π⊥z]. (3.12) Since by Theorem 1.1, respectively Proposition 3.1, the operators $${\cal T}(z)$$, $${\rm d} \Psi_L(z) : h^0_c \to H^{0}_c$$ are invertible, so is $${\cal T}(z)^{-1} {\rm d} \Psi_L(z)= {\rm Id} + {\cal T}(z)^{- 1}{\cal R}(z)$$, implying that   dΨL(z)−1=(Id+T(z)−1R(z))−1T(z)−1,=T(z)−1−T(z)−1R(z)S(z), (3.13) where   S(z):=(Id+T(z)−1R(z))−1T(z)−1∈L(Hc0,hc0). (3.14) Furthermore, by Theorem 1.1   T(z)−1=(dΨnls(ΠSz))−1=dΦnls(ΠSz)=Fnls+dAnls(Ψnls(ΠSz)). Altogether, it follows that for any $$z \in {\cal V} \cap h^0_r$$, the operator $${\cal A}_L(z) = {\rm d} \Psi_L(z)^{-1} - F_{\rm nls} : h^0_c \to H^{0}_c$$ can be written as   AL(z)=dAnls(Ψnls(ΠSz))−T(z)−1R(z)S(z). (3.15) Finally we note that by $$(L4)$$ of Proposition 3.1, $$ {\cal A}_L = {\rm d} A_L \circ \Psi_L$$ is one smoothing. More precisely, for any $$s \in \mathbb Z_{\ge 1}$$, the restriction of $$ {\cal A}_L$$ to $${\cal V} \cap h^s_r$$ is a real analytic map,   AL:V∩hrs→L(Hcs,hcs+1),z↦AL(z). Proposition 3.2. (Tame estimates for $${\cal A}_L$$). After shrinking, if necessary, the radius of the ball $$\cal V_\bot$$ in $${\cal V} = {\cal V}_S \times {\cal V}_\bot$$ of Proposition 3.1, the map $$ {\cal A}_L$$ satisfies for any $$z \in {\cal V} \cap h^0_r$$, $$\widehat w \in h^0_c$$,   ‖AL(z)[w^]‖0≲‖w^‖0 and for any $$k \in \mathbb Z_{\geq 1}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^0_c$$,   ‖dk(AL(z)[w^])[z^1,…,z^k]‖0≲k‖w^‖0∏j=1k‖z^j‖0. Furthermore, for any $$s \in \mathbb Z_{\ge 1}$$, $$z \in {\cal V} \cap h^s_r$$, $$\widehat w \in H^s_c$$,   ‖AL(z)[w^]‖s+1≲s‖z⊥‖s‖w^‖0+‖w^‖s (3.16) and for any $$k \ge 1,$$$$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk(AL(z)[w^])[z^1,…,z^k]‖s+1≲s,k(‖z⊥‖s‖w^‖0+‖w^‖s)∏j=1k‖z^j‖0+‖w^‖0∑j=1k‖z^j‖s∏i≠j‖z^i‖0. (3.17) □ Proof First we prove estimate (3.16). The starting point is formula (3.15) for $${\cal A}_L(z)$$. The two terms $$d A^{\rm nls}(\Psi^{\rm nls}(\Pi_S z))$$ and $${\cal T}(z)^{- 1} {\cal R}(z){\cal S}(z)$$ are estimated separately. By Theorem 1.1, $$\{ \Psi^{\rm nls}(\Pi_S z ) \, | \, z \in {\cal V} \cap h^0_r \}$$ is a relatively compact subset of $$ H^s_r$$ for any $$s \in \mathbb Z_{\geq 0}$$, and $$ A^{\rm nls}$$, $$ B^{\rm nls}$$ are one smoothing maps. It implies that for any $$s \in \mathbb Z_{\geq 1}$$,   ‖dAnls(Ψnls(ΠSz))[w^]‖s+1≲s‖w^‖s,∀z∈V∩hr0,∀w^∈Hcs. (3.18) Since $$\| \Pi_S \widehat z\|_s \lesssim_s \| \Pi_S \widehat z\|_0$$ for any $$z \in h^0_c$$, the linear operator $${\cal R}(z)$$, defined in (3.12), satisfies   ‖R(z)[z^]‖s+1≲s‖z⊥‖s‖ΠSz^‖s≲s‖z⊥‖s‖ΠSz^‖0 ∀z∈V∩hr0,∀z^∈hc0. (3.19) Furthermore, also by Theorem 1.1, one has for any $$s \in \mathbb Z_{\geq 0}$$,   ‖T(z)−1[w^]‖s≲s‖w^‖s,∀z∈V∩hr0,∀w^∈Hcs. (3.20) Combining (3.18–3.20), formula (3.15) leads to the estimate   ‖AL(z)[w^]‖s+1≤‖dAnls(Ψnls(ΠSz))[w^]‖s+1+‖T(z)−1R(z)S(z)[w^]‖s+1≲s‖w^‖s+‖z⊥‖s‖ΠSS(z)[w^]‖0. (3.21) It remains to estimate $$\| {\cal S}(z)[\widehat w]\|_0$$. Recall that by(3.14), $$ {\cal S}(z) = \big( {\rm Id} + {\cal T}(z)^{- 1}{\cal R}(z) \big)^{- 1} {\cal T}(z)^{-1}$$. By Theorem 1.1 there exists $$C_0 > 0$$ so that   ‖T(z)−1R(z)[z^]‖0≤C0‖z⊥‖0‖ΠSz^‖0∀z∈V∩hr0,∀z^∈hc0. (3.22) Shrinking the radius of the ball $${\cal V}_\bot$$ in $$ h^0_{\bot c}$$, if necessary, so that $$C_0 \| z_\bot\|_0 \leq 1/2$$ for any $$z_\bot \in {\cal V}_\bot$$, the Neumann series of the operator $$\big( {\rm Id} + {\cal T}(z)^{- 1}{\cal R}(z) \big)^{- 1}$$ absolutely converges in $${\cal L}(h^0_c, h^0_c)$$ and the operator norm of $$\big( {\rm Id} + {\cal T}(z)^{- 1}{\cal R}(z) \big)^{- 1}$$ in $${\cal L}(h^0_c, h^0_c)$$ is bounded by $$2$$. Hence   ‖S(z)[w^]‖0≲s‖w^‖0,∀z∈V∩hr0,∀w^∈Hc0, (3.23) implying together with (3.21) the claimed estimate (3.16). Finally let us prove the estimate (3.17) for the derivatives of $${\cal A}_L(z)$$. By formula (3.15) for any $$k, s \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V } \cap h^s_r$$, $$\widehat w \in H^s_c$$, and $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk(AL(z)[w^])[z^1,…,z^k]‖s+1 ≤‖dk(dAnls(Ψnls(ΠSz))[w^])[z^1,…,z^k]‖s+1 +‖dk(T(z)−1R(z)S(z)[w^])[z^1,…,z^k]‖s+1. (3.24) By Theorem 1.1, one concludes that   ‖dk(dAnls(ΨL(ΠSz))[w^])[z^1,…,z^k]‖s+1≲s,k‖w^‖s∏j=1k‖z^j‖0. (3.25) Furthermore   ‖dk(T(z)−1[w^])[z^1,…,z^k]‖s≲s,k‖w^‖s∏j=1k‖z^j‖0, (3.26)  ‖dk(R(z)[z^]) [z^1,…,z^k]‖s+1≲s,k‖z⊥‖s‖z^‖0∏j=1k‖z^j‖0+‖z^‖0∑j=1k‖z^j‖s∏i≠j‖z^j‖0, (3.27) an   ‖dk(S(z)[w^])[z^1,…,z^k]‖0≲s‖w^‖0∏j=1k‖z^j‖0. (3.28) Combining the estimates (3.26–3.28) and using the product rule implies that   (3.29) The three estimates (3.24), (3.25), (3.29) together yield (3.17). ■ In the remaining part of this section we describe the pullback $$\Psi_L^* \Lambda$$ by $$\Psi_L$$ of the standard symplectic form $$\Lambda$$ on $$H^0_r$$, introduced in (2.3). It turns out that $$\Psi_L^* \Lambda$$ is not the symplectic form $$\Lambda_M$$ of (2.7), making it necessary to construct the symplectic corrector $$\Psi_C$$ (see Section 4). Given a bounded linear operator $${\cal P} : h^0_c \to h^0_c$$, its transpose $${\cal P}^t : h^0_c\to h^0_c$$ is defined to be the operator determined by   (P[z^],z^′)r=(z^,Pt[z^′])r,∀z^,z^′∈hc0, (3.30) where the bilinear form $$(\cdot, \cdot)_r$$ on $$h^0_c$$ is defined in (2.4). Similarly, for a bounded linear operator $${\cal Q} : h^0_c \to H^0_c$$, we denote its transpose by $${\cal Q }^t : H^0_c \to h^0_c$$, determined by   ⟨Q[z^],w^⟩r=(z^,Qt[w^])r,∀z^∈hc0,w^∈Hc0, (3.31) where the bilinear form $$\langle \cdot, \cdot \rangle_r$$ on $$H^0_c$$ is the one introduced in (2.2). We now compute the pullback $$\Psi_L^*\Lambda(z)$$ at $$z = (z_S, z_\bot) \in h^0_c$$ applied to $$ \widehat z = ( \widehat z_S, \widehat z_\bot)$$, $$ \widehat z' = ( \widehat z_S', \widehat z_\bot')$$. By the definition of the pullback and the one of $$\Lambda$$ in (2.3) we have   ΨL∗Λ(z)[z^,z^′]=Λ(ΨL(z))[dΨL(z)[z^],dΨL(z)[z^′]]=i⟨JdΨL(z)[z^],dΨL(z)[z^′]⟩r. (3.32) By formula (3.5) for $$d \Psi_L(z)$$,   dΨL(z)[z^]=dΨnls(ΠSz)[z^]+dS(d⊥Ψnls(ΠSz)[z⊥])[z^S], one gets   ΨL∗Λ(z)[z^,z^′]=(I)+(II)+(III)+(IV), (3.33) where   (I):=i⟨JdΨnls(ΠSz)[z^],dΨnls(ΠSz)[z^′]⟩r=((Ψnls)∗Λ)(ΠSz)[z^,z^′], (3.34)  (II):=i⟨JdΨnls(ΠSz)[z^],dS(d⊥Ψnls(ΠSz)[z⊥])[z^S′]⟩r. (3.35) Writing $$d \Psi^{\rm nls}(\Pi_S z)[\widehat z]$$ as $$d_S \Psi^{\rm nls}(\Pi_S z)[\widehat z_S] + d_\bot \Psi^{\rm nls}(\Pi_S z)[\widehat z_\bot]$$ one gets   (II) =i⟨JdSΨnls(ΠSz)[z^S],dS(d⊥Ψnls(ΠSz)[z⊥])[z^S′]⟩r +i⟨Jd⊥Ψnls(ΠSz)[z^⊥],dS(d⊥Ψnls(ΠSz)[z⊥])[z^S′]⟩r. (3.36) Similarly one has   (III):=i⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],dΨnls(ΠSz)[z^′])⟩r =i⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],dSΨnls(ΠSz)[z^S′]⟩r +i⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],d⊥Ψnls(ΠSz)[z^⊥′]⟩r (3.37) and finally   (IV):=i⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],dS(d⊥Ψnls(ΠSz)[z⊥])[z^S′]⟩r. (3.38) Since by Theorem 1.1, $$\Psi^{\rm nls}$$ is symplectic, one has $$(\Psi^{\rm nls})^* \Lambda = \Lambda_M$$. Hence for any $$z \in {\cal V}$$, $$\Psi_L^* \Lambda(z)$$ can be written as   ΨL∗Λ(z)=ΛM+ΛL(z),ΛL(z)[z^,z^′]:=(L(z)[z^],z^′)r, (3.39) where $$L(z) : \mathbb C^S \times \mathbb C^S \times h^0_{\bot c} \to \mathbb C^S \times \mathbb C^S \times h^0_{\bot c}$$ is the linear operator of the form   L(z)=(LSS(z)LS⊥(z)L⊥S(z)0). (3.40) By the computations above, $$L_S^S(z) : \mathbb C^S \times \mathbb C^S \to \mathbb C^S \times \mathbb C^S$$, $$L_S^\bot(z) : h^0_{ \bot c} \to \mathbb C^S \times \mathbb C^S$$, and $$L_\bot^S(z) : \mathbb C^S \times \mathbb C^S \to h^0_{ \bot c}$$ are the linear operators defined by ($$z \in {\cal V} \cap h^0_r, \, \widehat z_S \in \mathbb C^S \times \mathbb C^S, \, \widehat z_\bot \in h^0_{\bot c}$$)   LSS(z)[z^S]:=i((〈JdSΨnls(ΠSz)[z^S],∂xjd⊥Ψnls(ΠSz)[z⊥]〉r)j∈S(〈JdSΨnls(ΠSz)[z^S],∂yjd⊥Ψnls(ΠSz)[z⊥]〉r)j∈S)+i((〈JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂xjΨnls(ΠSz)〉r)j∈S(〈JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂yjΨnls(ΠSz)〉r)j∈S)+i((〈JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂xjd⊥Ψnls(ΠSz)[z⊥]〉r)j∈S(〈JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂yjd⊥Ψnls(ΠSz)[z⊥]〉r)j∈S) (3.41) and similarly   LS⊥(z)[z^⊥] :=i((⟨Jd⊥Ψnls(ΠSz)[z^⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r)j∈S(⟨Jd⊥Ψnls(ΠSz)[z^⊥],∂yjd⊥Ψnls(ΠSz)[z⊥]⟩r)j∈S), (3.42)  L⊥S(z)[z^S] :=i((⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂xjΨnls(ΠSz)⟩r)j∈S⊥(⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂yjΨnls(ΠSz)⟩r)j∈S⊥). (3.43) The operator valued map $$z \mapsto L(z)$$ has the following properties: Lemma 3.2. The map $$L : {\cal V} \cap h^0_r \to {\cal L}(h^0_c, h^0_c)$$, $$z \mapsto L(z)$$ is real analytic. For any $$z \in {\cal V} \cap h^0_r$$, $$\widehat z \in h^0_c$$,   ‖L(z)[z^]‖0≲‖z⊥‖0‖z^‖0 and for any $$k \in \mathbb Z_{\geq 1}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^0_c$$,   ‖dk(L(z)[z^])[z^1,…,z^k]‖0≲k‖z^‖0∏j=1k‖z^j‖0. Furthermore, the map $$L$$ is one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, $$L : {\cal V} \cap h^s_r \to {\cal L}(h^0_c, h^{s + 1}_c)$$, $$z \mapsto L(z)$$ is real analytic and satisfies the following estimates: for any $$z \in {\cal V} \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖L(z)[z^]‖s+1≲s‖z⊥‖s‖z^‖0 (3.44) and for any $$k \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V} \cap h^s_r$$, $$ \widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk(L(z)[z^])[z^1,…,z^k]‖s+1≲s,k‖z^‖0∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z^‖0‖z⊥‖s∏j=1k‖z^j‖0. (3.45) In particular, $$L(z) = 0$$ for any $$z \in {\cal V} \cap h^0_r$$ with $$z_\bot = 0$$. Finally, $$L(z) = - L(z)^t$$ or, more explicitly, for any $$ z \in {\cal V} \cap h^0_r,$$  LSS(z)t=−LSS(z),LS⊥(z)t=−L⊥S(z),L⊥S(z)t=−LS⊥(z). (3.46) □ Proof The analyticity of $$L$$ follows by Theorem 1.1, using again that $$d_Sd_\bot\Psi^{\rm nls} = d_S d_\bot B^{\rm nls}$$. Since $${\mathbb J}^t = - {\mathbb J}$$, one reads off from the expressions (3.41–3.43) that (3.46) holds. The estimates (3.44) and (3.45) follow from Theorem 1.1 by differentiating the expressions in the definitions of $$L_S^S(z)$$, $$L_S^\bot(z)$$, and $$L_\bot^S(z)$$ with respect to $$z$$. ■ 4 The Symplectic Corrector $$\Psi_C$$ In this section we construct the coordinate transformation $$\Psi_C$$ on $${\cal V} \cap h^0_r$$ so that the composition $$\Psi_L \circ \Psi_C$$ is symplectic. As mentioned in the introduction, we follow Kuksin’s scheme of proof in [16], which uses arguments of Moser and Weinstein in the given infinite-dimensional setup. The map $$\Psi_C$$ will be defined as the time-one flow of an appropriately chosen non-autonomous vector field.In the sequel, $${\cal V}$$ denotes the neighbourhood of $${\cal K}\times {0}$$, given by Propositions 3.1 and 3.2. For any $$z \in {\cal V}$$ define the following two- and one-forms on $$h^0_c$$,   Λ0:=ΛM,Λ1(z):=ΨL∗Λ(z)=ΛM+ΛL(z), (4.1)  λ0:=λM,λ1(z):=ΨL∗λ(z). (4.2) 4.0 Analysis of the two-form $$\Lambda_1 (z)$$ Note that $$d \lambda_i = \Lambda_i$$, $$i = 0,1$$, and   Λ1−Λ0=ΛL=d(λ1−λ0). (4.3) In particular, the two-form $$\Lambda_L$$ is closed. By (2.7), (3.39) one has   Λ1(z)[z^,z^′]=(L1(z)[z^],z^′)r,L1(z):=J−1+L(z). For any $$\tau \in [0, 1]$$, define the two-form $$\Lambda_\tau = \Lambda_\tau(z)$$,   Λτ:=τΛ1+(1−τ)Λ0, (4.4) which can be written as   Λτ(z)[z^,z^′]=(Lτ(z)[z^],z^′)r,Lτ(z)=J−1+τL(z). (4.5) It turns out that for any $$\tau \in [0, 1]$$ and $$z \in {\cal V} \cap h^0_r$$, the map $${\cal L}_\tau(z)$$ is invertible and one smoothing. More precisely, the following holds: Lemma 4.1. After shrinking the ball $${\cal V}_\bot \subset h^0_{\bot c}$$ in $${\cal V} = {\cal V}_S \times {\cal V}_\bot$$, if necessary, one has that for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V} \cap h^s_r$$, and $$\tau \in [0, 1]$$, the operator $${\cal L}_\tau(z) : h^s_c \to h^s_c$$ is invertible and for any $$k \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V} \cap h^0_r$$, $$\widehat z, \widehat z_1, \ldots, \widehat z_k \in h^0_c$$,   ‖(Lτ(z)−1−J)[z^]‖0≲‖z⊥‖0‖z^‖0,‖dk(Lτ(z)−1[z^])[z^1,…,z^k]‖0≲k‖z^‖0∏j=1k‖z^j‖0. Moreover for any $$s \in \mathbb Z_{\geq 1}$$ and $$\tau \in [0, 1]$$, the map   Lτ−1−J:V∩hrs→L(hcs,hcs+1),z↦Lτ(z)−1−J is real analytic and the following tame estimates hold: for any $$k \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V} \cap h^s_r$$, $$\widehat z, \widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖(Lτ(z)−1−J)[z^]‖s+1≲s‖z⊥‖s‖z^‖0,‖dk(Lτ(z)−1[z^])[z^1,…,z^k]‖s+1≲s,k‖z^‖0∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z^‖0‖z⊥‖s∏j=1k‖z^j‖0. □ Proof For any $$\tau \in [0, 1]$$, we write   Lτ(z)=J−1(Id+Lτ(z)),Lτ(z):=τJL(z). By (3.40) and Theorem 1.1, the operator $$L_\tau (z)$$ satisfies the estimate $$\| L_\tau(z)[\widehat z]\|_0 \leq C_0 \| z_\bot\|_0 \| \widehat z\|_0$$, for any $$z \in {\cal V} \cap h^0_r$$ and $$\widehat z \in h^0_c$$ for some constant $$C_0 > 0$$. By shrinking the ball $${\cal V}_\bot$$, if necessary, one has that for any $$z_\bot \in {\cal V}_\bot$$, $$C_0\| z_\bot\|_0 \leq 1/2$$, implying that the operator $${\cal L}_\tau(z)$$ is invertible and its inverse $${\cal L}_\tau(z)^{- 1}$$ is given by the Neumann series   Lτ(z)−1=J+∑n≥1(−1)nLτ(z)nJ. (4.6) By Lemma 3.2, for any $$s, n \in \mathbb Z_{\geq 1}$$ and $$\tau \in [0, 1]$$, one has   ‖Lτ(z)nJ[z^]‖s+1 ≤C(s)‖z⊥‖s‖Lτ(z)n−1J[z^]‖0≤C(s)(C0‖z⊥‖0)n−1‖z⊥‖s‖z^‖0 (4.7) for some constant $$C(s) > 0$$. Since $$C_0 \| z_\bot\|_0 \leq 1/2$$, one gets   ‖(Lτ(z)−1−J)[z^]‖s+1≲s‖z⊥‖s‖z^‖0. The estimates for the derivatives $$d^k \big({\cal L}_\tau(z)^{- 1}[\widehat z]\big)$$ follow by differentiating the expression (4.6) with respect to $$z$$ and applying the estimates for $$d^k\big( L(z)[\widehat z]\big)$$ of Lemma 3.2. ■ Since by (4.3), the two-form $$\Lambda_L = \Lambda_1 - \Lambda_0$$ is closed and by Lemma 3.2, for any $$z \in {\cal V} \cap h^0_r$$, $$\Lambda_L (\Pi_S z) = 0$$, we can apply Lemma A.1 in Appendix 1. It says that the one-form   λL(z)[z^]:=∫01ΛL(zS,tz⊥)[(0,z⊥),(z^S,tz^⊥)]dt (4.8) satisfies $$d \lambda_L = \Lambda_L$$. By (3.39), (3.40), the one-form $$\lambda_L(z)$$ can be written as   λL(z)[z^]=∫01(L(zS,tz⊥)(0,z⊥),(z^S,tz^⊥))rdt=∫01LS⊥(zS,tz⊥)[z⊥]⋅z^Sdt. Moreover, using that by (3.42), $$L_S^\bot(z_S, t z_\bot) = t L_S^\bot(z_S, z_\bot)$$, it turns out that   λL(z)[z^]=(E(z),z^)r,E(z):=(ES(z),0)∈CS×CS×h⊥c0, (4.9) where   ES(z):=12LS⊥(z)[z⊥]=i2((⟨Jd⊥Ψnls(ΠSz)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r)j∈S(⟨Jd⊥Ψnls(ΠSz)[z⊥],∂yjd⊥Ψnls(ΠSz)[z⊥]⟩r)j∈S). (4.10) One of the features of $$\lambda_L(z)$$ is that it is quadratic in $$z_\bot$$. In more detail, we have the following Lemma 4.2. For any $$s \in \mathbb Z_{\geq 0}$$, the map $$E : {\cal V} \cap h^0_r \to h^s_r$$ is real analytic and satisfies the following tame estimates: for any $$z \in {\cal V} \cap h^0_r$$, $$\widehat z \in h^0_c$$,   ‖E(z)‖s≲s‖z⊥‖02,‖dE(z)[z^]‖s≲s‖z⊥‖0‖z^‖0, and any $$k \geq 2$$, $$\widehat z_1, \ldots, \widehat z_k \in h^0_c$$,   ‖dkE(z)[z^1,…,z^k]‖s≲s,k∏j=1k‖z^j‖0. □ Proof The lemma follows by the properties of the map $$\Psi^{\rm nls}$$, stated in Theorem 1.1, and the fact that $$E = \Pi_S E$$, $$\|\Pi_S z \|_s \lesssim_s \| z \|_0$$ for any vector $$z \in h^0_c$$, and $${\cal V}_\bot \subset h^0_{\bot c}$$ is a ball of radius smaller than $$1$$. ■ 4.1 Outline of the construction of $$\Psi_C$$ Following arguments of Moser and Weinstein, our candidate for $$\Psi_C$$ is $$\Psi^{0, 1}_X$$ where $$X \equiv X(z, \tau) \in h^0_r$$ is a non-autonomous vector field with well-defined flow $$\Psi^{\tau_0, \tau}_X$$, $$0 \le \tau_0, \tau \le 1$$, so that $$(\Psi^{0, 1}_X)^* \Lambda_1 = \Lambda_0$$. Here $$z \in {\cal V}$$ and the flow is normalized by $$\Psi^{\tau_0, \tau_0}_X (z) = z$$. To see how to choose $$X(z, \tau)$$, consider the pullback of the two-form $$\Lambda_\tau$$ by $$\Psi^{0, \tau}_X$$, $$(\Psi^{0, \tau}_X)^* \Lambda_\tau$$. Since $$(\Psi^{0, 0}_X)^* = Id$$, one has $$(\Psi^{0, 0}_X)^* \Lambda_0 = \Lambda_0$$. The desired identity $$(\Psi^{0, 1}_X)^* \Lambda_1 = \Lambda_0$$ then follows provided that $$(\Psi^{0, \tau}_X)^* \Lambda_\tau$$ is independent of $$\tau$$, that is, $$\partial_\tau \big( (\Psi^{0, \tau}_X)^* \Lambda_\tau \big) = 0$$. Since $$\partial_\tau \Lambda_\tau = \Lambda_1 - \Lambda_0 = d\lambda_L$$, it turns out that the latter identity holds if $$\lambda_L + \Lambda_\tau [ X(\cdot , \tau), \, \cdot \,] = 0$$. When expressed in terms of the bilinear form $$( \cdot , \cdot )_r$$ and taking into account the representation (4.5) of $$\Lambda_\tau$$ and (4.9) of $$\lambda_L$$, the latter identity reads   (E(z),z^)r+(Lτ(z)[X(z,τ)],z^)r=0. (4.11) We choose the vector field $$X(z, \tau)$$ so that (4.11) is satisfied. 4.2 Vector field $$X(z, \tau)$$ and its flow Motivated by (4.11), the non-autonomous vector field $$X(z, \tau)$$ is defined by   X(z,τ):=−Lτ(z)−1E(z),z∈VS×V⊥,τ∈[0,1]. (4.12) Lemmata 4.1 and 4.2 lead to the following Lemma 4.3. The vector field $$X : ({\cal V} \cap h^0_r) \times [0, 1]\to h^0_r$$ is real analytic and one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$  X:(V∩hrs)×[0,1]→hrs+1 is real analytic. In addition, the following tame estimates hold: for any $$\tau \in [0, 1]$$, $$z \in {\cal V} \cap h^0_r$$, $$\widehat z \in h^0_c$$,   ‖X(z,τ)‖0≲‖z⊥‖02,‖dX(z,τ)[z^]‖0≲‖z⊥‖0‖z^‖0 (4.13) and for any $$k \geq 2$$, $$\widehat z_1, \ldots , \widehat z_k \in h^0_c$$,   ‖dkX(z,τ)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0. Moreover, for any $$s \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V} \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖X(z,τ)‖s+1≲s‖z⊥‖s‖z⊥‖0,‖dX(z,τ)[z^]‖s+1≲s‖z⊥‖0‖z^‖s+‖z⊥‖s‖z^‖0 (4.14) and for any $$k \geq 2$$, $$\widehat z_1, \ldots , \widehat z_k \in h^s_c$$,   ‖dkX(z,τ)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof The lemma follows from Lemmata 4.1 and 4.2. ■ We now want to study the flow of the non-autonomous differential equation   ∂τz=X(z,τ). (4.15) Recall that for any $$ r > 0$$, we denote by $${\cal V}_{\bot }(r)$$ the ball in $$h^0_{\bot c}$$ of radius $$r$$, centred at $$0$$, and for any $$\tau_0, \tau \in [0, 1]$$ by $$\Psi^{\tau_0, \tau}_{X}$$ the flow map of the differential equation (4.15), satisfying $$\Psi^{\tau_0, \tau_0}_{X} (z) = z$$. By a standard contraction argument, there exists an open neighbourhood $${\cal V}_S' \subseteq {\cal V}_S$$ of $${\cal K}$$ in $$\mathbb C^S \times \mathbb C^S$$ and $$\delta > 0$$ with $${\cal V}_{\bot}(2 \delta) \subset {\cal V}_\bot$$ such that for any $$\tau, \tau_0 \in [0, 1]$$  ΨXτ0,τ:Vδ′∩hr0→V2δ∩hr0,Vδ′:=VS′×V⊥(δ),V2δ:=VS×V⊥(2δ) (4.16) is well defined and real analytic. In the next lemma we state the smoothing estimates for $$ \Psi^{\tau_0, \tau}_{X} - { \iota d}$$ where $${ \iota d}$$ denotes the identity map on $${\cal V}_\delta' \cap h^0_r$$. Lemma 4.4. By choosing $$ 0 < \delta < 1 $$ smaller, if necessary, it follows that for any $$\tau, \tau_0 \in [0, 1]$$, the map $$\Psi^{\tau_0, \tau}_{X} - {\iota d} : {\cal V}_\delta' \cap h^0_r \to h^{0}_r$$ is one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, the map   ΨXτ0,τ−ιd:Vδ′∩hrs→hrs+1 is real analytic. Furthermore, the following tame estimates hold: for any $$ z \in {\cal V}_\delta' \cap h^0_r$$, $$\widehat z \in h^0_c$$,   ‖ΨXτ0,τ(z)−z‖0≲‖z⊥‖02,‖(dΨXτ0,τ(z)−Id)[z^]‖0≲‖z⊥‖0‖z^‖0 (4.17) and for any $$k \geq 2$$, $$ \widehat z_1, \ldots , \widehat z_k \in h^0_c$$,   ‖dkΨXτ0,τ(z)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0 whereas for any $$s \in \mathbb Z_{\geq 1}$$, $$ z \in {\cal V}_\delta' \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖ΨXτ0,τ(z)−z‖s+1≲s‖z⊥‖s‖z⊥‖0,‖(dΨXτ0,τ(z)−Id)[z^]‖s+1≲s‖z⊥‖0‖z^‖s+‖z⊥‖s‖z^‖0 (4.18) and for any $$k \geq 2$$, $$ \widehat z_1, \ldots , \widehat z_k \in h^s_c$$,   ‖dkΨXτ0,τ(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof For any $$\tau_0, \tau \in [0, 1]$$ and $$z \in {\cal V}_\delta' \cap h^0_r$$, the flow $$ \Psi^{\tau_0, \tau}_{X}(z)$$ satisfies the integral equation   ΨXτ0,τ(z)=z+∫τ0τX(ΨXτ0,t(z),t)dt. (4.19) In view of the estimate (4.14) of the vector field $$X(z, \tau)$$, we first estimate $$\| \Pi_\bot \Psi^{\tau_0, \tau}_{X}(z)\|_{s}$$ for $$z \in {\cal V}_\delta' \cap h^s_r$$ with $$s \in \mathbb Z_{\geq 0}$$. Applying the operator $$\Pi_\bot$$ to both sides of the identity (4.19), one gets   Π⊥ΨXτ0,τ(z)=Π⊥z+∫τ0τΠ⊥X(ΨXτ0,t(z),t)dt. By Lemma 4.3, for any $$\tau, \tau_0 \in [0, 1]$$, one has   ‖Π⊥ΨXτ0,τ(z)‖s≤‖z⊥‖s+C(s)|∫τ0τ‖Π⊥ΨXτ0,t(z)‖s‖Π⊥ΨXτ0,t(z)‖0dt| (4.20) for some constant $$C(s) > 0$$, only depending on $$s$$. Then by shrinking $$\delta > 0$$, if necessary, so that for $$z_\bot \in {\cal V}_\bot(\delta)$$, we have $$\sup_{\tau_0, \tau \in [0, 1]} \| \Pi_\bot \Psi^{\tau_0, \tau}_{X}(z)\|_0\leq 1$$, the above estimate becomes   ‖Π⊥ΨXτ0,τ(z)‖s≤‖z⊥‖s+C(s)|∫τ0τ‖Π⊥ΨXτ0,t(z)‖sdt|. (4.21) By the Gronwall inequality one then gets   supτ0,τ∈[0,1]‖Π⊥ΨXτ0,τ(z)‖s≲s‖z⊥‖s,∀z∈Vδ′∩hrs. (4.22) Now let us prove (4.18). By (4.19), using again Lemma 4.3, one gets for any $$s \in \mathbb Z_{\geq 1}$$, $$\tau_0, \tau \in [0, 1]$$, and $$z \in {\cal V}_\delta' \cap h^s_r$$  ‖ΨXτ0,τ(z)−z‖s+1 ≤|∫τ0τ‖X(ΨXτ0,t(z),t)‖s+1dt|≲ssupt∈[0,1]‖Π⊥ΨXτ0,t(z)‖ssupt∈[0,1]‖Π⊥ΨXτ0,t(z)‖0 ≲s(4.22)‖z⊥‖s‖z⊥‖0, (4.23) which is the first claimed inequality in (4.18). To prove the one for the differential $${\rm d} \Psi^{\tau_0, \tau}_{X} - {\rm Id}$$, differentiate (4.19) with respect to $$z$$. Using the chain rule one gets   dΨXτ0,τ(z)[z^]=z^+∫τ0τdX(ΨXτ0,t(z),t)[dΨXτ0,t(z)[z^]]dt. (4.24) By applying the estimates of $${\rm d} X(\cdot, \tau)$$ of Lemma 4.3, it follows that for any $$s \in \mathbb Z_{\geq 0}$$ there is a constant $$C(s) > 0$$ such that    ‖dΨXτ0,τ(z)[z^]‖s ≤‖z^‖s+C(s)|∫τ0τ(‖Π⊥ΨXτ0,t(z)‖s‖dΨXτ0,t(z)[z^]‖0+‖Π⊥ΨXτ0,t(z)‖0‖dΨXτ0,t(z)[z^]‖s)dt| ≤(4.22)‖z^‖s+C1(s)|∫τ0τ(‖z⊥‖s‖dΨXτ0,t(z)[z^]‖0+‖z⊥‖0‖dΨXτ0,t(z)[z^]‖s)dt|. (4.25) for some constant $$C_1(s) > C(s) > 0$$. For $$s= 0$$, using that $$\|z_\bot \|_0 \leq \delta < 1$$, (4.25) becomes   ‖dΨXτ0,τ(z)[z^]‖0≤‖z^‖0+2C1(0)|∫τ0τ‖dΨXτ0,t(z)[z^]‖0dt| and hence by the Gronwall inequality   ‖dΨXτ0,τ(z)[z^]‖0≲‖z^‖0. For $$s \in \mathbb Z_{\geq 1}$$, substitute the latter estimate into (4.25) to get, again using that $$\|z_\bot\|_0 < \delta < 1$$  ‖dΨXτ0,τ(z)[z^]‖s ≤‖z^‖s+C2(s)‖z⊥‖s‖z^‖0+C2(s)|∫τ0τ‖dΨXτ0,t(z)[z^]‖sdt| (4.26) for some constant $$C_2(s) > C_1(s)$$. Then using again the Gronwall inequality one concludes that for any $$0 \le \tau_0 \le 1,$$  supτ∈[0,1]‖dΨXτ0,τ(z)[z^]‖s≲s‖z^‖s+‖z⊥‖s‖z^‖0. (4.27) We are now ready to prove the second estimate in (4.18). By (4.24) and the smoothing estimates on $${\rm d} X(\cdot, \tau)$$ of Lemma 4.3, one gets that for any $$s \in \mathbb Z_{\geq 1}$$, $$0 \le \tau_0 \le 1,$$  ‖(dΨXτ0,τ(z)−Id)[z^]‖s+1 ≲ssupt∈[0,1]‖Π⊥ΨXτ0,t(z)‖ssupt∈[0,1]‖dΨXτ0,t(z)[z^]‖0 +supt∈[0,1]‖Π⊥ΨXτ0,t(z)‖0supt∈[0,1]‖dΨXτ0,t(z)[z^]‖s ≲s(4.22),(4.27)‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s, where we used again that $$\| z_\bot\|_0 < \delta < 1$$. Hence the claimed estimate for $${\rm d} \Psi^{\tau_0, \tau}_{X}(z) - {\rm Id}$$ in (4.18) is established. The estimates for the higher-order derivatives $${\rm d}^k \Psi^{\tau_0, \tau}_{X}$$, $$k \geq 2$$, follow by similar arguments, differentiating $$k$$-times the equation (4.19) with respect to $$z$$. ■ 4.3 Definition of $$\Psi_C$$ and its properties Our candidate for the symplectic corrector is the time-one flow map of $$X(z, \tau)$$,   ΨC:=ΨX0,1:Vδ′∩hr0→hr0. (4.28) Clearly, $$\Psi_C$$ is one to one and its inverse is given by the backward flow of the PDE (4.15), namely $$\Psi_C^{- 1} = \Psi_{X}^{1, 0}$$. Hence the maps $$\Psi_C^{\pm 1}$$ satisfy the estimates stated in Lemma 4.4. Furthermore, recall that for any $$\tau \in [0, 1]$$, the two-form $$\Lambda_\tau$$ admits the representation (4.5). Then the following Darboux lemma holds. Proposition 4.1. The map $$\Psi_C$$ is a symplectic corrector, that is, for any $$z \in {\cal V}_\delta' \cap h^0_r$$, $$ \Psi_C^* \Lambda_1(z) = \Lambda_0$$. □ Proof For any $$\tau \in [0, 1]$$, consider the two-form $$( \Psi^{0, \tau}_{X})^* \Lambda_\tau$$. Since $$\Psi_{X}^{0, 0} = {\rm Id}$$, one has $$(\Psi_{X}^{0, 0})^* \Lambda_0 = \Lambda_0$$ and hence it suffices to prove that the map $$\tau \mapsto ( \Psi^{0, \tau}_{X})^* \Lambda_\tau$$ is constant or, equivalently,   ∂τ((ΨX0,τ)∗Λτ)=0,∀τ∈[0,1]. By Cartan’s identity (see, e.g., Lemma 1.2 in [16]) and the fact that $$\Lambda_\tau$$ is closed, it follows that   ∂τ((ΨX0,τ)∗Λτ)=(ΨX0,τ)∗(∂τΛτ+d(Λτ[X(⋅,τ),⋅])). Since $$\partial_\tau \Lambda_\tau \stackrel{(4.4)}{=} \Lambda_1 - \Lambda_0 = \Lambda_L$$ and $$\Lambda_L \stackrel{(4.8)}{=} {\rm d} \lambda_L$$, it remains to prove that   d(λL+Λτ[X(⋅,τ),⋅])=0. By (4.5), (4.9), (4.12), one has for any $$\tau \in [0, 1]$$, $$z \in {\cal V}_\delta' \cap h^0_r$$, and $$\widehat z \in h^0_c$$  λL(z)[z^]+Λτ[X(z,τ),z^]=(E(z),z^)r−(Lτ(z)Lτ(z)−1E(z),z^)r=0. It means that   λL+Λτ[X(⋅,τ),⋅]=0,∀τ∈[0,1], proving the proposition. ■ As a consequence of Lemma 4.4 we get the following. corollary 4.1. $$(i)$$ For any $$s \in \mathbb Z_{\geq 0}$$, the map $$\Psi_C : {\cal V}_\delta' \cap h^s_r \to h^s_r$$ is a real analytic diffeomorphism onto its image and its nonlinear part is one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, the map $$B_C := \Psi_C - { \iota d} : {\cal V}_\delta' \cap h^s_r \to h^{s + 1}_r$$ is real analytic. Furthermore, $$B_C$$ satisfies the following tame estimates: for any $$ z \in {\cal V}_\delta' \cap h^0_r$$, $$\widehat z \in h^0_c$$,   ‖BC(z)‖0≲‖z⊥‖02,‖dBC(z)[z^]‖0≲‖z⊥‖0‖z^‖0 and for any $$k \geq 2$$, $$\widehat z_1, \ldots , \widehat z_k \in h^0_c$$,   ‖dkBC(z)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0, whereas for any $$s \in \mathbb Z_{\geq 1}$$, $$ z \in {\cal V}_\delta' \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖BC(z)‖s+1≲s‖z⊥‖s‖z⊥‖0,‖dBC(z)[z^]‖s+1≲s‖z⊥‖0‖z^‖s+‖z⊥‖s‖z^‖0 and for any $$k \geq 2$$, $$\widehat z, \widehat z_1, \ldots , \widehat z_k \in h^s_c$$,   ‖dkBC(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. (ii) The map $$A_C :=\Psi_C^{- 1} - { \iota d} : \Psi_C({\cal V}_\delta') \cap h^0_r \to h^{0}_r$$ is real analytic and satisfies the following tame estimates: for any $$ z \in \Psi_C({\cal V}_\delta') \cap h^0_r$$, $$\widehat z \in h^0_c$$,   ‖AC(z)‖0≲‖z⊥‖02,‖dAC(z)[z^]‖0≲‖z⊥‖0‖z^‖0 and for any $$k \geq 2$$, $$ \widehat z_1, \ldots , \widehat z_k \in h^0_c$$,   ‖dkAC(z)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0. Furthermore, for any $$s \in \mathbb Z_{\geq 1}$$, $$A_C : \Psi_C({\cal V}_\delta') \cap h^s_r \to h^{s + 1}_r$$ is real analytic and satisfies the following tame estimates: for any $$ z \in \Psi_C({\cal V}_\delta') \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖AC(z)‖s+1≲s‖z⊥‖s‖z⊥‖0,‖dAC(z)[z^]‖s+1≲s‖z⊥‖0‖z^‖s+‖z⊥‖s‖z^‖0 and for any $$k \geq 2$$, $$ \widehat z_1, \ldots , \widehat z_k \in h^s_c$$,   ‖dkAC(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof The claimed results are a special case of Lemma 4.4, since $$\Psi_C = \Psi^{0, 1}_{X}$$ and $$\Psi_C^{- 1} = \Psi_{X}^{1, 0}$$. ■ An immediate consequence of Corollary 4.1 is the following result, needed in Section 5.2. corollary 4.2. The Taylor expansion of the map $$B_C = \Psi_C - {\iota d}$$ around $$\Pi_S z$$ up to order three is of the form   BC(z)=B2C(z)+B3C(z),z∈Vδ′∩hr0, where   B2C(z):=12d2BC(ΠSz)[Π⊥z,Π⊥z] (4.29) and $$B^C_{3}(z)$$ is the Taylor remainder term   B3C(z):=12∫01(1−t)2d3BC(ΠSz+tΠ⊥z)[Π⊥z,Π⊥z,Π⊥z]dt. (4.30) The maps $$B^C_i : {\cal V}_\delta' \cap h^0_r \to h^{0}_r$$, $$i = 2, 3$$, are real analytic and $$B^C_{3}$$ satisfies the following estimates: for any $$ z \in {\cal V}_\delta' \cap h^0_r$$, $$\widehat z , \widehat z_1, \widehat z_2 \in h^0_c$$,   ‖B3C(z)‖0≲‖z⊥‖03,‖dB3C(z)[z^]‖0≲‖z⊥‖02‖z^‖0,‖d2B3C(z)[z^1,z^2]‖0≲‖z⊥‖0‖z^1‖0‖z^2‖0 and for any $$k \geq 3$$, $$ \widehat z_1, \ldots , \widehat z_k \in h^0_c$$,   ‖dkB3C(z)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0. Furthermore, for any $$s \in \mathbb Z_{\geq 1}$$, $$B^C_{i} : {\cal V}_\delta' \cap h^s_r \to h^{s + 1}_r$$, $$i = 2, 3$$, are real analytic and $$B^C_3$$ satisfies the following tame estimates: for any $$ z \in {\cal V}_\delta' \cap h^s_r$$, $$\widehat z , \widehat z_1, \widehat z_2 \in h^s_c$$,   ‖B3C(z)‖s+1≲s‖z⊥‖s‖z⊥‖02,‖dB3C(z)[z^]‖s+1≲s‖z⊥‖02‖z^‖s+‖z⊥‖s‖z⊥‖0‖z^‖0,‖d2B3C(z)[z^1,z^2]‖s+1≲s‖z⊥‖0(‖z^1‖0‖z^2‖s+‖z^1‖s‖z^2‖0)+‖z⊥‖s‖z^1‖0‖z^2‖0 and for any $$k \geq 3$$, $$\widehat z, \widehat z_1, \ldots , \widehat z_k \in h^s_c$$,   ‖dkB3C(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof. By Corollary 4.1, $$B_C(\Pi_S z) = 0$$ and $${\rm d} B_C(\Pi_S z) = 0$$. Thus $$B_C(z) = B^C_2(z) + B^C_3(z)$$ is the Taylor expansion of $$B_C$$ around $$\Pi_S z$$ with Taylor remainder term given by (4.30). The claimed analyticity and tame estimates follow from Corollary 4.1. ■ 5 Proof of Theorem 1.2 In this section we prove Theorem 1.2. First we introduce and discuss our new canonical coordinates and then express the Hamiltonian of the dNLS equation in the new coordinates. New canonical coordinates Our candidate of the canonical transformation is the map   Ψ:=ΨL∘ΨC:Vδ′→Hc0, (5.1) where $${\cal V}_\delta'$$ is the neighbourhood introduced in (4.16). Proposition 5.1. By shrinking $$0 < \delta < 1$$, if necessary, it follows that for any $$s \in \mathbb Z_{\geq 0}$$, $$\Psi : {\cal V}_\delta' \cap h^s_r \to H^s_r$$ is a real analytic symplectic diffeomorphism onto its image with the property that its nonlinear part $$B := \Psi - F_{\rm nls}^{- 1}: {\cal V}_\delta' \cap h^0_r \to H^{0}_r$$ satisfies the following estimates: for any $$k \in \mathbb Z_{\geq 1}$$, $$ z \in {\cal V}_\delta' \cap h^0_r$$, $$ \widehat z_1, \ldots , \widehat z_k \in h^0_c$$,   ‖B(z)‖0≲1,‖dkB(z)[z^1,…,z^k]‖0≲k∏j=1k‖z^j‖0. Furthermore, $$B$$ is one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, the map $$B: {\cal V}_\delta' \cap h^s_r \to H^{s + 1}_r$$ is real analytic, and it satisfies the following tame estimates: for any $$k \in \mathbb Z_{\geq 1}$$, $$ z \in {\cal V}_\delta' \cap h^s_r$$, and $$ \widehat z_1, \ldots , \widehat z_k \in h^s_c$$,   ‖B(z)‖s+1≲s1+‖z⊥‖s,‖dkB(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof. By Proposition 3.1 and Corollary 4.1 one has that for any $$s \in \mathbb Z_{\geq 0}$$, the map $$\Psi = \Psi_L \circ \Psi_C : {\cal V}_\delta' \cap h^s_r \to H^s_r$$ is real analytic and   Ψ∗Λ=(ΨL∘ΨC)∗Λ=ΨC∗ΨL∗Λ=(4.1)ΨC∗Λ1=Proposition4.1Λ0=(4.1)ΛM, (5.2) implying that $$\Psi$$ is symplectic. Recalling that $$\Psi_L = F_{\rm nls}^{- 1} + B_L$$ ((3.7)) and using that, by Corollary 4.1, $$\Psi_C = { \iota d} + B_C$$, a direct calculation shows that for any $$z \in {\cal V}_\delta' \cap h^0_r$$  B(z)=Ψ(z)−Fnls−1(z) =Fnls−1(BC(z))+BL(ΨC(z)). (5.3) The claimed estimates for $$B$$ then follow from the estimates of Proposition 3.1 and the ones of Corollary 4.1. ■ Substituting formula (3.7) for $$B_L$$ one gets   Ψ(z)=Fnls−1(z)+Fnls−1(BC(z))+Bnls(ΠSz+ΠSBC(z))+d⊥Bnls(ΠSz+ΠSBC(z))[z⊥+π⊥BC(z)], (5.4) where according to Corollary 4.2,   BC(z)=12d2BC(ΠSz)[Π⊥z,Π⊥z]+12∫01(1−t)2d3BC(ΠSz+tΠ⊥z)[Π⊥z,Π⊥z,Π⊥z]dt. Next, we state and prove the one smoothing property and tame estimates for the map   A(z):=dΨ(z)−1−Fnls,z∈Vδ′∩hr0. (5.5) By the chain rule,   dΨ(z)−1=dΨC(z)−1(dΨL(ΨC(z)))−1. (5.6) By Corollary 4.1,   dΨC(z)−1=dΨC−1(ΨC(z))=Id+dAC(ΨC(z)), and that by (3.10), $${\rm d} \Psi_L(z)^{- 1} = F_{\rm nls} + {\cal A}_L(z)$$. Hence (5.6) can be written as   dΨ(z)−1=Fnls+A(z),A(z):=AL(ΨC(z))+dAC(ΨC(z))dΨL(ΨC(z))−1. (5.7) Proposition 5.2. (Tame estimates for $${\cal A}$$). For any $$s \in \mathbb Z_{\geq 1}$$, the map $${\cal A} : {\cal V}_\delta' \cap h^s_r \to {\cal L}(H^s_c , h^{s + 1}_c)$$ is real analytic and satisfies the following tame estimates: for any $$z \in {\cal V}_\delta' \cap h^0_r$$, $$\widehat w \in H^0_c$$,   ‖A(z)[w^]‖0≲‖w^‖0 and for any $$k \ge 1$$, $$\widehat z_1, \ldots, \widehat z_k \in h^0_c$$,   ‖dk(A(z)[w^])[z^1,…,z^k]‖0≲k ‖w^‖0∏j=1k‖z^j‖0. Moreover, for any $$s \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V}_\delta' \cap h^s_r$$, $$w \in H^s_c$$,   ‖A(z)[w^]‖s+1≲s‖z⊥‖s‖w^‖0+‖w^‖s and for any $$k \ge 1$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk(A(z)[w^])[z^1,…,z^k]‖s+1≲s,k (‖z⊥‖s‖w^‖0+‖w^‖s)∏j=1k‖z^j‖0+‖w^‖0∑j=1k‖z^j‖s∏i≠j‖z^i‖0. □ Proof. The claimed estimates for $${\cal A}$$ follow by Lemma 3.2 and Corollary 4.1 by the chain and product rules. ■ 5.2 The dNLS Hamiltonian in new coordinates In this subsection we prove the expansion of $${\cal H}^{\rm nls} \circ \Psi$$, stated in $$(C3)$$ of Theorem 1.2. Recall from (1.2) that the Hamiltonian of the dNLS equation is given by   Hnls(w)=∫01(∂xu∂xv+u2v2)dx,w=(u,v)∈Hr1. By Theorem 1.1, $$H^{\rm nls} := {\cal H}^{\rm nls} \circ \Psi^{\rm nls}$$ only depends on the actions. By a slight abuse of notation we write   Hnls=Hnls(I),I=(Ik)k∈Z∈ℓ+1,2,Ik≡Ik(z)=|zk|2/2=(xk2+yk2)/2∀k∈Z (5.8) and denote by $$ \omega_k^{\rm nls} (I)$$ the dNLS frequencies,   ωknls(I):=∂IkHnls(I),k∈Z. (5.9) The properties of the frequency map $$ I \mapsto \omega (I) := (\omega_k(I))_{k \in \mathbb Z}$$ , needed in the sequel, are summarized in the following. Proposition 5.3. (dNLS frequencies) The map   ℓ+1,2→ℓ∞, (Ik)k∈ℤ↦(ωnnls(I)−4π2n2)n∈ℤ (5.10) is real analytic and bounded. □ Proof. See, for example, Theorem 3.2 in [2]. ■ With the notation introduced above, the $$L^2$$-gradient $$\nabla H^{\rm nls}(z)$$ is then given by   ∇Hnls(z)=Ωnls(I)[z],z∈hr1,I≡I(z)=(In(z))n∈Z, where for any $$I \in \ell^{1,2}_+$$, $${ \Omega}^{\rm nls}(I ): h^1_r \to h^{-1}_r$$ is the diagonal operator   Ωnls(I):=(diagk∈Zωknls(I)00diagk∈Zωknls(I)). (5.11) Further note that since $$H^{\rm nls}(z) = {\cal H}^{\rm nls}(\Psi^{\rm nls}(z))$$ one has by the chain rule   Ωnls(I)[z]=∇Hnls(z)=(dΨnls(z))t∇Hnls(Ψnls(z)),∀z∈V∩hr1, (5.12) where $${\cal V}$$ is the neighbourhood of $$h^0_r$$ in $$h^0_c$$ of Theorem 1.1, $${\cal V} = \Phi^{\rm nls}({\cal W})$$. For later use we record that (5.12), evaluated at $$z$$ with $$z = \Pi_S z$$, reads   Ωnls(IS,0)[ΠSz]=(dΨnls(ΠSz))t∇Hnls(Ψnls(ΠSz)) implying that   Π⊥(dΨnls(ΠSz))t∇Hnls(Ψnls(ΠSz))=0. (5.13) The equations of motion, associated with the Hamiltonian $$H^{\rm nls}$$ are given by   ∂tz=JΩnls(I)[z],J=(0−IdId0). (5.14) According to the splitting $$z = (z_S, z_\bot) \in \mathbb C^S \times \mathbb C^S \times h^0_{ \bot c}$$, we can decompose the equation (5.14) as   {∂tzS=JΩSnls(I)[zS]∂tz⊥=JΩ⊥nls(I)[z⊥], (5.15) where   ΩSnls(I):=(diagk∈Sωknls(I)00diagk∈Sωknls(I)),  Ω⊥nls(I):=(diagk∈S⊥ωknls(I)00diagk∈S⊥ωknls(I)). (5.16) Similarly, by a slight abuse of terminology, we identify $$I = (I_k)_{k \in \mathbb Z}$$ with $$(I_S, I_\bot)$$,   I=(IS,I⊥),IS:=(Ik)k∈S,I⊥:=(Ik)k∈S⊥. (5.17) Although the frequencies $$\omega_k(I)$$ are functions of all the action variables $$I_n$$, $$n \in Z$$, the system (5.15) decouples since the action variables are invariant in time and depend only on the initial data. Now let us assume that $$z(t) = (z_S(t), 0)$$ is a solution of (5.15) with initial data $$z (0) = (z_S^{(0)}, 0)$$ and consider the equation obtained from (5.15) by linearizing it along $$(z_S(t), 0)$$ with initial data given by $$\widehat z^{(0)} = ( 0, \widehat z_\bot^{(0)})$$ and $$\widehat z_\bot^{(0)} \in h^1_{\bot r}$$ and denote by $$\widehat z(t)$$ the corresponding solution which evolves in $$h^1_r$$. By a straightforward computation one verifies that the differential of $$\Omega^{\rm nls}(I )$$ at $$(z_S^{(0)}, 0)$$ in direction $$( 0, \widehat z_\bot^{(0)})$$ vanishes, implying that $$\widehat z(t) = (0, \widehat z_\bot(t) )$$ where $$\widehat z_\bot(t)$$ is the solution of   ∂tz^⊥(t)=JΩ⊥(IS,0)[z^⊥(t)],z^⊥(0)=z^⊥(0). (5.18) Since by Theorem 1.1, $$\Psi^{\rm nls}: h^1_r \to H^1_r$$ is symplectic it follows that   w^(t):=dΨnls(zS(t),0)[(0,z^⊥(t))]=d⊥Ψnls(zS(t),0)[z^⊥(t)] (5.19) is a solution of the equation obtained by linearizing the dNLS equation along $$\Psi^{\rm nls}(z_S(t), 0)$$. More precisely,   ∂tw^(t)=iJd∇Hnls(Ψnls(zS(t),0))[w^(t)],w^(0)=dΨnls(zS(0),0)[(0,z^⊥(0))]. (5.20) On the other hand, by differentiating formula (5.19) with respect to $$t$$, one gets   ∂tw^(t) =d⊥Ψnls(zS(t),0)[∂tz^⊥(t)]+dS(d⊥Ψnls(zS(t),0)[z^⊥(t)])[∂tzS(t)] =d⊥Ψnls(zS(t),0)[JΩ⊥nls(IS,0)z^⊥(t)]+dS(d⊥Ψnls(zS(t),0)[z^⊥(t)])[JΩSnls(IS,0)zS(t)]. (5.21) Comparing (5.20) and (5.21) one gets   iJd∇Hnls(Ψnls(zS(t),0)) [d⊥Ψnls(zS(t),0)z^⊥]=d⊥Ψnls(zS(t),0)[JΩ⊥nls(IS,0)z^⊥(t)] +dS(d⊥Ψnls(zS(t),0)[z^⊥(t)])[JΩSnls(IS,0)zS(t)]. (5.22) The latter identity implies that for any $$z_S \in \mathbb R^S \times \mathbb R^S$$, $$\widehat z_\bot \in h^1_{\bot r}$$,   iJd∇Hnls(Ψnls(zS,0)) [dΨnls(zS,0)[(0,z^⊥)]]=dΨnls(zS,0)JΩnls(IS,0)[(0,z^⊥)] +dS(d⊥Ψnls(zS,0)[z^⊥])[JΩSnls(IS,0)zS]. (5.23) Solving for $$J \Omega^{\rm nls}(I_S, 0)[(0, \widehat z_\bot)] $$, one gets   JΩnls(IS,0)[(0,z^⊥)] =(dΨnls(zS,0))−1iJd∇Hnls(Ψnls(zS,0))[dΨnls(zS,0)(0,z^⊥)] −(dΨnls(zS,0))−1dS(d⊥Ψnls(zS,0)[z^⊥])[JΩSnls(IS,0)zS]. (5.24) Since $$ \Psi^{\rm nls}$$ is symplectic, one has   (dΨnls(zS,0))−1iJ=J(dΨnls(zS,0))t,(dΨnls(zS,0))−1=J(dΨnls(zS,0))tiJ and hence (5.24) reads   Ωnls(IS,0)[(0,z^⊥)] =(dΨnls(zS,0))td∇Hnls(Ψnls(zS,0))dΨnls(zS,0)[(0,z^⊥)]−R(1)(zS)[z^⊥], (5.25) where $$ {\cal R}^{(1)}(z_S) : h^{0}_{\bot c} \to h^{0}_{ c}$$ is the bounded linear operator, defined by   R(1)(zS)[z^⊥]:=dΨnls(zS,0)tiJdS(d⊥Ψnls(zS,0)[z^⊥])[JΩSnls(IS,0)zS]. (5.26) For later use we record the following estimates for $${\cal R}^{(1)}(z_S)$$. Lemma 5.1. The map $${\cal V}_S \cap (\mathbb R^S \times \mathbb R^S) \to {\cal L}(h^{0}_{\bot c}, h^{0}_c)$$, $$z_S \mapsto {\cal R}^{(1)} (z_S)$$ is real analytic and bounded. Moreover it is one smoothing, meaning that for any $$s \in \mathbb Z_{\geq 1}$$, $${\cal V}_S \cap (\mathbb R^S \times \mathbb R^S) \to {\cal L}(h^{s}_{\bot c}, h^{s + 1}_c)$$, $$z_S \mapsto {\cal R}^{(1)}(z_S)$$ is real analytic. Furthermore, for any $$s \in \mathbb Z_{\geq 1}$$, $$\alpha, \beta \in \mathbb Z_{\geq 0}^S$$, $$z_S \in {\cal V}_S \cap (\mathbb R^S \times \mathbb R^S)$$,   ‖∂Sα,βR(1)(zS)‖L(h⊥c0,hc0)≲α,β1,‖∂Sα,βR(1)(zS)‖L(h⊥cs,hcs+1)≲s,α,β1. □ Proof By Theorem 1.1, $$\Psi^{\rm nls} = F_{\rm nls}^{- 1} + B^{\rm nls}$$ and hence $${\rm d}_S \big( {\rm d}_\bot \Psi^{\rm nls}(z_S, 0)[\widehat z_\bot] \big)= {\rm d}_S \big( {\rm d}_\bot B^{\rm nls}(z_S, 0)[\widehat z_\bot] \big)$$. The claimed statements then follow from Theorem 1.1. ■ We also need to record some properties of the operator $$\Omega_\bot^{\rm nls}(I)$$ for $$I = (I_S, 0)$$. Write   Ω⊥nls(IS,0)=D⊥2+Ω⊥(0)(IS,0), (5.27) where   D⊥:=(diagn∈S⊥(2πn)00diagn∈S⊥(2πn)), (5.28) and   Ω⊥(0)(IS,0):=(diagn∈S⊥(ωnnls(IS,0)−4π2n2)00diagn∈S⊥(ωnnls(IS,0)−4π2n2)). (5.29) Lemma 5.2. For any $$s \in \mathbb Z_{\geq 0}$$, the map $${\cal V}_S \cap (\mathbb R^S \times \mathbb R^S) \to {\cal L}(h^s_{\bot c}, h^s_{\bot c})$$, $$z_S \mapsto \Omega_\bot^{(0)}(I_S(z_S), 0)$$ is real analytic and bounded. □ Proof The lemma is a straightforward application of Proposition 5.3, since for any $$\alpha, \beta \in \mathbb Z_{\geq 0}^S$$  supn∈S⊥|∂Sα,β(ωnnls(IS,0)−4π2n2)|≲α,β1 and   ‖∂Sα,βΩ⊥(0)(IS,0)‖L(h⊥cs,h⊥cs)≲supn∈S⊥|∂Sα,β(ωnnls(IS,0)−4π2n2)|≲α,β1 uniformly on $${\cal V}_S \cap (\mathbb R^S \times \mathbb R^S)$$. ■ After this preliminary discussion, we can now study the transformed Hamiltonian $${\cal H}^{\rm nls} \circ \Psi$$ where $$\Psi = \Psi_L \circ \Psi_C$$ is the symplectic transformation introduced in Section 5.1. We split the analysis into two parts. First we expand $${\cal H}^{(1)} := {\cal H}^{\rm nls} \circ \Psi_L$$ and then we analyse $${\cal H}^{(2)} = {\cal H}^{(1)} \circ \Psi_C$$. 5.2.1 Expansion of $${\cal H}^{\rm nls} \circ \Psi_L$$ To expand $${\cal H}^{\rm nls} \circ \Psi_L$$, it is useful to write $${\cal H}^{\rm nls}$$ in the form   Hnls(w)=H2nls(w)+H4nls(w), (5.30) where   ∂tw=iJ∇Hnls(w),J=(0−IdId0),∇Hnls=(∇uHnls,∇vHnls), (5.31) and the operator $${\cal D}_2$$ is defined as   D2:=(0−∂xx−∂xx0). Note that $${\cal D}_2 = {\cal D}^t_2$$. The Hamiltonian equations associated with (5.30) can be written as   ∂tw=iJ∇Hnls(w),J=(0−IdId0),∇Hnls=(∇uHnls,∇vHnls), (5.32) where   ∇Hnls(w)=D2w+∇H4nls(w),d∇Hnls(w)=D2+d∇H4nls(w). (5.33) The Taylor expansion of $${\cal H}^{\rm nls}$$ around $$\Psi^{\rm nls}(\Pi_S z)$$ up to order three reads   Hnls(Ψnls(ΠSz)+w) =Hnls(Ψnls(ΠSz))+⟨∇Hnls(Ψnls(ΠSz)),w⟩r +12⟨d∇Hnls(Ψnls(ΠSz))[w],w⟩r+T3(1)(zS,w), (5.34) where $${\cal T}_{3}^{(1)}(z_S, w) $$ is the Taylor remainder term of order three, given by   T3(1)(zS,w) :=12∫01(1−t)2d3Hnls(Ψnls(ΠSz)+tw)[w,w,w]dt =(5.30),(5.31)12∫01(1−t)2d3H4nls(Ψnls(ΠSz)+tw)[w,w,w]dt. (5.35) For later use we record that the third derivative of $$ {\cal H}^{\rm nls}_4$$ at $$w_0 = (u_0, v_0) \in H^1_r$$ in direction $$w = (u, v)$$ in $$H^1_r$$ can be computed as   d3H4nls(w0)[w,w,w]=12∫01(u0uv2+u2v0v)dx. (5.36) Substituting for $$w$$ the function $${\rm d}_\bot \Psi^{\rm nls}(\Pi_S z)[z_\bot]$$$$({=}{\rm d} \Psi^{\rm nls}(\Pi_S z)[\Pi_\bot z])$$ and taking into account that by (3.3), $$\Psi_L(z) = \Psi^{\rm nls}(\Pi_S z) + {\rm d}_\bot \Psi^{\rm nls}(\Pi_S z)[z_\bot]$$ yields   H(1)(z)=Hnls(ΨL(z)) =Hnls(Ψnls(ΠSz))+⟨∇Hnls(Ψnls(ΠSz)),dΨnls(ΠSz)[Π⊥z]⟩r +12⟨d∇Hnls(Ψnls(ΠSz))[dΨnls(ΠSz)[Π⊥z]],dΨnls(ΠSz)[Π⊥z]⟩r +T3(1)(zS,dΨnls(ΠSz)[Π⊥z]). Writing the right-hand side of the latter identity in a more convenient form one gets   H(1)(z) =Hnls(Ψnls(ΠSz))+(Π⊥(dΨnls(ΠSz))t∇Hnls(Ψnls(ΠSz)),Π⊥z)r +12(Π⊥(dΨnls(ΠSz))td∇Hnls(Ψnls(ΠSz))dΨnls(ΠSz)[Π⊥z],Π⊥z)r +T3(1)(zS,dΨnls(ΠSz)[Π⊥z]). (5.37) Recall that $$H^{\rm nls} ={\cal H}^{\rm nls} \circ \Psi^{\rm nls}$$. Hence by Theorem 1.1 one gets   Hnls(Ψnls(ΠSz))=Hnls(IS,0). (5.38) Furthermore by (5.13),   Π⊥(dΨnls(ΠSz))t∇Hnls(Ψnls(ΠSz))=0. (5.39) Next, the term in (5.37), which is quadratic in $$z_\bot$$, can be written as   12(Π⊥(dΨnls(ΠSz))td∇Hnls(Ψnls(ΠSz))dΨnls(ΠSz)[Π⊥z],Π⊥z)r =(5.25)12(Ωnls(IS,0)[Π⊥z],Π⊥z)r+12(R(1)(zS)[z⊥],z⊥)r. (5.40) Substituting (5.38–5.40) into (5.37) then yields   H(1)(z)=Hnls(IS,0)+12(Ω⊥nls(IS,0)[z⊥],z⊥)r+P2(1)(z)+P3(1)(z), (5.41) where   P2(1)(z):=12(R(1)(zS)[z⊥],z⊥)r,P3(1)(z):=T3(1)(zS,dΨnls(ΠSz)[Π⊥z]). (5.42) Lemma 5.3. $$(i)$$ For any $$s \in \mathbb Z_{\geq 0}$$, $${\cal P}_2^{(1)} : {\cal V}\cap h^s_r \to \mathbb R$$ is real analytic and the following estimates hold: for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V} \cap h^s_r$$,   ‖∇P2(1)(z)‖s≲s‖z⊥‖s and for any $$k \in \mathbb Z_{\geq 1}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P2(1)(z)[z^1,…,z^k]‖s≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. (ii) For any $$s \in \mathbb Z_{\geq 0}$$, $${\cal P}_3^{(1)} : {\cal V}\cap h^s_r \to \mathbb R$$ is real analytic and the following estimates hold: for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V} \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖∇P3(1)(z)‖s≲s‖z⊥‖s‖z⊥‖0,‖d∇P3(1)(z)[z^]‖s≲s‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s and for any $$k \in \mathbb Z_{\geq 2}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P3(1)(z)[z^1,…,z^k]‖s≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof Item (i) follows from Lemma 5.1 and item (ii) from (5.42), (5.35), (5.36), and Theorem 1.1. ■ 5.2.2 Expansion of $${\cal H}^{(2)} := {\cal H}^{(1)} \circ \Psi_C$$ To study the expansion of the composition $${\cal H}^{(2)}= {\cal H}^{(1)} \circ \Psi_C$$ of the Hamiltonian $${\cal H}^{(1)}$$ with the symplectic corrector $$\Psi_C$$, constructed in Section 4, we separately expand the compositions of the terms on the right-hand side of the identity (5.41) with $$\Psi_C$$. In addition to the projectors $$\Pi_S, \Pi_\bot$$, defined in (3.1), (3.2), we also introduce the following versions of them,   πS:CS×CS×h⊥c0→CS×CS,z=(zS,z⊥)→zS, (5.43)  π⊥:CS×CS×h⊥c0→h⊥c0,z=(zS,z⊥)→z⊥. (5.44) 5.2.2.1 Term $$H^{\rm nls}(I_S, 0)$$. It is convenient to define   hnls(zS):=Hnls(IS,0), (5.45) where we recall that by (5.8), (5.17)   IS=IS(zS)=(12(xj2+yj2))j∈S,zS=((xj)j∈S,(yj)j∈S)∈RS×RS. By Corollaries 4.1, 4.2 $$\Psi_C(z)$$, defined for $$z \in {\cal V}_\delta' \cap h^0_r ,$$ is of the form $$\Psi_C(z) = z + B_C(z) = z + B_2^C(z) + B_3^C(z)$$. Hence the Taylor expansion of $$h^{\rm nls}(\pi_S\Psi_C(z))$$ around $$z_S$$ reads   hnls(πSΨC(z)) =hnls(zS)+∇Shnls(zS)⋅πSB2C(z)+P3(2a)(z), (5.46) where $${\cal P}^{(2a)}_3(z)$$ is the Taylor remainder term of order three, given by   P3(2a)(z) :=∇Shnls(zS)⋅πSB3C(z)+∫01(1−t)dS∇Shnls(zS+tπSBC(z))[πSBC(z)]⋅πSBC(z)dt. (5.47) In the next lemma we provide estimates for the Hamiltonian $${\cal P}^{(2a)}_3(z)$$. Lemma 5.4. For any $$s \in \mathbb Z_{\geq 0}$$, $${\cal P}^{(2a)}_3 \circ \Psi_C : {\cal V}'_\delta \cap h^s_r \to \mathbb R$$ is real analytic. Furthermore, $$\nabla {\cal P}^{(2a)}_3$$ satisfies the following tame estimates: for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V}'_\delta \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖∇P3(2a)(z)‖s≲s‖z⊥‖s‖z⊥‖0,‖d∇P3(2a)(z)[z^]‖s≲s‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s and for any $$k \in \mathbb Z_{\geq 2}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P3(2a)(z)[z^1,…,z^k]‖s≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof. The lemma follows by differentiating $${\cal P}^{(2a)}_3$$ and applying the estimates of Corollaries 4.1 and 4.2. ■ 5.2.2.2 Term $${\cal H}_\Omega(z) := \frac12 \big( \Omega_\bot^{\rm nls}(I_S, 0) z_\bot, z_\bot \big)_r $$ To begin with let us point out that the expansion of the composition of the term $${\cal H}_\Omega(z)$$ with the transformation $$\Psi_C$$ needs special care. To explain this in more detail, write $$2{\cal H}_\Omega(z)$$ in the form   (Ω⊥nls(IS,0)z⊥,z⊥)r= (D⊥2z⊥,z⊥)r+ (Ω⊥(0)(IS,0)z⊥,z⊥)r, where $$D_\bot$$ is the diagonal operator defined in (5.28). When composed with $$\Psi_C = { \iota d} + B_C$$, the term $$ \big( D_\bot^2z_\bot, z_\bot \big)_r$$ becomes    (D⊥2[z⊥+π⊥BC(z)],z⊥+π⊥BC(z))r=(D⊥2[z⊥],z⊥)r+(D⊥2[z⊥],π⊥BC(z))r +(D⊥2[π⊥BC(z)],z⊥)r+(D⊥2[π⊥BC(z)],π⊥BC(z))r, (5.48) where $$\pi_\bot$$ is defined in (5.44). By (4.29) and (4.30)}, it then follows that the difference   12(D⊥2[z⊥+π⊥BC(z)],z⊥+π⊥BC(z))r−12(D⊥2[z⊥],z⊥)r belongs to the error term $${\cal P}_3(z)$$ in Theorem 1.2. Since $$B_C$$ is only one smoothing, the two terms   (D⊥2[z⊥],π⊥BC(z))r,(D⊥2[π⊥BC(z)],z⊥)r could prevent that $${\cal P}_3$$ satisfies the estimates (1.7), stated in Theorem 1.2. To proceed, recall that $$\Psi_C = \Psi^{0, 1}_{X}$$ where $$\Psi_{X}^{\tau_0, \tau}$$ is the flow map, defined in (4.16). We have   HΩ(ΨC(z))=HΩ(z)+P3(2b)(z),P3(2b)(z):=HΩ(ΨC(z))−HΩ(z). (5.49) Using the mean value theorem and recalling (4.16), one has   P3(2b)(z)=∫01PΩ(ΨX0,τ(z),τ)dτ, (5.50) where for any $$\tau \in [0, 1]$$, the Hamiltonian $${\cal P}_{\Omega}(z, \tau)$$ is defined by   PΩ(z,τ):=(∇HΩ(z),X(z,τ))r. (5.51) One has that   (∇HΩ(z),X(z,τ))r =12∇SHΩ(z)⋅πSX(z,τ)+(Ω⊥nls(IS,0)z⊥,π⊥X(z,τ))r. (5.52) By (4.12), the vector field $$X(z, \tau)$$ was chosen to be   X(z,τ)=−Lτ(z)−1E(z), where $$E(z)$$ is given by (4.10) and $${\cal L}_\tau(z)^{- 1}$$ by the Neumann series (4.6) in Lemma 4.1. Hence   X(z,τ)=−Lτ(z)−1E(z)=−JE(z)−∑n≥1(−1)nτn(JL(z))nJE(z)=−JE(z)+τJL(z)∑n≥0(−1)nτn(JL(z))nJE(z)=−JE(z)+τJL(z)X(z,τ). (5.53) Since $$E = \Pi_S E$$ and $$J^t = - J$$, the last term in (5.52) becomes   (Ω⊥nls(IS,0)z⊥,π⊥X(z,τ))r=(Ω⊥nls(IS,0)z⊥,π⊥τJL(z)X(z,τ))r=−τ(JΩ⊥nls(IS,0)z⊥,π⊥L(z)X(z,τ))r. (5.54) By (3.40), the component $$L_\bot^\bot(z)$$ of $$L(z)$$ vanishes. Hence using the projections introduced in (3.1) and (3.2), one has   π⊥L(z)X(z,τ)=L⊥S(z)πSX(z,τ). Substituting the latter expression into (5.54) then leads to   (Ω⊥nls(IS,0)z⊥,π⊥X(z,τ))r =−τ(JΩ⊥nls(IS,0)z⊥,L⊥S(z)πSX(z,τ))r =−τL⊥S(z)tJΩ⊥nls(IS,0)z⊥⋅πSX(z,τ) =(3.46)τLS⊥(z)JΩ⊥nls(IS,0)z⊥⋅πSX(z,τ). (5.55) By the definition (3.42),   LS⊥(z)JΩ⊥nls(IS,0)z⊥=((i⟨Jd⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r)j∈S(i⟨Jd⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥],∂yjd⊥Ψnls(ΠSz)[z⊥]⟩r)j∈S). (5.56) Let us take a closer look at the expression   d⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥]=dΨnls(ΠSz)[JΩnls(IS,0)(0,z⊥)]. Substituting for $$J \Omega^{\rm nls}(I_S, 0)(0, z_\bot)$$ the right-hand side of the identity (5.25), one gets   dΨnls(ΠSz)[JΩnls(IS,0)(0,z⊥)] =dΨnls(ΠSz)J(dΨnls(ΠSz))td∇Hnls(Ψnls(ΠSz)dΨnls(ΠSz)[(0,z^⊥)] −dΨnls(ΠSz)JR(1)(zS)[z^⊥]. The first term on the right-hand side of the latter identity can be simplified. Since $$\Psi^{\rm nls}$$ is symplectic,   dΨnls(ΠSz)J(dΨnls(ΠSz))t=iJ, one has   dΨnls(ΠSz)J(dΨnls(ΠSz))td∇Hnls(Ψnls(ΠSz))dΨnls(ΠSz)[(0,z^⊥)] =iJd∇Hnls(Ψnls(ΠSz))dΨnls(ΠSz)[(0,z⊥)]=iJd∇Hnls(Ψnls(ΠSz))d⊥Ψnls(ΠSz)[z⊥]. (5.57) Combining the above identities, the component $${\rm i} \, \big\langle {\mathbb J} {\rm d}_\bot \Psi^{\rm nls}(\Pi_S z)[ J \Omega^{\rm nls}_\bot(I_S, 0) z_\bot ] , \partial_{x_j} {\rm d}_\bot \Psi^{\rm nls}(\Pi_S z)[ z_\bot] \big\rangle_r$$ on the right-hand side of (5.56) becomes, for $$j \in S$$ arbitrary,   i⟨Jd⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r =⟨d∇Hnls(Ψnls(ΠSz))d⊥Ψnls(ΠSz)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r −i⟨JdΨnls(ΠSz)JR(1)(zS)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r (5.58) which in view of $${\rm d} \nabla {\cal H}^{\rm nls}(w) = {\cal D}_2 + {\rm d} \nabla {\cal H}_4^{\rm nls}(w)$$ (cf. (5.33)) leads to   i⟨Jd⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r =⟨D2d⊥Ψnls(zS,0)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r +⟨d∇H4nls(Ψnls(ΠSz))d⊥Ψnls(ΠSz)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r −i⟨JdΨnls(ΠSz)JR(1)(zS)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r. (5.59) Since $${\cal D}_2 = {\cal D}_2^t$$, the first term on the right-hand side on the latter identity can be written as   ⟨D2d⊥Ψnls(ΠSz)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r=12∂xj⟨D2d⊥Ψnls(ΠSz)[z⊥],d⊥Ψnls(ΠSz)[z⊥]⟩r =12∂xj⟨D2dΨnls(ΠSz)[(0,z⊥)],dΨnls(ΠSz)[(0,z⊥)]⟩r =12∂xj((dΨnls(ΠSz))tD2dΨnls(ΠSz)[(0,z⊥)],(0,z⊥))r, (5.60) which can be further transformed as follows: using $$ {\cal D}_2 \, = \, {\rm d} \nabla {\cal H}^{\rm nls} - {\rm d} \nabla {\cal H}_4^{\rm nls} $$ (cf. (5.33)) and taking into account that by (5.25),   (dΨnls(zS,0))td∇Hnls(Ψnls(zS,0))dΨnls(zS,0)[(0,z⊥)]=Ωnls(IS,0)[(0,z⊥)]+R(1)(zS)[z⊥] one is lead to   12∂xj((dΨnls(ΠSz))tD2dΨnls(ΠSz)[(0,z⊥)],(0,z⊥))r=12∂xj(Ωnls(IS,0)[(0,z⊥)],(0,z⊥))r +12∂xj(R(1)(zS)[z⊥],(0,z⊥))r−12∂xj⟨d∇H4nls(Ψnls(ΠSz))]d⊥Ψnls(zS,0)[z⊥],d⊥Ψnls(ΠSz)[z⊥]⟩r. Let us analyse $$ \partial_{x_j} \big( \Omega^{\rm nls}(I_S, 0) [( 0, z_\bot)] , \, ( 0, z_\bot) \big)_r = \big( \partial_{x_j}\Omega_\bot^{\rm nls}(I_S, 0) z_\bot, \, z_\bot \big)_r $$ in more detail. Substituting for $$\Omega_\bot^{\rm nls}(I_S, 0)$$ the expression $$D^2_\bot + \Omega_\bot^{(0)}(I_S, 0)$$ (cf. (5.27)) and using that $$\big( \partial_{x_j} D^2_\bot z_\bot, \, z_\bot \big)_r =0 $$ for any $$j \in S,$$ one concludes that   (∂xjΩ⊥nls(IS,0)z⊥,z⊥)r=(∂xjΩ⊥(0)(IS,0)z⊥,z⊥)r,∀j∈S. The above identities then imply that (5.60) becomes    ⟨D2d⊥Ψnls(ΠSz)[z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r=12(∂xjΩ⊥(0)(IS,0)z⊥,z⊥)r +12(∂xjR(1)(wS)z⊥,(0,z⊥))r−12∂xj⟨d∇H4nls(Ψnls(ΠSz))d⊥Ψnls(ΠSz)[z⊥],d⊥Ψnls(ΠSz)[z⊥]⟩r. (5.61) With (5.60) and (5.61), the identity (5.59) becomes   i⟨Jd⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥],∂xjd⊥Ψnls(ΠSz)[z⊥]⟩r=(Rxj(zS)[z⊥],z⊥)r, (5.62) where for any $$j \in S$$, $${\cal R}_{x_j}(z_S): h^{0}_{\bot c} \to h^{0}_{\bot c} $$ is the linear operator defined by   12∂xjΩ⊥(0)(IS,0)+12π⊥∂xjR(1)(zS)−12∂xj((d⊥Ψnls(zS,0))td∇H4nls(Ψnls(ΠSz))d⊥Ψnls(ΠSz)) +(∂xjd⊥Ψnls(ΠSz))td∇H4nls(Ψnls(ΠSz))d⊥Ψnls(ΠSz) −i(∂xjd⊥Ψnls(ΠSz))tJdΨnls(ΠSz)JR(1)(zS). (5.63) Arguing similarly as above one obtains   i⟨Jd⊥Ψnls(ΠSz)[JΩ⊥nls(IS,0)z⊥],∂yjd⊥Ψnls(ΠSz)[z⊥]⟩r=(Ryj(zS)[z⊥],z⊥)r, (5.64) where $${\cal R}_{y_j}(z_S): h^{0}_{\bot c} \to h^{0}_{\bot c}$$ is given by   12∂yjΩ⊥(0)(IS,0)+12π⊥∂yjR(1)(zS)−12∂yj((d⊥Ψnls(zS,0))td∇H4nls(Ψnls(ΠSz))d⊥Ψnls(ΠSz))+ (∂yjd⊥Ψnls(ΠSz))td∇H4nls(Ψnls(ΠSz))d⊥Ψnls(ΠSz)−i(∂yjd⊥Ψnls(ΠSz))tJdΨnls(ΠSz)JR(1)(zS). (5.65) In the next lemma we state estimates for the operators $${\cal R}_{x_j}(z_S)$$ and $${\cal R}_{y_j}(z_S)$$. Lemma 5.5. For any $$j \in S$$ and $$s \in \mathbb Z_{\geq 0}$$, the maps   Rxj:VS∩(RS×RS)→L(h⊥cs,h⊥cs),zS↦Rxj(zS),Ryj:VS∩(RS×RS)→L(h⊥cs,h⊥cs),zS↦Ryj(zS) are real analytic and bounded. Furthermore, for any $$\alpha, \beta \in \mathbb Z_{\ge 0}^S$$,   ‖∂Sα,βRxj(zS)‖L(h⊥cs,h⊥cs),‖∂Sα,βRyj(zS)‖L(h⊥cs,h⊥cs)≲s,α,β1. □ Proof The lemma follows from Theorem 1.1 and Lemmata 5.1, 5.2. ■ Finally, by (5.51), (5.52), (5.55), (5.56), (5.62), (5.64) and writing   πSX(z,τ)=((Xj,+(z,τ))j∈S,(Xj,−(z,τ))j∈S)∈RS×RS one sees that the Hamiltonian $${\cal P}_{\Omega} (z, \tau)$$, defined by (5.51), can be written in the form   12∇SHΩ(z)⋅πSX(z,τ)+∑j∈SXj,+(z,τ)(Rxj(zS)[z⊥],z⊥)r+∑j∈SXj,−(z,τ)(Ryj(zS)[z⊥],z⊥)r. (5.66) In the next lemma we state estimates for the Hamiltonian $${\cal P}_3^{(2b)}$$, defined in (5.50). Lemma 5.6. For any $$s \in \mathbb Z_{\geq 0}$$, the Hamiltonian $${\cal P}_3^{(2b)} : {\cal V}'_\delta \cap h^s_r \to \mathbb R$$ is real analytic. Moreover, it satisfies the following tame estimates: for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V}'_\delta \cap h^s_r$$, $$\widehat z, \widehat z_1, \widehat z_2 \in h^s_c$$,   ‖∇P3(2b)(z)‖s≲s‖z⊥‖s‖z⊥‖02,‖d∇P3(2b)(z)[z^]‖s≲s‖z⊥‖s‖z⊥‖0‖z^‖0+‖z⊥‖02‖z^‖s,‖d2∇P3(2b)(z)[z^1,z^2]‖s≲s‖z⊥‖s‖z^1‖0‖z^2‖0+‖z⊥‖0(‖z^1‖s‖z^2‖0+‖z^1‖0‖z^2‖s), and for any $$k \in \mathbb Z_{\geq 3}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P3(2b)(z)[z^1,…,z^k]‖s+1≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof The lemma follows by (5.50), (5.66), and Lemmata 4.3, 4.4, 5.2, 5.5. ■ 5.2.2.3 Term $${\cal P}_2^{(1)}$$. Recall that the Hamiltonian $${\cal P}_2^{(1)}$$ was introduced in (5.42). For $$z \in {\cal V}'_\delta \cap h^0_r$$ one has $$\Psi_C(z) = z + B_C(z)$$ and hence the Taylor expansion of $$ {\cal P}_2^{(1)}(\Psi_C(z))$$ around $$z$$ reads   P2(1)(ΨC(z))=P2(1)(z)+P3(2c)(z),P3(2c)(z):=∫01(∇P2(1)(z+tBC(z)),BC(z))rdt. (5.67) The following lemma holds: Lemma 5.7. For any $$s \in \mathbb Z_{\geq 0}$$, the Hamiltonian $${\cal P}_2^{(1)} \circ \Psi_C : {\cal V}'_\delta \cap h^s_r \to \mathbb R$$ is real analytic. Moreover, the Hamiltonian $${\cal P}_3^{(2c)}$$, defined in (5.67), satisfies the following tame estimates: for any $$s \in \mathbb Z_{\geq 0}$$, $$z \in {\cal V}'_\delta \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖∇P3(2c)(z)‖s≲s‖z⊥‖s‖z⊥‖0,‖d∇P3(2c)(z)[z^]‖s≲s‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s, and for any $$k \in \mathbb Z_{\geq 2}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P3(2c)(z)[z^1,…,z^k]‖s≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof The lemma follows by differentiating $${\cal P}_3^{(2c)}$$ and applying Corollary 4.1 and Lemma 5.3(i). ■ 5.2.2.4 Term $${\cal P}_3^{(1)} $$. By (5.42), $${\cal P}_3^{(1)}$$ is given by $$ {\cal T}_3^{(1)}\big(z_S, \, {\rm d} \Psi^{\rm nls}(\Pi_S z)[\Pi_\bot z] \big)$$ where $${\cal T}_3^{(1)}$$ is the Taylor remainder term of order three, introduced in (5.35). Using the estimates of $${\cal P}_3^{(1)}$$ of Lemma 5.3(ii), the Hamiltonian $${\cal P}_3^{(1)} \circ \Psi_C$$ can be estimated as follows: Lemma 5.8. For any $$s \in \mathbb Z_{\geq 0}$$, $${\cal P}_3^{(1)} \circ \Psi_C : {\cal V}'_\delta \cap h^s_r \to \mathbb R$$ is real analytic. Moreover, the following tame estimates hold: for any $$s \in \mathbb Z_{\geq 1}$$, $$z \in {\cal V}'_\delta \cap h^s_r$$, $$\widehat z \in h^s_c$$,   ‖∇(P3(1)∘ΨC)(z)‖s≲s‖z⊥‖s‖z⊥‖0,‖d∇(P3(1)∘ΨC)(z)[z^]‖s≲s‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s, and for any $$k \in \mathbb Z_{\geq 2}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇(P3(1)∘ΨC)(z)[z^1,…,z^k]‖s≲s,k∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof The lemma follows by differentiating the Hamiltonian $${\cal P}_3^{(1)} \circ \Psi_C$$ and applying Corollary 4.1 and Lemma 5.3(ii). ■ By (5.41), (5.46), (5.49), (5.67) one gets that the Hamiltonian $${\cal H}^{(2)} := {\cal H}^{(1)} \circ \Psi_C = {\cal H}^{\rm nls} \circ \Psi_L \circ \Psi_C$$ has the form   H(2)(z)=Hnls(IS,0)+12(Ω⊥nls(IS,0)[z⊥],z⊥)r+P2(z)+P3(z), (5.68) where for any $$z \in {\cal V}'_\delta \cap h^0_r$$,   P2(z):=∇Shnls(zS)⋅πSB2C(z)+P2(1)(z), (5.69)  P3(z):=P3(2a)(z)+P3(2b)(z)+P3(2c)(z)+P3(1)(ΨC(z)). (5.70) Note that $${\cal P}_2 $$ is quadratic with respect to $$z_\bot$$, whereas $${\cal P}_3$$ is a remainder term of order three in $$z_\bot$$. Being quadratic with respect to $$z_\bot$$, $${\cal P}_2$$ can be written as   P2(z)=12(d⊥(∇⊥P2(ΠSz))[z⊥],z⊥)r. We prove the following Lemma 5.9. The Hamiltonian $${\cal P}_2$$ vanishes on $$ {\cal V}'_\delta \cap h^0_r$$. □ Proof By Corollary 4.1, $$\Psi_C(\Pi_S z) = \Pi_S z$$ and $${\rm d} \Psi_C(\Pi_S z) = {\rm Id}$$. Hence by the chain rule and formula (3.5), the map $$\Psi = \Psi_L \circ \Psi_C$$ satisfies   dΨ(ΠSz)=dΨL(ΠSz)=dΨnls(ΠSz). (5.71) Recall that we denoted by $$\widehat w(t)$$ the solution of equation (5.20), obtained by linearizing the dNLS equation along $$w(t) = \Psi^{\rm nls}(\Pi_S z(t))$$ with initial data $$\widehat w(0) = {\rm d}\Psi^{(nls)}(\Pi_S z (t)) (0, \widehat z_\bot^{(0)})$$ and by $$\widehat z(t) = (0, \widehat z_\bot(t))$$ the one of the equations obtained by linearizing the dNLS equation, expressed in Birkhoff coordinates (cf. (5.14)), along $$(z_S(t), 0) = \Pi_S z(t)$$ with initial data $$(0, \widehat z_\bot^{(0)})$$. Since $$\Psi^{\rm nls}$$ is symplectic, $$\widehat w(t) = {\rm d}\Psi^{\rm nls}(\Pi_S z(t))[\widehat z(t)]$$. We remark that $$(z_S(t), 0) = \Pi_S z(t)$$ is also a solution of the Hamiltonian equation $$\partial_t z^{(2)} = J \nabla {\cal H}^{(2)}(z^{(2)})$$ with $${\cal H}^{(2)}$$ given by (5.68). Denote by $$\widehat z^{(2)}(t) = (0, \widehat z^{(2)}_\bot(t))$$ the solution of the equation obtained by linearizing $$\partial_t z^{(2)} = J \nabla {\cal H}^{(2)}(z^{(2)})$$ along $$\Pi_S z(t)$$ with the same initial data $$(0, \widehat z_\bot^{(0)})$$ as above. Since $$\Psi$$ is symplectic, $$\widehat w(t) = {\rm d}\Psi(\Pi_S z(t))[\widehat z^{(2)}(t)]$$, implying together with $${\rm d} \Psi(\Pi_S z) = {\rm d} \Psi^{\rm nls}(\Pi_S z)$$ (cf. (5.71) above) that $$\widehat z^{(2)}(t) = \widehat z(t)$$ for any $$t$$. By (5.18), $$\widehat z_\bot (t)$$ satisfies   ∂tz^⊥(t)=JΩ⊥nls(IS,0)[z^⊥(t)] (5.72) whereas by (5.68), one has   ∂tz^⊥(2)(t)=Jd⊥∇H(2)(ΠSz(t))[z⊥(2)(t)]=JΩ⊥nls(IS,0)[z^⊥(2)(t)]+Jd⊥∇⊥P2(ΠSz(t))[z^⊥(2)(t)]. (5.73) In particular, it follows that $${\rm d}_\bot \nabla_\bot {\cal P}_2(\Pi_S z(0))[\widehat z^{(0)}_\bot] = 0$$. Since $${\cal P}_2(z)$$ is quadratic in $$z_\bot$$ and the initial data $$z_S(0) \in \pi_S ({\cal V}'_\delta \cap h^0_r)$$, $$\widehat z^{(0)}_\bot \in h^0_{\bot c} $$ are arbitrary, it follows that $$ {\cal P}_2(z) = 0$$ for any $$z \in {\cal V}'_\delta \cap h^0_r$$, which proves the claimed statement. ■ As a consequence of Lemma 5.9, formula (5.68) becomes   H(2)(z)=Hnls(IS,0)+12(Ω⊥nls(IS,0)[z⊥],z⊥)r+P3(z). (5.74) The Hamiltonian $${\cal P}_3$$, introduced in (5.70), satisfies the following tame estimates. Lemma 5.10. (Tame estimates of $${\cal P}_3$$) For any $$s \in \mathbb Z_{\geq 0}$$, the Hamiltonian $${\cal P}_3 : {\cal V}'_\delta \cap h^s_r \to \mathbb R$$ is real analytic and satisfies the following tame estimates: for any $$z \in {\cal V}_\delta' \cap h^s_r$$, $$\widehat z \in h^s_c$$,    ‖∇P3(z)‖s≲s‖z⊥‖s‖z⊥‖0,‖d∇P3(z)[z^]‖s≲s‖z⊥‖s‖z^‖0+‖z⊥‖0‖z^‖s and for any $$k \in \mathbb Z_{\geq 2}$$, $$\widehat z_1, \ldots, \widehat z_k \in h^s_c$$,   ‖dk∇P3(z)[z^1,…,z^k]‖s≲s∑j=1k‖z^j‖s∏i≠j‖z^i‖0+‖z⊥‖s∏j=1k‖z^j‖0. □ Proof. The claimed statements follow from Lemmata 5.4 and 5.6–5.8. ■ 5.3 Summary of the proof of Theorem 1.2 Theorem 1.2 is a direct consequence of Propositions 5.1, 5.2, formula (5.74), and Lemma 5.10. The neighbourhood $${\cal V}$$ in the statement of the theorem is given by $${\cal V}_\delta'$$, introduced in (4.16). 6 Proof of Theorem 1.3 Within this proof, it is convenient to use complex Birkhoff coordinates, given by $$\zeta_n := (x_n - {\rm i} y_n) / \sqrt 2,$$$$n \in \mathbb Z.$$ A solution $$z(t) = (x(t), y(t))$$ of the dNLS equation in Birkhoff coordinates then satisfies the equations   ∂tζn=−iωnnlsζn,n∈Z, (6.1) where   ωnnls≡ωnnls(IS,I⊥)=∂InHnls(IS,I⊥). Linearize (6.1) at a solution $$\zeta(t)$$ of the form $$( \zeta_S(t), 0)$$. For initial data of the form $$\widehat \zeta(0) = (0, \widehat \zeta_\bot(0))$$, the corresponding solution $$\widehat \zeta(t) = (\widehat \zeta_S (t), \widehat \zeta_\bot(t))$$ of the linearized equation satisfies   ζ^S(t)≡0,∂tζ^n(t)=−iωnnls(IS,0)ζ^n(t),n∈S⊥. The latter equation is reduced to constant coefficients and hence   ζ^⊥(t)=(e−iωn(IS,0)tζ^n(0))n∈S⊥. Since $$\Psi^{\rm nls}$$ is symplectic, the solution of the equation, obtained by linearizing the dNLS equation along $$\Psi^{\rm nls}(z_S(t), 0)$$, with initial data $${\rm d}\Psi^{\rm nls}(0, \widehat \zeta_\bot(0))$$, is given by   w^(t)=dΨnls(zS(t),0)[0,ζ^⊥(t)]. We now consider the special solutions $$\widehat \zeta^{\pm , j}(t) = \,{\rm e}^{\pm {\rm i}\omega_j(I_S, 0) t} \,\widehat \zeta^{\pm , j}(0)$$, $$j \in S^\bot$$, corresponding to the initial data   ζ^±,j(0)=(e(1,j)±ie(2,j))/2,e(1,j)=((δnj)n∈Z,0),e(2,j)=(0,(δnj)n∈Z). These solutions are periodic in time and that $${\rm d}\Psi^{\rm nls}(z(t))[\widehat \zeta^{\pm , j}(t)] $$ can be written as   w^±,j(t)=e±iωj(IS,0)tdΨnls(zS(t),0)[ζ^±,j(0)]. In the terminology of [16], $$\widehat w^{+, j}(t), \,\, \widehat w^{-, j}(t)$$, $$j \in S^\bot$$, are Floquet solutions with Floquet exponents $$\pm \omega_j(I_S, 0)$$. By Theorem 1.1 one then concludes that up to normalizations (cf. Appendix 2) and natural identifications (such as the identifications of action angle with Birkhoff coordinates), the map $$\Phi_1$$, obtained by applying the scheme of construction of [16] to the dNLS equation, coincides with the map   RS×RS→L(h⊥rs,Hrs),zS↦dΨnls(zS,0)|h⊥rs. Since according to [16], the map $$\Phi(z)$$ can be chosen of the form $$\Psi^{\rm nls}(z_S, 0) + \Phi_1(z)$$ and since the symplectic corrector $$\Psi_C$$ is constructed following the scheme in [16], one concludes that again up to normalizations and natural identifications, $$\Psi = \Psi_L \circ \Psi_C$$ coincides with the map $$\Phi \circ \phi$$ obtained by applying the scheme of [16] to the dNLS equation. $$\square$$ Remark 6.1. In the terminology of [16], the system of the Floquet exponents $$\pm \omega_j(I_S, 0)$$, $$j \in S^\bot$$, is nonresonant—see for example [2] where the relevant properties of the dNLS frequencies are discussed. □ 7 Restrictions of $$\Psi$$ In this section we present results concerning potentials $$\varphi \in H^0_r$$ which are even, odd, or real valued. To describe them, introduce the operator $$T: H^0(\mathbb T, \mathbb C) \to H^0(\mathbb T, \mathbb C)$$ where for any $$u \in H^0(\mathbb T, \mathbb C),$$$$T(u)$$ is given by   T(u)(x):=u(−x),x∈R a.e. (Here and in the sequel we identify an element in $$H^0(\mathbb T, \mathbb C)$$ with a representative $$f: \mathbb R \to \mathbb C$$ of its lift, obtained by extending $$f: [0, 1) \to \mathbb C$$ periodically in $$x$$ to $$\mathbb R$$, $$f(x + n) = f(x) $$, $$n \in \mathbb Z$$. If this element is in $$ H^s(\mathbb T, \mathbb C)$$ with $$s \in \mathbb Z_{\ge 1}$$, then $$f$$ will be chosen to be the representative of period $$1$$ which is in $${\cal C}^{s -1}(\mathbb R, \mathbb C)$$.) Let $$T_1, T_2, T_3: H^s_c(\mathbb T, \mathbb C) \to H^s_c(\mathbb T, \mathbb C)$$, $$s \in \mathbb Z_{s \ge 0}$$, denote the involutions,   T1(u,v):=(T(u),T(v)),T2(u,v):=−(T(u),T(v)),T3(u,v):=(v,u) and $$H^s_{r, j}$$ the following subspaces of $$H^s_{r}$$,   Hr,1s:={(u,u¯)∈Hrs:T(u)=u},Hr,2s:={(u,u¯)∈Hrs:T(u)=−u}, and   Hr,3s:={(u,u¯)∈Hrs:u real valued}. For any $$1 \le j \le 3$$ and $$s \in \mathbb Z_{s \ge 0}$$, $$T_j(u, v) = (u, v)$$ on $$H^s_{r, j}$$. It is straightforward to verify that $${\cal H}^{\rm nls}$$ is left invariant by $$T_j$$,   Hnls(Tju)=Hnls(u)∀ u∈Hr1,1≤j≤3, (7.1)$$T_1$$, $$T_2$$ are canonical, and hence the subspaces $$H^s_{r, 1},$$$$H^s_{r, 2}$$ are symplectic. In contrast, $$T_3$$ is not canonical and the subspace $$H^s_{r, 3}$$ Lagrangian. To describe how the involutions $$T_j$$ act on Birkhoff coordinates, we define the operator $$\tilde T : h^0_{\mathbb C} \to h^0_{\mathbb C},$$ defined for $$x = (x_k)_{k \in \mathbb Z} \in h^0_{\mathbb C}$$ by $$(\tilde Tx)_k := x_{-k}$$, $$k \in \mathbb Z$$, and introduce the involutions $$\tilde T_j$$ on $$h^s_r$$, $$s \in \mathbb Z_{\ge 0}$$, given by   T~1(x,y):=(T~(x),T~(y)),T~2(x,y):=−(T~(x),T~(y)),T~3(x,y):=(T~(x),−T~(y)) as well as the subspaces $$h^s_{r, j}$$ of $$h^s_{r}$$, defined by   hr,js:={(x,y)∈hrs:T~j(x,y)=(x,y)}. For any $$1 \le j \le 3$$ and $$s \in \mathbb Z_{\ge 0}$$, it follows from Theorem 1.2 in [8] that $$\Phi^{\rm nls} \circ T_j = \tilde T_j \circ \Phi^{\rm nls}$$ on $$H^s_r$$ implying that on $$h^s_r$$,   Ψnls∘T~j=Tj∘Ψnls. (7.2) Since elements in the subspaces $$H^s_{r, j}$$ and $$h^s_{r, j}$$ are kept fixed by the corresponding involutions introduced above, one then concludes from Theorem 1.1 that for any such $$j$$ and $$s$$, $$\Psi^{\rm nls} : h^s_{r, j} \to H^s_{r, j}$$ is a real analytic diffeomorphism. Furthermore, by (7.1) and (7.2), and the fact that $$I(\tilde T_j (z)) = \tilde T (I(z))$$ for any $$1 \le j \le 3$$, it follows that $$H^{\rm nls} (\tilde T (I)) = H^{\rm nls} ( I)$$ for any $$I \equiv I(z) = ((x_k^2 + y_k^2)/2)_{k \in \mathbb Z}$$ with $$z= \big((x_k)_{k \in \mathbb Z}, (y_k)_{k \in \mathbb Z} \big) \in h^1_r$$. Thus for any $$ k \in \mathbb Z,$$  ωknls(T~(I))=(∂IkHnls)(T~(I))=∂I−k(Hnls(T~(I)))=∂I−k(Hnls(I))=ω−k(I). (7.3) In particular, $$h^s_{r, 1}$$ and $$h^s_{r, 2}$$ are left invariant by the dNLS flow (in Birkhoff coordinates). We remark that $$h^s_{r, 3}$$ is left invariant by the flow of the defocusing mKdV equation (cf. e.g., [13]). Furthermore note that for any $$u \in H^0(\mathbb T, \mathbb C)$$, the Fourier coefficients $$(T(u))_{n}$$, $$n \in \mathbb Z$$, of $$T(u)$$ satisfy   (T(u))n=u−n=(T~((uk)k∈Z))n. The following proposition will be applied in subsequent work: Proposition 7.1. In addition to the setup of Theorem 1.2, assume that $$S \subset \mathbb Z$$ is symmetric, $$S = - S$$, and $$1 \le j \le 3$$ and that the complex neighbourhood $${\cal V} \subset h^0_c$$ of Theorem 1.2 is invariant with respect to $$\tilde T_j$$, $$\tilde T_j({\cal V}) = {\cal V}.$$ Then for any $$s \in \mathbb Z_{\ge 0}$$,   Ψ∘T~j=Tj∘Ψon V∩hrs. As a consequence $$\Psi : {\cal V} \cap h^s_{r, j} \to H^s_{r, j}$$ is a real analytic diffeomorphism on to its image. Furthermore, on $$ {\cal V} \cap h^1_r$$, the Hamiltonian $${\cal H} = {\cal H}^{\rm nls} \circ \Psi$$ is invariant under $$\tilde T_j$$, $${\cal H} \circ \tilde T_j = {\cal H}$$, and in the expansion (1.6),   H(z)=Hnls(IS,0)+∑n∈S⊥ωnnls(IS,0)In(z)+P3(z), the three terms on the right-hand side, when restricted to $${\cal V} \cap h^1_r$$, are in view of (7.3) invariant under $$\tilde T_j$$. Since in the case $$1 \le j \le 2$$, $$\tilde T_j$$ is canonical, it then follows that the Hamiltonian vector fields $$X_{{\cal H}}$$ and $$X_{{\cal P}_3}$$ (cf. (2.6)) satisfy on $$ {\cal V} \cap h^1_r$$ and for $$1 \le j \le 2$$  XH(T~j(z))=T~jXH(z),XP3(T~j(z))=T~jXP3(z). (7.4) It implies that for $$s \in \mathbb Z_{\ge 1}$$,   XH:V∩hr,js+2→hr,js,XP3:V∩hr,js→hr,js. (7.5) □ Proof. Since the proofs for $$j=1,$$$$2$$, and $$3$$ are similar, we concentrate on the case $$j = 1$$ only. Recall that $$\Psi = \Psi_L \circ \Psi_C$$ where the maps $$\Psi_L$$ and $$\Psi_C$$ were introduced in Sections 3 and 4, respectively. In the latter section, the neighbourhood $$\cal V$$ in the statement of Theorem 1.2 is actually denoted by $${\cal V}_\delta'$$ (cf. Section 5.3). By (4.16), $${\cal V}_\delta'$$ is contained in the neighbourhood $$\cal V$$, which was introduced in Section 3. In the course of this proof we use the notation established in these two sections. By (1.8), $$\Psi_L(z_S, z_\bot) = \Psi^{\rm nls}(z_S, 0) + {\rm d} \Psi^{\rm nls}(z_S, 0)[0, z_\bot]$$ for any $$z = (z_S, z_\bot)$$ in $${\cal V}$$. Since $$S = - S $$, $$T_1({\cal V}) = {\cal V}$$, (7.2) applies to $$\Psi^{\rm nls}(z_S, 0)$$ and $${\rm d} \Psi^{\rm nls}(z_S, 0)[0, z_\bot]$$ and hence   T1∘ΨL=ΨL∘T~1. (7.6) Next we show that $$\tilde T_1 \circ \Psi_C = \Psi_C \circ \tilde T_1$$. Recall that $$\Psi_C$$, defined on $$ {\cal V}_\delta' \cap h^0_r$$ by (4.28), is given by the time-one flow of the non-autonomous vector field $${ X}( z, \tau) = - {\cal L}_\tau(z)^{- 1} E(z)$$, introduced in (4.12). Here $$0 \le \tau \le 1$$ and for any $$z \in {\cal V}\cap h^0_{r}$$, the operators $${\cal L}_\tau(z) = J^{- 1} + \tau L(z): \mathbb C^S \times \mathbb C^S \times h^0_{\bot c} \to \mathbb C^S \times \mathbb C^S \times h^0_{\bot c}$$ and $$L(z) : \mathbb C^S \times \mathbb C^S \times h^0_{\bot c} \to \mathbb C^S \times \mathbb C^S \times h^0_{\bot c}$$ are defined in (4.5), respectively (3.39), and the element $$E(z) = (E_S(z), 0)\in \mathbb C^S \times \mathbb C^S \times h^0_{\bot c}$$ in (4.9) and (4.10). In a first step we prove that   L(T~1z)∘T~1=T~1∘L(z)∀z∈V∩hr0. (7.7) Recall from (3.40) that $$L(z)$$ is of the form   L(z)=(LSS(z)LS⊥(z)L⊥S(z)0), where the operators $$L_S^S(z),$$$$L_{S}^\bot(z),$$ and $$L_\bot^S(z)$$ are defined in (3.41), (3.42), and (3.43), respectively. It is to show that for any $$z = (z_S, z_\bot) \in {\cal V}\cap h^0_{r}$$ and $$\widehat z = (\widehat z_S, \widehat z_\bot)$$ in $$h^0_{r},$$$$L_S^S(z)[\widehat z_S]$$, $$L_{S}^\bot(z)[\widehat z_\bot],$$ and $$L_\bot^S(z)[\widehat z_S]$$ satisfy the symmetry conditions required for $$L(\tilde T_1 z)[\tilde T_1 \widehat z] = \tilde T_1 (L(z)[\widehat z])$$ to hold. Since the arguments for each of the vectors $$L_S^S(z)[\widehat z_S]$$, $$L_{S}^\bot(z)[\widehat z_\bot],$$ and $$L_\bot^S(z)[\widehat z_S]$$ are similar, we consider only $$L_\bot^S(z)[\widehat z_S]$$. By (3.43), it is given by   L⊥S(z)[z^S]=i((⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂xjΨnls(ΠSz)⟩r)j∈S⊥(⟨JdS(d⊥Ψnls(ΠSz)[z⊥])[z^S],∂yjΨnls(ΠSz)⟩r)j∈S⊥). The two components of $$ L_\bot^S(z)[\widehat z_S] $$ can be analysed in the same way, so it suffices to look at the first one. Write $$z = (x, y)$$. Further introduce $$G^{(1)}(x, y)[\widehat z_S]:= {\mathbb J} {\rm d}_S \big({\rm d}_\bot \Psi^{\rm nls}(\Pi_S z) [z_\bot] \big)[\widehat z_S]$$ and $$G^{(2)}(x, y):= \Psi^{\rm nls}(\Pi_S z)$$, which are both elements in $$H^0_r$$, and define for any $$j \in S^\bot$$  aj:=⟨G(1)(x,y),∂xjG(2)(x,y)⟩r,bj:=⟨G(1)(T~1(x,y))[T~1z^S],(∂xjG(2))(T~1(x,y))⟩r. We show that $$b_j = a_{-j}$$ for any $$j \in S^\bot.$$ Indeed, by [8] one has   G(1)(T~1(x,y))[T~1z^S]=T1(G(1)(x,y)[z^S]),G(2)(T~1(x,y))=T1(G(2)(x,y)). (7.8) Apply $$\partial_{x_{j}}$$ to both sides of the latter identity to get   (∂xjG(2))(T~1(x,y))=∂x−j(G(2)(T~1(x,y)))=∂x−jT1(G(2)(x,y))=T1(∂x−jG(2)(x,y)). When combined with (7.8), one then concludes that for any $$j \in S^\bot,$$  bj=⟨T1(G(1)(x,y)[z^S]),T1(∂x−jG(2)(x,y))⟩r=⟨G(1)(x,y)[z^S],∂x−jG(2)(x,y)⟩r=a−j. We thus have verified (7.7). Since $${\cal L}_\tau(z) = J^{- 1} + \tau L(z)$$, $$0 \le \tau \le 1$$, with $$J^{-1} = \left( {0 \atop -\mathrm{Id}} \quad {\mathrm{Id} \atop 0}\right)$$, $${\cal L}_\tau(z)$$ is invertible (cf. Lemma 4.1), $$ \tilde T_1^{-1} = \tilde T_1$$, and $$\tilde T_1 ({\cal V}\cap h^0_{r}) = {\cal V}\cap h^0_{r}$$ (by assumption), one then also has $${\cal L}_\tau^{-1} (\tilde T_1 z) \circ \tilde T_1 = \tilde T_1 \circ {\cal L}^{-1}_\tau(z)$$ for any $$z$$ in $${\cal V}\cap h^0_{r}$$. Furthermore, $$E(z) = ( \frac12 L_S^\bot(z)[z_\bot] , 0)$$ (cf. (4.10)) satisfies $$E(\tilde T_1 z) = \tilde T_1 E(z)$$. Altogether we conclude that the vectorfield $${ X}( z, \tau) = - {\cal L}_\tau(z)^{- 1} E(z)$$ (cf. (4.12)) has the property that for any $$z \in {\cal V}\cap h^0_{r}$$ and $$0 \le \tau \le 1$$, $$X(\tilde T_1 z, \tau) = \tilde T_1 { X}( z, \tau)$$. Hence by Lemma 4.4, $$\Psi_C,$$ given by the time-one flow of the vector field $${ X}( z, \tau)$$, satisfies $$\tilde T_1 \circ \Psi_C = \Psi_C \circ \tilde T_1$$ on $${\cal V}_\delta' \cap h^0_{r}$$. When combined with the identity (7.6), we therefore have proved that $$ T_1 \circ \Psi = \Psi \circ \tilde T_1$$ on $${\cal V}_\delta' \cap h^0_{r}$$. Clearly, the corresponding identity also holds on $${\cal V}_\delta' \cap h^s_{r}$$ for any $$s \in \mathbb Z_{\ge 1}$$. Concerning (7.4), note that in view of the definition (2.6) of a Hamiltonian vector field,   XH(T~j(z))=J(∇H)(T~j(z))=JT~j∇(H(T~j(z)))=JT~j(∇H(z))=T~jJ(∇H(z))=T~jXH(z). A similar computation shows that $$X_{{\cal P}_3}(\tilde T_j(z)) = \tilde T_j X_{{\cal P}_3}(z)$$. The remaining statements of the proposition are an immediate consequence of the proved identities. ■ Funding This work was supported in part by the Swiss National Science Foundation. Appendix 1: A Version of the Poincaré Lemma We follow the general approach of [18, Chapter V], and restrict to the finite-dimensional setup as the extension to infinite dimension is straightforward by restriction, see [16, Lemma 1.1]. Let $$E = \mathbb R^n$$ and denote by $$L_a^r(E)$$ the space of multilinear continuous alternating forms of degree $$0 \le r \le n$$. Let $$U \subseteq E$$ be an open nonempty set and consider   ω:U→Lar(E). For any $$z \in U$$, denote by   ω(z)[ξ1,…,ξr]∈R the value of $$\omega(z)$$ when evaluated at $$\xi_1, \ldots, \xi_r \in E$$. Similarly, if $$\xi_j = \xi_j(z) \in E$$, $$j = 1, \ldots, r$$, are vector fields on $$U$$, then we denote by $$\omega[\xi_1, \ldots, \xi_r]$$ the function   U→R,z↦ω(z)[ξ1(z),…,ξr(z)]. Furthermore, we denote by $$\omega'(z) \cdot \xi$$, $$\xi \in E$$, the alternating $$r$$-form   ∂ε∣ε=0ω(z+εξ)∈Lar(E). (A.1) The exterior differential $${\rm d} \omega$$ of $$\omega$$, evaluated at $$z \in U$$, $$\xi_1, \ldots, \xi_{r + 1} \in E$$, is then given by the formula   ∑j=1r+1(−1)j+1ω′(z)⋅ξj[ξ1,…,ξj−1,ξj+1,…,ξr+1], (A.2) also referred to as Cartan's formula. Let us now consider the case where   E=Rn1×Rn2,n=n1+n2,n2≥1,U=U1×U2⊆Rn1×Rn2, and $$U_2$$ is a ball in $$\mathbb R^{n_2}$$ centred at $$0$$. We denote the elements of $$U$$ by $$z = (x, y)$$ and the ones of $$E$$ by $$\xi = (v, w) \in \mathbb R^{n_1} \times \mathbb R^{n_2}$$. For any $$r$$-form $$ \omega$$ on $$U$$, denote by $$ \omega_{\cal C}$$ the $$(r - 1)$$-form on $$U$$, obtained by the cone construction: for any $$x \in U_1,$$$$y \in U_2$$, $$v_1, \dots , v_{r-1} \in \mathbb R^{n_1}$$, and $$w_1, \dots , w_{r-1} \in \mathbb R^{n_2}$$,   ωC(x,y)[(v1,w1),…,(vr−1,wr−1)]=∫01ω(x,ty)[(0,y),(v1,tw1),…,(vr−1,twr−1)]dt. (A.3) Since $$U_2$$ is a ball in $$\mathbb R^{n_2}$$, centred at $$0$$, for any $$0 \le t \le 1$$, $$(x, t y)$$ is in $$U_1 \times U_2$$ and hence $$ \omega(x, t y)$$ in (A.3) is well defined. Lemma A.1 (Poincaré lemma). Assume that $$\omega$$ is a $$r$$-form on $$U = U_1 \times U_2$$, with $$1 \le r \leq n$$ and $$n_2 \geq 1$$, satisfying   ω(x,0)[(v1,0),…,(vr,0)]=0,∀x∈U1,∀v1,…,vr∈Rn1. (A.4) Then   d(ωC)+(dω)C=ω. (A.5) In particular, if in addition $$\omega$$ is closed, $${\rm d} \omega = 0$$, then $$ {\rm d} ( \omega_{\cal C} ) = \omega$$. □ Appendix 2: Formulas for $${\rm d}\Psi^{\rm nls}(z_S, 0) [ (0, z_\bot) ]$$ Note that for $$z = (z_S, z_\bot)$$ with $$z_S \in \mathbb R^S \times \mathbb R^S$$ and $$z_\bot = ( (x_j)_{j \in S^\bot}, (y_j)_{j \in S^\bot}) \in h^0_{\bot r}$$,   dΨnls(zS,0)[(0,z⊥)]=∑j∈S⊥xjdΨnls(zS,0)[e(1,j)]+∑j∈S⊥yjdΨnls(zS,0)[e(2,j)], where for any $$j \in S^\bot,$$  e(1,j)=((δnj)n∈Z,0),e(2,j)=(0,(δnj)n∈Z). It turns out that for $$j \in S^\bot$$, $${\rm d}\Psi^{\rm nls}(z_S, 0) [{\rm e}^{(1, j)}]$$ and $${\rm d}\Psi^{\rm nls}(z_S, 0) [{\rm e}^{(2, j)}]$$ can be computed quite explicitly. Consider the Hamiltonian equation with Hamiltonian given by the coordinate function $$x_j$$, $$\partial_t w = {\rm i} {\mathbb J} \partial x_j$$, and denote by $$w(t)$$ its solution with initial data $$w(0) = \Psi^{\rm nls}(z_S, 0)$$. Then $$z(t):= \Phi^{\rm nls}(w(t))$$ solves   ∂tz=dΦnls(w(t))∂tw(t)=dΦnls(w(t))iJ∂xj. (B.1) Since by Theorem 1.1, $$\Phi^{\rm nls}$$ is symplectic, one has $$\partial_t z = J \,{\rm e}^{(1, j)} = \,{\rm e}^{(2,j)}$$. When combined with (B.1) it implies that $${\rm d} \Psi^{\rm nls} (z(t))[{\rm e}^{(2, j)}] = {\rm i} {\mathbb J} \partial x_j$$. Similarly, one derives the corresponding identity for the coordinate function $$y_j$$. When evaluated at $$t=0$$ we then obtain   dΨnls(zS,0)[e(2,j)]=iJ∂xj=(−i∂vxj,i∂uxj),dΨnls(zS,0)[e(1,j)]=iJ∂yj=(−i∂vyj,i∂uyj). By the definition of $$x_j,$$$$y_j$$ in [10, p. 113], one has for a potential $$w \in H^0_r$$ with Birkhoff coordinates $$(z_S, 0)$$ (referred to as $$S$$-gap potential)   xj=ξj8(eiβjzj+e−iβjzj),yj=ξj8i(eiβjzj−e−iβjzj), where $${\frak z}_j^\pm = \gamma_j \,{\rm e}^{\pm {\rm i} \eta_j}$$ if $$\gamma_j \ne 0$$ and $${\frak z}_j^\pm = 0$$ otherwise. We refer to [10] for the definitions of $$\xi_j$$, $$ \eta_j$$, and $$\beta_j$$. Since $$w$$ is assumed to be a $$S-$$gap potential, it follows that for any $$j \in S^\bot,$$  ∂xj=ξj8(eiβj∂zj++e−iβj∂zj−),∂yj=ξj8i(eiβj∂zj+−e−iβj∂zj−), where by formula (17.3) in [10],   ∂zj±=2(∂τj−∂μj)±(i2δ(μj)∂ϕj+2ϕj(i∂δ∣λ=μj+iδ˙(μj)∂μj)). We refer to [10] for the definitions of the various quantities as well as for formulas of the gradients in the latter expression. Each of the two components of these gradients are shown to be a linear combination of quadratic expressions in the entries of the fundamental solution $$M = M(x, \lambda)$$ of the Zakharov Shabat operator   L:=(i00−i)∂x+(0uu¯0),w=(u,v)=(u,u¯). In fact, in [11], it has been proved that   ∂zj±=((Kj2±iHj2)2,(Kj1±iHj1)2) where   Hj=(Hj1,Hj2)=1‖M1+M2‖L2(M1+M2)∣λ=μj denotes the $$L^2$$-normalized eigenfunction of $$L$$ for the Dirichlet eigenvalue $$\mu_j$$, $$M_1,$$$$M_2$$ are the two columns of $$M$$, and $$K_j = (K_{j1}, K_{j2}) $$ is the $$L^2$$-normalized solution of $$LF = \mu_j F,$$ which is $$L^2$$-orthogonal to $$H_j$$ and satisfies the additional normalization condition $$-{\rm i} (K_{j1}(0) - K_{j2}(0)) > 0. $$ Communicated by Prof. Jonatan Lenells References [1] Baldi P. Berti M. and Montalto. R. “KAM for autonomous quasi-linear perturbations of KdV.” Annales de l’Institut H. Poincaré (C) Analyse Non Linéaire . https://doi.org/10.1016/j.anihpc.2015.07.003. [2] Berti M. Kappeler T. and Montalto. R. “Large KAM tori for semilinear perturbations of the defocusing NLS equation.” ( 2016), arXiv:1603.09252. [3] Berti M. and Montalto. R. “Quasi-periodic standing wave solutions of gravity capillary standing water waves.” ( 2016), arXiv:1602.02411. [4] Bourgain J. “Nonlinear Schrödinger equations.” In Hyperbolic Equations and Frequency Interactions , edited by Cafferelli L. and Weinan. E. IAS/Park City Mathematics Series, Providence, American Mathematical Society, 1999. Google Scholar CrossRef Search ADS   [5] Craig W. and Wayne. C. “Periodic solutions of nonlinear Schrödinger equations and Nash Moser method” In Hamiltonian Mechanics, Toruñ, 1993 , edited by Seimenis J. NATO Advanced Science Institutes Series B Physics, 331, 103– 22, New York, Plenum Press, 1994. Google Scholar CrossRef Search ADS   [6] Geng J. and You. J. “A KAM theorem for the one dimensional Schrödinger equation with periodic boundary conditions.” Journal of Differential Equations  209 ( 2005): 1– 56. Google Scholar CrossRef Search ADS   [7] Geng J. and You. J. “A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces.” Communications in Mathematical Physics  262 ( 2006): 343– 72. Google Scholar CrossRef Search ADS   [8] Grébert B. and Kappeler. T. “Symmetries of the nonlinear Schrödinger equation.” Bulletin de la Société Mathématique de France  130, no. 4 ( 2002): 603– 18. Google Scholar CrossRef Search ADS   [9] Grébert B. and Kappeler. T. “Perturbations of the defocusing nonlinear Schrödinger equation.” Milan Journal of Mathematics  71 ( 2003): 141– 74. Google Scholar CrossRef Search ADS   [10] Grébert B. and Kappeler. T. The Defocusing NLS Equation and Its Normal Form . Zürich, European Mathematical Society, 2014. Google Scholar CrossRef Search ADS   [11] Grébert B. Kappeler T. and Pöschel. J. “Normal form theory for the NLS equation: a preliminary report.” Unpublished notes, 2003. [12] Kappeler T. and Liang. Z. “A KAM theorem for the defocusing NLS equation.” Journal of Differential Equations  252, no. 6 ( 2012): 4068– 113. Google Scholar CrossRef Search ADS   [13] Kappeler T. and Molnar. J. On the wellposedness of the defocusing mKdV equation below  $$L^2$$, ( 2016), arXiv: 1606.07052. [14] Kappeler T. Schaad B. and Topalov. P. “Semi-linearity of the nonlinear Fourier transform of the defocusing NLS equation.” International Mathematics Research Notices  ( 2016), https://doi.org/10.1093/imrn/rnv397. [15] Krichever I. “Perturbation theory in periodic problems for two-dimensional integrable systems.” Soviet Scientific Reviews. Section C, Mathematical Physics  9 ( 1991): 1– 103. [16] Kuksin S. Analysis of Hamiltonian PDEs , Oxford, Oxford University Press, 2000. [17] Kuksin S. and Pöschel J. “Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation.” Annals of Mathematics  143 ( 1996): 149– 79. Google Scholar CrossRef Search ADS   [18] Lang S. Differentiable Manifolds . Addison-Wesley, 1972. [19] Molnar J. “New estimates of the nonlinear Fourier transform for the defocusing NLS equation.” International Mathematics Research Notices  2015, no. 17 ( 2015): 8309– 52. Google Scholar CrossRef Search ADS   © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]

Bacher, Roland;de la Harpe, Pierre

doi: 10.1093/imrn/rnw282pmid: N/A

Abstract It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions and permutations. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural “generalized Ramanujan congruences”, as we call them, for which we give numerical evidence in Appendix 3. Pour le parfait flâneur, pour l'observateur passionné, c'est une immense jouissance que d'élire domicile dans le nombre, dans l'ondoyant dans le mouvement, dans le fugitif et l'infini. (Baudelaire, in Le peintre de la vie moderne [6].) 1 Explicit conjugation growth series Let G be a group generated by a set S. For g∈G, the word length ℓG,S(g) is defined to be the smallest non-negative integer n for which there are s1,s2,…,sn∈S∪S−1 such that g=s1s2⋯sn, and the conjugacy length κG,S(g) is the smallest integer n for which there exists h in the conjugacy class of g such that ℓG,S(h)=n. For n∈N, denote by γG,S(n)∈N∪{∞} the number of conjugacy classes in G consisting of elements g with κG,S(g)=n (we agree that 0∈N). Assuming that the pair (G,S) satisfies the condition   γG,S(n) is finite for all n∈N, (Fin) we define the conjugacy growth series  CG,S(q)=∑n=0∞γG,S(n)qn=∑g∈Conj(G)qκG,S(g)∈N[[q]]. Here Σg∈Conj(G) indicates a summation over a set of representatives in G of the set of conjugacy classes of G. The exponential rate of conjugacy growth is HG,Sconj=limsupn→∞logγG,S(n)n. note that exp(−HG,Sconj) is the radius of convergence of the series CG,S(q). In case G is generated by a finite set S, Condition (Fin) is obviously satisfied, so that the formal series CG,S(q) and the number HG,Sconj are well defined; they have recently been given some attention, see for example, [2], [9], [16], [20], [26], [30, Chap. 17], [33], [34]. The subject is related to that of counting closed geodesics in compact Riemannian manifolds [4], [13], [25], [27], [31]. When S is finite, denote for n∈N by σG,S(n)∈N the number of elements g∈G with ℓG,S(g)=n. In this situation, it is tempting to compare the series CG,S to the growth series  LG,S(q)=∑n=0∞σG,S(n)qn=∑g∈GqℓG,S(g)∈N[[q]]. For finite series, for example, for finite groups, we rather write “conjugacy growth polynomial” and “growth polynomial”. The first purpose of the present article is to observe that there are groups G which are not finitely generated, and yet have interesting series CG,S(q) for appropriate infinite generating sets S. Groups of concern here are locally finite infinite symmetric groups, some of their wreath products, and infinite alternating groups. We are also led to compute and compare polynomials CG,S and LG,S for finite symmetric and alternating groups, for various generating sets S. For a non-empty set X, we denote by Sym(X) the finitary symmetric group of X, that is, the group of permutations of X with finite support. The support of a permutation g of X is the subset sup(g)={x∈X|g(x)≠x} of X. Two permutations of X are disjoint if their supports are disjoint (below, this will be used mainly for cycles). It is convenient to agree that, for g,h∈Sym(X),   we denote by gh the result of the permutation h followed by g, such that (gh)(x)=g(h(x)) for all x∈X. For example, for X=N, we have (1,2)(2,3)=(1,2,3), and not (1,3,2) as with the other convention. The conjugacy class   TX={(x,y)∈Sym(X)∣x,y∈X are distinct}⊂Sym(X) of all transpositions in Sym(X) is a generating set of Sym(X). We consider also other generating sets, in particular for X=N  SNCox={(i,i+1)∣i∈N}, which makes Sym(N) look like an infinitely generated irreducible Coxeter group of type A. When X is finite, Sym(X) is the usual symmetric group of X. For n≥1 and X={1,2,…,n}, we write Sym(n). The sets of transpositions   SnCox={(1,2),(2,3),⋯,(n−1,n)}, Tn={(i,j)∣1≤i,j≤n, i<j} are particular cases for Sym(n) of generating sets which are standard for finite Coxeter groups. In the following proposition, we collect a sample of equalities that appear again in Proposition 8 in a more general situation. Proposition 1. Let S⊂Sym(N) be a generating set such that SNCox⊂S⊂TN. For every n≥1, let Sn⊂Sym(n) be a generating set such that SnCox⊂Sn⊂Tn. Then (i) CSym(N),S(q)=∑m=0∞p(m)qm=∏j=1∞11−q j, in particular the sequence of coefficients of CSym(N),S(q) if of intermediate growth, (ii) CSym(n),Sn(q)=∑k=0n−1pn−k(n)qk, (iii) ∑n=0∞CSym(n),Sn(q)tn=∏j=1∞11−qj−1tj, where the partition function p(n) and the second equality of (i) are as recalled in Appendix 2.1, and the number pn−k(n) of partitions of n with n−k positive parts as in Appendix 2.2. Moreover, (iv) when n→∞, the polynomials CSym(n),Sn(q) of (ii) converge coefficientwise towards the series CSym(N),S(q) of (i).    □ For example,   CSym(2),S2(q)=1+q,CSym(3),S3(q)=1+q+q2,CSym(4),S4(q)=1+q+2q2+q3,CSym(5),S5(q)=1+q+2q2+2q3+q4,CSym(6),S6(q)=1+q+2q2+3q3+3q4+q5,CSym(n),Sn(q)=1+q+2q2+⋯+⌊n/2⌋qn−2+qn−1 (n≥5). The main ingredients for the proof of Proposition 1 are the classical Observation 2 and Lemma 3. We use the following standard notation: for an integer n≥0, we denote by λ=(λ1,λ2,⋯,λk)⊢n a partition of weight n=λ1+λ2+⋯+λk, with k≥0 and λ1≥λ2≥⋯≥λk≥1. In Proposition 18 of Section 2, we come back to the convergence of CSym(n),Sn(q) to CSym(N),S(q). Observation 2. Let X be a non-empty set, finite or infinite. Denote by |X| its cardinality. Conjugacy classes in Sym(X) are in natural bijection with appropriate sets of partitions. More precisely, for each pair (L,k) of non-negative integers with L+k≤|X|, there is a bijection between the set of partitions of the form   λ=(λ1,…,λk)⊢L (2.a) on the one hand, and conjugacy classes in Sym(X) of elements of the form   g=c1c2⋯ck∈Sym(X) where ci is a cycle of some length λi+1≥2 for i=1,2,…,k, ci and ci′ are disjoint for i≠i′, and therefore  |sup(g)|−k=L=∑i=1kλi, (2.b) on the other hand. In this article,   the length of a cycle is at least 2, unless otherwise stated; we always make it explicit when we want to consider fixed points as cycles of length 1.    □ Lemma 3. Consider two integers L,k≥0, a set X of cardinal at least L+k (possibly infinite), an element g∈Sym(X) product of k disjoint cycles with |sup(g)|=L+k, and the corresponding partition λ⊢L in k parts, as in Observation 2. (i) There exist transpositions s1,…,sL∈Sym(X) such that g=s1⋯sL and sup(sl)⊂sup(g) for all l∈{1,…,L}. (ii) There exist transpositions t1,…,tM∈Sym(X) such that g=t1⋯tM if and only if M≥L and M−L is even. Suppose, moreover that X is given together with trees T1,…,Tk with the following properties: for i∈{1,…,k}, the vertex set of Ti is a subset of X of cardinality λi+1, and these subsets are disjoint from each other. Let {{x1,x′1},…,{xL,x′L}} be an enumeration of the edges of the forest ∩i=1kTi. (iii) The product h=(x1,x′1)⋯(xL,x′L) is conjugate to g in Sym(X).    □ We postpone until Section 2 the proofs of these, and of further propositions in the present section. Before we can state more general cases of some of the equalities of Proposition 1, we introduce two definitions and provide examples. Definition 4. For a set S of transposition of a set X, the transposition graph Γ(S) has vertex set X and edge set those pairs {x,y}⊂X for which the transposition (x,y) is in S.    □ But for their names, these graphs appear in [35]. It is well-known and easy to check (Lemma A.32) that   the group Sym(X) is generated by Sif and only if the graph Γ(S) is connected. (GC) Definition 5. For a set X, a set S of transpositions of X is partition-complete if it satisfies the following condition:   the transposition graph Γ(S) is connected and,for every partition λ=(λ1,…,λk)⊢L such that L+k≤|X|,Γ(S) contains a forest consisting of k treeshaving respectively λ1+1,…,λk+1 vertices. (PC) The graph Γ(S) itself is partition-complete when S is so.    □ Example 6. When X={1,…,n}, sets of transpositions satisfying Condition (PC) include sets S such that SnCox⊂S⊂Tn, and also those for which Γ(S) is one of the Dynkin graphs D2n+1 with n≥2, or E7 or E8. But if S is such that Γ(S) is one of D2n with n≥2, or E6, then S does not satisfy Condition (PC), because D2n does not contain n disjoint trees with two vertices each, and E6 does not contain two disjoint trees with three vertices each. When X is finite, S is partition-complete as soon as the graph Γ(S) is semi-hamiltonian; recall that a graph is semi-hamiltonian [respectively hamiltonian] if it contains a path [respectively a cycle] containing every vertex exactly once. Condition (PC) for a graph can be seen as a weakening of the property of being semi-hamiltonian. When X is infinite, Condition (PC) is equivalent to (PC∞):   S generates Sym(X) and,forall n≥1,the graph Γ(S) contains a disjoint unionof n trees with at least n vertices each. PC∞ When X=N, here are two families of examples of sets S satisfying Condition (PC∞). The first is that of sets of transpositions of which the transposition graph contains arbitrarily long segments; this family contains sets S such that SNCox⊂S⊂TN. For a set of the second family, choose an increasing sequence (kn)n≥1 of positive integers such that kn+2−kn+1>kn+1−kn for all n≥1; define then S as the set of transpositions (0,kn) and (kn,j) for all n≥1 and j with kn+1≤j≤kn+1−1, so that Γ(S) is obtained from a star with centre 0 and infinitely many neighbours kn by attaching kn+1−kn−1 vertices to each vertex kn; thus Γ(S) is a tree of diameter 4, with all vertices but one (the origin) of finite degrees. On the contrary, the set SN0={(0,n)∣n≥1} does not satisfy Condition (PC∞). Proposition 9 provides the conjugacy growth series for the pair (Sym(N),SN0). We ignore the existence of a simple criterion for graphs or trees to be partition complete.    □ Using Definition 5, we reformulate Lemma 3(iii) and generalize Proposition 1 as follows: Lemma 7. Let X be a non-empty set and S a partition-complete set of transpositions of X. Let g=c1⋯ck∈Sym(X) be a product of disjoint cycles of non-increasing lengths; denote these lengths by λ1+1,…,λk+1, and set L=∑i=1kλi, so that |sup(g)|=L+k. Then   κSym(X),S(g)=L.     □ Proposition 8. Let X be an infinite set and S⊂Sym(X) a partition-complete set of transpositions. (a) The equalities of (i) in Proposition 1 hold true. In particular the series CSym(X),S(q) does not depend on the cardinality of X, as long as X is infinite. For every n≥1, let Sn⊂Sym(n) be a partition-complete set of transpositions. (b) Claims (ii), (iii), and (iv) in Proposition 1 hold true.    □ For the next proposition, we consider the generating sets of transpositions   SN0={(0,n)∈Sym(N)∣n≥1}⊂Sym(N),Sn0={(0,i)∣1≤i≤n−1}⊂Sym(n)=Sym({0,1,…,n−1}), which do not satisfy Condition (PC). Proposition 9. Let SN0⊂Sym(N) and, for every n≥1, let Sn0⊂Sym(n) be as above. Then (i) CSym(N),SN0(q)=1+∑k=1∞q3k−2∏j=1k11−q j=1+q+q2+q3+2q4+2q5+3q6+4q7+5q8+6q9+9q10+10q11+13q12+17q13+21q14+25q15+33q16+39q17+49q18+60q19+73q20+88q21+110q22+130q23+158q24+⋯, (ii) CSym(n),Sn0(q)=1+∑k=1⌊n/2⌋q2k−2∑j=knpk(j)qj. Moreover, when n→∞, the polynomials CSym(n),Sn0 of (ii) converge coefficientwise towards the series CSym(N),SN0(q) of (i).    □ At the day of writing, the sequence   of coefficients of the series CSym(N),SN0(q):=∑n=0∞cnqn of (i) does not appear in [32]. The equality of (ii) is repeated in Proposition 22 below. Numerically, the series of Proposition 9(i) converges in the unit disc, and shows two roots of smallest absolute value, near −0.53±0.68i. This makes it unlikely that the series of Proposition 9 has such a nice product expansion like that of Proposition 1(i). Let X be an infinite set and H a finite group. Let W=H≀XSym(X) be the corresponding permutational wreath product. Let S be a generating set of W containing a set of transpositions SX of X generating Sym(X) and satisfying Condition (PCwr) of Section 3. Denote by M the number of conjugacy classes of H. Proposition 10 (see Proposition 19 below). Let W=H≀XSym(X), S and M be as above. Then   CW,S(q)=(CSym(X),SX(q))N=∏k=1∞1(1−qk)M.     □ The finitary alternating group of N is the subgroup Alt(N) of Sym(N) of permutations of even signature. Consider its generating set   SNA={(i,i+1,i+2)∈Alt(N)∣i∈N}, as well as the subset TNA:=∪g∈Alt(N)gSNAg−1 of all 3-cycles. Proposition 11 is the analogue for the finitary alternating group of N of Proposition 1(i) for the finitary symmetric group. Proposition 11. Let S⊂Alt(N) be a generating set such that SNA⊂S⊂TNA. Then   CAlt(X),S(q)=∑u=0∞p(u)qu∑v=0∞pe(v)qv=12∏j=1∞1(1−qj)2+12∏j=1∞11−q2j=1+q+3q2+5q3+11q4+18q5+34q6 +55q7+95q8+150q9+244q10+⋯, where pe(v) denotes the number of partitions of v∈N involving an even number of positive parts, as in Appendix 2.3.    □ Observation 12. For the series of Proposition 11, set   CAlt(N),S(q)=∑n=0∞pA(n)qn. The coefficients pA(n) satisfy the following congruence relations:   pA(5n+3)≡0 (mod5),pA(10n+7)≡0 (mod5),pA(10n+9)≡0 (mod5),pA(25n+23)≡0 (mod25). Moreover, conjecturally:   pA(49n+17)≡0 (mod7),pA(49n+31)≡0 (mod7),pA(49n+38)≡0 (mod7),pA(49n+45)≡0 (mod7),pA(121n+111)≡0 (mod11). See Proposition 30 for the first four relations. The conjectured relations have been verified numerically for pA(m) when m≤5000, as discussed in Section 6 and Appendix 3.    □ Remark 13. (i) Let G be a group generated by a subset T. Then   κG,T(g)=ℓG,T(g) for all g∈G κT = ℓT if and only if T is closed by conjugation, as it is straightforward to check. (ii) Suppose that G is also generated by a subset S, and assume that T=∩h∈GhSh−1. Then   κG,T(g)≤κG,S(g) for all g∈G, κT ≤ κS but equality need not hold. For example, if G=Sym(4) and S={(1,2),(2,3,4)}, then   κG,T((1,2)(3,4))=2<κG,S((1,2)(3,4))=4. (iii) It is remarkable that we have   κG,T(g)=κG,S(g) for all g∈G, κT = κS in many cases of interest here, including – G=Sym(N) and S as in Proposition 1(i), so that T=TN, – G=Sym(n) and S=Sn as in Proposition 1(ii), so that T=Tn, – G=Alt(N) and S as in Proposition 11, so that T=TNA. In these cases, it follows that   CG,T(q)=CG,S(q). CT = CS However, in the case of G=Sym(N) and S=SN0, and therefore T=TN, the series CG,S(q) of Proposition 9 and CG,T(q) of Proposition 1(i) are different, so that the equalities (κT = κS) and (CT = CS) do not hold.    □ Overview Section 2 contains proofs of Propositions 1, 8, 9 and Lemmas 3, 7. In Section 3, we write and prove formulas for conjugacy growth series of wreath products, see Propositions 10 and 19. Suppose that G is a finite symmetric group Sym(n), and S a system of generators. When S is either SnCox or Tn, the polynomial LG,S(q) is well-known, and is recalled in Proposition 20 below. Indeed, these polynomials make sense and are explicitely known for all finite Coxeter systems; they appear in many places, for example [38] and [8, exercises of Section IV.1], as well as [36]. In Section 4, we compute CSym(n),S(q), and compare these polynomials with those for another generating set, the set Sn0 defined above; this uses lemmas of Section 2, as well as some facts on derangements recalled in Appendix 2. In Section 5, we present results of analogous computations for finitary alternating groups, and in particular the proof of Proposition 11. In the final Section 6, we discuss the context of Observation 12. There is a short Appendix 1 with three lemmas on symmetric and alternating groups, and a longer Appendix 2 that is a reminder of various definitions and identities involving partitions and derangements. Finally, in Appendix 3, we define a generalization of Ramanujan congruences and we record a large number of these for the coefficients p(n)(e1,e2,e3,…) of the power series   ∑n=0∞p(n)(e1,e2,e3,…)qn=∏n=1∞1(1−qn)e1(1−q2n)e2(1−q3n)e3⋯, where (e1,e2,e3,…) is a finite sequence of non-negative integers. Some of these congruences are established in the literature, but most are (as far as we know) conjectural only, based on our numerical evidence. 2 Proof of Lemma 3 and 7, and Propositions 1, 8, and 9 We will moreover state and prove a sharpening of Proposition 1(iv), in Proposition 18. 2.1 Proof of Lemmas 3 and 7 As a preliminary step for the proof, consider a cycle   c=(x1,…,xμ+1)∈Sym(X), where 1≤μ≤|X|−1. By Lemma A.31 applied μ−1 times (see Appendix 1), the cycle c can be written as a product of μ transpositions with supports in sup(c). Let g∈Sym(X) and λ=(λ1,…,λk)⊢L+k be as in Lemma 3. Write g=c1⋯ck, where c1,…,ck are disjoint cycles of lengths λ1+1,…,λk+1 respectively. For i∈{1,…,k}, it follows from the preliminary step that ci can be written as a product of λi transpositions with supports in sup(g). Hence g can be written as a product of L=∑i=1kλi transpositions with supports in sup(g). This proves (i) of Lemma 3. With the extra ingredient of Lemma A.32, this also proves (iii) of Lemma 3 and Lemma 7. Consider now g=t1⋯tM as in (ii) of Lemma 3. For i=1,…,k, write ci=(x1i,x2i,…,xλi+1i). Define a multigraph G=G(t1,…,tM) as follows: its vertex set is VG:=∩ν=1Msup(tν), and there is one edge between the two vertices of sup(tν) for each ν∈{1,…,M}. Observe that VG⊃sup(g)=∩i=1ksup(ci). Erasing from the product t1⋯tM those tν contributing to connected components of G disjoint from sup(g) does not change this product. We can therefore assume that each connected component of G intersects sup(g). For each i∈{1,…,k} and j∈{1,…,λi+1}, the connected component of G containing xji contains sup(ci); it follows that each connected component of G contains at least one of the sup(ci) 's, and therefore that the number of connected components of G, say γG, is at most k. Given any finite multigraph with v vertices, e edges, and γ connected components, e≥v−γ, with equality if and only if the multigraph is a forest. For the multigraph G, we have therefore   M≥|VG|−γG≥|sup(g)|−k=∑i=1kλi. Moreover, M and L have the same parity, which is also the signature of g. Conversely, for every M≥L with M−L even, g can be written as a product of M transpositions, for example the L transpositions of (i) and (M−L)/2 times the product s1s1. This proves (ii) of Lemma 3.    □ 2.2 Proof of Propositions 1 and 8 We prove the equalities of Proposition 1 in the more general case of Proposition 8. (i) Let X be an infinite set and S⊂Sym(X) a partition-complete set of transpositions. The series CSym(X),S(q) is a sum over partitions λ⊢L as in (2.a) of Observation 2, and the contribution of such a partition is qL by Lemma 7. Hence CSym(X),S(q)=∑L=0∞p(L)qL. Equality with ∏j=1∞11−qj is Euler's identity (EP1) recalled in Appendix 2.1. (ii) Consider a positive integer n and a partition-complete set Sn⊂Sym(n). Conjugacy classes in Sym(n) are now in bijection with partitions of n as follows: a partition (μ1,…,μk)⊢n with exactly k positive parts corresponds to a permutation g=c1⋯ck where cj is a cycle of length μj, and “cycles” of length 1, that is, fixed points of g, are now allowed (this is why we use μ here rather than λ as above). By Lemma 7, the Sn-conjugacy length of such a g is κSym(n),Sn(g)=∑j=1k(μj−1)=n−k. Hence the polynomial CSym(n),Sn(q) is a sum over partitions of n (where n is fixed) with exactly k parts (where k ranges from 1 (long cycles) to n (identity)), and each such partition contributes by qn−k. Hence CSym(n),Sn(q)=∑k=1npk(n)qn−k=∑k=0n−1pn−k(n)qk. (iii) Exchanging product and sum, we have   ∏k=1∞11−qk−1tk=∏k=1∞∑ℓk=0∞qℓk(k−1)tℓkk=∑ℓ1,ℓ1,ℓ3,…≥0∏k=1∞qℓk(k−1)tℓkk=∑n=0∞(∑ℓ1,ℓ2,ℓ3,…≥0ℓ1+2ℓ2+⋯+kℓk+⋯=nq∑k=1∞ℓk(k−1))tn. For n≥0, there is a contribution to the coefficient of tn for each sequence (ℓ1,ℓ2,ℓ3,…) of non-negative integers such that ℓ1+2ℓ2+3ℓ3+⋯=n, equivalently for each partition 1ℓ12ℓ23ℓ3⋯ of n, with ℓ1 parts 1, and ℓ2 parts 2, and ℓ3 parts 3, …, equivalently for each conjugacy class in Sym(n). Since 0ℓ1+1ℓ2+2ℓ3+⋯ is the Sn-length of such a conjugacy class, the contributions to the coefficient of tn add up precisely to CSym(n),Sn(q). (iv) The polynomials of (ii) converge coefficientwise towards the series of (i) because pn−k(n)=p(k) when 2k≤n. See ( EP4′) in Appendix 2.2).    □ 2.3 A computation of lengths For the next two lemmas, we agree that Sym(n) denotes the group of permutations of {0,1,…,n−1}, and we consider the generating set Sn0 defined just before Proposition 9. Lemma 14. Let g=c1c2…ck∈Sym(n), where c1,…,ck are disjoint cycles, each of length at least 2; set m=|sup(g)|.   ℓSym(n),Sn0(g)≤{m+kifg(0)=0,m+k−2ifg(0)≠0.     □ Proof. Choose i∈{1,…,k}. Let μi denote the length of ci, and write ci=(x1,x2,…,xμi). If sup(ci) does not contain 0, then   ci=(0,x1)(0,xμi)(0,xμi−1)⋯(0,x2)(0,x1) and ℓSym(n),Sn0(ci)≤μi+1. If sup(ci) contains 0, say x1=0, (this occurs for at most one value of i), then   ci=(0,xμi)(0,xμi−1)(0,xμi−2)⋯(0,x2) and ℓSym(n),Sn0(ci)≤μi−1. Since ℓSym(n),Sn0(g)≤∑i=1kℓSym(n),Sn0(ci), the lemma follows.    ▪ Lemma 15. Let g=c1c2…ck∈Sym(n) and m=|sup(g)| be as in the previous lemma. Then   ℓSym(n),Sn0(g)={m+kifg(0)=0,m+k−2ifg(0)≠0,κSym(n),Sn0(g)= m+k−2 as soon as g≠id.     □ Proof. Set L=ℓSym(n),Sn0(g); there exist r1,…,rL∈Sn0 such that g=r1r2⋯rL. For i∈{1,…,k}, there are distinct elements x1i,…,xμii∈{0,1,…,n−1} such that ci=(x1i,x2i,…,xμii); and μ1+⋯+μk=m. Observe that, for all i∈{1,…,k} and j∈{1,…,μi}, the transposition (0,xji) occurs in the list r1,…,rL, at least once. Suppose first that 0∉sup(g). We know from Lemma 14 that L≤m+k. If one had L<m+k, there would exist i∈{1,…,k} such that (0,x) occurs only one time in the list r1,…,rL for each x∈sup(ci); but this is not possible since 0∉sup(ci). Hence L=m+k. Suppose now that 0∈sup(g); we can assume that x11=0. We know from Lemma 14 that L≤m+k−2. If one had L<m+k−2, at least one of the two following situations would hold: (a) there exists i∈{2,…,k} such that (0,x) occurs only one time in the list r1,…,rL for each x∈sup(ci), (b) there exists j∈{2,3,…,μ1} such that the transposition (0,xj1) does not occur in the list r1,…,rL; but this is not possible. Hence L=m+k−2, and the formula for ℓSym(n),Sn0(g) follows. For all g≠id in Sym(n), there exists a conjugate h of g such that h(0)≠0 to which the same computation applies. The formula for κSym(n),Sn0(g) follows.    ▪ Similarly: Lemma 16. Let g=c1c2⋯ck∈Sym(N), where c1,…,ck are disjoint cycles, each of length at least 2; set m=|sup(g)|. Then ℓSym(N),SN0(g) and κSym(N),SN0(g) are given by the formulas of the previous lemma.    □ 2.4 Proof of Proposition 9 We record a minor variation of Observation 2, as follows. Given m≥2 and k≥1, there is a bijection between (a) the set of partitions of m with k parts, all at least 2, (i.e., partitions of the form μ=(μ1,…,μk)⊢m with μ1≥⋯≥μk≥2), and (b) the set of conjugacy classes of elements g≠1 in Sym(N) or Sym(n), with |sup(g)|=m, which are products of k disjoint cycles, where moreover m≤n in the case of Sym(n) (i.e., of elements of the form g=c1⋯ck with length(ci)=μi). For each μ as in (a), set ν=(ν1,…,νk):=(μ1−1,⋯,μk−1)⊢m−k. which is a partition in k positive parts. The relevant length of the conjugacy class of g as in (b) is m+k−2, by Lemmas 15 and 16. For (i) of Proposition 9, it follows that   CSym(N),SN0(q)=∑m=0∞γSym(N),SN0(m)qm=1+∑m=2∞∑k=1⌊m/2⌋pk(m−k)qm+k−2=1+∑k=1∞q2k−2∑m=2k∞pk(m−k)qm−k=1+∑k=1∞q2k−2∑n=k∞pk(n)qn=1+∑k=1∞q3k−2∏j=1k11−qj where the last equality holds by (EP2) of Appendix 2.2. (ii) Similarly:   CSym(n),Sn0(q)=1+∑m=2n∑k=1⌊m/2⌋pk(m−k)qm+k−2=1+∑k=1⌊n/2⌋q2k−2∑m=2knpk(m−k)qm−k=1+∑k=1⌊n/2⌋q2k−2∑j=knpk(j)qj. (Note: ∑j=knpk(j)qj=∑j=0npk(j)qj.) It is now clear that these polynomials converge coefficientwise to 1+∑m=2∞∑k=1⌊m/2⌋pk(m−k)qm+k−2, that is, to CSym(N),SN0(q).    □ We end this section with a sharpening of Claim (iv) of Proposition 1; this applies more generally to the situation of Proposition 8. Let S⊂Sym(N) be a partition-complete set of transpositions, and let L be a non-negative integer. Set   KL(S)={g∈Sym(N)∣κSym(N),S(g)=L}. Observe that KL(S) is a union of conjugacy classes in Sym(N). For g∈Sym(N), we denote by kg the number of disjoint cycles of which g is the product. Lemma 17. Let S, L, and KL(S) be as above. (i) Let g∈KL(S). Then |sup(g)|=L+kg≤2L for all g∈KL(S). Equality kg=L holds if and only if g is a product of L disjoint transpositions. (ii) Let s∈N be such that 0≤s≤L/2. Then KL(S) contains exactly p(s) conjugacy classes of elements g such that |sup(g)|=2L−s.    □ Proof. (i) Let g∈KL(S) be written as a product c1⋯ckg of disjoint cycles of decreasing sizes. For i∈{1,…,kg}, denote by λi+1 the length of ci; set λ=(λ1,…,λkg), so that λ⊢L by Lemma 7. Since kg≤L, we have |sup(g)|=L+kg≤2L. If |sup(g)|=2L, then λi=1 for i=1,…,kg, and every ci is a transposition. (ii) Let s be such that 0≤s≤L/2. We proceed to establish a bijection between the set of partitions of s on the one hand, and the set of conjugacy classes of elements g∈Sym(N) such that g∈KL(S) and |sup(g)|=2L−s on the other hand; this will end the proof. As Claim (i) covers the case s=0, we could assume that s≥1. Choose a partition μ=(μ1,…,μm)⊢s. Since s≤L/2, we have L−s≥m. Set   λ=(λ1,…,λL−s)=(μ1+1,…,μm+1,1,…1), a partition of L with L−(s+m) parts 1. Let g∈Sym(N) be a product of disjoint cycles of lengths λ1+1,…,λL−s+1. Then   κSym(N),S(g)=∑j=1L−sλj=(∑j=1mμj)+L−s=L, in particular g∈KL(S), and   |sup(g)|=∑j=1L−s(λj+1)=2L−s. Conversely, choose g∈KL(S) with |sup(g)|=2L−s. Let λ1+1,…,λL−s+1 be the lengths, in decreasing order, of the disjoint cycles of which g is the product; note that λ=(λ1,…,λL−s)⊢L. Define a partition μ=(μ1,…,μm) by m=max{j∈{1,…,L−s}|λj≥2}, and μj=λj−1 for j∈{1,…,m}. Then μ⊢L−(L−s)=s.    ▪ Here is the announced sharpening, see Propositions 1 and 8. Proposition 18. Let S be a partition-complete set of transpositions in Sym(N) and, for each m≥1, let Sm be a partition-complete set of transpositions in Sym(m). Write C∞(q) for CSym(N),S(q) and Cm(q) for CSym(m),Sm(q). Then:   limn→∞1qn+1(C∞(q)−C2n+1(q))=∑i=0∞p(≤2i)qi,limn→∞1qn+1(C∞(q)−C2n(q))=∑i=0∞p(≤(2i+1))qi, where p(≤j):=p(0)+p(1)+…+p(j) for all j∈N.    □ Proof. First, for L,m,k∈N, a conjugacy class in KL(S) of elements g such that |sup(g)|=L+k intersects Sym(m) if and only if L+k≤m. Let n≥1. Choose an integer k such that 1≤k≤n+43. Let C be a conjugacy class in Sym(N) such that C⊂Kn+k(S). Suppose that C contributes to the coefficient of qn+k in C∞(q) and not to the coefficient of qn+k in C2n+1(q). Equivalently, suppose that, for every g∈C, we have |sup(g)|≥2n+2; if s≥0 is defined by |sup(g)|=2(n+k)−s, this means that s≤2k−2. Since k≤n+43, i.e. 3k−42≤n2, we have s≤3k−42+k2≤n+k2, so that C is one of the ∑s=02k−2p(s) classes which appear in Lemma 17(ii). It follows that the coefficient of qn+k in C∞(q)−C2n+1(q) is p(≤(2k−2)), so that the coefficient of qk−1 in 1qn+1(C∞(q)−C2n+1(q)) is p(≤(2k−2)) for k with 1≤k≤n+43. Consequently, for given i∈N, the coefficient of qi in 1qn+1(C∞(q)−C2n+1(q)) is p(≤2i) as soon as n is large enough. Similarly, suppose that C contributes to the coefficient of qn+k in C∞(q) and not to the coefficient of qn+k in C2n(q). A similar argument shows that C is one of the ∑s=02k−1p(s) classes which appear in Lemma 17(ii), and finally that, for i∈N, the coefficients of qi in 1qn+1(C∞(q)−C2n(q)) is p(≤(2i+1)) for n large enough.    ▪ 3 Some wreath products Consider a non-empty set X, a group H, and the permutational wreath product H≀XSym(X):=H(X)⋊Sym(X). Here, H(X) denotes the group of functions from X to H having finite support, for the pointwise multiplication, and the semi-direct product “ ⋊” refers to the natural action of Sym(X) on H(X), that is, to f∈Sym(X) acting on ψ∈H(X) by ψ↦f(ψ):=ψ∘f−1. The multiplication in this wreath product is given by (φ,f)(ψ,g)=(φf(ψ),fg), for φ,ψ∈H(X) and f,g∈Sym(X). There is a natural action of the group H≀XSym(X) on the set H×X, for which (φ,f) acts by (h,x)↦(φ(f(x))h,f(x)); this action is faithful. For a∈H∖{1} and u∈X, denote by φua∈H≀XSym(X) the permutation that maps (h,x)∈H×X to (ah,u) if x=u, and to (h,x) otherwise; the support of φua is the set {(h,u)}h∈H. Observe that (φua)a∈H∖{1},u∈X generates the subgroup H(X), and that φua,φvb are conjugate in H≀XSym(X) if and only if a,b are conjugate in H. For u∈X, we denote by Hu the set of elements φua for a∈H∖{1}, and by TH the subset ∪u∈XHu of H(X); recall that TX is the subset of all transpositions in Sym(X). Consider subsets SH⊂TH and SX⊂TX, and define S to be the disjoint union SH⊔SX, inside H≀XSym(X). It is again elementary to check that   if Γ(SX) is connected and if SH={φu1a1,…,φurar}for some generating subset {a1,…,ar}⊂Hand some sequence u1,…,urof points of X.then the group H≀XSym(X) is generated by S. (GCwr) When X is infinite, we consider subsets of H≀XSym(X) of the form S=SH⊔SX that satisfy the following condition:   the transposition graph Γ(SX) is connected and,forall L≥0 and partition λ=(λ1,…,λk)⊢L,Γ(SX) contains a forest of k trees T1,…,Tk,with Ti having λi vertices,including one of them,say x(i),such thatφx(i)a∈SHfor all a∈H∖{1}. (PCwr) (The conditions “for all a∈H∖{1}” could be replaced by “for all a in a set of representatives of the conjugacy classes in H distinct from {1}”.) Proposition 19. Let H be a finite group; denote by M the number of conjugacy classes in H. Consider an infinite set X, the wreath product W=H≀XSym(X), and a generating subset S that satisfies Condition (PCwr). Then   CW,S(q)=∏k=1∞1(1−qk)M.     □ Set ∏k=1∞1(1−qk)M=∑n=0∞p(n)(M)qn. For low values of the integer M, the sequences (p(n)(M))n=0,1,2,… are well documented. For example, with A000041 and other similar numbers referring to those of [32], we have:   See also Section 6 and Appendix 3 for some congruence relations satisfied by the coefficients p(n)(M). Proof of Proposition 19. In this proof, we write G for Sym(X) and W for H≀XSym(X)=H(X)⋊G, Preliminary Remark. There are several ways to associate a conjugacy class in a symmetric group to a partition. For example, when X=N, in Observation 2 above and many other places of this article, the conjugay class associated to a partition such as (3,3,1)⊢7 is that of   (1,2,3,4)(5,6,7,8)(9,10)∈Sym(N). In other places, in particular at some point of the present proof, some fixed points of permutations are counted as parts of size 1, so that the conjugacy class associated to the same partition is that of   (1,2,3)(4,5,6)∈Sym(N). (At this point, it could be more consistent to include some fixed points in cycle decompositions of permutations, and thus to write (1,2,3)(4,5,6)(7)(8)(9)⋯∈Sym(N).) This is the reason for which we use below one symbol, λ, for a partition indexed by 1∈H* and a different symbol, μ, for a partition indexed by η≠1 in H*. First step: reminder on the conjugacy classes of W. The set of conjugacy classes of W is in bijection with the set of H*-decorated partitions, as we now describe, much as in [29]. Here, H* denotes the set of conjugacy classes of H; we write 1∈H* rather than {1}∈H* for the class {1}⊂H. Let w=(φ,f)∈H(X)⋊XSym(X). We proceed to associate a H*-indexed family of partitions   (λ(1),(μ(η))η∈H*∖1) (†) to w. Let X(w) be the finite subset of X that is the union of the supports of φ and f. Denote by c1,…,ck the disjoint cycles of which f is the product. Here, we include a cycle of length 1 for each point x∈X such that x∈sup(φ) and x∉sup(f), so that we have a disjoint union X(w)=⊔1≤i≤ksup(ci). For i∈{1,…,k}, there are points xji in X(w), with 1≤j≤νi:=length(ci), such that ci=(x1(i),x2(i),…,xνi(i)). Define η*w(ci)∈H* to be the conjugacy class of the product φ(xνi(i))φ(xνi−1(i))⋯φ(x1(i))∈H. Observe that the product itself is not well-defined by ci, since the xj(i) are well-defined up to cyclic permutation only, but that its conjugacy class is well-defined. Observe also that, if νi=1, then η*w(ci)≠1. For η∈H* and ℓ≥1, let mℓw,η denote the number of cycles c in {c1,…,ck} that are of length ℓ and are such that η*w(c)=η. Let μw,η be the partition with mℓw,η parts equal to ℓ, for all ℓ≥1; let nw,η be the sum of the parts of this partition, so that μw,η⊢nw,η. We have ∑η∈H*nw,η=∑η∈H*,ℓ≥1ℓmℓw,η=|X(w)|. We define the pretype of w as the family (μw,η)η∈H*. By a routine argument, it can now be checked that (i) for all w=(φ,f)∈W and g∈Sym(X), the pretypes of w and (1,g)w(1,g−1) coincide; (ii) for all w=(φ,f)∈W and ψ∈H(X), the pretypes of w and (ψ,1)w(ψ−1,1) coincide; hence conjugate elements in W have the same pretype. Moreover, (iii) two elements in W that have the same pretype are conjugate. For details, we refer to [29, Appendix I.B, No. 3]. For w=(φ,f)∈W, observe that the partition μw,1 does not have parts of size 1. With the same notation as above, denote by λw,1 the partiton with mℓw,1 parts equal to ℓ−1. We define the type of w as the family (λw,1,(μw,η)η∈H*∖1). Then (i)–(iii) hold with “type” instead of “pretype”. Moreover, (iv) every H*-indexed family of partitions, i.e., (λ(1),(μ(η))η∈H*∖1) as in (†), is the type of one conjugacy class in W. Second step: proof of the formula for CW,S(q). Consider a H*-index family of partitions (λ(1),(μ(η))η∈H*∖1) as in (†) and the corresponding conjugacy class in W. Denote by n(1),n(η) the sum of the parts and by k(1),k(η) the number of the parts of λ(1),μ(η), respectively. Choose a representative w=(φ,f) of this class, with f of the form f=∏i=1kci=∏i=1k(x1(i),x2(i),…,xμi(i)) and   φ(xj(i))=1∈H for all j∈{1,…,μi}when η*w(ci)=1φ(xj(i))={   1for all j∈{1,…,μi−1}h≠1for j=μiwhen η*w(ci)≠1. Recall that η*w(ci)≠1 when μi=1, and observe that   k=k(1)+∑η∈H*,η≠1k(η)|X(w)|=n(1)+k(1)+∑η∈H*,η≠1n(η). The contribution of (φ|sup(ci),ci) to κW,S(q) is μi−1 if η*w(ci)=1, and μi if η*w(ci)≠1. Hence, the contribution of the type (λ(1),(μ(η))η∈H*∖1) to CW,S(q) is qn(1)∏η∈H*,η≠1qn(η). It follows that   CW,S(q)=(∑n1=0∞p(n1)qn1)∏η∈H*,η≠1(∑nη=0∞p(nη)qnη)=∏k=1∞1(1−qk)|H*|. This ends the proof of Proposition 19.    ▪ 4 A sample of growth polynomials and conjugacy growth polynomials for finite symmetric groups The purpose of the present section is to compute for Sym(n) growth polynomials LSym(n),S(q) and conjugacy growth polynomials CSym(n),S(q), with respect to a sample of generating sets S. Our computations rely partly on Lemmas 3 of Section 1 and 15 of Section 2. Before this, we review part of what is known in the broader and classical setting of finite Coxeter groups. Though we will not recall precise statements, this is strongly related to the topology of connected compact Lie groups and their homogenous spaces. Let (W,S) be a finite Coxeter system; set l=|S|. Denote the corresponding Coxeter exponents by m1,…,ml; they are positive integers. The growth polynomial is known to be   LW,S(q)=∏k=1l(1+q+⋯+qmk). LW,S This has received much attention; see for example, [38] and [8, exercises of Section IV.1 and VI.4]. As Solomon observes, the computation of LW,S for the particular case of the symmetric groups goes back to Rodrigues, in the first half of XIXth century (with a different formulation). Set T=∪w∈WwSw−1. The word length ℓW,T is sometimes called the reflection length [10] and the corresponding growth polynomial is known to be   LW,T(q)=∏k=1l(1+mkq). LW,T For a group W⊂GL(V) generated by reflections, define ρ:W→N by   ρ(w)=dim(V)−dim({v∈V∣w(v)=v}) and set RW(q)=∑w∈Wqρ(w). Then RW(q)=∏k=1l(1+mkq); this is a special case of [36, Number 5.3], verified there by inspection, and shown again more conceptually in [37]. For a finite Weyl group, it is easy to show that ρ(w)=ℓW,S(w), see for example, [10, Lemma 2], so that LW,T=RW, and (LW,T) holds; this carries over to every finite Coxeter group, see for example, [28]. Other avatars of these polynomials are discussed in [5]. We do not know whether the companion polynomials CW,S,CW,T have already been given any attention. In the next proposition, we particularize LW,S(q) and LW,T(q) to W=Sym(n), and we provide expressions for the corresponding conjugacy growth polynomials. In the special case of finite symmetric groups, there is an ad hoc proof for (LW,T) in Remark 21 and one for (LW,S) in [24]. Proposition 20. Consider an integer n≥1, the symmetric group Sym(n) and its generating sets   SnCox={(1,2),(2,3),⋯,(n−1,n)},Tn={(i,j)∣1≤i,j≤n, i<j}, as in Proposition 1. The corresponding growth polynomial and conjugacy growth polynomial are   LSym(n),SnCox(q)=∏k=1n−1(1+q+⋯+qk),LSym(n),Tn(q)=∏k=1n−1(1+kq),CSym(n),SnCox(q)=CSym(n),Tn(q)=∑k=0n−1pn−k(n)qk, where pn−k(n) is as in Appendix 2.2.    □ Proof. The equalities involving the two products are particular cases of (LW,S) and (LW,T), since the Coxeter exponents of (Sym(n),SnCox) are 1,2,…,n−1. The equality for CSym(n),SnCox(q) is that of Proposition 1(ii), and CSym(n),Tn(q) is the same polynomial, see Remark 13.    ▪ The polynomials CSym(n),Tn(q) for small n 's are given by   CSym(2),T2(q)=1+q,CSym(3),T3(q)=1+q+q2,CSym(4),T4(q)=1+q+2q2+q3,CSym(5),T5(q)=1+q+2q2+2q3+q4,CSym(6),T6(q)=1+q+2q2+3q3+3q4+q5. (Compare with the polynomials written after Proposition 22.) Remark 21. (i) The second polynomial of Proposition 20 can also be written   LSym(n),Tn(q)=1+∑m=2n(nm)∑k=1⌊m/2⌋dk(m)qm−k, where dk(m) is as in Appendix 2.4. (ii) It is easy to check directly from (i) that we have also   LSym(n),Tn(q)=∏k=1n−1(1+kq), as in Proposition 20.    □ Proof. (i) For m∈{0,1,…,n}, there are (nm) subsets of size m in {1,2,…,n}. For each such subset, say A, and each k∈{0,1,…,n}, there are dk(m) permutations in Sym(n) with support A which are products of k disjoint cycles, and these elements have Tn-word length m−k, by Lemma 3. The growth polynomial of the situation is therefore   ∑m=0n(nm)∑k=0ndk(m)qm−k. To end this computation, we observe that the contribution of m=0 is 1, that of m=1 is 0, and dk(m)=0 for 2k>n. (ii) We proceed by induction on n. There is nothing to check for n=1; we assume now that n≥2, and that the statement holds for n−1. Consider an element g∈Sym(n) which is not in Sym(n−1). There is a unique pair consisting of i∈{1,…,n−1} and h∈Sym(n−1) such that g=(i,n)h. This implies that   CSym(n),Tn(q)=CSym(n−1),Tn−1(q)+(n−1)qCSym(n−1),Tn−1(q). Hence   CSym(n),Tn(q)=CSym(n−1),Tn−1(q)(1+(n−1)q)=∏i=1n−1(1+kq) by the induction hypothesis.    ▪ The final proposition of this section shows polynomials L and C for finite symmetric groups and a third generating set Sn0, essentially distinct from the generating sets SnCox and Tn of Proposition 20 for n≥4. It is convenient to see Sym(n) as the symmetric group of {0,1,…,n−1}; the generating set Sn0 is that already considered in Lemmas 14 and 15. Proposition 22. Consider an integer n≥1, the symmetric group Sym(n) and the generating set Sn0={(0,i)∣1≤i≤n−1}. The corresponding growth polynomial and conjugacy growth polynomial are   LSym(n),Sn0(q)=1+∑m=2n−1(n−1m)∑k=1⌊m/2⌋dk(m)qm+k+∑m=2n(n−1m−1)∑k=1⌊m/2⌋dk(m)qm+k−2,CSym(n),Sn0(q)=1+∑k=1⌊n/2⌋q2k−2∑j=knpk(j)qj(as in Proposition 9).     □ For example:   LSym(4),S40(q)=1+3q+6q2+9q3+5q4,LSym(5),S50(q)=1+4q+12q2+30q3+44q4+26q5+3q6,LSym(6),S60(q)=1+5q+20q2+70q3+170q4+250q5+169q4+35q7, and   CSym(4),S40(q)=1+q+q2+q3+q4,CSym(5),S50(q)=1+q+q2+q3+2q4+q5,CSym(6),S60(q)=1+q+q2+q3+2q4+2q5+2q6+q7. (Compare with the polynomials written after Proposition 20.) Proof. Let us deal with the polynomial L. Consider first elements g∈Sym(n) with 0∉sup(g). For each m∈{0,1,…,n−1}, there are (n−1m) subsets of size m in {1,2,…,n−1}. For each such subset, say A, and each k∈{0,1,2,…,m}, there are dk(m) elements with support A which are products of k cycles, and these elements have Sn0-word length m+k, by Lemma 15. The contribution to the growth polynomial of elements with 0∉sup(g) is therefore   ∑m=0n−1(n−1m)∑k=0mdk(m)qm+k. (0 ∉ sup) The contribution of m=0 is 1 and that of m=1 is zero; for m≥2, the contributions of terms with k=0 or k>m/2 is also zero. Consider now elements g∈Sym(n) with 0∈sup(g). For each m∈{1,2,…,n}, there are (n−1m−1) subsets of size m in {0,1,…,n−1} containing 0. For each such subset, say B, and each k∈{1,2,…,m}, there are dk(m) elements with support B which are products of k cycles, and these elements have Sn0-word length m+k−2. The contribution of these elements is therefore   ∑m=1n(n−1m−1)∑k=1mdk(m)qm+k−2. (0 ∈ sup) As above, the contributions of terms with m=1 or k>m/2 vanish. The formula for LSym(n),Sn0(q) follows. That for CSym(n),Sn0(q) is a repetition of part of Proposition 9.    ▪ 5 Alternating groups For a non-empty set X, we denote by Alt(X) the finitary alternating group of X, that is, the subgroup of Sym(X) of permutations of even signature. Set   TXA={(x,y,z)∈Alt(X)∣x,y,z∈X are distinct},UXA={(x,y)(z,u)∈Alt(X)∣x,y,z,u∈X are distinct}. Recall from the introduction that, when X=N, we have defined   SNA={(i,i+1,i+2)∈Alt(N)∣i∈N}, and we consider also   RNA={(1,i,i+1)∈Alt(N)∣i≥2}. When X={1,…n} for some n≥3, we write   Alt(n)=Alt({1,2,…,n}),SnA={(i,i+1,i+2)∈Alt(n)∣1≤i≤n−2},RnA={(1,i,i+1)∈Alt(n)∣2≤i≤n−1}. When X is either N or {1,…,n} for some n≥1, we write SXA to denote the relevant set, either SNA or SnA, and similarly for RXA. The following lemma is well-known, even if we did not find a convenient reference. Lemma 23. With the notation above: for all n≥3, the sets SnA and RnA both generate Alt(n); the sets SNA and RNA both generate Alt(N); and the set TXA generates Alt(X).    □ Proof. Let Hn denote the subgroup of Alt(n) generated by SnA; we claim that Hn=Alt(n). The case of n=3 is obvious; we proceed by induction on n, assuming that n≥4 and that the claim holds for n−1. The group Hn acts transitively on {1,…,n}, because it contains the 3-cycle (n−2,n−1,n) as well as Hn−1=Alt(n−1). Hence the order of Hn is n times the index of the isotropy group {h∈Hn∣h(n)=n}, that is |Hn|=n12(n−1)!=12n!. It follows that Hn=Alt(n). As a consequence, RnA also generates Alt(n), since (1,i+1,i)(1,i+2,i+1)(1,i,i+1)=(i,i+1,i+2) for all i∈{2,…,n−1}. The claims for SNA, RNA and TXA follow.    ▪ Note that TXA∪UXA is the set of products of two distinct elements of the generating set SX of Sym(X). It follows that   κAlt(X),TXA∪UXA(g)=12κSym(X),SX(g) for all g∈Alt(X). Since, for X infinite, two elements of Alt(X) are conjugate in Alt(X) if and only if they are conjugate in Sym(X), we obtain the following straightforward consequence of Proposition 8: Proposition 24. Let X be an infinite set, and TXA,UXA as above. Then   CAlt(X),TXA∪UXA(q)=∑m=0∞p(2m)qm=12∏k≥111−qk/2+12∏k≥111−(−1)kqk/2=1+2q+5q2+11q3+22q4+42q5+77q6+135q7+231q8+385q9+627q10+1002q11+1575q12+⋯, where the numerical coefficients are those of the series [32, A058696].    □ Let X be a set containing at least 5 elements. It is easy to check that UXA generates Alt(X), and it can be shown that   κAlt(X),UXA(g)={κAlt(X),TXA∪UXA(g)if g=id or|sup(g)|>3,  2if g is a 3-cycle. It follows that Proposition 25. Let X be an infinite set, and UXA as above. Then   CAlt(X),UXA(q)=q2−q+∑m=0∞p(2m)qm.     □ Remark. For the generating set   VNA:={(i,i+1,i+2)∈Alt(N)∣i≥0}∪{(i,i+1)(i+2,i+3)∈Alt(N)∣i≥0}, it can be shown that   CAlt(N),VNA(q)=CAlt(N),TNA∪UNA(q). Our next target is to identify CAlt(X),TXA(q). Lemma 26. Let g∈Alt(X) and g=t1⋯tL a writing of g as a word of minimal length L=ℓAlt(X),TXA(g) in the generators of TXA. Then tj≠ti±1, equivalently sup(ti)≠sup(tj), for all i,j∈{1,…,L} with i≠j.    □ Proof. Let g=u1⋯uM be a writing of g as a word in the generators of TXA. Suppose first that there exist j,k∈{1,…,M} with j<k such that uk=uj−1. If k=j+1, then deleting ujuk produces a new TXA-word of length M−2 representing g; if k≥j+2, then g can be written as   u1⋯uj−1(ujuj+1uj−1)⋯(ujuk−1uj−1)uk+1⋯um, i.e. g can again be written as a TXA-word of length M−2 representing g. Suppose now that there exist j,k∈{1,…,M} with j<k such that uk=uj. If k=j+1, then replacing ujuk by uj−1 produces a new TXA-word of length M−1 representing g; if k≥j+2, then g can be written as   u1⋯uj−1ujuj+1⋯uk−1uj−1uj−1uk+1⋯uM and the previous procedure provides a TXA-word representing g of length M−1. The lemma follows.    ▪ For g∈Alt(X) a product of disjoint cycles, we denote by k′g the number of cycles of odd lengths ≥3 and by 2k″g the number of cycles of even lengths ≥2. Note that kg=k′g+2k″g for kg as in Lemma 17. Lemma 27. Let X be a set and S a generating set of Alt(X). Let g∈Alt(X) be a product of disjoint cycles, with k′g,k″g as above. Suppose either that S=TXA or that X is one of N, {1,…,n} for some n≥1, and that SXA⊂S⊂TXA. We have   ℓAlt(X),TXA(g)=κAlt(X),S(g)=12(|sup(g)|−k′g).     □ In the proof below, we write ℓ for ℓAlt(X),TXA and κ for κAlt(X),S. Proof of the upper bounds κ(g),ℓ(g)≤12(|sup(g)|−k′g). We show the bound for ℓ(g), and leave it to the reader to check that a minor modification of the same argument shows the bound for κ(g). Whenever convenient, we write k′,k″ rather than k′g,k″g. Consider a cycle of odd length, say   cα=(x1,x2,…,x2p+1) for x1,…,x2p+1∈X. We have   cα=(x1,x2,x3)(x3,x4,x5)(x5,x6,x7)⋯(x2p−1,x2p,x2p+1) and therefore ℓ(cα)≤p=12(|sup(cα)|−1). Consider a pair of disjoint cycles of even lengths, say   cβcγ=(x1,x2,…,x2r)(y1,y2,…,y2s) for x1,…,x2r,y1…,y2s∈X (where we consider an appropriate conjugate of g and 2r+2sconsecutive integers y1,y2,…,y2s,x1,x2,…,x2r for the case of κ(g)). We have   cβcγ=(y1,y2,y3)(y3,y4,y5)⋯(y2s−3,y2s−2,y2s−1)(y2s−1,y2s,x1)  (x1,x2,x3)(x3,x4,x5)⋯(x2r−3,x2r−2,x2r−1)(x2r−1,x2r,y2s) and therefore ℓ(cβcγ)≤r+s=12(|sup(cβ)|+|sup(cγ)|). For g=c1c2⋯ck′ck′+1ck′+2⋯ck′+2k″, where c1,…,ck′+2k″ are disjoint cycles, cν of odd length for 1≤ν≤k′ and of even length for k′+1≤ν≤k′+2k″, it follows that   ℓ(g)≤∑ν=1k′+2k″ℓ(cν)≤12(∑α=1k′(|sup(cα)|−1)+∑β=k′+12k″|sup(cβ)|)       =12(|sup(g)|−k′), as was to be shown.    ▪ Proof of the lower bounds ℓ(g),κ(g)≥12(|sup(g)|−k′g). For g≠id in Alt(X) such that |sup(g)|≤3, we have obviously 1=ℓ(g)=κ(g)≥12(|sup(g)|−k′g)=12(3−1). We consider from now on an element g in Alt(X) with |sup(g)|>3, and therefore with ℓ(g)>1 and κ(g)>1. As above, we continue and deal with ℓ(g) only. Suppose by contradiction that there exists g∈Alt(X) with |sup(g)|>3 and   ℓ(g)<12(|sup(g)|−k′g); (♭) suppose moreover that ℓ(g) is minimal for the elements for which (♭) holds. We can write   g=t1⋯tL (♭♭) for some t1,…,tL∈TXA with 1<L=ℓ(g)<12(|sup(g)|−k′g). By Lemma 26, we know that the supports sup(ti) are pairwise distinct. For each i∈{1,…,L}, let xi,yi,zi∈X be such that ti=(xi,yi,zi). Set Yi=sup(ti)={xi,yi,zi} and Zi=∪1≤j≤L, j≠iYj. Claim: We have   |Yi∩Zi|≥2 for all i∈{1,…,L}. (♯) Upon conjugating g by ti+1⋯tL, we can assume that i=L for the proof of the claim. Let us first check that |YL∩ZL|≥1. Indeed, otherwise, set   h=∏i=1L−1ti. (‡) Observe that ℓ(h)≤L−1. We have |sup(h)|=|sup(g)|−3, and also k′h=k′g−1, since the cycle of odd length ti has been deleted in the product defining h. It follows that ℓ(h)<12(|sup(h)|−k′h). This contradicts the minimality hypothesis on g made above; hence |YL∩ZL|≥1. Let us now show that |YL∩ZL|≥2. Indeed, otherwise, |YL∩ZL|=1. Let again h be defined by (‡); observe again that ℓ(h)≤L−1, and that |sup(h)|=|sup(g)|−2; it can be shown that k′h=k′g (details below). It follows that ℓ(h)<12(|sup(h)|−k′h). This contradicts again the minimality hypothesis above; hence |YL∩ZL|≥2. Here are the announced details. Let x,y,z∈X be such that YL∩ZL={x} and tL=(x,y,z); Then x is contained in the support of a cycle d of h of length ℓ≥2, and also by Lemma A.31 in the support of a cycle c=dtL of g=htL of length ℓ+2. Hence k′h=k′g. This ends the proof of the Claim. Lemma 26 and the claim just proven imply that, for each i∈{1,…,L}, there are xi,yi,zi∈X such that ti=(xi,yi,zi), yi,zi∈Zi. Consider the product of 2L transpositions, equal to g, obtained from the product (♭♭) by changing each ti to (xi,zi)(xi,yi), say   g=s1s2⋯s2L−1s2L. Set S={s1,…,s2L}; define Γ˜(S) to be the multigraph with vertex set V:=∪j=12Lsup(sj), and one edge connecting x,y∈V for every j∈{1,…,2L} with sj=(x,y); here, “multigraph” means that Γ˜(S) may have multiple edges. On the one hand, the number of vertices of this graph is bounded below by |sup(g)|; on the other hand, what we have shown so far implies that the degree of each vertex of Γ˜(S) is at least 2; it follows that the number of edges of this graph, which is at least twice its number of vertices, is bounded below by |sup(g)|; in other words, L≥12|sup(g)|. This is strongly in contradiction with (♭); hence the inequality of (♭) is not true, and this ends the proof of the lemma.    ▪ Remark concerning the claim of the previous proof. Consider an element g∈Alt(X) which is a word g=t1⋯tL in the letters of TXA of minimal length L=ℓ(g), now with 2≤L≤12(sup(g)|−k′g). The cardinality |Y1∩Z1| can be any of 0,1,2,3, as the following examples show:   g0=(1,2,3)(4,5,6) for which  L=2, |sup(g0)|−k′g0=6−2, and Y1∩Z1=Ø,g1=(1,4,5)(1,2,3)=(1,2,3,4,5) for which  L=2, |sup(g1)|−k′g1=5−1, and Y2∩Z2={1},g2=(5,6,7)(2,3,4)(1,4,7)=(1,2,3,4,5,6,7) for which  L=3, |sup(g2)|−k′g2=7−1, and Y3∩Z3={4,7},g3=(1,8,9)(5,6,7)(2,3,4)(1,4,7)=(1,2,3,4,5,6,7,8,9) for which  L=4, |sup(g3)|−k′g3=9−1, and Y4∩Z4={1,4,7}. Proposition 28 is a minor generalization of Proposition 11. Recall from Appendix 2.3 that pe(n) denotes the number of partitions of n∈N involving an even number of positive parts. Proposition 28. Let X be an infinite set and S a generating set of Alt(X). Suppose either that S=TXA or that X=N and that SNA⊂S⊂TNA. Then   CAlt(X),S(q)=∑u=0∞p(u)qu∑v=0∞pe(v)qv=12∏j=1∞1(1−qj)2+12∏j=1∞11−q2j=1+q+3q2+5q3+11q4+18q5+34q6 +55q7+95q8+150q9+244q10+⋯.     □ Proof. We write κ for κAlt(X),S. Let g∈Alt(X) be written as a product of disjoint cycles, say k′ of them of odd lengths and 2k″ of them of even lengths. Denote by go the product of the cycles of odd lengths and by ge the product of the cycles of even lengths, so that g=goge. Let λ(g)=(λ1(g),…,λk′(g))⊢u and ν(g)=(ν1(g),…c,ν2k″(g))⊢v be the partitions such that go is the product of cycles of lengths 2λ1(g)+1,…,2λk′(g)+1, and ge the product of cycles of lengths 2ν1(g),…,2ν2k″(g); note that |sup(go)|=2u+k′ and |sup(ge)|=2v. By Lemma 27, we have   κ(go)=u, κ(ge)=v, and κ(g)=κ(go)+κ(ge)=u+v. The set of conjugacy classes in Alt(X) is naturally parametrized by pairs (λ,ν) of partitions such that ν has an even number of positive parts. (It is important here that the set X is infinite, otherwise some pairs correspond to two conjugacy classes in the alternating group). The contribution to CAlt(X),S(q) of classes of elements such that g=go is therefore ∑u=0∞p(u)qu=∏i=1∞11−qi; the contribution of classes of elements such that g=ge is ∑v=0∞pe(v)qv=12∏j=1∞11−qj+12∏j=1∞11+qj (this uses Proposition A.34); finally CAlt(X),S is the product of these two contributions. Remark 29. (i) Recall from Observation 12 that we denote by   (pA(n))n≥0=(1,1,3,5,11,18,34,55,95,150,244,…) the sequence of coefficients of the series of Proposition 28. At the day of writing, this sequence does not appear in [32]. (ii) The sums and products in the previous proposition converge again for q complex with |q|<1. Numerically, the roots of smallest absolute value of CAlt(N),TNA(q) are simple and located at ~0.67±0.43i. (iii) As in the case of CSym(X),S(q), see Proposition 8(a), it can be observed that the series CAlt(X),TXA(q) does not depend on the cardinality of X, as long as X is infinite.    □ 6 Congruences à la Ramanujan for the coefficients of the series of Proposition 28 Ramanujan, and later Watson, Atkin, Andrews, and others, have discovered remarkable congruence properties for the partition function, including   p(5n+4)≡0 (mod5),p(7n+5)≡0 (mod7),p(11n+6)≡0 (mod11),p(25n+24)≡0 (mod52),p(125n+99)≡0 (mod53),p(49n+47)≡0 (mod72),p(121n+116)≡0 (mod112). See for example [22], or [7] and references there. Consider a finite group H with M conjugacy class, an infinite set X, the permutational wreath product W=H≀XSym(X), a generating set S that satisfies Condition (PCwr), and the corresponding conjugacy growth series   CW,S(q)=∏k=1∞1(1−qk)M=∑n=0∞p(n)(M)qn as in Proposition 19. There is an important literature on congruence properties of the sequences (p(n)(M))n=0,1,2,… of so-called multipartition numbers. In particular:   p(5n+3)(2)≡0 (mod5),p(11n+4)(8)≡0 (mod11), (Gandhi)  p(5n+B)(2)≡0 (mod5) for B∈{2,3,4}, (Andrews)  p(25n+23)(2)≡0 (mod25). (CDHS) See [17], a particular case of Theorem 1 in [1], and Formula (1.17) in [12], respectively. Like the partition numbers p(n) and the multipartition numbers p(n)(M), the coefficients of the conjugacy growth series   CAlt(X),S(q)=12∏j=1∞1(1−qj)2+12∏j=1∞11−q2j=∑n=0∞pA(n)qn of Proposition 28 verify intriguing congruence relations, as was recorded in Observation 12 of the Introduction. With the notation of Appendix 3, the coefficients of this series can be written as   pA(n)=12(p(n)(2)+p(n)(0,1)). Proposition 30. With the notation above, we have   pA(5n+3)≡0 (mod5),pA(10n+7)≡0 (mod5),pA(10n+9)≡0 (mod5),pA(25n+23)≡0 (mod25).     □ Proof. On the one hand, as recorded above in (Gandhi), it is known that p(5n+3)(2)≡0 (mod5) for all n≥0. On the other hand, it follows from the definitions that   p(k)(0,1)={p(m)if k=2m0if k is odd. Since p(5n+4)≡0 (mod5) for all n≥0, we have also p(5n+3)(0,1)≡0 (mod5) for all n≥0. Hence pA(5n+3)=12(p(4n+3)(2)+p(4n+3)(0,1))≡0 (mod5) for all n≥0. Similarly, since p(n)(0,1)=0 for all odd n, the congruences for pA(10n+7) and pA(10n+9) follows from (Andrews), and for pA(25n+23) from (CDHS).    ▪ On the conjectured relations of Observation 12. For pA(⋅), Proposition 30 contains the established part of Observation 12. The remaining congruences of this observation follow from the congruences   p(49n+17)(2)≡0 (mod7),p(49n+33)(1)≡0 (mod7),p(49n+31)(2)≡0 (mod7),p(49n+40)(1)≡0 (mod7),p(49n+38)(2)≡0 (mod7),p(49n+19)(1)≡0 (mod7),p(49n+45)(2)≡0 (mod7),p(49n+47)(1)≡0 (mod7),p(121n+111)(2)≡0 (mod11),p(121n+116)(1)≡0 (mod11). For what we know, the congruences of the left-hand side are conjectural, with numerical evidence recorded in our Appendix 3. The congruences on the right-hand side are all established, and are indeed particular cases of the classical congruences p(7n+5)≡0 (mod7) and p(11n+6)≡0 (mod11). Appendix 1. Three lemmas on symmetric and alternating groups For reference elsewhere, we state here three elementary facts. Recall from the introduction that, for a,b∈Sym(X), we agree that ab denotes bfollowed by a. The first lemma is straightforward: Lemma A.31. Let X be a set with at least 3 elements, and a,b∈Sym(X) two cycles such that their supports have exactly one element in common. Then ab is a cycle and sup(ab)=sup(a)∪sup(b). More precisely, if a=(x1,…,xr) and b=(xr,…,xr+s−1), then ab=(x1,…,xr+s−1).    □ The next lemma is well-known. See e.g. [18, Lemmas 3.10.1 and 3.10.2], where the proof of (2) is left as an exercise. Lemma A.32. Let X be a non-empty set, S a set of transpositions of X, and Γ(S) the transposition graph, as in Definition 4. (1) S generates Sym(X) if and only if Γ(S) is connected. (2) Suppose that X is finite, say of cardinality n, and that Γ(S) is a tree. Let s1,s2,…,sn−1 be an enumeration of the elements of S. Then the product s1s2⋯sn−1 is a cycle of length n.    □ Proof. (1) Denote by G the subgroup of Sym(X) generated by S. Suppose that Γ(S) is not connected. Choose a connected component of Γ(S), denote by X1 its vertex set, and set X2=X∖X1. Then G is a subgroup of the proper subgroup Sym(X1)×Sym(X2) of Sym(X), hence S does not generate Sym(X). Assume that Γ(S) is connected. We have to show that G=Sym(X). Since this is trivial when |X|≤2, we assume that |X|≥3. Let x,y,z be three distinct elements in X; observe that (y,z)(x,y)(y,z)=(x,z). For two distinct elements u,v in X, it follows that (u,v)∈G by induction on the length of a path connecting u and v in Γ(S). Hence G contains all transpositions of elements of X, and therefore G=Sym(X). (2) We proceed by induction on n. The lemma is obvious for n=2; suppose that n>2, and that the lemma holds up to n−1. Choose a leaf x of Γ(S). There is a unique i(x)∈{1,…,n−1} such that x∈sup(si(x)). Upon replacing the product s1⋯sn−1 by a conjugate element, we can assume that si(x)=sn−1. By the induction hypothesis, the product s1⋯sn−2 is now a cycle c′ of length n−1. By Lemma A.31, s1⋯sn−2sn−1=c′sn−1 is a cycle of length n.    ▪ The third lemma is a cheap confirmation of the fact that most pairs of elements of Sym(n) generate either Alt(n) or Sym(n) [3]. Lemma A.33. Let X be a non-empty set with at least 3 elements, a,b∈Sym(X) two cycles, respectively of lengths ℓ,m≥2, such that their supports have exactly one element in common (as in Lemma A.31). Let G be the subgroup of Sym(X) generated by {a,b}. Then G is isomorphic to the alternating group Alt(ℓ+m−1) if ℓ,m are both odd, and to Sym(ℓ+m−1) otherwise.    □ Proof. Denote by x the element in sup(a)∩sup(b); set y=a−1(x) and z=b−1(x). The commutator a−1b−1ab is the 3-cycle c:=(x,y,z). By Lemma 23 for Rℓ+1A, the conjugates of c by the powers of a generate Alt(sup(a)∪{z}); similarly the conjugates of c by the powers of b generate Alt({x}∪sup(b)). Observe that the intersection Alt(sup(a)∪{z})∩Alt({x}∪sup(b)) contains c, and the union Alt(sup(a)∪{z})∪Alt({x}∪sup(b)) contains a set of 3-cycles similar to Sℓ+m−1A. By Lemma 23 again, this time for Sℓ+m−1A, the group G contains Alt(sup(a)∪sup(b)), isomorphic to Alt(ℓ+m−1). If ℓ and m are both odd, every element in G has an even signature, hence G=Alt(sup(a)∪sup(b))≃Alt(ℓ+m−1). Otherwise, G is a subgroup of Sym(sup(a)∪sup(b)) in which Alt(sup(a)∪sup(b)) is a proper subgroup, hence G=Sym(sup(a)∪sup(b))≃Sym(ℓ+m−1).    ▪ This lemma implies for example that the set   {(0,1,2),(2,3,4),(4,5,6),…,(2i,2i+1,2i+2),…} generates Alt(N). It is a proper subset of the generating set SNA introduced in the beginning of Section 5. Appendix 2. Reminder on partitions and derangements Appendix 2.1. The partition function For n∈N, let p(n) denote the number of partitions of n. The first values are given by the table   (more values in [32, A000041]). In our context p(n) is the number of conjugacy classes in the finite symmetric group Sym(n), alternatively the number of conjugacy classes in Sym(N) of elements of supports of size at most n. For this reason, the partition function appears already in Propositions 1 and 9. It is known since Euler that the generating series for p(n) has a product expansion   ∑n=0∞p(n)qn=∏k=1∞11−qk. EP1 See [14, Caput XVI], as well as, for example, [23, Section 19.3]. The equality can be viewed either between formal expressions, or between absolutely converging sum and product for q∈C with |q|<1. There is an asymptotic formula for n→∞  p(n)=143(n−124) exp(π23(n−124))+O(1(n−124)3/2 exp(π23(n−124))) due to Hardy and Ramanujan [21, Formula (1.41)]. For this and more on p(n) when n→∞, see e.g. [11, Chapter VII] and [22, Chapters VI and VIII]. This shows in particiular that the sequence (p(n))n≥0 has intermediate growth, that is, that its growth is superpolynomial and subexponential. Appendix 2.2. Partitions with k parts For n,k∈N, we denote by pk(n) the number of partitions of n in exactly k positive parts, equivalently the number of partitions of n with largest part k, equivalently the number of partitions of n−k in k non-negative parts. Whenever needed, we set pk(n)=0 for all n∈N and k<0. Numbers pk(⋅) appear in connection with finite symmetric groups, in Propositions 1, 9, 20, and 22. We have classically   p0(0)=1 and p0(n)=0 for all n≥1,p1(0)=0 and p1(n)=1 for all n≥1,p2(n)=⌊n/2⌋ for all n≥0,p3(n)=⌊112(n2+6)⌋ for all n≥0 [32, A069905],⋯   ⋯pn−2(n)=2 for all n≥4,pn−1(n)=pn(n)=1 for all n≥2,pk(n)=0 for all k>n≥0,pk(n)=pk(n−k)+pk−1(n−1) for all n≥k≥1,∑k=0npk(n)=∑k=1npk(n)=p(n) for all n≥1 , and the generating function   ∑n≥0pk(n)qn=qk∏i=1k11−qi for all k≥0 . EP2 (Observe that ∑n≥0pk(n)qn=∑n≥kpk(n)qn.) Up to the notation, Equality (EP2) is contained in Number 312 of [14, Caput XVI]. Moreover, if P(n,t):=∑k=0npk(n)tk, then   ∑n=0∞P(n,t)qn=∏j=1∞11−tq j . EP3 This appears in Number 304 of [14, Caput XVI], and is used in the proof of our Proposition A.34. For n,ℓ∈N with n≤2ℓ, every partition of n−ℓ has at most ℓ parts. Thus every partition of n−ℓ can be obtained from a unique partition of n in ℓ parts by substracting 1 from each part. Consequently   pℓ(n)=p(n−ℓ) for integers n,ℓ such that 0≤ℓ≤n≤2ℓ , EP4 or, setting k=n−ℓ,   The double sequence (pk(n))n≥0, 0≤k≤n gives rise to a generalized Pascal triangle of which the first rows are:   Appendix 2.3. Partitions with even or odd numbers of parts We denote by pe(n), respectively po(n), the number of partitions of a non-negative integer n involving an even, respectively odd, number of non-zero parts. Working with conjugate partitions, we see that pe(n), respectively po(n), is equivalently given by the number of partitions of n having an even largest part, respectively an odd largest part. We have the trivial identity p(n)=pe(n)+po(n). These numbers pe(n) appear in Proposition 28. Their values for n≤15 are given by   see A027187 and A027193 of [32]. Proposition A.34. (1) The generating series of the sequence pe(n) is   ∑n=0∞pe(n)qn=∑k=0∞q2k∏j=12k11−q j=12(∏j=1∞11−q j+∏j=1∞11+q j)=∏j=1∞11−q j∑m=0∞(−q)m2. (2) The generating series of the sequence po(n) is   ∑n=0∞po(n)qn=∑k=0∞q2k+1∏j=12k+111−q j=12(∏j=1∞11−q j−∏j=1∞11+q j)=−∏j=1∞11−q j∑m=1∞(−q)m2.     □ Proof. (1) Using (EP2), we have   ∑n=0∞pe(n)qn=∑n=0∞∑k=0⌊n/2⌋p2k(n)qn=∑k=0∞∑n=0∞p2k(n)qn=∑k=0∞q2k∏j=12k11−q j. Also, if P(n,t):=∑k=0npk(n)tk as in (EP3), then   ∑n=0∞pe(n)qn=12(∑n=0∞P(n,1)qn+∑n=0∞P(n,−1)qn)=12(∏j=1∞11−q j+∏j=1∞11+q j). For the third equality in (1), one way is to refer to [15]: see there Equation (7.324), p. 6, and also Example 7, p. 39. The proof of (2) is similar. Here is an alternative to citing [15]. We have   ∑n=0∞(pe(n)−po(n))qn=2∑n=0∞pe(n)qn−∑n=0∞p(n)qn=∏j=1∞11+q j =∏j=1∞1−q j1−q2j=∏j=1∞(1−q2j−1)=∏j=1∞11−q j ∏k=1∞(1−q2k−1)2(1−q2k) . The Jacobi triple product identity reads   ∏k=1∞(1−x2k)(1+x2k−1y2)(1+x2k−1y2)=∑n=−∞∞xn2y2n (see e.g., [23, Theorem 352]). For x=q and y=−1 it reduces to   ∏k=1∞(1−q2k−1)2(1−q2k)=∑n=−∞∞(−q)n2 , hence   ∑n=0∞(pe(n)−po(n))qn=∏j=1∞11−q j ∑n=−∞∞(−q)n2 . Finally:   ∑n=0∞pe(n)qn=12∑n=0∞(pe(n)−po(n))qn+12∑n=0∞(pe(n)+po(n))qn=12∏j=1∞11−q j ∑n=−∞∞(−q)n2+12∏j=1∞11−q j=∏j=1∞11−q j ∑n=0∞(−q)n2 , as was to be shown.    ▪ Observation A.35. We have   (∑n=0∞pe(n)qn)2−(∑n=0∞po(n)qn)2=∑n=0∞p(n)q2n=∏j=1∞11−q2j.     □ Proof. The left-hand side can be written as   14(∏j=1∞11−q j+∏j=1∞11+q j)2−14(∏j=1∞11−q j−∏j=1∞11+q j)2 =∏j=1∞11−q2j , and the claim follows.    ▪ Appendix 2.4. Derangements that are products of k cycles A derangement is a fixed point free permutation. For n,k∈N, denote by dk(n) the number of derangements of {1,2,…,n} that are products of k disjoint cycles. These numbers appear in Remark 21 and Proposition 22. Lemma A.36. With the notation above, we have (i) d0(0)=1 ; (ii) dk(1)=0 for all k∈N ; (iii) dk(n)=0 for all n,k∈N with k=0 or 2k>n ; For all n≥2 and k≥1, we have (iv) dk(n)=(n−1)(dk(n−1)+dk−1(n−2)) ; (v) dk(n)=∑a=2n(n−1a−1)(a−1)! dk−1(n−a).    □ Proof. Claims (i) to (iii) are obvious. For (iv), consider a derangement g of {1,…,n} product of k cycles. Either n is in the support of a cycle (x1,…,xℓ−1,n) of length at least 3. Replacing it by the cycle (x1,…,xℓ−1) produces a derangement of {1,…,n−1} product of k cycles, and each of the latter is obtained n−1 times in this way. This explains the contribution (n−1)dk(n−1) of the right-hand side. Or n is in the support of a transposition, say (i,n) with i∈{1,…,n−1}, so that g is the product of (i,n) with a derangement h of {1,…,n−1}∖{i} product of k−1 cycles. For each of the n−1 possible values of i, there are dk−1(n−2) such permutations h, and this explains the contribution (n−1)dk−1(n−2). For (v), a permutation contributing to dk(n) is the product of a cycle c of length a≥2, with n∈sup(c), and there are (n−1a−1)(a−1)! such cycles, with a derangement of {1,…,n}∖sup(c) which is a product of k−1 cycles.    ▪ Remark A.37. (i) The double sequence (dk(n))n≥0, 0≤k≤n gives rise to a generalized Pascal triangle of which the first rows are:   (ii) Besides the relations of Lemma A.36, we have also   ∑m=0n(nm)∑k=0mdk(m)=n! for all n∈N, Σd which is useful to check numerical values. Indeed, each of the n! permutations g of {1,…,n} induces a derangement of sup(g). For m∈{0,1,…,n}, there are (nm) subsets of {1,…,n} of size m. Since there are ∑k=0mdk(m) derangements of each of these subsets, we obtain the left-hand side. Relation (Σd) reduces to d0(0)=1 for n=0, and to d0(0)+d0(1)+d1(1)=1+0+0=1 for n=1. Otherwise, it can be written   1+∑m=2n(nm)∑k=1⌊m/2⌋dk(m)=n! for all n≥2. Σd′ The sum d(m):=∑k=0mdk(m)=∑k=0⌊m/2⌋dk(m) is the number of derangements of m objects, and there is a classical formula:   d(m)=∑k=0mdk(m)=m! (1−11!+12!−13!+⋯+(−1)m1m!) for all m≥0; it follows that we have the relations   d(m)=md(m−1)+(−1)m for all m≥1 ,d(m)=(m−1)(d(m−1)+d(m−2)) for all ≥2 ; see for example, [39, Example 2.2.1]. The sequence   (d(m))m≥0=(1,0,1,2,9,44,265,1854,14833,133496,1334961,...) is A000166 in [32]. (iii) Numbers dk(n) have some flavour of Stirling numbers. For n,k∈N with 0≤k≤n, recall that the unsigned Stirling number of the fist kind [nk] counts the number of ways to arrange n objects into k cycles (here, cycles of length 1 are included, unlike elsewhere in this article, and this is why entries in (PTd) are smaller or equal than entries in (PTStir). When n≥1, we have [nk]=(n−1)[n−1k]+[n−1k−1]. See for example, [19, p. 245] and [32, A132393]. The generalized Pascal triangle for ([nk])n≥0, 0≤k≤n is   10101102310611610245035101012027422585151 (PTStir) We have   [nk]=∑j=0k(nj)dk−j(n−j) . Indeed, in the right-hand side, the term with a given value of j counts the number of contributions to [nk] with j fixed points.    □ Appendix 3. Generalized Ramanujan congruences This appendix is partly experimental. It grew out of our desire to understand the reasons for the congruences for the numbers pA(n) described in Observation 12 and Section 6. All our experimental observations have now been proved by T. Cotron, R. Dicks, and S. Fleming [40]. Appendix 3.1. Definitions Definition A.38. Given a sequence e=(e1,e2,e3,…)∈Z(1,2,3,…) of integers with ed=0 for d large enough, the corresponding generalized partition numbers p(n)e are the coefficients of the power series   ∑n=0∞p(n)eqn=∏n=1∞∏d=1∞1(1−qdn)ed=∏n=1∞1(1−qn)e1(1−q2n)e2(1−q3n)e3⋯ . (A.1)     □ Remark A.39. As a shorthand, we also write a sequence e as above as (e1,e2,…,ek) when ek≠0 and ed=0 for all d≥k+1. For example:   ∑n=0∞p(n)(0,3)qn=∏n=1∞1(1−q2n)3 . (A.2) For a sequence of the form (e1,…,ej,0,…,0,ek) with ek≠0, and ed=0 when j<d<k or d>k, we also write (e1,…,ej,(ek)k). For example:   ∑n=0∞p(n)(0,1,28)qn=∏n=1∞1(1−q2n)(1−q8n)2 . (A.3) For a positive integer M, the numbers p(n)(M) arising as coefficients of the series defined by ∏n=1∞1(1−qn)M are called multi-partition numbers in the literature, since p(n)(M) counts the number of ways of writing n as a sum of parts, each coloured in one of M colours. More generally, for e∈Z(1,2,3,…) as above, p(n)e can be interpreted as multi-partition numbers which constraints on the parts; for example, the coefficient p(n)(0,1,28) of (A.3) counts the number of partitions of the form   n=λ1+⋯+λi+μ1+⋯+μj+ν1+⋯+νk where   λ1≥⋯≥λi≥1 and λ1,…,λi are even,μ1≥⋯≥μj≥1 and μ1,…,μj are multiples of 8,ν1≥⋯≥νk≥1 and ν1,…,νk are multiples of 8.     □ Definition A.40. A generalized Ramanujan congruence is - a sequence e=(e1,e2,e3,…)∈Z(1,2,3,…) as above, - an arithmetic progression (An+B)n≥0 with A≥2 and 1≤B≤A−1 - a prime power ℓf, with ℓ prime and f≥1, such that   p(An+B)e≡0 (modℓf) for all n≥0. (A.4)     □ Observation A.41. (1) Let p(An+B)e≡0 (modℓf) be a generalized Ramanujan congruence as above, and let m≥2. Define a sequence e′ by e′d=ed/m if m divides d and e′d=0 otherwise. Then we have   p(mAn+mB)e′≡0 (modℓf)  for all n≥0,p(mn+B′)e′=0  for all n≥0 and B′∈{1,2,…,m−1}. (A.5) Observe that the integers of the support {d≥1∣e′d≠0} of e′ have a common divisor m≥2. A generalized Ramanujan congruence is primitive if the integers in its support are coprime. All examples of generalized Ramanujan congruences appearing below are primitive. (2) In lists of examples involving congruences modulo ℓ (and not ℓf with f≥2), we write shortly p(ℓn+B)e for p(ℓn+B)e≡0 (modℓ). The Ramanujan congruences of this sort in Section 6 can therefore be written   p(5n+4)(1),  (7n+5)(1),  p(11n+6)(1),  p(5n+B)(2),  p(11n+4)(2). (With B∈{2,3,4}.) Moreover, we also write   p(ℓn+B)e, e′, …, e″ as a shorthand for p(ℓn+B)e, p(ℓn+B)e′, …, p(ℓn+B)e″. This shorthand notation will be used systematically in the lists of Subsections 3.2 to 3.6. (3) When we consider below generalized Ramanujan congruence involving a prime ℓ (and not a prime power ℓf with f≥2), it suffices to consider sequences e=(e1,e2,e3,…) with 0≤ed≤ℓ−1 for all d≥0. This is a corollary of the following standard proposition, for which we did not find a convenient reference.    □ Proposition A.42. Let ℓ be a prime, S(q)=∑n=0∞snqn,T(q)=∑n=0∞tnqn two power series in Z[[q]], and (pn+B)n≥0 an arithmetic progression of common difference ℓ and first term B≥1 not divisible by ℓ. Set   U(q)=S(q)(T(q))ℓ=∑n=0∞unqn. Assume that sℓn+B≡0 (modℓ) for all n≥0. Then uℓn+B≡0 (modℓ) for all n≥0.    □ Proof. For the binomial coefficients, we have the well-known congruences   (ℓj)≡0 (modℓ) for all j≥0 with j≢0 (modℓ). Hence the power series (T(q))ℓ=∑n=0∞t′nqn and T(qℓ)=∑n=0∞tnqℓn have coefficients that are congruent modulo ℓ; in particular, t′n≡0 (modℓ) for all n≥0 with n≢0 (modℓ). In particular, if sℓn+B≡0 (modℓ) for all n≥0, then uℓn+B≡0 (modℓ) for all n≥0.    ▪ Corollary A.43. Consider a sequence e=(e1,e2,e3,…)∈Z(1,2,3,…), an arithmetic progression (An+B)n≥0 with A≥2 and 1≤B≤A−1, a prime ℓ, and another sequence e′=(e′1,e′2,e′3,…)∈Z(1,2,3,…). Assume that e′d≡ed (modℓ) for all d≥0. If p(An+B)e≡0 (modℓ) for all n≥0 (as in Definition A.40), then p(An+B)e′≡0 (modℓ) for all n≥0.    □ We now proceed to indicate a list of examples of generalized Ramanujan congruences. We found these examples using computations on series modulo qN for a fairly large value of N. Subsequently, T. Cotron, R. Dicks, S. Fleming, have given proofs for all these identities [40]. We use the shorthand notation explained in Remark A.41(2). Appendix 3.2. Some examples of the form p(3n+B)e≡0 (mod3)   p(3n+2)(1,1), (2,1,0,2), (2,1,0,1,2,110,120), (1,1,0,2,1,110,220) . Appendix 3.3. Some examples of the form p(5n+B)e≡0 (mod5) For ℓ=5 and when ed=0 for all d≥3, we find the Ramanujan congruences   p(5n+2)(2), (3,1), (1,3), p(5n+3)(2), (4), (3,1), p(5n+4)(1), (2), (4), (2,2), (1,3) . When ed=0 for all d not dividing 4, we find moreover the Ramanujan congruences   When ed=0 for all d not dividing 6, we find moreover   When ed=0 for all d not dividing 8, we find moreover   Appendix 3.4. Some examples of the form p(7n+B)e≡0 (mod7)   Appendix 3.5. Some examples of the form p(11n+B)e≡0 (mod11)     Appendix 3.6. Some examples of the form p(13n+B)e≡0 (mod13) An incomplete list of (conjectural) primitive examples modulo 13 involving only unit-roots of order at most 4 is given by:     Appendix 3.7. Computational aspects We outline here briefly the discovery of the (conjectural) generalized Ramanujan congruences previously described. The computations where done in two steps. In a first step, we used series expansions of ∑n=0∞p(n)qn (with coefficients reduced modulo a small fixed prime l) and its powers up to degree N~200 in order to guess them. In a second step, we redid the computations up to degree N=5000 for the discovered examples (we did not encounter false positives, they should be rare since the probability for a false positive should naively be close to l−N/l for examples of the kind considered here.) Conjectural examples where guessed by considering all possible exponents ei∈{0,…,p−1} for i ranging over the set D(a) of all divisors of a small integer a (we considered mainly a∈{2,3,4,6,8}). We wrote a small Maple-program generating all l|D(a)| possible series ∏i∈D(a)Aiei up to order N over Fl where Aj=∑n=0∞P(n)qjn (with coefficients reduced modulo l and working only up to degree N), and checking for generalized Ramanujan congruences up to order N. Examples of generalized Ramanujan congruences seem surprisingly abundant, it is not hard to find them, they come in large numbers and seem to be very common. Moreover, the proofs by Cotron, Dicks, and Fleming [40] of the above identities can easily be adapted to additional cases. Funding This work was supported by the Swiss National Science Foundation. Communicated by Prof. Igor Rivin References [1] Andrews G. E. A Survey of Multipartitions: Congruences and Identities , Survey in Number Theory, 1– 19, vol. 17, Developments in Mathematics. New York, Springer, 2008. [2] Antolín Y. Ciobanu L. “ Formal conjugacy growth in acylindrically hyperbolic groups.” International Mathematicics Research Notices ( 2016): 1– 37. Google Scholar CrossRef Search ADS   [3] Babai L. “ The probability of generating the symmetric group.” Journal of Combinatorial Theory. Series A  52, no. 1 ( 1989): 148– 53. Google Scholar CrossRef Search ADS   [4] Babenko I. “ Closed geodesics, asymptotic volume and the characteristics of growth of groups.” Mathematics of the USSR-Izvestiya  33 ( 1989): 1– 37. [Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 52, no. 4 (1988): 675–711]. Google Scholar CrossRef Search ADS   [5] Barcelo H. Goupil A. “Combinatorial aspects of the Poincaré polynomial associated with a reflection group.” in “Jerusalem Combinatorics '93”, Contemporary Mathematics 178 (Amer, Math. Soc. 1994), 21–44. [6] Baudelaire C. Le peintre de la vie moderne, Le Figaro, November 26, 1863, http://baudelaire.litteratura.com/ressources/pdf/oeu_29.pdf (accessed on November 13, 2016). [7] Berndt B. C. Number theory in the spirit of Ramanujan . Student Mathematical Library 34, Providence, RI, American Mathematical Society, 2006. Google Scholar CrossRef Search ADS   [8] Bourbaki N. Groupes et algèbres de Lie, chapitres 4, 5 et 6 , Paris, Hermann, 1968. [9] Breuillard E. de Cornulier Y. Lubotzky A. Meiri C. “ On conjugacy growth of linear groups.” Mathematical Proceedings of the Cambridge Philosophical Socitey  154, no. 2 ( 2013): 261– 77. Google Scholar CrossRef Search ADS   [10] Carter R. “ Conjugacy classes in the Weyl group.” Compositio Mathematica  25 ( 1972): 1– 59. [11] Chandrasekharan K. Arithmetical Functions , New York, Berlin, Springer, 1970. Google Scholar CrossRef Search ADS   [12] Chen W. Y. C. Du D. K. Hou Q. Sun L. H. “ Congruences of multipartition functions modulo powers of primes.” Ramanujan Journal  35, no. 1 ( 2015): 1– 19. Google Scholar CrossRef Search ADS   [13] Coornaert M. Knieper G. “ An upper bound for the growth of conjugacy classes in torsion-free word hyperbolic groups.” International Journal of Algebra and Computation  14 ( 2004): 395– 401. Google Scholar CrossRef Search ADS   [14] Euler L. Introductio in analysin infinitorum, M.-M. Bousquet & Socios, Lausannæ, MDCCXLVIII. [Available on ETH Bibliothek web page: http://www.e-rara.ch/doi/10.3931/e-rara-8740. English translation by Ian Bruce: http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm.] (accessed on June 1, 2016). [15] Fine N. J. Basic Hypergeometric Series and Applications , vol. 27, Mathematic Surveys and Monographs, Providence, RI, American Mathematical Society, 1988. Google Scholar CrossRef Search ADS   [16] Fink E. “ Conjugacy growth and conjugacy width of certain branch groups.” International Journal of Algebra and Computation  24 ( 2014): 1213– 31. Google Scholar CrossRef Search ADS   [17] Gandhi J. M. “ Congruences for pr(n) and Ramanujan's τ function.” American Mathematical Monthly  70 ( 1963): 265– 74. Google Scholar CrossRef Search ADS   [18] Godsil C. Royle G. Algebraic Graph Theory . New York, Springer, 2001. Google Scholar CrossRef Search ADS   [19] Graham R. L. Knuth D. E. Patashnik O. Concrete Mathematics , Reading, MA, Addison-Wesley, 1989. [20] Guba V. Sapir M. “ On the conjugacy growth functions of groups.” Illinois Journal of Mathematics  54, no. 1 ( 2010): 301– 13. [21] Hardy G. H. Ramanujan S. “ Asymptotic formulae in combinatory analysis.” Proceedings of the London Mathematical Society (2)  17 ( 1918): 75– 115. Google Scholar CrossRef Search ADS   [22] Hardy G. H. Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work , Chelsea, 1959 [originally published by Cambridge Univ. Press, 1940]. [23] Hardy G. H. Wright E. M. An Introduction to the Theory of Numbers , 5th ed. New York, Oxford Univ. Press, 1979. [24] de la Harpe P. Ghys É. Haefliger A. Verjovsky A. “ An invitation to Coxeter groups.” in “ Group theory from a geometrical viewpoint (Trieste, 1990)”, River Edge, NJ, World Scientific Publishing Co. ( 1991): 193– 253. [25] Huber H. “ Über eine neue Klasse automorpher Funktionen und eine Gitterpunktproblem in der hyperbolischen Ebene. I.” Commentarii Mathematici Helvetici  30 ( 1956): 20– 62. Google Scholar CrossRef Search ADS   [26] Hull M. Osin D. “ Conjugacy growth of finitely generated groups.” Advances in Mathematics  235 ( 2013): 361– 89. “Corrigendum” in Advances in Mathematics 294 (2016): 857–59. Google Scholar CrossRef Search ADS   [27] Knieper G. “ Das Wachstum der Äquivalenzklassen geschlossener Geodätischer in kompakten Mannigfaltigkeiten.” Archiv der Mathematik (Basel)  40, no. 6 ( 1983): 559– 68. Google Scholar CrossRef Search ADS   [28] Lehrer G. I. “ On the Poincaré series associated with Coxeter group actions on complements of hyperplanes.” Journal of the London Mathematical Society  36, no. 2 ( 1987): 275– 94. Google Scholar CrossRef Search ADS   [29] Macdonald I. G. Symmetric Functions and Hall Polynomials , Second ed., New York, Oxford Univ. Press, 1995. [30] Mann A. How Groups Grow . Cambridge, Cambridge Univ. Press, 2012. [31] Margulis G. “ Certain applications of ergodic theory to the investigation of manifolds of negative curvature.” Functional Analysis and its Applications  3, no. 4 ( 1969): 335– 6. Google Scholar CrossRef Search ADS   [32] The on-line encyclopedia of integer sequences (OEIS), https://oeis.org (accessed on June 1, 2016). [33] Parkkonen J. Paulin F. “ On the hyperbolic orbital counting problem in conjugacy classes.” Mathematische Zeitschrift  279 ( 2015): 1175– 96. Google Scholar CrossRef Search ADS   [34] Rivin I. “ Growth in free groups (and other stories) – twelve years later.” Illinois Journal of Mathematics  54 ( 2010): 327– 70. [35] Sergiescu V. “ Graphes planaires et présentations des groupes de tresses.” Mathematische Zeitschrift  214 ( 1993): 477– 90. Google Scholar CrossRef Search ADS   [36] Shephard G. C. Todd J. A. “ Finite unitary reflection groups.” Canadian Journal of Mathematics  6 ( 1954): 274– 304. Google Scholar CrossRef Search ADS   [37] Solomon L. “ Invariants of finite reflection groups.” Nagoya Mathematical Journal  22 ( 1963): 57– 64. Google Scholar CrossRef Search ADS   [38] Solomon L. “ The orders of the finite Chevalley groups.” Journal of Algebra  3 ( 1966): 376– 93. Google Scholar CrossRef Search ADS   [39] Stanley R. P. Enumerative Combinatorics , vol. 1, Cambridge, Cambridge Univ. Press, 1997. Google Scholar CrossRef Search ADS   [40] Cotron T. Dicks R. Fleming S. Asgmptotics and congruences for partition functions which arise from finitary permutation groups. Preprint (2016) https://arxiv.org/abs/1606.09074 (accessed on November 13, 2016). To appear in Research in Number Theory. © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. 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Shkredov, Ilya D;Zhelezov, Dmitrii

doi: 10.1093/imrn/rnw291pmid: N/A

Abstract We prove that finite sets of real numbers satisfying $$|AA| \leq |A|^{1+\epsilon}$$ with sufficiently small $$\epsilon > 0$$ cannot have small additive bases nor can they be written as a set of sums $$B+C$$ with $$|B|, |C| \geq 2$$. The result can be seen as a real analog of the conjecture of Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible. 1 Introduction The duality between the additive and multiplicative structure of an arithmetic set, now called “sum–product phenomena” has been extensively studied since the seminal paper of Erdős and Szemerédi [5]. In its classical formulation the duality is expressed in the fact that for a finite set of integers or reals $$A$$ either the set of pairwise sums $$A+A$$ or pairwise products $$AA$$ is significantly larger than the original set, unless $$A$$ is close to a subring of the ambient ring, see [22] for details. In the current paper we consider an intrinsic version of the sum–product phenomenon in the following setting. Assume that a set $$A \subset \mathbb{R}$$ has small multiplicative doubling, so that $$|AA| \leq |A|^{1+\epsilon}$$ for some sufficiently small $$\epsilon > 0$$ (we assume that $$\epsilon$$ is fixed and $$|A| > C(\epsilon)$$ is large). Recall that a set $$B$$ is called a basis (All bases in the present paper are assumed to be of order two.) for $$A$$ if each element in $$A$$ can be represented as a sum of two elements in $$B$$ or simply $$A \subseteq B+B$$. First, we show that if $$A$$ is multiplicatively structured in the above sense, then it does not admit small additive bases of order $$|A|^{1/2+c}$$ for an explicit constant $$c > 0$$. Theorem 1.1. There is an $$\epsilon > 0$$ such that the following holds: if $$B$$ is an arbitrary additive basis for a real set $$A$$ with $$|AA| \leq |A|^{1+\epsilon}$$ then   |B|≫|A|1/2+1/442−o(1). □ Theorem 1.1 implies a “power-saving” estimate for the multiplicative energy $$E_\times$$ of a sumset or a difference set, significantly improving on the sub-exponential bound of [12]. Corollary 1. There is an effective constant $$\epsilon_0 > 0$$ such that for any set $$B$$ of real numbers holds   E×(B±B)≤|B|6−ϵ0. □ In fact, we prove an even stronger statement than that of Theorem 1.1, replacing the condition $$A \subseteq B + B$$ by a weaker assumption that the number of pairs $$(b_1, b_2) \in B \times B : b_1 + b_2 \in A$$ is large, see the upcoming Lemma 3.3 for details. While slightly more technical, such a reformulation may be useful if one wishes to take into account the number of ways an element in $$A$$ can be represented as a sum $$b_1 + b_2$$, see Remark 1. Second, we show that $$A$$ is additively irreducible, that is, it cannot be written as a set of sums $$B+C$$ unless one of the sets consists of a single element. Theorem 1.2. There is an $$\epsilon > 0$$ such that for all sufficiently large $$A \subset \mathbb{R}$$ with $$|AA| \leq |A|^{1+\epsilon}$$ there is no decomposition $$A = B + C$$ with $$|B|, |C| \geq 2$$. □ Both theorems substantially extend the results of Roche-Newton and the second author [12], where a much more restrictive condition $$|AA| \leq K|A|$$ with $$K$$ fixed, was assumed. In particular, in the case at hand deep structural results such as the Freiman-type theorem of Sanders (see, e.g., [14]) and the Subspace theorem of Evertse et al. [6] are no longer available. Instead, we rely solely on techniques of additive combinatorics, which we see as the main innovation of the present paper. On the other hand, Theorem 1.2 extends the result of the first author [19], which is the specialisation of Theorem 1.2 to the case $$C = -B$$. Clearly, Theorem 1.2 has the following sum–product result as a corollary. Corollary 2. There is an absolute constant $$c > 0$$ such that for any real sets $$B, C$$ with $$|B|, |C| \geq 2$$ holds   |(B+C)(B+C)|≫|B+C|1+c. □ Both questions considered in the paper arise as a natural extension of classical problems concerning thin additive bases and additive reducibility of integer sequences to the finite setting. One of the basic questions, posed by Erdős and Newman [4] in 1976, is to estimate the minimal basis size for a given set $$A$$ and, in particular, to decide if there is a very thin basis for $$A$$ of size of order $$|A|^{1/2}$$. They noted that while for a randomly picked set $$A$$ one should expect that no such basis exists, it is in general hard to prove such a claim for a specific set $$A$$. The question of additive irreducibility of integer sequences was posed by Ostmann [10] back in 1956. Perhaps the most notable conjecture of Ostmann, still wide open, is that the set of prime numbers cannot be written as a non-trivial set of sums $$B+C$$ even if one allows to discard a finite number of elements. We refer the reader to [2] for the history of the problem and [3] for the state of the art partial results (see also [7]). Sárközy [15] (see also [8] and references therein) extended such problems to the finite field setting, perhaps led by the intuition that in general “multiplicatively structured” sets should be additively irreducible (though this intuition may sometimes be misleading, as showed by Elsholtz in [1]). As a special case of this program, Sárközy conjectured that multiplicative subgroups of finite fields of prime order are additively irreducible, in particular the set of quadratic residues modulo a prime. Despite some progress (see, e.g., [18] and references therein), the conjecture of Sárközy is an important open problem with rich connections (to, e.g., multiplicative character sums). From this point of view, Theorem 1.2 can be seen as a real analog of Sárközy’s conjecture for small subgroups, and in fact the proof can be transferred to the finite field setting except a certain sum–product type estimate not currently available in finite fields. We discuss this matter in more detail at the end of the paper. The notation is briefly explained in the next section and is relatively standard in additive combinatorics. We recommend the reader to consult [22] for further details when needed. 2 Notation The following notation is used throughout the paper. The expressions $$X \gg Y$$, $$Y \ll X$$, $$Y = O(X)$$, $$X = \Omega(Y)$$ all have the same meaning that there is an absolute constant $$c > 0$$ such that $$|Y| \leq c|X|$$. The expressions $$X \gtrsim Y$$ and $$Y \lesssim X$$ both mean that $$|Y| = O(|X| \log^C |X|)$$ for some absolute constant $$C > 0$$. For a graph $$G$$, $$E(G)$$ denotes the set of edges and $$V(G)$$ denotes the set of vertices. If $$X$$ is a set then $$|X|$$ denotes its cardinality. For sets of numbers $$A$$ and $$B$$ the sumset $$A + A$$ is the set of all pairwise sums $$\{ a + a' : a, a' \in A \}$$, and similarly $$AA$$, $$A-A$$ denotes the set of products and differences, respectively. If $$G$$ is some graph with the vertex set identified with $$A$$ then $$A\stackrel{G}{+}A$$ denotes the set of sums $$\{ a + a' : (a, a') \in E(G) \}$$ (with the obvious generalisation to all arithmetic operations). For a bipartite graph $$G$$ with parts $$(V_1, V_2)$$ we will occasionally use the term bipartite density for the quantity $$\frac{|E(G)|}{|V_1||V_2|}$$. For a general graph, the edge density is defined as the quotient $$\frac{|E(G)|}{|V(G)|^2}$$, where the standard notation $$E(G), V(G)$$ is used for the set of edges and vertices, respectively. For a vertex $$v \in V(G)$$, $$N(v)$$ denotes the set of neighbours of $$v$$. The additive energy $$E_+(A)$$ (see [22], Definition 2.8) denotes the number of additive quadruples $$(a_1, a_2, a_3, a_4)$$ such that $$a_1 + a_2 = a_3 + a_4$$. The multiplicative energy $$E_\times$$ is defined similarly as the number of multiplicative quadruples. 3 Proof of Theorem 1.1 In what follows we will always assume that $$0 \notin A$$ but $$1 \in A$$ which one can do without loss of generality. We will use the following result of the first author (Theorem 5.4 in [17]). Theorem 3.1. Let $$A \subset \mathbb{R}$$ be such that $$|AA| = M|A|$$. Then   E+(A)≪M14/13|A|32/13log71/65⁡|A|. □ Combining with the Plünnecke–Ruzsa inequality [22, Corollary 6.29], we have the following corollary. Corollary 3. For any $$c < 1/26$$ there is $$\epsilon > 0$$ such that for any $$A \subset \mathbb{R}$$ with $$|AA| \leq |A|^{1+\epsilon}$$ holds   E+(AA/A)≪|A|5/2−c. □ Recall the following graph–theoretic lemma of Gowers (see, e.g., [22, Lemma 6.19]). Lemma 3.2. Let $$G$$ be a bipartite graph on $$(B_1, B_2)$$ where $$|B_1| = |B_2| = n$$ and $$0 < \alpha = E(B_1, B_2)/n^2$$. Let $$0 < \epsilon < 1$$ be fixed. Then there is a subset $$B'_1 \subseteq B_1$$ with $$|B'_1| \geq \alpha n/2$$ such that for at least $$(1-\epsilon)|B'_1|^2$$ of the ordered pairs of vertices $$(v_1, v_2) \in B'_1 \times B'_1$$ the following holds:   |N(v1)∩N(v2)|≥ϵα2n2. □ In what follows, we use a weaker definition of a basis, merely assuming that the number of pairs $$(b_1, b_2) \in B \times B: b_1 + b_2 \in A$$ is large. First, it allows one to take into account “multiplicities,” that is, elements in $$A$$ which have many representations as a sum. Second, it gives additional flexibility by relaxing the condition that $$B+B$$ contains all elements in $$A$$. For this matter, let us call a set $$B$$ an $$(L, K)$$-basis for $$A$$, with $$K,L \ge 1$$, if $$|B| = K|A|^{1/2}$$ and   |{(b1,b2):b1+b2∈A}|=L−1|A|. We will also make a use of the containment graph$$G$$, which is a bipartite graph $$G$$ on $$(B, B)$$ such that $$(b_1, b_2) \in E(G)$$ if and only if $$b_1 + b_2 \in A$$. In particular, if $$B$$ is an $$(L, K)$$-basis, one has $$|E(G)| = L^{-1}|A| = |B|^2/LK^2$$, so the bipartite edge density of $$G$$ is $$L^{-1}K^{-2}$$. Lemma 3.3 (The set of popular differences contains an almost closed difference set) Let $$B$$ be an $$(L, K)$$-basis for $$A$$. Then there is $$B' \subset B$$ with $$|B'| \gg L^{-1}K^{-2}|B|$$ such that for at least $$0.99 |B'|^2$$ of the ordered pairs $$(b_1, b_2) \in B' \times B'$$ the equation   b1−b2=a−a′ (1) has at least $$\Omega(L^{-2}K^{-3}|A|^{1/2})$$ solutions $$(a, a') \in A \times A$$. □ Proof. Applying Lemma 3.2 to the containment graph $$G$$ we obtain a set $$B'$$ of size $$\Omega(L^{-1}K^{-2}|B|)$$ such that for $$0.99 |B'|^2$$ of the ordered pairs $$(b_1, b_2) \in B' \times B'$$ holds   |N(b1)∩N(b2)|≥Ω(L−2K−4|B|). On the other side, if $$b \in N(b_1) \cap N(b_2)$$ then by construction $$b + b_1 = a$$ and $$b + b_2 = a'$$ for some $$a, a' \in A$$, which after rearranging gives   b1−b2=a−a′. Since, $$b + b_1$$ are all distinct as $$b$$ varies, we obtain at least $$|N(b_1) \cap N(b_2)|$$ distinct solutions of (1) for a fixed pair $$(b_1, b_2)$$ (and $$(b_2, b_1)$$ as well). The result follows. ■ Another ingredient is a beautiful incidence result, originally proved by Jones [9] with a shorter proof discovered by Roche-Newton [11]. Lemma 3.4. Let $$A \subset \mathbb{R}$$. Then the number of solutions to   (a−b)(a′−c′)=(a−c)(a′−b′) (2) such that $$a, a', b, b', c, c' \in A$$ is $$O(|A|^4 \log |A|)$$. □ The following lemma is crucial for the proof. Lemma 3.5 (Generating a large set of popular differences). Let $$B$$ be an $$(L, K)$$-basis for $$A$$. Then there is a set $$R$$ such that the following holds. (i) $$R \subseteq A/A$$. (ii) $$|R| \gtrsim L^{-8}K^{-14}|A|$$. (iii) For any $$x \in R$$, the equation   1−x=α1−α2 has at least $$\Omega(L^{-2}K^{-3}|A|^{1/2})$$ distinct solutions $$(\alpha_1, \alpha_2)$$ with $$\alpha_1, \alpha_2 \in A/A$$. □ Proof. Let $$B'$$ be the set given by Lemma 3.3. Let us call a pair $$(b_1, b_2) \in B' \times B'$$rich if $$|N(b_1) \cap N(b_2)| \gg L^{-2}K^{-3}|A|^{1/2}$$ in the containment graph $$G$$, so that at least 99% of pairs are rich. Take   R={b2+bb1+b:(b1,b2) is rich,b∈N(b1)∩N(b2)}. The first claim follows from the construction of the containment graph, so it remains to prove (ii) and (iii). For $$x \in R$$, define   n(x):=|{(b2,b1,b):x=b2+bb1+b,(b1,b2) is rich,b∈N(b1)∩N(b2)}| and   Q:=|{(b2,b1,b,b2′,b1′,b′)∈B6:b2+bb1+b=b2′+b′b1′+b′}|. We get by the Cauchy–Schwartz inequality   L−2K−3|B′|2|A|1/2≪∑x∈Rn(x)≤|R|1/2(∑x∈Rn2(x))1/2≤|R|1/2Q1/2. But by Lemma 3.4 we have $$Q \ll |B|^4 \log |B|$$ and combining with the bound $$|B'| \gg L^{-1}K^{-1}|A|^{1/2}$$ we obtain   L−8K−14|A|≲|R|. For the third bullet, for an $$x \in R$$ fix $$(b_1, b_2, b)$$ such that $$x = (b_2 + b)/(b_1 + b)$$ and observe that   1−b2+bb1+b=b1−b2b1+b=b1+b′b1+b−b2+b′b1+b. (3) If we vary over $$b' \in N(b_1) \cap N(b_2)$$, the pairs $$(b_1 + b')/(b_1 + b), (b_2 + b')/(b_1 + b)$$ are all distinct, and the claim follows since $$b_1 + b', b_2 + b', b_1 + b$$ are all in $$A$$ by construction. ■ he next lemma is simple but powerful. Lemma 3.6 (Generating a larger set additive quadruples). Assume $$1 \in X$$. Let $$R \subset X$$ such that for any $$x \in R$$ the equation   1−x=α1−α2 has at least $$N$$ solutions $$(\alpha_1, \alpha_2) \in X \times X$$. Then for any set $$Y$$ one has the estimate   E+(YX)≥N|Y||R|. □ Proof. By definition, $$E_+(YX)$$ is the number of additive quadruples $$(y_1, y_2, y_3, y_4)$$ such that $$y_i \in YX$$ and $$y_1 - y_2 = y_3 - y_4$$. If $$1 - x = \alpha_1 - \alpha_2$$ with $$x, \alpha_1, \alpha_2 \in X$$, then clearly $$(y, yx, y\alpha_1, y\alpha_2)$$ is an additive quadruple with elements in $$YX$$ (remember that $$1 \in X$$ so $$y \in YX$$). It remains to check that such quadruples are all distinct. But if   (y,yx,yα1,yα2)=(y′,y′x′,y′α1′,y′α2′) then $$y = y'$$ and $$(x, \alpha_1, \alpha_2) = (x', \alpha'_1, \alpha'_2)$$, therefore the number of additive quadruples is at least $$|Y|$$ times the number of distinct triples $$(x, \alpha_1, \alpha_2)$$, which is at least $$N|R|$$. ■ It remains to put everything together. Proof. (of Theorem 1.1) Let $$B$$ be an $$(L, K)$$-basis. Applying Lemma 3.5 and then Lemma 3.6 with $$Y = A$$ and $$X = R$$ from Lemma 3.5 we obtain   E+(AA/A)≫L−2K−3|A|3/2|R|≳L−10K−17|A|5/2. (4) On the other hand, by Corollary 3   E+(AA/A)≲|A|5/2−c provided $$|AA|/|A|$$ is small enough. We then have   L10K17≳|A|c. (5) In particular, if $$B$$ is a basis for $$A$$ then   |B|≳|A|1/2+c/17. ■ We record an immediate Corollary of Theorem 1.1. Corollary 4. There is an $$\epsilon > 0$$ and an absolute constant $$c > 0$$, such that for any $$A \subset \mathbb{R}$$ with $$|AA| \leq |A|^{1+\epsilon}$$ the following holds. If $$A \subset B+C$$ for some real sets $$B, C$$ then   max(|B|,|C|)≫|A|1/2+c. If $$B + C \subset A$$ for some real sets $$B, C$$ then   min(|B|,|C|)≪|A|1/2−c. □ Proof. The first claim follows immediately from Theorem 1.1 and the fact that $$B \cup C$$ is a basis for $$A$$ of size $$O(\max(|B|, |C|))$$. In the second case, assume $$|B| \leq |C|$$ and let $$B'$$ be an arbitrary subset of $$C$$ of size $$|B|$$. Then $$B \cup B'$$ is a basis for $$A' := B+B'$$ of size at most twice $$\min(|B|, |C|)$$. Applying Lemma 3.5 and then Lemma 3.6 we obtain similarly to (4)   E+(A′A′/A′)≳|B′|2|B′||B′|2=|B′|5. But clearly $$A' \subset A$$, so $$E_+(A'A'/A') \leq E_+(AA/A)$$. On the other hand, by Corollary 3   E+(AA/A)≲|A|5/2−c and the claim follows. ■ The proof of Corollary 1 is a simple application of the Balog–Szemerédi–Gowers theorem (BSG) (see [22]) so we present it in a sketchy manner. Let $$X = B \pm B$$. Since trivially $$E_{\times}(X) \ll |X|^3$$ we may assume that $$|X| \gg |B|^{2 - \epsilon}$$ with $$\epsilon$$ to be defined in due course. If now one assumes that   E×(X)≫|X|3−ϵ, then by BSG there is $$X' \subset X$$ such that $$|X'X'| \leq |X'|^{1+\epsilon'}$$ and $$|X'| \gg |X|^{1-\epsilon'}$$ with $$\epsilon'$$ depending polynomially on $$\epsilon$$. But $$X' \subset X = B \pm B$$ and whence $$B \cup (-B)$$ is a basis for $$X'$$ of size $$O(|X'|^{1/2 + \epsilon''})$$ with polynomial dependence of $$\epsilon''$$ on $$\epsilon$$. Taking $$\epsilon$$ small enough, we contradict Theorem 1.1 and the claim follows. Remark 1. The following question was posed in [19]. Assume that $$A\subseteq B-B$$, $$|AA| \le |A|^{1+\varepsilon}$$, where $$\varepsilon > 0$$ is small. Is it true that there is $$\delta= \delta(\varepsilon)$$ such that   σA(B):=∑x∈A|{b1−b2=x : b1,b2∈B}|≪|B|2−δ? It is easy to see that our methods allow one to resolve the problem in the affirmative. Indeed, put $$\sigma = \sigma_A (B)$$ and consider the graph with $$B$$ as the vertex set and two vertices $$x,y$$ adjacent if and only if $$x-y \in A$$. Then the edge density of the graph is $$\alpha := \sigma / |B|^2$$. Next, repeat the arguments of Section 3 and obtain that for a sufficiently small $$\varepsilon$$ holds   |A|5/2−c≳α2|B||A||R|≳α10|B|3|A| and hence since $$A\subseteq B-B$$  σ≲|B|2⋅|A|−c/10⋅(|A|3/2|B|3)1/10≤|B|2⋅|A|−c/10. (6) The asymmetric case with two different sets $$B,C$$ is considered in the next section. In this case one can obtain an analog of the upper bound (6) of the form   ∑x∈A|{b+c=x : b∈B,c∈C}|≲|B||C|⋅|A|−c/10. □ Remark 2. It is highly probable that the exponent $$1/2+1/442$$ in Theorem 1.1 can be improved, perhaps even to $$1-o(1)$$. We pose a much more modest question, which however does not seem to follow from the results of the present paper. Is it true that there exists $$c, \epsilon > 0$$ such that for all sufficiently large real sets $$B$$ such that $$A \subset B+B $$ with $$|AA| \leq |A|^{1+\epsilon}$$ holds   |B+B|≫|A|1+c? □ 4 Additive Irreducibility of Multiplicative Sets In this section we prove Theorem 1.2. From now on we assume for the sake of contradiction that $$A = B+C$$ with $$|B| \geq |C| \geq 2$$ and $$|AA| \leq |A|^{1+\epsilon}$$ with $$\epsilon$$ small enough. First, note that it follows from Lemma 29 of [19] that for $$\alpha \neq 0$$ and $$A$$ such that $$|AA| \leq M|A|$$ holds   |A∩(A+α)|≤M4/3|A|2/3. (7) For any $$c_1 \neq c_2 \in C$$ one has $$(B + c_1) \subseteq A \cap (A + (c_1 - c_2))$$, and thus by (7) and the trivial bound $$|B||C| \geq |A|$$ one concludes that $$|B|, |C| \gg |A|^{1/3 - \epsilon}$$, say. Hence, we can safely assume that both $$|B|, |C|$$ are large. An inspection of the proof of Theorem 1.1 reveals that if we put   X={b1+cb2+c:b1,b2∈B,c∈C} and   Y={c1+bc2+b:b∈B,c1,c2∈C} then one has that   E+(AA/A)≫min (|A||X||C|,|A||Y||B|). Thus, by Corollary 3 one has   |X||C|≪|A|3/2−c and   |Y||B|≪|A|3/2−c (8) for some explicit $$c > 0$$. However, sufficiently good lower bounds for $$|X|$$ and $$|Y|$$ are not readily available. Let $$T(X, Y, Z)$$ be the number of collinear triples of distinct points $$(x, y, z)$$ with $$x \in X \times X$$, $$y \in Y \times Y$$, $$z \in Z \times Z$$. Following the lines of Lemma 3.5 we have by the Cauchy–Schwarz inequality   |X|≫|B|4|C|2T(B,B,−C), (9) and   |Y|≫|C|4|B|2T(C,C,−B), (10) Indeed, without loss of generality it suffices to check that if, say,   y:=c1+bc2+b=c1′+b′c2′+b′ then either the points $$(c_1, c'_1), (c_2, c'_2), (-b, -b')$$ are all distinct and collinear or $$y \in \{0, 1, \infty \}$$. But we can simply exclude such degenerate values from $$X$$ and $$Y$$ and thus justify (9) and (10). We prove the following estimate with $$|C| \leq |B|$$ which improves on Lemma 3.4 when $$|B|$$ is significantly larger than $$|C|$$. Lemma 4.1 (Bounding collinear triples for different sets). For sets $$C \times C$$ and $$B \times B$$ with $$|B| \geq |C|$$ holds   T(C,C,B)≪|B|4/3|C|8/3log2⁡|B|. □ Proof. Write $$L_{i, j}$$, $$i,j\ge 0$$ for the set of lines $$\ell$$ such that $$2^{i} \leq |\ell \cap (C \times C)| < 2^{i+1}$$ and $$2^{j} \leq |\ell \cap (B \times B)| < 2^{j+1}$$. Then   T(C,C,B)≪∑i=0log⁡|C|∑j=0log⁡|B||Li,j|22i2j. Since the number of summands is at most $$\log^2 |B|$$ it is enough to bound each term by $$|B|^{4/3}|C|^{8/3}$$. For the sake of notation, denote $$k = 2^i$$ and $$l = 2^j$$, $$L = L_{i,j}$$ so that out task is to estimate $$|L|k^2l$$ where $$L$$ is the set of lines intersecting $$C \times C$$ in $$k$$ (up to a factor of two) points and $$B \times B$$ in $$l$$ points (again, up to a factor of two). By the Szemerédi–Trotter theorem [22], for $$k \geq 2, l \geq 2$$ holds   |L|≪min(|C|4k3+|C|2k,|B|4l3+|B|2l), so   T:=k2l|L|≪min(l|C|4k+kl|C|2,k2|B|4l2+k2|B|2):=min(M1,M2). Before we proceed, let us rule out the cases not covered by the Szemerédi–Trotter theorem as stated above. Since each line contains at least two distinct points in $$C \times C$$, only the case $$l = 1$$ should be considered separately. In this case we have   T≪|C|4k+k|C|2≪|C|4≤|B|4/3|C|8/3, which is the desired bound. Next, since always $$k \leq |C|$$, $$l \leq |B|$$, we obtain that $$\frac{ l |C|^4}{k} \gg M_1$$ and $$\frac{k^2 |B|^4}{l^2} \gg M_2$$, so   T3≪M12M2≪(l|C|4k)2⋅k2|B|4l2=|C|8|B|4 or   T≪|B|4/3|C|8/3. This finishes the proof. ■ Combining Lemma 4.1, estimate (10), and the trivial bounds $$|B||C| \geq |A|, |B| \geq |A|^{1/2}$$, we have $$T(C, C, B) \lesssim |C|^{8/3}|B|^{4/3} $$ and   |Y||B|≳|C|4/3|B|2/3|B|=(|C||B|)4/3|B|1/3≥|A|4/3+1/6=|A|3/2, which contradicts (8) if $$|A|$$ is large enough. Thus, $$A$$ is additively irreducible. Remark 3. Instead of using the graph approach as it was done in Sections 3 and 4 one can apply in the proof of Theorem 1.2 the following analog of the formula (3)   1−b1+cb2+c=b2+c′b2+c(1−b1+c′b2+c′), which holds for any $$b_1,b_2\in B$$ and all $$c,c'\in C$$. It means in particular, in the notation of Section 4, that taking an arbitrary $$x\in X$$, we have at least $$|C|$$ solutions to the equation   1−x=y(1−x∗), where $$y\in Y$$ and $$x_* \in X$$ are fixed. Of course, one can replace $$X$$ and $$Y$$ in the last formula. It immediately gives that $$E_{+} (AA/A) \ge |X||C||A|$$ and $$E_{+} (AA/A) \ge |Y||B||A|$$. □ 5 Discussion First, let us note that the only feature of the reals we have used, which is not available in an arbitrary field, is the Szemerédi–Trotter theorem (which is also used in the proof of Theorem 3.1). Thus, our results can be readily extended to the sets of complex numbers thanks to the extensions of the Szemerédi–Trotter theorem to the complex plane due to Zahl [25] and Toth [23]. However, when we replace $$A$$ with a sufficiently small multiplicative subgroup of $$\mathbb{F}_p$$, the situation becomes more subtle. For multiplicative subgroups Theorem 3.1 can be substituted with a similar energy bound, beating the exponent $$5/2$$, as was shown by the first author [16]. Next, it follows from the result of Shkredov and Vyugin [20] (see also the paper [21] by Shparlinski) that if a multiplicative subgroup $$G$$ is written as a non-trivial sumset $$B+C$$, then necessarily $$|G|^{1/2+o(1)} \ll |B|, |C| \ll |G|^{1/2+o(1)}$$. Further, a slightly weaker form of the Szemerédi–Trotter bound for the number of incidences for Cartesian products in $$\mathbb{F}^2_p$$ is now available, see [24]. Thus, the only remaining obstacle for translating Theorem 1.2 to subgroups is to find a substitute for Lemma 3.4, which we were unable to do. However, as was shown in [24], see also [13] the bound $$O(|A|^4 \log |A|)$$ of Lemma 4 can indeed be replaced with $$O(|A|^{9/2})$$, which only barely falls short of the desired bound. Finally, let us mention that similarly to Section 3 one can slightly weaken the hypothesis of Theorem 1.2 and assume only that the number of pairs $$(b, c) \in B \times C : b + c \in A$$ is large. We leave the details for the interested reader. Funding This work was supported by Russian Scientific Foundation grant 14-11-00433 to I.D.S. Acknowledgements The authors would like to thank Tomasz Schoen, Misha Rudnev, and Oliver Roche-Newton for useful discussions and the anonymous referees for valuable suggestions. The second author would like to acknowledge the hospitality of Adam Mickiewicz University in Poznan at which the work on the present paper was partially conducted. References [1] Elsholtz, C. “Multiplicative decomposability of shifted sets.” Bulletin London Mathematical Society  40, no. 1 ( 2008): 97– 107. Google Scholar CrossRef Search ADS   [2] Elsholtz, C. “Additive decomposability of multiplicatively defined sets.” Functiones et Approximatio Commentarii Mathematici  35, no. 1 ( 2006): 61– 77. Google Scholar CrossRef Search ADS   [3] Elsholtz, C., and Harper. A. “Additive decompositions of sets with restricted prime factors.” Transactions of the American Mathematical Society  367, no. 10 ( 2015): 7403– 27. Google Scholar CrossRef Search ADS   [4] Erdős, P., and Newman. D. J. “Bases for sets of integers.” Journal of Number Theory  9, no. 4 ( 1977): 420– 5. Google Scholar CrossRef Search ADS   [5] Erdős, P., and Szemerédi. E. Sums and Products of Integers . Studies in Pure Mathematics. Basel: Birkhäuser, 1983. Google Scholar CrossRef Search ADS   [6] Evertse, J.-H., Schlickewei, H. P. and Schmidt. W. M. “Linear equations in variables which lie in a multiplicative group.” Annals of Mathematics  155, no. 3 ( 2002): 807– 36. Google Scholar CrossRef Search ADS   [7] Green, B., and Harper. A. “Inverse questions for the large sieve.” Geometric and Functional Analysis  24, no. 4 ( 2014): 1167– 203. Google Scholar CrossRef Search ADS   [8] Gyarmati, K., Mauduit, C. and Sárközy. A. “On reducible and primitive subsets of $$\mathbb{F}_p$$, I.” Integers 15A  ( 2015): A6, 21. [9] Jones, T. G. F. “New quantitative estimates on the incidence geometry and growth of finite sets.” PhD Thesis,  University of Bristol, arXiv:1301.4853, 2013. [10] Ostmann, H.-H. Additive Zahlentheorie , l. 2, Berlin: Springer, 1956. [11] Roche-Newton, O. “A short proof of a near-optimal cardinality estimate for the product of a sum set.” 31st International Symposium on Computational Geometry,  vol. 34, 74– 80, Wadern, Schloss Dagstuhl. Leibniz-Zent. Inform., 2015. [12] Roche-Newton, O., and Zhelezov. D. “A bound on the multiplicative energy of a sum set and extremal sum-product problems.” Moscow Journal of Combinatorics and Number Theory  5, no. 1 ( 2015): 53– 70. [13] Roche-Newton, O., Rudnev, M. and Shkredov. I. D. “New sum-product type estimates over finite fields.” Advances in Mathematics 293 ( 2016): 589– 605. [14] Sanders, T. “The structure theory of set addition revisited.” Bulletin of the American Mathematical Society (N.S.)  50, no. 1 ( 2013): 93– 127. Google Scholar CrossRef Search ADS   [15] Sárközy, A. “On additive decompositions of the set of quadratic residues modulo $$p$$.” Acta Arithmetica  155, no. 1 ( 2012): 41– 51. Google Scholar CrossRef Search ADS   [16] Shkredov, I. D. “Some new inequalities in additive combinatorics.” Moscow Journal of Combinatorics and Number Theory 3,  no. 3–4 ( 2013): 237– 88. [17] Shkredov, I. D. “Some new results on higher energies.” Transactions of the Moscow Mathematical Society  74 ( 2013): 31– 63. Google Scholar CrossRef Search ADS   [18] Shkredov, I. D. “Sumsets in quadratic residues.” Acta Arithmetica  164, no. 3 ( 2014): 221– 43. Google Scholar CrossRef Search ADS   [19] Shkredov, I. D. “Difference sets are not multiplicatively closed.” Discrete Analysis  no. 17 ( 2016): 20 pp. [20] Shkredov, I. D. and Vyugin. I. V. “On additive shifts of multiplicative subgroups.” Matematicheskii Sbornik  203, no. 6 ( 2012): 81– 100. Google Scholar CrossRef Search ADS   [21] Shparlinski, I. “Additive decompositions of subgroups of finite fields.” SIAM Journal on Discrete Mathematics  27, no. 4 ( 2013): 1870– 9. Google Scholar CrossRef Search ADS   [22] Tao, T., and Vu. V. Additive Combinatorics . Cambridge, Cambridge University Press, 2006. Google Scholar CrossRef Search ADS   [23] Tóth, C. “The Szemerédi-Trotter theorem in the complex plane}.” Combinatorica  35, no. 1 ( 2015): 95– 126. Google Scholar CrossRef Search ADS   [24] Yazici, A. S., Murphy, B. Rudnev, M. and Shkredov. I. D. “Growth estimates in positive characteristic via collisions.” International Mathematics Research Notices  2016 ( 2016): 1– 42. [25] Zahl, J. “A Szemerédi-Trotter type theorem in $$\Bbb{R}4$$.” Discrete & Computational Geometry  54, no. 3 ( 2015): 513– 72. Google Scholar CrossRef Search ADS   © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. 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