PSL(2, ℂ ), the exponential and some new free groups

PSL(2, ℂ ), the exponential and some new free groups Abstract We prove a normal form result for the groupoid of germs generated by PSL(2,ℂ) and the exponential map. We discuss three consequences of this result: (1) a generalization of a result of Cohen about the group of translations and powers, which gives a positive answer to a problem posed by Higman; (2) as proof that the subgroup of Homeo(ℝ,+∞) generated by the positive affine maps and the exponential map is isomorphic to an HNN-extension; (3) a finitary version of the immiscibility conjecture of Ecalle–Martinet–Moussu–Ramis. 1. Introduction 1.1. Normal forms We recall some basic concepts and terminology from the theory of groupoids (see for example, [4]). A groupoid is a category G whose objects Obj(G) form a set and in which every morphism is an isomorphism. For each x,y∈Obj(G), we denote by G(x,y) the set of morphisms in G from x to y. We denote also by G the disjoint union of G(x,y), for all x,y∈Obj(G). The composition of morphism is written multiplicatively: if f∈G(x,y) and g∈G(y,z), then these morphisms can be composed and its composition is the morphism gf∈G(x,z). From now on, when we write the expression gf for two morphisms f,g, we are tacitly assuming that these morphism can be composed. The symbol 1 will generally denote the identity morphism and f−1∈G(y,z) will denote the inverse of the morphism f∈G(x,y). We will say that morphism f has source x=s(f) and target y=t(f) if f∈G(x,y). The group G(x,x) will be called vertex group at x and will be denoted simply by G(x). A finite sequence of morphisms [f1,f2,…,fn] in G is called a path if s(fi)=t(fi+1). Given such a path, we will say that f=f1⋯fn∈G is the morphism defined by the path. The operation of concatenation in the set of paths is defined in the obvious way, taking into account the source/target compatibility. A path [f1,f2,…,fn] is called reduced if: no two consecutive morphisms fi, fi+1 are mutually inverse; if some fi is the identity morphism then n=1 and f=[1].We can give a groupoid structure to the set of reduced paths. The operation of composition of two paths is defined as follows: first concatenate the paths and then successively eliminate all consecutive terms which are mutually inverses. The resulting groupoid is called the free groupoid on the graph of G ([4, Section 8.2]). Given a differentiable manifold M, let G(M) denote the Haefliger groupoid over M (see for example, [20, Section 5.5]). We recall that, by definition, the set of objects Obj(G(M)) is the set of points of M and, for each p,q∈M, G(M)(p,q) is the set of all germs of diffeomorphisms (M,p)→(M,q). In order to keep the traditional naming, we will refer to the morphisms of G(M) simply as germs. Given a map f:U→V, where U,V⊂M are open sets and f is a local diffeomorphism (that is, locally invertible), we denote by Germ(f)⊂G(M) the smallest wide subgroupoid containing all the germs f,p of f at all points p of its domain. We recall that a subgroupoid G1 of a groupoid G2 is called wide if Obj(G1)=Obj(G2). More generally, given an arbitrary collection C of local diffeomorphisms as above, we denote by Germ(C)⊂G(M) the smallest subgroupoid containing Germ(f), for all f∈C. From now on, we shall assume that M=P1(C) and that all maps are holomorphic. As a basic object, we will frequently consider the groupoid GExp=Germ(exp) associated to the usual exponential map. This is the groupoid whose germs at each point are given by finite compositions f=f1⋯fn of the following germs:   {1,p}∪{exp,p:ifp∈C}∪⋃k∈Z{lnk,p:ifp∈C*},where 1,p is the identity germ at p and lnk,p is the germ at p of the kth-branch of the logarithm, that is, the map   lnk:C*⟶Jk={x+iy:y∈](2k−1)πi,(2k+1)πi]}z⟼ln(∣z∣)+iargk(z),where argk:C→](2k−1)πi,(2k+1)πi] is the kth-branch of the argument function. In general, we have the relation   exp,qlogk,p=1,qfor all p∈C and q=logk(p). On the other hand,   logk,qexp,p:z↦z+2πi(k−s)for all p∈Js and q=exp(p). In particular, notice that germ corresponding to the translation by 2πi lies in GExp.   In what follows, we are going to simplify the notation and omit the subscripts ,p when referring to the germ of a local diffeomorphism at a point p of its domain. Thus, the same symbol, say exp, will denote both the exponential map and the germ at each point of its domain. In the situation where we want to emphasize that we are considering its germ at a specific point p, we will simply write that s(exp)=p. We also introduce the following symbols for the (germ of) exponential map and the zeroth branch of the logarithm:   e:z↦exp(z),l:z↦ln0(z).Another important object for us is the groupoid   GPSL(2,C)=Germ(PSL(2,C)).We recall that the group PSL(2,C) is generated by the subgroups   W={1:z↦z,w:z↦1/z},T={ta:z↦z+a,a∈C}S={sα:z↦αz,α∈C*}which are, respectively, the involution, the translations and the scalings. We denote by Aff⊂PSL(2,C) the subgroup of affine maps, that is, Aff=T⋊S. Following the above notational convention, the same symbols ta,sα and w will be used to denote the corresponding germs in GPSL(2,C). Our main result is a normal form for elements in the groupoid   GPSL(2,C),Exp=Germ(PSL(2,C)∪{exp}).In order to state this result, consider the subgroups H0,H1⊂PSL(2,C) given by   H0=T⋊{s−1},andH1=S⋊{w}.For the next definition, we recall that, given a group G and a subgroup H⊂G, a right transversal for H is a subset T⊂G of representatives for the right cosets {Hg:g∈G} which contains the identity of G. Definition 1.1 Let T0,T1⊂PSL(2,C) be right transversals for H0,H1, respectively. A (T0,T1)-normal form in GPSL(2,C),Exp is a path   g=[g0,h1,g1,…,hn,gn],n≥0such that the following conditions hold: The germ g0 lies in GPSL(2,C). For each 1≤i≤n, hi∈{e,l}. If hi=e then gi∈T0. If hi=l then gi∈T1. There are no subpaths of the form [e,1,l] or [l,1,e].We denote by NFT0,T1 the set of all (T0,T1)-normal forms. The path [1] will be called the identity normal form.There is an obvious mapping   φ:NFT0,T1→GPSL(2,C),Expwhich associates to each normal form g=[g0,h1,…,gn] the germ φ(g)=g0h1⋯gn. The main goal of this paper will be to study the surjectivity and injectivity properties of this mapping. Remark 1.2 As we shall see in Lemma 2.1, a possible choice of transversals T0,T1 for H0,H1, respectively, is as follows:   T0={sρ:ρ∈Ω}∪{sρwtb:ρ∈Ω,b∈C}T1={tb:b∈C}∪{tawtb:a∈Ω,b∈C*⧹{−1/a}}∪{tcw:c∈C*},where Ω={α:Re(α)>0}∪{α:Re(α)=0,Im(α)>0} is the region shown in Fig. 1. (In fact, we could define similar transversals by choosing any region in C* which is a fundamental domain for the Z2-action z↦−z and contains 1.) Figure 1. View largeDownload slide The region Ω. Figure 1. View largeDownload slide The region Ω. From now on, in order to simplify the exposition, we shall fix the choice of transversals T0,T1 as described in Remark 1.2, and write NFT0,T1 simply as NF. Each result that we are going to discuss in the remainder of the paper can be appropriately translated to different choices of transversals. In order to state the Main Theorem, we need define certain special normal forms. To simplify the notation, we shall frequently omit the square braces and write a path [g1,…,gn] simply as g1⋯gn. For each α∈C*, the power map with exponent α is the germ defined by   pα=esαl,that is, the germ of power map z↦zα obtained by choosing the zeroth branch of the logarithm. A normal form a∈NF will be called an algebraic path (resp. rational path) of length n≥0 if it has the form   a=θ0pα1θ1⋯pαnθn,where αi are exponents in Ω∩Q (resp. αi∈Ω∩Z), and   θ0∈PSL(2,C),θn∈T1andθ1,…,θn−1∈T1⧹{1}.We will say that a is of affine type if θi is an affine map for each 0≤i≤n−1. All paths of length n=0 are of affine type. Notice that each path g∈NF can be decomposed as   g=a0γ1a1⋯γmam,m≥0,where each ai is an algebraic path and each γi is either e, l or a power germ pαi with an exponent αi∈Ω⧹Q. This decomposition is unique if we further require that there are no subpaths of the form   γiaiγi+1=esαl.In other words, we assume that each subpath esαl is grouped together into written as the power map pα. The above unique decomposition of g will be called the algebro-transcendental decomposition. Each ai will be called a maximal algebraic subpath of g. The natural number m will be called the height of g and noted height(g). Hence, normal forms of height zero correspond to algebraic paths. Given symbols η1,η2∈{e,l,p}, we will say that the maximal algebraic subpath ailies in a [γ,η]-segment, if γi=η1 and γi+1=η2. Example 1.3 The path g=es2llt1es2lt1wew is a normal form with algebro-transcendental decomposition   g=1︸a0p2︸γ11︸a1l︸γ2t1p2t1w︸a2e︸γ3w︸a3The maximal algebraic subpaths a1 and a2 lie in [p,l] and [l,e] segments, respectively. Notice that maximal algebraic subpaths can be the identity, as it is the case of a0 and a1. We will say that a normal form g∈NF is tame if Either height (g)=0 and g is an algebraic path of affine type. Or height (g)≥1 and each maximal algebraic subpath lying in a segment of type   [e,l],[l,e],[l,p],[p,e],or[p,p]is of affine type.For instance, the normal form of the previous example is tame. We shall denote by NFtame the subset of tame normal forms. Main Theorem (Normal form in GPSL(2;ℂ),Exp) The mapping  φ:NF→GPSL(2,C),Expis surjective. Moreover, this mapping is injective when restricted to NFtame. The study of non-tame normal forms puts into play some difficult problems concerning the study of finite coverings P1(C)→P1(C) with imprimitive monodromy groups. This issue is strongly related to the well-known Ritt's decomposability theorem [23], which fully describes monoid structure of the polynomials under the composition operation. The following example shows that we cannot expect the map ρ:NF→GPSL(2,C),Exp to be bijective without further restrictions. Example 1.4 Each Chebyshev polynomial Tn(x) lies in GPSL(2,C),Exp, as it can be defined by the identity   Tn=φpnφ−1,where pn(z)=zn and φ(z)=z+1/z is given explicitly by   φ=s−1t2wt−1/4p2t−1/2wt1.On the other hand, we can also express T2(z)=z2−2 as t−2p2. Hence, the relation   T2(z+1z)=z2+1z2is equivalent to say that the normal form   s−1t2wt−1/4p2t−1/2wt1p2t−1wt1/2p1/2t1/4wt−2p1/2t2defines the identity germ. Of course, this normal form is not tame. Remarks 1.5 (1) Some readers will probably notice the similarities between the above normal form and Britton's normal form for HNN-extensions (see for example, [16, IV.2]). Indeed, there is a general notion of HNN-extension for groupoids ([4, Section 8.4.1]) which generalizes the usual notion for groups (see Section 1.4). At a first view, one could expect to prove that GPSL(2,C),Exp is isomorphic to the HNN-extension of GPSL(2,C), with the exponential e acting as the stable letter, that is, somehow conjugating the subgroupoids GH0 and GH1. This cannot hold (at least not in such a naïve way). In fact, consider the subgroupoids L0 and L1 obtained by restricting the groupoids GH0 and GH1 to the subdomains C and C*, respectively. Then, the exponential map indeed defines a morphism of groupoids by   Θ:L0⟶L1h⟼ehe−1,where the germ e−1 is chosen in such a way that s(h)=t(e−1). At the level of objects, this induces the mapping Θ:C→C*, Θ(p)=exp(p). However, Θ is not an isomorphism of groupoids, since it annihilates all germs t2πik, with k∈Z; and identifies each two points in C which differ by an integer multiple of 2πi. As a matter of fact, Θ establishes an isomorphism between the groupoid L1 and the quotient groupoid L0/Ker(Θ), which is simply the groupoid with the object set C/2πiZ and morphisms given by the action of {ta:a∈C} and s−1 modulo 2πiZ. (2) It is easy to see that GPSL(2,C),Exp coincides with GT,Exp, that is, the groupoid generated only by the translations and the exponential. Indeed, one easily constructs the subgroups S and W by defining   sα=etln0(α)l,andw=e2tiπl2,for all α∈C*. The Normal form theorem could be formulated solely in terms of paths in GT,Exp. However, this would lead to a much more complicated enunciation and to the loss of the analogy with the theory of HNN-extensions. (3) Notice that GPSL(2,C) has a natural Lie groupoid structure, which is inherited from étale groupoid structure of G(P1(C)) (see [20, Section 5.5]). Some readers may be wondering which is the relation between GPSL(2,C) and the so-called semi-direct product Lie groupoid  PSL(2,C)⋉P1(C),which is naturally defined by the action of PSL(2,C) on P1(C) (see [20, Section 5.1]). One can show that PSL(2,C)⋉P1(C) and GPSL(2,C) are isomorphic as groupoids, but not as Lie groupoids. Indeed, the source fibers of GPSL(2,C) (that is, the sets s−1(p), p∈P1(C)) have a discrete topology, while all source fibers of PSL(2,C)⋉P1(C) are manifolds diffeomorphic to PSL(2,C). (4) Another interesting construction can be obtained by combining the groupoids Germ(PSL(2,C)), Germ(exp) and Germ(℘,℘′), where   ℘:C/Λ→P1(C)is the Weierstrass function associated to a period lattice Λ⊂C. In this case, the resulting groupoid G would contain a rich class of rational maps, the so-called finite quotients of affine maps (see [19]), that is, rational maps f of degree two or more which fit into commutative diagrams of the form   where l(z)=az+b is an affine map defined on C/Λ and Θ:C/Λ→P1(C) is a finite covering. For instance (see [18, Problem 7-f]), for Λ=Z⊕iZ and l(z)=(1+i)z, the germ ℘l℘−1 is (up to a conjugation by a Möebius map) the quadratic rational map h(z)=(z+1/z)/2i. 1.2. Powers and affine maps As a consequence of the Main Theorem, we are going to obtain a generalization of a result of Cohen. Let R be an arbitrary multiplicative subgroup of C* and let PowR be the set of germs determined by all the branches of the power maps   C*∋z↦zr,withr∈R.Clearly, the associated groupoid Germ(PowR) is simply obtained by taking the union of PowR with the identity germs 1 at 0 and ∞. As above, for each r∈R, we denote by   pr=esrlthe germ of power map obtained by choosing the zeroth branch of the logarithm. Initially motivated by a question of Friedman, several authors (cf. [12]) considered the groupoid   GAff,PowR=Germ(Aff,PowR)whose elements are obtained by finite compositions of germs of affine and power maps. In particular, they studied the following property: Definition 1.6 We will say that GAff,PowR has the amalgamated structure property if each element GAff,PowR can be uniquely defined by a path   [g0,pr1,ta1,…,prn,tan]for some n≥0, where g0∈Germ(Aff), ri∈R⧹{1} and ai∈C for i=1,…,n, such that ai is non-zero for 1≤i≤n−1. In particular, this property implies that, given n≥1 and two sequences of constants r1,…,rn∈R⧹{1} and a1,…,an∈C with ai non-zero for 1≤i≤n−1, the germ defined by   z↦(a1+(a2+⋯+(an+x)rn⋯)r2)r1(where we choose arbitrary branches for the power maps) cannot be the identity. Building upon a method originally introduced by White [28], Cohen proved in [5] that GAff,PowQ>0 has the amalgamated structure property (that is, one takes R equal to Q>0). Using our normal form Theorem, we prove the following: Theorem 1.7 The groupoid GAff,PowRhas the amalgamated structure property if R∩Q<0=∅. Equivalently, we assume that for each r∈R, the ray rQ<0 does not intersect R.   Remark 1.8 Assume R is the multiplicative subgroup of C* generated by exp(2πiλ1),…,exp(2πiλn), for some collection of complex numbers λ1,…,λn. Then, the condition R∩Q<0=∅ is equivalent to the following non-resonance condition:   (12+iln(Q>0))∩(Z+λ1Z+⋯+λnZ)=∅,where ln denotes the principal branch of the logarithm function. 1.3. Generalized Witt algebras We describe another consequence of the Normal Form Theorem. Let M be an additive sub-monoid of C (that is, a subset M⊂C which is closed under addition and contains zero). Following [1], we define the generalized Witt algebra W(M) as the C-vector space with a basis {wg:g∈M}, subject to the Lie multiplication   [wg,wh]=(g−h)wg+h.Each basis element can be represented by a (possibly multivalued) complex vector field on P1(C) given by   wg=zg(z∂∂z)whose flow at time a is given by the multivalued map   exp(awg)=z↦{(−ag+z−g)−1/g,ifg≠0exp(a)z,ifg=0.Following the conventions of the first subsection, we are going to denote also by exp(awg) the germs in G(P1(C)) obtained by taking all possible determinations of the maps z↦(−ag+z−g)−1/g at all points of its domain of definition. Example 1.9 For M=Z we obtain the classical Witt algebra W(Z). The subalgebra W(Z≤0)⊂W(Z) plays an important role in holomorphic dynamics. The flow maps in this subalgebra can be written as   z↦exp(aw−k)(z)=z(1−akz−k)1/k,∀k∈Z≥0,∀a∈Cand they generate a well-known subgroup of the group Diff(C,∞) of germs of holomorphic diffeomorphisms fixing the infinity. Theorem 1.10 Let Mbe an arbitrary additive sub-monoid of C. Then, for all n≥1, all scalars a1,…,an∈C⧹{0}and all elements g1,…,gn∈M⧹{0}such that gi+1/gi∉Q<0∪{1}, the germ  z↦exp(a1wg1)⋯exp(anwgn)(z)cannot be the identity. Remark 1.11 The condition gi+1/gi≠1 must be imposed due to the trivial relation   exp(awg)exp(bwg)=exp((a+b)wg).Moreover, there are numerous counter-examples to the above result if drop the assumption gi+1/gi≠−1. For instance, given a∈C*, consider the so-called two parabolic group Ga⊂PSL(2,C), which is the group generated by the time a flows maps of w−1 and w1, namely   z↦exp(aw−1)(z)=z+a,z↦exp(aw1)(z)=z1+az.Following [15], we say that a is a free point if Ga is a free group. There are plenty of non-free points. For instance, Ree showed in [21] that the real segment ]−2,2[ is contained in an open set where the non-free points are densely distributed. Assume that a is a non-free point. Then, by definition, there exist a n≥1 and non-zero integers p1,q1,…,pn,qn such that   exp(p1aw1)exp(q1aw−1)⋯exp(pnaw1)exp(qnaw−1)=1.Clearly, each relation of this type would give a counter-example to the above theorem if the assumption gi+1/gi+1≠−1 were dropped. Our next goal is to state a normal form result for the groupoid   GM=Germ({exp(awg):g∈M,a∈C}).For this, given g∈M, and a∈C, we define the following germ:   Φa,g={p−1/gt−agp−g,ifg≠0sexp(a),ifg=0,where ta and sα are the translation and scalings germs, respectively; and the power map pr is defined as in Section 1.2. In other words, Φa,g∈GM is simply the germ obtained from the (multivalued) flow map exp(awg) by choosing the zeroth branch of the logarithm in the definition of the power maps x↦x−g and x↦x−1/g. The phenomena described in the previous remark leads us to define the following concept. We say that an additive sub-monoid M of C has no rational antipodal points if   M∩(MQ≤0)={0}.Using our Main Theorem, we shall prove the following: Theorem 1.12 (Normal form in GM) Suppose that Mhas no rational antipodal points. Then, each element of the groupoid GMis uniquely defined by a path  [Φa0,0,Φa1,g1,…,Φan,gn]for some n≥0, gi∈Mand ai∈Csuch that: gi∈M⧹{0}and ai∈C⧹{0}, for 1≤i≤n, Im(a0)∈]−πi,πi], gi≠gi+1, for 1≤i≤n−1. Remark 1.13 For sub-monoids M having antipodal points, it follows from Remark 1.11 that a normal form result as above would depend on precise characterization of the set of free points. This seems to be a very difficult problem. As a hint, we refer to Fig. 2, reproduced from [11]. It shows a numerically computed representation the set of free points in the plane Cλ, where λ=a2/2. Figure 2. View largeDownload slide The known free points are unshaded. Figure 2. View largeDownload slide The known free points are unshaded. 1.4. HNN-extensions in Homeo(R,+∞) Going in another direction, we can consider the analogous problem for the group Homeo(R,+∞) of germs at +∞ of real homeomorphisms defined in open intervals of the form {x:x>x0} and which go to infinity as x goes to infinity. The group operation being the composition. Consider the following subgroups of Homeo(R,+∞),   T={ta:x↦x+a,a∈R},S+={sα:x↦αx,α∈R>0}Exp={e:x↦exp(x),e−1:x↦ln(x)},where ln:R*→R is obviously taken as the real branch of the logarithm. Let Aff+=T⋊S+ denote the subgroup of real positive affine maps. As it is well known, the conjugation by the exponential map defines an isomorphism   θ:T⟶S+ta⟼etae−1=sexp(a)and we can consider the group Aff+⋆θ derived from (T,Aff+,θ) by HNN extension. We recall (see for example, [24, 1.4]) that given a group G with presentation G=⟨F∣R⟩ and an isomorphism θ:H→K between two subgroups H,K⊂G, the HNN extension derived from (H,G,θ) is a group G⋆θ with presentation   G⋆θ=⟨F,𝚔∣R,𝚔h𝚔−1=θ(h),∀h∈H⟩.The new generator 𝚔 is called stable letter. Consider now the subgroup GAff+,Exp of Homeo(R,+∞) generated by Aff+∪Exp. From the universal property of the HNN extensions, we know that there is a unique morphism   ϕ:Aff+⋆θ⟶GAff+,Expwhich is the identity when restricted to Aff+ and which maps the stable letter to the exponential map. We claim that GAff+,Exp contains no other relations besides the one expressing that exp conjugates T to S+. In other words, Theorem 1.14 ϕ:Aff+⋆θ→GAff+,Expis an isomorphism. Remark 1.15 Based on the above result, we can obtain a quite economic presentation for the group GAff+,Exp, namely   GAff+,Exp=⟨R,𝚔∣(𝚔a𝚔−1)b(𝚔a𝚔−1)−1=exp(a)b,∀a,b∈R⟩,where R is equipped with its usual additive group structure. For instance, the multiplicative structure of S+ is easily obtained by defining sexp(a)≔𝚔a𝚔−1. As another consequence, we obtain a large collection of (apparently new) free subgroups inside Homeo(R,+∞). Indeed, consider the family of subgroups {Tn}n∈Z*⊂Homeo(R,+∞) given by   T0=T,Tn=θn(T0)=enT0e−n,∀n∈Z*,where for n>0 (resp. n<0), en denotes the n-fold composition of e (resp. e−1). Notice that S+=T1. We define   An=θn(Aff+)=Tn⋊Tn+1,∀n∈Z,where Tn+1 acts on Tn by conjugation (exactly as S+ acts on T). Corollary 1.16 The subgroup of GAff+,Expgenerated by ⋃n∈ZAnis isomorphic to the infinite free amalgamated product given by the following diagram:  where the north-east and north-west arrows are, respectively, the identity inclusions and the monomorphism S+=θ(T). In [12], Glass attributes to Higman the following question: Do T and Pow+={x↦xr:r∈R>0}generate their free product? The above corollary allows us to answer this question affirmatively. Indeed, as A1=S+⋊Pow+, the above diagram shows that the subgroup GAff+,Pow+ of Homeo(R,+∞) generated by T∪S+∪Pow+ has the presentation   GAff+,Pow+=(T⋊S+)⋆S+(S+⋊Pow+),where the amalgam is obviously made over S+. 1.5. Transseries and a finitary version of Lemma 1 We follow the notation from [9]. Let T=R[[[x]]] be the real ordered field of well-based transseries and P⊂T be the subset large positive transseries. Then, P is a group under the composition operation and there is a injective homomorphism   T:GAff+,Exp→Pwhich associates to each element g∈GAff+,Exp its transseries at infinity. Indeed, each germ in GAff+,Exp defines element in the Hardy field H(Ran,exp) (see for example, [26]), and therefore this homomorphism is a direct consequence of the embedding of H(Ran,exp) into T (see [26, Corollary 3.12]). In this subsection, we shall be concerned with the following property (see for example, [6, 13]): Definition 1.17 Given an element ϕ∈P and a subgroup H⊂P, we shall say that H and ϕ are immiscible if the subgroup generated by H∪ϕHϕ−1 is isomorphic to the free product H⋆H. For each integer k≥1, let Gk⊂P denote the subgroup real formal series at +∞ which are tangent to identity to order k, that is, the group of transseries of the form   x↦x+∑j≥k−1bjx−j,withbj∈R.The following problem is stated in [7] (see also [6, Section 1.4]): Immiscibility problem. Prove that G2and ϕ are immiscible in the following cases:   ϕ:x↦exp(x),ϕ:x↦x+ln(x),orϕ:x↦xλ,where λ∈R>0⧹Q>0. Remarks 1.18 (1) The immiscibility problem naturally appears in the study of the Poincaré first return map in the vicinity of an elementary polycycle. Such study is an essential ingredient in the proofs of the Finiteness theorem of limit cycles for polynomial vector fields in the plane (see in [6, 13]). According to the strategy sketched in [7, 8], one expects that a positive answer to the immiscibility problem would allow us to significantly simplify these proofs. (2) The immiscibility problem has an obvious negative answer if G2 is replaced by G1. Indeed, given an arbitrary non-identity element f∈G1 and a scalar a∈R*, consider the series   g=sexp(a)fsexp(−a)which is also an element of G1. Then, using the identity eta=sexp(a)e one can rewrite   g=etae−1fet−ae−1.Since the translation ta is an element of G1, one obtains the following relation in the subgroup generated by G1∪eG1e−1:   etae−1fet−ae−1=sexp(a)fsexp(−a),which shows that this subgroup is not isomorphic to the free product G1⋆G1. Notice that an element f∈Gk can be written as the limit of a (Krull convergent) sequence {fn}n≥k⊂P given by   f0=1,fn=T(exp(anw−n))fn−1,wherew−n=x−n(x∂∂x)with constants an∈R uniquely determined by f and the flow maps exp(anw−n) being given by Example 1.9. This motivates us to consider the subgroup Gk,finite⊂Gk of those elements f which can be expressed as finite words, namely   f=T(exp(a1w−k1)⋯exp(anw−kn))for some n≥0, ai∈R and ki∈Z≥k. Notice that each Gk,finite is indeed defined by an analytic function in a neighborhood of infinity and lies in the image of the morphism T considered above. It also lies in the Hardy field H(Ran,exp) (cf. [26, Section 3.11]). In order to formulate our next result, let Λ be the subset of all non-identity elements g∈GAff+,Exp of the form   g=g1⋯gnfor some n≥1 and gi∈{e,l}∪{pr:r∈R>0⧹Q>0}. As a consequence of the previous theorem and the Normal form Theorem, we obtain the following finitary immiscibility property: Theorem 1.19 Let g be an arbitrary element in Λ. Then, G2,finiteand ϕ=T(g)are immiscible. Remarks 1.20 (1) Notice that the result includes the case where g is given by towers of exponentials and powers, such as   g:x→eexr,r∈R>0⧹Q>0.However, it does not include the so-called inverse log-Lambert map,   L:x↦x+ln(x)which is a solution of the differential equation   (x1+xddx)L=1.The map L plays an important role in proof of the finiteness of limit cycles. Indeed, it constitutes one of the building blocks in the construction of the Dulac transition map near a hyperbolic saddle or a saddle-node. We believe that it is possible to adapt our proof to include this function in the statement of the above theorem. (2) The passage from G2,finite to G2 in the immiscibility problem seems to be outside the reach of the tools developed in this paper. A possible strategy of proof could consist in appropriately identifying G2 to some subset of ends in the Bass–Serre tree defined by the HNN-extension Aff+⋆θ. 2. Formal theory in GPSL(2,C),Exp In this section, we will start our proof of the Main Theorem. As a first step, we recall some basic universal constructions in groupoid theory, following closely [4]. 2.1. Free product and quotient of groupoids Let G and H be groupoids, and let j1:G→K, j2:H→K be morphisms of groupoids. We say that these morphisms present K as the free product of G and H if the following universal property is satisfied: if g:G→L and h:H→L are morphisms of groupoids which agree on Obj(G)∩Obj(H), then there is a unique morphism k:K→L such that kj1=g, kj2=h. Such free product always exists (see [4, Section 8.1]) and will be noted G∗H. If the groupoids G and H have no common morphism except the identity, the elements of G∗H are can be identified with the set of paths   [g1,g2,…,gn]which are either equal to [1] or where each gi belongs to either G or H, no gi is the identity, and gi, gi+1 do not belong to the same groupoid. We now recall the construction of the quotient of a groupoid by a set of relations. In a groupoid G, suppose given, for each object p, a set R(p) of elements of G(p) (the vertex group at p). The disjoint union R of the R(p) is called a set of relations in G. We define the normal closure N=N(R) of R as the following subgroupoid: Given an object x∈Obj(G), a consequence of R at x is either the identity at x or any morphism of the form   an−1ρnan⋯a1−1ρ1a1for which ai∈G(x,xi) and ρi, or ρi−1, is an element of R(xi). The set of all consequences at a point x, which we note N(x), is a subgroup of G(x) and the disjoint union N of all N(x) has the structure of a totally disconnected normal subgroupoid of G (see [4], Section 8.3), where by totally disconnected groupoid we mean a groupoid where each morphism has its source equal to its target. It can be shown that N is the smallest wide normal groupoid of G which contains R. Let G/N(R) be the quotient groupoid (see [4, Theorem 8.3.1]). The projection π:G→G/N(R) has the following universal property: for each morphism of groupoids f:G→H which annihilates R, there exists a unique morphism f′:G/N(R)→H such that f=f′π. 2.2. Product normal form in GPSL(2,C)∗ΓGExp The essence of our Normal Form theorem is to present GPSL(2,C),Exp as the quotient of a free product of groupoids by some explicit set of relations. For this, we consider the groupoids   GPSL(2,C)=Germ(PSL(2,C)),GExp=Germ({exp})and let ΓGExp denote the free groupoid on the graph of GExp, that is, the groupoid defined by the set of reduced paths on GExp (see Section 1.1). Let F=GPSL(2,C)∗ΓGExp be the free product of these groupoids. A first necessary step to obtain a normal form in F is to describe the normal forms in GPSL(2,C) and ΓGExp. We need two preparatory Lemmas: Lemma 2.1 Each element g∈PSL(2,C)can be written as one of the following expressions:   g=sαtawtborg=sαtbfor some uniquely determined constants α∈C*and a,b∈C. Moreover, if we consider the region Ω⊂Cgiven by  Ω={α:Re(α)>0}∪{α:Re(α)=0,Im(α)>0}(see Fig. 1, at the Introduction), the following holds: Each right coset of H0=T⋊{s−1}in PSL(2,C)contains a unique element of the form  g=sρwtb,org=sρfor some constants b∈Cand ρ∈Ω. Each right coset of H1=S⋊{w}in PSL(2,C)contains a unique element of the form  g=tawtb,g=tcw,org=tbfor some constants c∈C⧹{0}, b∈Cand a∈Ωsuch that b≠−1/a. Proof The first part of the lemma follows from the well-known presentation of PSL(2,C) (see for example, [14, XI]). In particular, we recall the following relation in PSL(2,C):   1a+1z=−1a2(−a+1z+1a),∀a∈C⧹{0},z∈C,or, equivalently,   wtaw=s−1/a2t−awt1/a.Now, in order to prove items (i) and (ii), it suffices to study the orbit of sαtb and sαtawtb under the left multiplication by H0 and H1, respectively. For instance, given sαtawtb∈PSL(2,C) such that a≠0, the above relation in PSL(2,C) allows us to write   sαtawtb≡(s−a2sαw)sαtawtb≡s−a2wtawtb≡t−awtb+1/a,where ≡ denotes the equivalence in H1⧹PSL(2,C). Therefore, the coset H1sαtawtb contains either an element of the form tawtb with a∈Ω and b≠−1/a or an element of the form tcw, with c≠0.□ To state the next result, we introduce the symbols   lk:z↦lnk(z),∀k∈Z,where we recall, lnk denotes the kth branch of the logarithm. Notice that l0=l. Lemma 2.2 Each element in ΓGExpis either the identity 1or a path of the form  [lk1,…,lkn,e,…,e︸s−times]for some positive integers n,s, not both zero, and integers k1,…,kn∈Zsuch that the rightmost germ lknand the leftmost germ ein the path are not mutually inverses. Proof Each germ g∈GExp is defined by a path   [g1,g2,…,gm],where each gi is either equal to e or to lk for some k∈Z. We transform this path to a reduced one by successively canceling out each two consecutive germs gi,gi+1 such that gigi+1=1. Recall now the following (unique) two relations in GExp (see the discussion at the Introduction),   (1)elk=1and(2)lke=1,ifs(e)∈Jk,for all k∈Z. Therefore, after performing all possible cancellations in the above path, we either obtain the identity path, or a path as above such that no germ lk has a germ e to its left; and furthermore, that no consecutive germs lk,e are mutually inverse. This is precisely a path of the form in the statement of the lemma.□ We now consider normal forms inside the free product groupoid F=GPSL(2,C)∗ΓGExp. Definition 2.3 A product normal form in F is a path of the form   g=[g0,h1,f1,g1,…,hn,fn,gn]for some n≥0, such that the following holds: The germ g0 lies in GPSL(2,C) (with possibly g0=1). For 1≤i≤n, hi is either equal to e or to lk, for some k∈Z. If hi=e then fi∈H0 and gi is given by item (i) of Lemma 2.1. If hi=lk then fi∈H1 and gi is given by item (ii) of Lemma 2.1. There are no subpaths [lk,1,1,e] or [e,1,1,lk] such that the germs lk and e are mutually inverse.We denote by PNF the set of all product normal forms. As a consequence of the definition of F and the previous two lemmas, we obtain the following. Proposition 2.4 Each morphism of Fcan be uniquely defined by an element of PNF. Proof By the definition of a free product, each non-identity element g∈F can be uniquely identified with a path   g=[g1,g2,…,gn]such that the following conditions hold: gi∈GPSL(2,C)∪ΓGExp, for i=1,…,n. No two consecutive morphisms gi, gi+1 belong to the same groupoid. No gi is the identity morphism.Given such a path, we can uniquely obtain a path in PNF. Indeed, proceeding from left to right, for i=1,…,n, we do the following: If gi∈ΓGExp, then we use Lemma 2.2 to write   gi=[lk1,…,lkn,e,…,e]and, in the expression of g, we replace gi by the subpath   [lk1,1,1,lk2,…,lkn,1,1,e,1,1,…,e]. If i≥1, gi belongs to GPSL(2,C) and gi−1 has an e as its last symbol, then we use Lemma 2.1 to write   gi=fg′,forsomef∈H0,andg′givenbyLemma2.1(i)and we replace gi by the subpath [f,g′] in the expression of g. If i≥1, gi belongs to GPSL(2,C) and gi−1 has an lk as its last symbol, then we use Lemma 2.1 to write   gi=fg′,forsomef∈H1,andg′givenbyLemma2.1(ii)and we replace gi by the subpath [f,g′] in the expression of g.This concludes the proof.□ Remark 2.5 Recall that the subgroup of translations by 2πiZ lies in the intersection GExp∩GPSL(2,C). Therefore, the normal forms in the free product groupoid GPSL(2,C)∗GExp are more subtle to describe than those in F. Let now NF be the set of normal form paths defined in Remark 1.2 of the Introduction. Clearly, there is a natural embedding of NF into PNF given by   [g0,h1,g1,…,hn,gn]∈NF⟶[g0,h1,1,g1,…,hn,1,gn]∈PNF.To simplify the notation, we will keep the symbol NF to denote the image of this embedding. 2.3. Quotienting GPSL(2,C)∗ΓGExp Now, we consider the following collection Rel of relations in F:   Rel{le=t−2πir,ifs(e)∈Jr.es−1=weeta=sexp(a)e,∀a∈C,where l=l0 is the 0th branch of the logarithm. Notice that, for simplicity, we have written these relations in the form of an equality of germs w=u, but this should be understood as saying that w composed with the inverse of u is a relation (in the sense of Section 2.1) at every point where the corresponding germs are defined. Let F/N(Rel) denote the quotient groupoid, as defined in the previous subsection, and let   π:F⟶F/N(Rel)be the canonical morphism. The following theorem will be proved in the next subsection. Theorem 2.6 Each element in the quotient F/N(Rel)is uniquely defined by a normal form in NF. We now observe that, by construction and the universal property of F, there is a uniquely defined groupoid epimorphism   ϕ:F→GPSL(2,C),Expwhich is induced by the inclusion morphisms GPSL(2,C)→GPSL(2,C),Exp and GExp→GPSL(2,C),Exp. Using the obvious relations between the exponential, the affine maps and the involution, we conclude that this morphism factors out through the canonical morphism π:F→F/N(Rel), that is, we have a commutative diagram   for a uniquely defined morphism φ:F/N(Rel)→GPSL(2,C),Exp. As an immediate consequence of this discussion and Theorem 2.6, we obtain: Corollary 2.7 The first statement of the Main Theorem is true. 2.4. Reduction to normal forms in F/N(Rel) This subsection is devoted to the proof of Theorem 2.6. For this, we briefly recall the basic concepts of reduction systems (see for example, [3]). An abstract reduction system is a pair (X,→) where the reduction → is a binary relation on the set X. Traditionally, we write x→y (or y←x) instead of (x,y)∈→. The binary relation →* is the reflexive transitive closure of →. In other words, x→*y if and only if there is x0,…,xn such that x=x0→x1→⋯→xn=y. The binary relation ↔* is the reflexive transitive symmetric closure of →. Equivalently, x↔*y if and only if there are z1,…,zn∈X such that   x↔z1↔z2⋯↔zn↔y,where ↔=←∪→. We also say that: x∈X is reducible if there is a y∈X such that x→y. x∈X is in normal form if it is not reducible. x∈X is a normal form of y∈X if y→*x and x is a normal form. x,y∈X are joinable if there is a z∈X such that x→*z←*y.A reduction system (X,→) is called terminating if there is no infinite descending chain x0→x1→⋯. In this case, each element x has at least one normal form. A reduction system (X,→) is called confluent if y1←*x→*y2 implies that the elements y1 and y2 are joinable. We say that (X,→) is Church–Rosser if x↔*y implies that x and y are joinable. These two properties are usually pictured by the following respective diagrams:   We shall use the following consequences of the definitions: if (X,→) is terminating and confluent, then every element has a unique normal form (see [3, Lemma 2.1.8]). The Church–Rosser and the confluent properties are equivalent (see [3, Theorem 2.1.5]).We are going to apply this formalism to the set X=PNF of product normal forms (see Definition 2.3). In order to simplify the notation, in the remainder of this subsection, we shall identify a path [f1,…,fn] with a word f1f2⋯fn in the letters f1,…,fn. We stress that the formal word f1f2⋯fn should not be confounded with the element of the groupoid GPSL(2,C),Exp defined by the corresponding path. The letter l0 will be written simply l. Moreover, the identity path [1] will be identified with the empty word ε. Thus, for instance, the path [e,1,1,e] will be written simply as ee. First of all, we introduce the following reduction rules (recall that both sides of the ⇒ relation should be seen as paths in F):   lk⇒t2πikl,∀k∈Z*el⇒ε,le⇒t−2πir,ifs(e)∈Jres−1⇒we,eta⇒sexp(a)e,∀a∈C,lw⇒s−1lifarg0(s(w))≠πlw⇒t2πis−1lifarg0(s(w))=πlsα⇒tbl,∀α∈C*,where, in this last rule, we define b∈C as follows:   b={ln0(α),if−π<arg0(α)+arg0(s(sα))≤πln−1(α),ifπ<arg0(α)+arg0(s(sα))≤2πln1(α),if−2π<arg0(α)+arg0(s(sα))≤−π.The reduction system (PNF,→) is now defined as follows: Given g,h∈PNF, we say that g→h if there exists some reduction rule u⇒v as above such that one can write   g=g′ug″,forsomeg′,g″∈PNF,and h∈PNF is the product normal form of the path g′vg″. Remark 2.8 Notice that the simple substitution g′ug″→g′vg″ would not map PNF into itself. For instance, if b=ln0(−2) then, applying the fifth substitution rule to g=etaetb, one would obtain etas−2e, which is not in PNF, since tas−2 should be decomposed in H0T0 as (tas−1)s2. Proposition 2.9 The reduction system (PNF,→)is terminating and confluent. Moreover, the set of normal forms of (PNF,→)is precisely the subset NF. Proof We claim that there can be no infinite sequence of reductions. To prove this, we define a well-order on PNF which will decrease after each reduction. First of all, recall that a germ f∈H0∪H1 is either the identity or can be uniquely expressed as follows: In H0: f=tas−1, f=ta or f=s−1, for some a∈C⧹{0}, In H1: f=sαw, f=sα or f=w, for some α∈C*⧹{1}.Accordingly, we define the h-length lh(f)∈{0,1,2} by   lh(f)=2iff∈{tas−1,sαw},lh(f)=1iff∈{ta,sα,s−1,w},and we put lh(f)=0 if f=1. Consider now a path g=g0h1f1g1⋯hnfngn in PNF. We define its h-length as the integer n-vector   lh(g)=(lh(fn),lh(fn−1),…,lh(f1))∈{0,1,2}n.We further define m(g) to be the total number of germs of type lk, for k∈Z*, and n(g) to be the total number of germs of type e or l in the expression of g. Finally, we define a total order in PNF by saying that g<g′ if   (m(g),n(g),lh(g))<lex(m(g′),n(g′),ln(g′)),where <lex is the lexicographical ordering in the set of positive integer vectors. By inspecting the rules in Rel, one sees that if g′→g then g<g′. Moreover, a path g∈PNF is not reducible if and only if the following holds: m(g)=0, lh(g)=(0,…,0), and g contains no subpath of the form el or le.According to Definition 1.1, this corresponds precisely to say that g∈NF. Thus, we have proved that (PNF,→) is terminating and that its set of normal forms is NF. In order to prove the confluence of the reduction system, we use Bendix–Knuth criteria as stated in [10, Lemma 6.2.4]. Thus, it suffices to consider all shortest paths for which at least two of the above reduction rules can be applied (that is, they overlap) and show that the paths obtained after applying these reductions are then joinable. For instance, one sees that   The computation for the other possible overlaps is straightforward but quite tedious. We omit this computation.□ We are now ready to prove Theorem 2.6: Proof of Theorem 2.6 We need to prove that each coset of F/N(Rel) contains exactly one element of NF. By the fact that (PNF,→) is terminating, we know that each coset of F/N(Rel) contains at least one element of NF. Now, the essential remark is that the equivalence relation ↔* on PNF defines precisely the cosets of the quotient groupoid F/N(Rel). Indeed, for each relation u=v in the list Rel given at Section 2.3, one sees that u↔*v. Reciprocally, for each reduction rule u⇒v, one sees that uv−1 belongs to N(Rel). Therefore, assume that there exist two elements g,g′ in NF such that π(g)=π(g′) (where π:F→F/N(Rel) is the quotient map). This is equivalent to say that g↔*g′. Since (PNF,→) is confluent, it is Church–Rosser. Therefore, g↔*g′ implies that g and g′ are joinable. But since both g and g′ are normal forms (and hence not reducible), we conclude that g=g′.□ Remark 2.10 (Word problem and decidability) One could ask if the reduction system (PNF,→) would allow us to algorithmically solve the word problem in F/N(Rel). Equivalently, one asks if, given an element g∈PNF, there exists an algorithm to decide if   g→*1.Notice that the reduction rules in (PNF,→) assume the existence of an oracle which, given a complex constant α∈C, will answer affirmatively or negatively to the question   Isα=0?Even assuming that the constants appearing in the initial path g are, say, rational numbers, this oracle will eventually need to test new constants which are exp–log expressions in these initial constants, such as   eee2log(3/4)+e−3ee10−eee2ln5−ln(ln(3/2)).The existence of an algorithm for the above oracle is strongly related to the decidability of (R,exp) and the known algorithms assume Schanuel's conjecture. On the positive side, using the results of [22, 25] (see also [17, Section 2.1]), one can prove the following: Assuming Schanuel’s Conjecture, the word problem is decidable for the groupoid  FQ/N(Rel),where FQdenotes the free product groupoid GPSL(2,Q)*ΓGexp. 3. From normal forms to field extensions Our present goal is to prove the second part of the Main Theorem. Many of the following constructions will be carried out for arbitrary normal forms, not necessarily satisfying the tameness property. We shall explicitly indicate the points where this assumption will be necessary. Given a point p∈P1(C), we denote by (M,∂) the differential field of meromorphic germs at p equipped with the usual derivation ∂=d/dz with respect to some arbitrary local coordinate z at p (with constant subfield Const(∂)=C). Given a normal form g∈NF, with source point p=s(g), our next goal is to construct a sequence of field extensions in M which will encode the necessary information to study the identity   φ(g)=id,where we recall that φ:NF→GPSL(2,C),Exp is the mapping which associates a germ in GPSL(2,C),Exp to each path in NF. 3.1. Algebraic paths and Cohen field In this subsection, we consider field extensions defined by algebraic paths. Let a∈NF be an algebraic path of length n≥0. Thus, we can uniquely write   a=θ0pα1θ1⋯pαnθn,where each exponent αi lies in Ω∩Q⧹{1}, θ0∈PSL(2,C), θi∈T1⧹{1}, for 1≤i≤n and θ1,…,θn−1 are not the identity. We consider the sequence of algebraic field extensions in M  En⊂En−1⊂⋯⊂E0=Einductively defined as follows. First, En=C(xn) is the field defined by the identity germ xn=φ(1). Then, for each i=0,…,n−1, we define   Ei=Ei+1(xi),where xi is a germ of solution of the algebraic equation   xiv=θ(xi+1)u,where we have written θ=θi+1 and αi+1=u/v for some co-prime integers u,v. Here, the branch of the vth-root is uniquely chosen accordingly to the source/target compatibility condition determined by a. We will say that the resulting field   E=C(x0,…,xn)is the Cohen field of a. In the seminal paper [5], Cohen has studied the Cohen field for algebraic normal forms of affine type. In what follows, we shall make essential use of the following immediate consequence of a result in [5]. Theorem 3.1 (cf. [5, Theorem 3.2]) Assume that a is an algebraic normal form of affine type and length n≥0, with associated Cohen field E=C(x0,…,xn). Then, we can write  E=C(x0,xn)that is, x1,…,xn−1are rational functions of x0and xn. Moreover, if a is not a rational path (Recall (see the Introduction) that an algebraic path is called rational if each power map in its basic decomposition has a exponent in Ω∩Z), then E is a strict algebraic extension of C(xn). Notice that the second statement of the theorem does not hold if a is not of affine type. Example 3.2 Consider the algebraic normal form   a=p1/2t−1/4p2θp2,where θ is a Möebius map such that θ(0)=−1/2 and θ(∞)=1/2. Then, we can write Ea=C(x,y,z), where x,y,z satisfy the relations   z2=−x2(x2+1)2andy=z2.Obviously, C(x,z) is not a strict algebraic extension of C(x). From now on, the elements y=xn and z=θ0(x0) (where θ0 is the Möebius part of a) will be called, respectively, the tail and the head elements of the Cohen field of a. Corollary 3.3 Assume that a is a non-identity algebraic normal form of affine type, with Cohen field E. Then, the head and tail elements z,y∈Ecannot satisfy the relation  yz=1. Proof We consider the following three possible cases: a is not a rational path. a is a rational path of length n≥1. a=θ is a Möebius map.In case (1), the result is an immediate consequence of Theorem 3.1. In case (2), we write the decomposition of a as   a=θ0pu1θ1⋯punθnfor some integers u1,…,un∈Ω∩Z. It follows that E is the function field of an algebraic curve C⊂(P1(C))n+1 which defined by the vanishing of the ideal n⊂C[X0,…,Xn] generated by the equations   xn−1=θn(xn)un,…,x0=θ1(x1)u1(after appropriately eliminating the denominators in the expression of the Möebius maps). In particular, the rational map y=xn defines a degree 1 unbranched covering C→P1(C), while the rational map z=θ(x0) defines a branched covering C→P1(C) of degree u=∣u1⋯un∣≥2. Therefore, y/z is a non-constant rational map on C. Finally, in the case (3), we have the identity x0=xn=y. Therefore, we can write   zy=θ(y)yand the right-hand side is non-constant because θ≠1. This concludes the proof.□ Remark 3.4 Example 1.4 shows that Corollary 3.3 is false if we drop the condition that a is of affine type. 3.2. The field associated to g∈NF We will now generalize the construction of the previous section to arbitrary normal forms. Assume that g∈NF has an algebro-transcendental decomposition   g=a0γ1a1⋯γmam,m≥0.Then, we inductively construct a chain of subfields in M,   Lm⊂Fm⊂⋯⊂L0⊂F0=F,as follows. First, we define Lm=C(ym) to be the field defined over C by the identity germ ym=φ(1). Then, we put: Fi=Li(zi), where zi∈M is the germ defined   zi=φ(aiγi+1ai+1⋯γmam),i=0,…,m. Li=Fi+1(yi), where yi∈M is the germ defined   yi=φ(γi+1ai+1⋯γmam),i=0,…,m−1.We will say that F=C(y0,z0,…,ym,zm) is the field associated to g. Notice that each field extension Fi+1⊂Li (described in item (2)) is obtained by adjoining to Fi+1 a (germ of) non-zero solution y to one of the following differential equations:   ∂(y)y=∂(z),∂(y)=∂(z)zor∂(y)y=α∂(z)z,where we have written write z=zi+1. These equations correspond respectively to the case where γi+1 is given by e, l or pα, for some α∈Ω⧹Q. On the other hand, each extension Li⊂Fi (described in item (1)) is algebraic, obtained by adjoining a germ of solution zi of a polynomial equation with coefficients in C(yi). More precisely, if we consider the Cohen field E associated to the maximal algebraic path ai (as defined in Section 3.1), and write E=C(x0,…,xn), then we can embed C(yi,zi) into E by setting   y=xn,andz=θ0(x0).In particular, the first part of Theorem 3.1 can be reformulated as follows: If g=a is an algebraic path of affine type, then the fields F and E coincide. Remark 3.5 It follows from the above discussion that the differential field (F,∂) associated to a normal form is a Liouvillian extension of (C(x),∂) (see [27, Section 1.5]). Furthermore, one always has   trdegCF≤m+1,where m=height(g).The second part of the Main Theorem will be a direct consequence of Corollary 3.3 and the following result. Theorem 3.6 Let g∈NFtamebe a tame normal form of height m≥1and associated field F=C(y0,z0,…,ym,zm). Then  trdegCF=m+1.Moreover, {y0,…,ym}forms a transcendence basis for F/C. 3.3. Three lemmas on twisted equations The proof of Theorem 3.6 uses induction on the height of a normal form and is essentially based on the three lemmas stated below, which treat special types of differential equation in (F,∂). For future reference, the equations (3.1), (3.2) and (3.3) below will be called, respectively, the first, second and third twisted equations. To fix the notation, we consider a tame normal form g∈NFtame of height m≥0 with algebro-transcendental decomposition   g=a0γ1a1⋯γmam,m≥0and associated differential field F=C(y0,z0,…,ym,zm) (equipped with the usual derivation ∂=d/dz induced from M). In the following statements, we use the following definition. Given a germ γ∈{e,l,pα:α∈Ω⧹Q}, the γ-augmention of g (or, shortly, the γ-augmented path) is the path   gaug=γa0γ1a1⋯γnanobtained by adjoining γ to the left of g (with the obvious compatibility condition that the source of the germ γ coincides with the target of the germ θ0). Note that gaug is not necessarily in normal form. We will say that gaug is a nice augmentation of g if gaug lies in NFtame and moreover the right-hand side of the above displayed equation is precisely the algebro-transcendental decomposition of gaug. Lemma 3.7 Assume {y0,…,ym}is a transcendence basis for F/Cand suppose that the e-augmented path   gaug=eθ0γ1θ1⋯γnθnis a nice augmentation of g. Let f∈Fbe a non-zero solution of the equation  (∂−μ∂(z0))f=0 (3.1)for some μ∈C. Then, necessarily μ=0and f∈C. Lemma 3.8 Assume that {y0,…,ym}is a transcendence basis for F/Cand suppose that the p-augmented path  gaug=pαθ0γ1θ1⋯γnθn,forsomeα∈Ω⧹{1}is a nice augmentation of g. Let f∈Fbe a non-zero solution of the equation  (∂−μ∂(z0)z0)f=0 (3.2)for some μ∈C⧹Q*. Then, μ=0and f∈C. Lemma 3.9 Assume that {y0,…,ym}is a transcendence basis for F/Cand suppose that the l-augmented path  gaug=lθ0γ1θ1⋯γnθnis a nice augmentation of g. Let f∈Fbe a non-zero solution of the equation  ∂f=c∂(z0)z0 (3.3)for some c∈C. Then, necessarily c=0and f∈C. We postpone the proofs of these lemmas to Section 3.7. 3.4. The proof of Theorem 3.6 Let us see how the previous results imply Theorem 3.6. Proof of Theorem 3.6 As we said above, the proof is by induction in m=height(g). Therefore, we start with the case m=0. By definition, g=a is an algebraic path of affine type, and the associated field K=C(y,z) has transcendency degree one over C. Now, given m≥0, we assume by induction that all tame normal forms of height ≤m satisfy the conclusions of the theorem. Let h∈NFtame be a normal form of height m+1. Then, we can write the decomposition   h=a0γ0gfor some algebraic path a0∈PSL(2,C), a germ γ0∈{e,l,pα:α∈Ω⧹Q}. Further,   g=a1γ2θ2⋯γn+1θn+1is a tame normal form of height m and the augmented path gaug=γ0g is a nice augmentation of g (see definition at Section 3.3). Let us denote by F and F′ the differential fields associated to g and h, and write   F=C(y1,z1,…,ym+1,zm+1),F′=C(y0,z0,…,ym+1,zm+1).Recall that, by construction, z0 is algebraic over C(y0). Therefore, to prove the induction step, it suffices to show that the element y0 is transcendental over F. To simplify the notation, from now on we will write y=y0 and z=z1. Let us assume for a contradiction, that y is algebraic over F. We choose a minimal polynomial f∈F[X] for y, say   f=f0+⋯+fd−1Xd−1+Xd,fk∈F,where we can assume that d≥1 and that f0 is non-zero. We discuss separately the cases where γ0=e (extension of exp type), γ0=l (extension of log type) and γ0=pα (extension of power type). Suppose that the extension F⊂F′ is of exp type. Recalling that y satisfies the equation ∂(y)=y∂(z), it follows from the equation ∂(f(y))=0 that the polynomial   p=∂(f0)+⋯+(∂(fd−1)+(d−1)∂(z)fd−1)Xd−1+d∂(z)Xdvanishes on y. As a consequence, the polynomial of degree at most d−1 given by q=p−d∂(z)f also vanishes y. By the minimality of f, this polynomial must vanish identically. This is equivalent to the collection of equations   (∂−(k−d)∂(z))fk=0,k=0,…,d−1.We claim that this implies d=0, which contradicts the definition of f. Indeed, assume for a contradiction that d≥1. Then, we are precisely in the hypothesis of Lemma 3.7, that is, each fk satisfies the first twisted equation with μ=(k−d). In particular, for k=0, one has   (∂−d∂(θ(x)))f0=0and the lemma implies that f0=0, which is absurd. Assume now that the extension F⊂F′ is of power type. Then, it follows from the relation ∂(y)=αy∂(z)/z that the polynomial   p=∂(f0)+⋯+(∂(fd−1)+α(d−1)∂(z)zfd−1)Xd−1+αd∂(z)zXdalso vanishes on y. Hence, by the minimality of d, the polynomial   q=p−αd∂(z)zfmust vanish identically. This corresponds to the collection of equations   (∂−α(d−k)∂(z)z)fk=0,k=0,…,d−1.We claim that this set of equations has no solution if d≥1. Indeed, in this case, each one of the above equations corresponds to a twisted equation as described in Lemma 3.8 with μ=α(d−k)∈C⧹Q. In particular, for k=0, we conclude from that lemma that f0=0, which is absurd. Finally, in case where the extension F⊂F′ is of log type, we have   ∂(y)=∂(z)z.Hence, the polynomial of degree at most d−1 given by   q=(∂(f0)+∂(z)zf1)+⋯+(∂(fd−1)+d∂(z)z)Xd−1also vanishes on y. By the minimality of d, this polynomial vanishes identically and this corresponds to say that the collection of equations   ∂fk=∂(z)z(k+1)fk+1,fork=0,…,d−1hold, where we put fd=1. Taking k=d−1, the rightmost equation is exactly the twisted equation from Lemma 3.9. From the lemma, it follows that d=0, which contradicts our assumption. This concludes the proof of the theorem.□ Remark 3.10 As some readers may have noticed, the above computations are very similar to the classical computations of the Picard–Vessiot extension K/k for the elementary linear differential equations   ∂(y)=a,or∂(y)=ay,witha∈kover a given differential field (k,∂) (see for example, [27, Examples 1.18 and 1.19]). In this simple setting, the computation of the differential Galois group of the extension K/k reduces to studying when these equations have no solution in the base field. 3.5. Resonance relations and Ax Theorem In this subsection, we recall some results about differential field extensions and differentials forms, following closely Wilkie's notes [29]. They are key ingredients in proof of the celebrated Ax's Theorem (cf. [2]). We observe that the results described here are completely independent of Theorem 3.6 and Lemmas 3.7, 3.8 and 3.9. In this subsection, k⊆K will denote arbitrary fields of characteristic zero, and k will be assumed to be algebraically closed. We will say that elements x1,…,xn∈K satisfy a power resonance relation (over k) if there exist integers r1,…,rn, not all zero, such that   ∏i=1nxiri∈k.Similarly, we will say that y1,…,yn∈K satisfy a linear resonance relation (over k) if there exist integers r1,…,rn, not all zero, such that   ∑i=1nriyi∈k.Since k is supposed algebraically closed, we can assume in both cases that the integers r1,…,rn are coprime. We will denote by Derk(K) the set of derivations δ:K→K whose constant subfield Const(δ) contains k. For each n∈N, Ωkn(K) denotes the K-vector space of alternating, K-linear n-forms on Derk(K) and   d:Ωkn(K)⟶Ωkn+1(K)is the total differential map. The space Ωk1(K) is the dual of Derk(K), and it is usually called the space of Kähler differentials of K over k. The space Ωk0(K) is identified to K. We say that a 1-form ω∈Ωk1(K) is closed (resp. exact) if dω=0 (resp. ω=du for some u∈K). Finally, given an intermediate field k⊂K0⊂K, we say that a form ω∈Ωk1(K) is defined over K0 if ω=∑aidbi, for some ai,bi∈K0. Lemma 3.11 Suppose that K0is a field such that k⊂K0⊂Kand tr.deg.k(K0)=nfor some n≥1. Let δ∈Derk(K)be a derivation such that Const(δ)=k, and suppose that  ω1,…,ωn∈Ωk1(K)are closed 1-forms defined over K0satisfying ωi(δ)=0, for i=1,…,n. Then ω1,…,ωnare linearly dependent over k. The above statement and its proof can be found, for instance, in Wilkie's notes [29, Theorem 2]. Lemma 3.12 Suppose that there exists non-zero x1,…,xm∈Kand elements e1,…,em∈knot all zero such that the differential form  ∑i=1meidxixiis exact. Then, x1,…,xmsatisfy a power resonance relation over k. Moreover, assume that there exists a derivation δ∈Derk(K)and y1,…,ym∈Ksuch that  Const(δ)=k,andδ(xi)xi=δ(yi)for each 1≤i≤m. Then y1,…,ymsatisfy a linear resonance relation over k. Proof The first statement is proved in [29]. For the second statement, it suffices to remark that if the monomial m=∏xiri belongs to k, then the linear form l=∑riyi satisfies   δ(l)=∑riδ(yi)=∑riδ(xi)xi=δ(m)m=0which implies that l∈k.□ 3.6. Resonances and transcendental equations We now consider an arbitrary normal form g∈NF of height m≥0, with algebro-transcendental decomposition,   g=a0γ1a1⋯γmam,m≥0and associated differential field (F,∂), where we write   F=C(y0,z0,…,ym,zm)as in Section 3.2. Further, for each i=0,…,m, we denote by Fi=C(yi,zi) the field associated to the maximal algebraic subpath ai. Proposition 3.13 Assume that a non-zero element f∈Fsatisfies one of the following three equations:   (1)∂(f)f=∂(z0),(2)∂(f)=∂(z0)z0or(3)∂(f)f=β∂(z0)z0for some β∈C⧹Q. Then, the elements  y0,z0,…,ym,zm∈Fsatisfy either a power resonant relation or a linear resonant relation over C. Proof Based on the algebraic-transcendental expansion of g written above, we consider the following subsets of {1,…,m}  Ie={j:γj=e},Il={j:γj=l},andIp={j:γj=pαj,forsomeαj∈Ω⧹Q},and define a collection of closed 1-forms ω1,…,ωm∈ΩC1(F) as follows:   ωj={dyj−1/yj−1−dzj,ifj∈Iedyj−1−dzj/zj,ifj∈Ildyj−1/yj−1−αjdzj/zj,ifj∈Ip,where, in this last case, αj denotes the exponent in the power map γj=pαj. Similarly, we define the closed 1-form ω0 as   ω0={df/f−dz0,iffsatisfies(1)df−dz0/z0,iffsatisfies(2)df/f−βdz0/z0,iffsatisfies(3).Then, by Lemma 3.11, since ωj(∂)=0 for all 0≤j≤m, and trdegC(F)≤m+1, there exist constants c0,…,cm∈C, not all zero, such that the relation   c0ω0+⋯+cmωm=0holds. We consider, first of all, the case where f satisfies equation (2). Then, by suitably regrouping the terms in the above relation, we obtain a 1-form   c0dz0z0+∑j∈Iecidyi−1yi−1+∑j∈Ilcjdzjzj+∑j∈Ipcj(dyj−1yj−1+αjdzjzj)which is exact. Therefore, we can apply Lemma 3.12 to conclude that there exists a monomial   M=z0v0∏j∈Ieyj−1uj−1∏j∈Ilzjvj∏j∈Ipyj−1uj−1zjvjwith integer exponents not all zero, which belongs to C. This proves the result. We consider now the case (1). Here, we conclude from the relation ∑cjωj=0 that the 1-form   c0dff+∑j∈Iecjdyi−1yi−1+∑j∈Ilcjdzjzj+∑j∈Ipcj(dyj−1yj−1−αjdzjzj)is exact. Hence, applying again Lemma 3.12, we show that there exists a monomial of the form   M=fw∏j∈Ieyj−1uj−1∏j∈Ilmzjvj∏j∈Ipyj−1uj−1zjvj∈C(for some integers w,vj,uj, not all zero) which belong to C. Notice that if w=0 we are done because this would give the desired relation. By the same reason, we would be done if c0=0. Hence, from now on, we can assume that w is non-zero and that c0=1. By computing the logarithmic derivative dM/M, we can write   dff=∑j∈Iempj−1dyj−1yj−1+∑j∈Ilmqjdzjzj+∑j∈Ipm(pj−1dyj−1yj−1+qjdzjzj)with pj=−uj/w and qj=−vj/w being rational numbers. We can now replace this expression for df/f in the relation ∑cjωj=0 and, again by suitably regrouping the terms, conclude that the 1-form   ∑j∈Ie(cj+pj−1)dyi−1yi−1+∑j∈Il(cj+qj)dzjzj+∑j∈Ip(cj+pj−1)dyj−1yj−1+(−cjαj+qj)dzjzjis exact. If at least one of the coefficients of this 1-form is non-zero, then we can apply again Lemma 3.12 in order to obtain a monomial satisfying the conditions in the statement. So, let us assume that all these coefficients vanish. Since αi∉Q for each i∈Ip, we conclude that   cj=pj−1=qj=0,forallj∈Ip.In particular, the monomial M has simply the form   M=fw∏j∈Ieyj−1uj−1∏j∈Ilmzjvj.If we consider the linear form   l=wz0+∑j∈Ieuj−1zj+∑j∈Ilvjyj−1it is easy to see that ∂(l)=∂(M)/M=0. Therefore, l∈Const(∂)=C. It remains to consider the case of equation (3). The treatment is similar to the previous case. Here, we obtain a 1-form   c0(dff−βdz0z0)+∑j∈Iecidyi−1yi−1+∑j∈Ilcjdzjzj+∑j∈Ipcj(dyj−1yj−1+αjdzjzj)which is exact, and hence there exists a monomial   M=fwz0v0∏j∈Ieyj−1uj−1∏j∈Ilmzjvj∏j∈Ipyj−1uj−1zjvj∈C(with exponents not all zero). Assuming that c0=1 and that w is non-zero (otherwise we are done), we can apply exactly the same reasoning as above to conclude that the 1-form   (q0−β)dz0z0+∑j∈Ie(cj+pj−1)dyi−1yi−1+∑j∈Il(cj+qj)dzjzj+∑j∈Ip(cj+pj−1)dyj−1yj−1+(−cjαj+qj)dzjzjis exact (where q0=−v0/w). Since β∉Q, the coefficient in front of dz0/z0 cannot vanish. Therefore, we can apply Lemma 3.12 in order to obtain a monomial which satisfies the desired relation. This concludes the proof.□ For later use, we need to establish a more precise statement about the existence of power/linear relations in the fields F0,…,Fm associated to the algebraic subpaths a0,…,am. Given a normal form g=a0γ1a1⋯γmam as in the beginning of the subsection and an equation for f∈F as in the statement of the previous proposition, we define the augmentation of g as the path   gaug=γ0a0γ1⋯γmamwhich is obtained by concatenating to g the symbol γ0=e (resp. l or pβ) if f satisfies equation (1) (resp. (2) or (3)). Further, given an index 0≤j≤m−1 and two symbols γ,γ′∈{e,l,p}, we will say that the algebraic subpath aj of gaug lies in a [γ,γ′]segment if   γj=γandγj+1=γ′. Corollary 3.14 Assume that {y0,…,ym}is a transcendence basis for F/C. Let f∈Fbe a nonzero solution of one of the equations (1), (2) or (3) from Proposition3.13. Then, there exists at least one index 0≤j≤m−1such that Either ajlies in a [e,l]segment and yj,zjsatisfy a linear resonance relation in Fj. Or ajlies in a [l,e], [l,p], [p,e]or [p,p]segment and yj,zjsatisfy a power resonance relation in Fj.In particular, if m=0then there is no non-zero element f∈Fsatisfying (1), (2) or (3). Proof The hypothesis imply that {dy0,…,dym} is a basis of ΩC1(F) and that the F-subspaces generated by ΩC1(F0),…,ΩC1(Fm) are F-linearly independent. Moreover, since each zj is algebraic over yj, the 1-form dzj lies in the one-dimensional F-subspace generated by dyj. From Proposition 3.13, we conclude that if a non-zero element f∈F satisfies (1), (2) or (3), then either there exists a monomial M=∏j=0mzjvjyjuj or a linear form l=∑j=0mvjzj+ujyj (with integers uj,vj not all zero) which belong to C. Taking the logarithmic derivative dM/M in the former case or the derivative dl in the later case, we obtain   ∑j=0mvjzjdzj+ujyjdyj=0,or∑j=0mvjdzj+ujdyj=0,respectively. Therefore, by the linear independency of dy0,…,dyn, either vjzjdzj+ujyjdyj=0 or vjdzj+ujdyj=0 for all 0≤j≤m. In the former case, we conclude that d(zvjyjuj)=0, while in the latter case d(vjzj+ujyj)=0. Now, to conclude the proof, it suffices to consider more carefully the expressions of m and l obtained in the proof of the previous proposition. For instance, we consider the case where   m=z0v0∏j∈Ieyj−1uj−1∏j∈Ilzjvj∏j∈Ipyj−1uj−1zjvj∈Cwhich, by the above argument, implies a collection of power resonance relations of the form d(zjvjyjuj)=0, for j=0,…,n. Notice that no relation of type d(zjvj)=0 (that is, with uj=0) or d(yjuj) (that is, with vj=0) can appear, since this would imply that zj or yj belong to C, contradicting the fact that both yj and zj are germs of invertible maps. Thus, there necessarily exists a monomial relation of the form zjvjyjuj∈C with exponents uj,vj both non-zero. But looking to the above expression for m, we conclude that this can only happen in the index j is such that j∈Il∪Ip and j+1∈Ip∪Ie. This is equivalent to say that aj lies in an [l,e], [l,p], [p,e] or [p,p] segment. The other cases can be treated in an analogous way.□ 3.7. Proofs of lemmas on twisted equations We now proceed to the proof of Lemmas 3.7, 3.8 and 3.9. We keep the notation introduced in Section 3.3. Proof of Lemma 3.7 Let us assume that μ≠0. By contradiction, we assume that there exists a non-zero f∈F such that   ∂(f)f=∂(z0).Writing the algebro-transcendental decomposition of the e-augmented path gaug as   gaug=ea0γ1a1⋯γmam,m≥0we let zj,yj denote the head and tail elements of the Cohen differential field Ej associated to the algebraic path aj, for j=0,…,m. Defining γ0=e, we can now apply Corollary 3.14 to conclude that there exists at least one index 0≤j≤m−1 such that Either γj=e, γj+1=l and yj,zj satisfy a linear resonance relation. Or γj∈{l,pα:α∈Ω⧹Q}, γj+1∈{e,pα:α∈Ω⧹Q} and yj,zj satisfy a power resonance relation.If m=0 we get our desired contradiction. If m≥1, we will deduce the contradiction using Corollary 3.3. For this, we treat cases (1) and (2) separately. To simplify the notation, we define   a=aj,y=yjandz=zjand write the expansion of the algebraic path a (of affine type) as   a=θ0p1θ1⋯pnθn,n≥0.In the case (1), y and z satisfy a relation of the form vz+uy=c, for some u,v∈Z* and c∈C. We consider then the modified algebraic path  a*=normalformreductionofs−v/ut−c/va.Explicitly, for a given as above, we can write   a*=θ0*p1θ1⋯pnθn,where the Möebius part of a* is given by θ0*=s−v/ut−c/vθ0. In particular, the assumption that a is an algebraic path of affine type implies that the same property holds for a*. Now, by the definition of a*, the head and tail elements z* and y* of the Cohen field E* associated to a* should satisfy the relation   y*z*=1.Hence, we will obtain the desired contradiction to Corollary 3.3 once we show that   a*≠1.To prove that this always holds, observe that a*=1 if and only if a=(s−v/ut−c/v)−1=tc/vs−u/v. Since a is a maximal algebraic subpath (lying in an [e,l] segment) of the augmented path gaug, this would contradict the hypothesis that gaug is a nice augmentation of g, as stated in Section 3.3. Indeed, if either c≠0 or −u/v∉Ω⧹{1}, then the subpath eal is certainly not in normal form. On the other hand, if c=0 and −u/v∈Ω⧹{1} then, according to our definition of algebro-transcendental decomposition, the corresponding subpath es−u/vl should instead be considered as a rational power map p−u/v. This concludes the proof of (1). Consider now the case (2). We write the corresponding power resonance relation as yuzv=c, for some u,v∈Z* and c∈C*. Since the algebraic path a lies in a [γ,η]-segment (with γ∈{l,p} and η∈{e,p}) and gaug is a nice augmentation of g, it follows from the definition of NF that its Möebius part θ0 necessarily lies in T1⧹{1}. We introduce now the modified algebraic path   a*=sc1/upv/ua,which is also a non-identity normal form by the discussion of the above paragraph. Similarly to the previous case, the head and tail elements z*,y* of the Cohen field E* associated to a* satisfy the relation y*z*=1 (up to a convenient choice of the branch of c1/u). Furthermore, a* is an algebraic normal form of affine type, and the above identity contradicts Corollary 3.3 when applied to a*. This concludes the proof of the lemma.□ Proofs of Lemmas 3.8 and 3.9 We follow exactly the same strategy of the previous proof. Namely, we assume for a contradiction that c≠0 and that there exists a non-zero element f∈F satisfying one of the following two equations:   ∂(f)f=μ∂(z0)z0or∂(f)=∂(z0)z0,where μ∈C⧹Q. Considering the algebraic transcendental decomposition of the augmented path gaug, the same alternatives (1) and (2) listed in the previous proof appear. By repeating the same reasoning, we obtain a contradiction.□ 4. Some consequences We proceed to prove the other results stated in the Introduction. Proof of Theorems 1.7, 1.10 and 1.12 We will only prove Theorem 1.7, since the other two results are immediate consequences. First of all, we remark that GAff,PowR is a subgroupoid of GPSL(2,C),Exp. Therefore, if we consider the free product groupoid F=GPSL(2,C)∗ΓGExp and the groupoid morphism   φ:F→GPSL(2,C),Expdefined in Section 2.3, then each germ lying in the GAff,PowR is the image of a (not necessarily unique) path in F. Further, we can assume that such path of the form   g=θ0pr1θ1⋯prnθn,n≥1,where each pri is a power map with exponent ri∈R and each θi is an affine map. Possibly making some simplifications, we can further assume that θ1,…,θn≠1 and that r1,…,rn≠1. As a consequence, g is a product normal form, that is, an element of the subset PNF⊂F given by Definition 2.3. Applying the reduction system (PNF,→) defined in Section 2.4, we can make the reduction   g→*g′,where g′ has the same form as g, but with the additional property that each affine map θ1,…,θnis a translation. The subset of paths in F satisfying these properties will be called normal forms of power-translation type, and noted NFPT. Notice that a path in NFPT is not necessarily an element of NF (see Definition 1.1), because the exponents r1,…,rn of the power maps do not necessarily lie in the region Ω described in Remark 1.2. However, the reduction from NFPT to NF can be easily obtained. Indeed, assuming that g∈NFPT is written as above, its normal form reduction g→*h gives the path   h=θ0wε1ps1θ1wε2ps2⋯wεnpnθn,where we define each pair (si,εi)∈Ω×{0,1} as follows:   (si,εi)={(ri,0)ifri∈Ω(−ri,1)if−ri∈Ω.We remark the following two facts: The normal form h lies in NFtame. If g,g′∈NFPT reduce to a same normal form h∈NFtame then necessarily g=g′.Indeed, the assumption R∩Q<0=∅ implies that a subpath of the form θiw appears in the expansion of h if and only if the power map pri+1 has an exponent in R⧹Q. Therefore, the algebro-transcendental decomposition of h can only contain maximal algebraic subpaths of affine type. This proves (1). The proof of (2) is immediate, since the original powers r1,…,rn∈R can be read out from the expression of the normal form. Based on these remarks, the result is now an immediate consequence of the second part of the Main Theorem.□ Proof of Theorems 1.14 and 1.19 We will only give the details of the proof of Theorem 1.14, since Theorem 1.19 is an immediate consequence of this result. Keeping the notation of Section 1.4, we want to prove that the homomorphism   ϕ:Aff*θ+→GAff+,Exp,is injective. Using Britton's normal form (see for example, [16, IV.2]), and the right transversals to T0,T1 to H0,H1 defined in Remark 1.2, it follows that we can (setwise) identify Aff*θ+ to a set BNF (so-called Britton normal forms) contained in the free product Aff+∗⟨𝚔i:i∈Z⟩ (where 𝚔 denotes the stable letter of the HNN-extension). By definition, each f∈BNF can be uniquely written as   f=θ0γ1θ1⋯γnθn,n≥0,where θ0,…,θn are affine maps, γi∈{𝚔,𝚔−1}, and If γi=𝚔 then θi∈S+. If γi=𝚔−1 then θi∈T. There are no subwords of the form 𝚔1𝚔−1 or 𝚔−11𝚔.The set BNF has a natural group structure which is inherited from the group structure of Aff*θ+. Using the above expansion for f∈BNF, we define maximal interval of existence If∈(R,+∞) of f as the largest open neighborhood of ∞ (of the form ]Af,+∞[ for some Af∈R) such that each one of the n+1truncations of the above normal form, namely   f[i]=θiγi+1θi+1⋯γnθnfori=0,…,nmaps under ϕ to a germ ϕ(f[i])∈GAff+,Exp which extends analytically to an invertible function defined on the interval If. To simplify the notation, we denote also by   ϕ(f):If→Rthe corresponding (uniquely determined) analytic function. Now, we consider a mapping ρ1:BNF→NFtame which sends an element f∈BNF to a tame normal form ρ1(f)∈NFtame. If f is written as above, this mapping is defined as follows: Each symbol θi is replaced by a corresponding germ of affine map. Each symbol 𝚔 (resp. 𝚔−1) is replaced by a germ of exponential (resp. principal branch of logarithm) map. The source point of the rightmost affine germ θn is chosen to be Af+1 (or 0 if Af=−∞).Notice that condition (3) uniquely determines the choice of all germs given in (1) and (2) due to the necessarily source/target compatibility conditions. Consequently, the mapping is well-defined by these conditions and, moreover, injective. Similarly, we consider the mapping   ρ2:BNF→GPSL(2,C),Expdefined as follows: given f∈BNF, we consider the analytic function ϕ(f):If→R and let ρ2(f)∈GPSL(2,C),Exp be the germ of ϕ(f) at the point Af+1 (or at 0 if Af=−∞). By construction, if φ:NF→GPSL(2,C),Exp denotes the mapping defined at the Main Theorem, the following diagram   is commutative. Now, we reason by contradiction assuming that there exists a non-identity Britton normal form f∈BNF lying in the kernel of ϕ. Then, it follows that ϕ(f):If→R is the identity map and, consequently, that ρ2(f) is the identity germ. On the other hand, ρ1(f) is a non-identity tame normal form and it follows from the Main theorem that φ◦ρ1(f) cannot be the identity germ. This is a contradiction.□ Funding This work has been partially supported by the ANR project STAAVF and by the CAPES/COFECUB project MA731-12. Acknowledgements I would like to thank Dominique Cerveau, Robert Roussarie, Robert Moussu, Jean-Jacques Risler, Jean-Philippe Rolin, Jean-Marie Lion, Bernard Teissier, Norbert A'Campo, Frank Loray, Étienne Ghys and Thomas Delzant for numerous enlightening discussions. References 1 R. K. Amayo and I. Stewart, Infinite-dimensional Lie Algebras , Noordhoff International Publishing, Leyden, 1974. Google Scholar CrossRef Search ADS   2 J. Ax, On Schanuel's conjectures, Ann. 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Ree, On certain pairs of matrices which do not generate a free group, Can. Math. Bull.  4 ( 1961), 49– 52. Google Scholar CrossRef Search ADS   22 D. Richardson, How to recognize zero, J. Symb. Comput . 24 ( 1997), 627– 645. Google Scholar CrossRef Search ADS   23 J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc.  23 ( 1922), 51– 66. Google Scholar CrossRef Search ADS   24 J.-P. Serre, Trees. Springer Monographs in Mathematics , Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. 25 J. R. Shackell, Symbolic Asymptotics, volume 12 of Algorithms and Computation in Mathematics , Springer-Verlag, Berlin, 2004. 26 L. van den Dries, A. Macintyre and D. Marker, Logarithmic-exponential series, Ann. Pure Appl. Log.  111 ( 2001), 61– 113. Google Scholar CrossRef Search ADS   27 M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, volume 328 of Grundlehren der Mathematischen Wissenschaften , Springer-Verlag, Berlin, 2003. Google Scholar CrossRef Search ADS   28 S. White, The group generated by x↦x+1 and x↦xp is free, J. Algebra  118 ( 1988), 408– 422. Google Scholar CrossRef Search ADS   29 A. Wilkie, Lecture Notes on Marker's Lecture Notes on Ax's Theorem. Logic Advanced Class, University of Oxford, Oxford, 2003. © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

PSL(2, ℂ ), the exponential and some new free groups

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Abstract We prove a normal form result for the groupoid of germs generated by PSL(2,ℂ) and the exponential map. We discuss three consequences of this result: (1) a generalization of a result of Cohen about the group of translations and powers, which gives a positive answer to a problem posed by Higman; (2) as proof that the subgroup of Homeo(ℝ,+∞) generated by the positive affine maps and the exponential map is isomorphic to an HNN-extension; (3) a finitary version of the immiscibility conjecture of Ecalle–Martinet–Moussu–Ramis. 1. Introduction 1.1. Normal forms We recall some basic concepts and terminology from the theory of groupoids (see for example, [4]). A groupoid is a category G whose objects Obj(G) form a set and in which every morphism is an isomorphism. For each x,y∈Obj(G), we denote by G(x,y) the set of morphisms in G from x to y. We denote also by G the disjoint union of G(x,y), for all x,y∈Obj(G). The composition of morphism is written multiplicatively: if f∈G(x,y) and g∈G(y,z), then these morphisms can be composed and its composition is the morphism gf∈G(x,z). From now on, when we write the expression gf for two morphisms f,g, we are tacitly assuming that these morphism can be composed. The symbol 1 will generally denote the identity morphism and f−1∈G(y,z) will denote the inverse of the morphism f∈G(x,y). We will say that morphism f has source x=s(f) and target y=t(f) if f∈G(x,y). The group G(x,x) will be called vertex group at x and will be denoted simply by G(x). A finite sequence of morphisms [f1,f2,…,fn] in G is called a path if s(fi)=t(fi+1). Given such a path, we will say that f=f1⋯fn∈G is the morphism defined by the path. The operation of concatenation in the set of paths is defined in the obvious way, taking into account the source/target compatibility. A path [f1,f2,…,fn] is called reduced if: no two consecutive morphisms fi, fi+1 are mutually inverse; if some fi is the identity morphism then n=1 and f=[1].We can give a groupoid structure to the set of reduced paths. The operation of composition of two paths is defined as follows: first concatenate the paths and then successively eliminate all consecutive terms which are mutually inverses. The resulting groupoid is called the free groupoid on the graph of G ([4, Section 8.2]). Given a differentiable manifold M, let G(M) denote the Haefliger groupoid over M (see for example, [20, Section 5.5]). We recall that, by definition, the set of objects Obj(G(M)) is the set of points of M and, for each p,q∈M, G(M)(p,q) is the set of all germs of diffeomorphisms (M,p)→(M,q). In order to keep the traditional naming, we will refer to the morphisms of G(M) simply as germs. Given a map f:U→V, where U,V⊂M are open sets and f is a local diffeomorphism (that is, locally invertible), we denote by Germ(f)⊂G(M) the smallest wide subgroupoid containing all the germs f,p of f at all points p of its domain. We recall that a subgroupoid G1 of a groupoid G2 is called wide if Obj(G1)=Obj(G2). More generally, given an arbitrary collection C of local diffeomorphisms as above, we denote by Germ(C)⊂G(M) the smallest subgroupoid containing Germ(f), for all f∈C. From now on, we shall assume that M=P1(C) and that all maps are holomorphic. As a basic object, we will frequently consider the groupoid GExp=Germ(exp) associated to the usual exponential map. This is the groupoid whose germs at each point are given by finite compositions f=f1⋯fn of the following germs:   {1,p}∪{exp,p:ifp∈C}∪⋃k∈Z{lnk,p:ifp∈C*},where 1,p is the identity germ at p and lnk,p is the germ at p of the kth-branch of the logarithm, that is, the map   lnk:C*⟶Jk={x+iy:y∈](2k−1)πi,(2k+1)πi]}z⟼ln(∣z∣)+iargk(z),where argk:C→](2k−1)πi,(2k+1)πi] is the kth-branch of the argument function. In general, we have the relation   exp,qlogk,p=1,qfor all p∈C and q=logk(p). On the other hand,   logk,qexp,p:z↦z+2πi(k−s)for all p∈Js and q=exp(p). In particular, notice that germ corresponding to the translation by 2πi lies in GExp.   In what follows, we are going to simplify the notation and omit the subscripts ,p when referring to the germ of a local diffeomorphism at a point p of its domain. Thus, the same symbol, say exp, will denote both the exponential map and the germ at each point of its domain. In the situation where we want to emphasize that we are considering its germ at a specific point p, we will simply write that s(exp)=p. We also introduce the following symbols for the (germ of) exponential map and the zeroth branch of the logarithm:   e:z↦exp(z),l:z↦ln0(z).Another important object for us is the groupoid   GPSL(2,C)=Germ(PSL(2,C)).We recall that the group PSL(2,C) is generated by the subgroups   W={1:z↦z,w:z↦1/z},T={ta:z↦z+a,a∈C}S={sα:z↦αz,α∈C*}which are, respectively, the involution, the translations and the scalings. We denote by Aff⊂PSL(2,C) the subgroup of affine maps, that is, Aff=T⋊S. Following the above notational convention, the same symbols ta,sα and w will be used to denote the corresponding germs in GPSL(2,C). Our main result is a normal form for elements in the groupoid   GPSL(2,C),Exp=Germ(PSL(2,C)∪{exp}).In order to state this result, consider the subgroups H0,H1⊂PSL(2,C) given by   H0=T⋊{s−1},andH1=S⋊{w}.For the next definition, we recall that, given a group G and a subgroup H⊂G, a right transversal for H is a subset T⊂G of representatives for the right cosets {Hg:g∈G} which contains the identity of G. Definition 1.1 Let T0,T1⊂PSL(2,C) be right transversals for H0,H1, respectively. A (T0,T1)-normal form in GPSL(2,C),Exp is a path   g=[g0,h1,g1,…,hn,gn],n≥0such that the following conditions hold: The germ g0 lies in GPSL(2,C). For each 1≤i≤n, hi∈{e,l}. If hi=e then gi∈T0. If hi=l then gi∈T1. There are no subpaths of the form [e,1,l] or [l,1,e].We denote by NFT0,T1 the set of all (T0,T1)-normal forms. The path [1] will be called the identity normal form.There is an obvious mapping   φ:NFT0,T1→GPSL(2,C),Expwhich associates to each normal form g=[g0,h1,…,gn] the germ φ(g)=g0h1⋯gn. The main goal of this paper will be to study the surjectivity and injectivity properties of this mapping. Remark 1.2 As we shall see in Lemma 2.1, a possible choice of transversals T0,T1 for H0,H1, respectively, is as follows:   T0={sρ:ρ∈Ω}∪{sρwtb:ρ∈Ω,b∈C}T1={tb:b∈C}∪{tawtb:a∈Ω,b∈C*⧹{−1/a}}∪{tcw:c∈C*},where Ω={α:Re(α)>0}∪{α:Re(α)=0,Im(α)>0} is the region shown in Fig. 1. (In fact, we could define similar transversals by choosing any region in C* which is a fundamental domain for the Z2-action z↦−z and contains 1.) Figure 1. View largeDownload slide The region Ω. Figure 1. View largeDownload slide The region Ω. From now on, in order to simplify the exposition, we shall fix the choice of transversals T0,T1 as described in Remark 1.2, and write NFT0,T1 simply as NF. Each result that we are going to discuss in the remainder of the paper can be appropriately translated to different choices of transversals. In order to state the Main Theorem, we need define certain special normal forms. To simplify the notation, we shall frequently omit the square braces and write a path [g1,…,gn] simply as g1⋯gn. For each α∈C*, the power map with exponent α is the germ defined by   pα=esαl,that is, the germ of power map z↦zα obtained by choosing the zeroth branch of the logarithm. A normal form a∈NF will be called an algebraic path (resp. rational path) of length n≥0 if it has the form   a=θ0pα1θ1⋯pαnθn,where αi are exponents in Ω∩Q (resp. αi∈Ω∩Z), and   θ0∈PSL(2,C),θn∈T1andθ1,…,θn−1∈T1⧹{1}.We will say that a is of affine type if θi is an affine map for each 0≤i≤n−1. All paths of length n=0 are of affine type. Notice that each path g∈NF can be decomposed as   g=a0γ1a1⋯γmam,m≥0,where each ai is an algebraic path and each γi is either e, l or a power germ pαi with an exponent αi∈Ω⧹Q. This decomposition is unique if we further require that there are no subpaths of the form   γiaiγi+1=esαl.In other words, we assume that each subpath esαl is grouped together into written as the power map pα. The above unique decomposition of g will be called the algebro-transcendental decomposition. Each ai will be called a maximal algebraic subpath of g. The natural number m will be called the height of g and noted height(g). Hence, normal forms of height zero correspond to algebraic paths. Given symbols η1,η2∈{e,l,p}, we will say that the maximal algebraic subpath ailies in a [γ,η]-segment, if γi=η1 and γi+1=η2. Example 1.3 The path g=es2llt1es2lt1wew is a normal form with algebro-transcendental decomposition   g=1︸a0p2︸γ11︸a1l︸γ2t1p2t1w︸a2e︸γ3w︸a3The maximal algebraic subpaths a1 and a2 lie in [p,l] and [l,e] segments, respectively. Notice that maximal algebraic subpaths can be the identity, as it is the case of a0 and a1. We will say that a normal form g∈NF is tame if Either height (g)=0 and g is an algebraic path of affine type. Or height (g)≥1 and each maximal algebraic subpath lying in a segment of type   [e,l],[l,e],[l,p],[p,e],or[p,p]is of affine type.For instance, the normal form of the previous example is tame. We shall denote by NFtame the subset of tame normal forms. Main Theorem (Normal form in GPSL(2;ℂ),Exp) The mapping  φ:NF→GPSL(2,C),Expis surjective. Moreover, this mapping is injective when restricted to NFtame. The study of non-tame normal forms puts into play some difficult problems concerning the study of finite coverings P1(C)→P1(C) with imprimitive monodromy groups. This issue is strongly related to the well-known Ritt's decomposability theorem [23], which fully describes monoid structure of the polynomials under the composition operation. The following example shows that we cannot expect the map ρ:NF→GPSL(2,C),Exp to be bijective without further restrictions. Example 1.4 Each Chebyshev polynomial Tn(x) lies in GPSL(2,C),Exp, as it can be defined by the identity   Tn=φpnφ−1,where pn(z)=zn and φ(z)=z+1/z is given explicitly by   φ=s−1t2wt−1/4p2t−1/2wt1.On the other hand, we can also express T2(z)=z2−2 as t−2p2. Hence, the relation   T2(z+1z)=z2+1z2is equivalent to say that the normal form   s−1t2wt−1/4p2t−1/2wt1p2t−1wt1/2p1/2t1/4wt−2p1/2t2defines the identity germ. Of course, this normal form is not tame. Remarks 1.5 (1) Some readers will probably notice the similarities between the above normal form and Britton's normal form for HNN-extensions (see for example, [16, IV.2]). Indeed, there is a general notion of HNN-extension for groupoids ([4, Section 8.4.1]) which generalizes the usual notion for groups (see Section 1.4). At a first view, one could expect to prove that GPSL(2,C),Exp is isomorphic to the HNN-extension of GPSL(2,C), with the exponential e acting as the stable letter, that is, somehow conjugating the subgroupoids GH0 and GH1. This cannot hold (at least not in such a naïve way). In fact, consider the subgroupoids L0 and L1 obtained by restricting the groupoids GH0 and GH1 to the subdomains C and C*, respectively. Then, the exponential map indeed defines a morphism of groupoids by   Θ:L0⟶L1h⟼ehe−1,where the germ e−1 is chosen in such a way that s(h)=t(e−1). At the level of objects, this induces the mapping Θ:C→C*, Θ(p)=exp(p). However, Θ is not an isomorphism of groupoids, since it annihilates all germs t2πik, with k∈Z; and identifies each two points in C which differ by an integer multiple of 2πi. As a matter of fact, Θ establishes an isomorphism between the groupoid L1 and the quotient groupoid L0/Ker(Θ), which is simply the groupoid with the object set C/2πiZ and morphisms given by the action of {ta:a∈C} and s−1 modulo 2πiZ. (2) It is easy to see that GPSL(2,C),Exp coincides with GT,Exp, that is, the groupoid generated only by the translations and the exponential. Indeed, one easily constructs the subgroups S and W by defining   sα=etln0(α)l,andw=e2tiπl2,for all α∈C*. The Normal form theorem could be formulated solely in terms of paths in GT,Exp. However, this would lead to a much more complicated enunciation and to the loss of the analogy with the theory of HNN-extensions. (3) Notice that GPSL(2,C) has a natural Lie groupoid structure, which is inherited from étale groupoid structure of G(P1(C)) (see [20, Section 5.5]). Some readers may be wondering which is the relation between GPSL(2,C) and the so-called semi-direct product Lie groupoid  PSL(2,C)⋉P1(C),which is naturally defined by the action of PSL(2,C) on P1(C) (see [20, Section 5.1]). One can show that PSL(2,C)⋉P1(C) and GPSL(2,C) are isomorphic as groupoids, but not as Lie groupoids. Indeed, the source fibers of GPSL(2,C) (that is, the sets s−1(p), p∈P1(C)) have a discrete topology, while all source fibers of PSL(2,C)⋉P1(C) are manifolds diffeomorphic to PSL(2,C). (4) Another interesting construction can be obtained by combining the groupoids Germ(PSL(2,C)), Germ(exp) and Germ(℘,℘′), where   ℘:C/Λ→P1(C)is the Weierstrass function associated to a period lattice Λ⊂C. In this case, the resulting groupoid G would contain a rich class of rational maps, the so-called finite quotients of affine maps (see [19]), that is, rational maps f of degree two or more which fit into commutative diagrams of the form   where l(z)=az+b is an affine map defined on C/Λ and Θ:C/Λ→P1(C) is a finite covering. For instance (see [18, Problem 7-f]), for Λ=Z⊕iZ and l(z)=(1+i)z, the germ ℘l℘−1 is (up to a conjugation by a Möebius map) the quadratic rational map h(z)=(z+1/z)/2i. 1.2. Powers and affine maps As a consequence of the Main Theorem, we are going to obtain a generalization of a result of Cohen. Let R be an arbitrary multiplicative subgroup of C* and let PowR be the set of germs determined by all the branches of the power maps   C*∋z↦zr,withr∈R.Clearly, the associated groupoid Germ(PowR) is simply obtained by taking the union of PowR with the identity germs 1 at 0 and ∞. As above, for each r∈R, we denote by   pr=esrlthe germ of power map obtained by choosing the zeroth branch of the logarithm. Initially motivated by a question of Friedman, several authors (cf. [12]) considered the groupoid   GAff,PowR=Germ(Aff,PowR)whose elements are obtained by finite compositions of germs of affine and power maps. In particular, they studied the following property: Definition 1.6 We will say that GAff,PowR has the amalgamated structure property if each element GAff,PowR can be uniquely defined by a path   [g0,pr1,ta1,…,prn,tan]for some n≥0, where g0∈Germ(Aff), ri∈R⧹{1} and ai∈C for i=1,…,n, such that ai is non-zero for 1≤i≤n−1. In particular, this property implies that, given n≥1 and two sequences of constants r1,…,rn∈R⧹{1} and a1,…,an∈C with ai non-zero for 1≤i≤n−1, the germ defined by   z↦(a1+(a2+⋯+(an+x)rn⋯)r2)r1(where we choose arbitrary branches for the power maps) cannot be the identity. Building upon a method originally introduced by White [28], Cohen proved in [5] that GAff,PowQ>0 has the amalgamated structure property (that is, one takes R equal to Q>0). Using our normal form Theorem, we prove the following: Theorem 1.7 The groupoid GAff,PowRhas the amalgamated structure property if R∩Q<0=∅. Equivalently, we assume that for each r∈R, the ray rQ<0 does not intersect R.   Remark 1.8 Assume R is the multiplicative subgroup of C* generated by exp(2πiλ1),…,exp(2πiλn), for some collection of complex numbers λ1,…,λn. Then, the condition R∩Q<0=∅ is equivalent to the following non-resonance condition:   (12+iln(Q>0))∩(Z+λ1Z+⋯+λnZ)=∅,where ln denotes the principal branch of the logarithm function. 1.3. Generalized Witt algebras We describe another consequence of the Normal Form Theorem. Let M be an additive sub-monoid of C (that is, a subset M⊂C which is closed under addition and contains zero). Following [1], we define the generalized Witt algebra W(M) as the C-vector space with a basis {wg:g∈M}, subject to the Lie multiplication   [wg,wh]=(g−h)wg+h.Each basis element can be represented by a (possibly multivalued) complex vector field on P1(C) given by   wg=zg(z∂∂z)whose flow at time a is given by the multivalued map   exp(awg)=z↦{(−ag+z−g)−1/g,ifg≠0exp(a)z,ifg=0.Following the conventions of the first subsection, we are going to denote also by exp(awg) the germs in G(P1(C)) obtained by taking all possible determinations of the maps z↦(−ag+z−g)−1/g at all points of its domain of definition. Example 1.9 For M=Z we obtain the classical Witt algebra W(Z). The subalgebra W(Z≤0)⊂W(Z) plays an important role in holomorphic dynamics. The flow maps in this subalgebra can be written as   z↦exp(aw−k)(z)=z(1−akz−k)1/k,∀k∈Z≥0,∀a∈Cand they generate a well-known subgroup of the group Diff(C,∞) of germs of holomorphic diffeomorphisms fixing the infinity. Theorem 1.10 Let Mbe an arbitrary additive sub-monoid of C. Then, for all n≥1, all scalars a1,…,an∈C⧹{0}and all elements g1,…,gn∈M⧹{0}such that gi+1/gi∉Q<0∪{1}, the germ  z↦exp(a1wg1)⋯exp(anwgn)(z)cannot be the identity. Remark 1.11 The condition gi+1/gi≠1 must be imposed due to the trivial relation   exp(awg)exp(bwg)=exp((a+b)wg).Moreover, there are numerous counter-examples to the above result if drop the assumption gi+1/gi≠−1. For instance, given a∈C*, consider the so-called two parabolic group Ga⊂PSL(2,C), which is the group generated by the time a flows maps of w−1 and w1, namely   z↦exp(aw−1)(z)=z+a,z↦exp(aw1)(z)=z1+az.Following [15], we say that a is a free point if Ga is a free group. There are plenty of non-free points. For instance, Ree showed in [21] that the real segment ]−2,2[ is contained in an open set where the non-free points are densely distributed. Assume that a is a non-free point. Then, by definition, there exist a n≥1 and non-zero integers p1,q1,…,pn,qn such that   exp(p1aw1)exp(q1aw−1)⋯exp(pnaw1)exp(qnaw−1)=1.Clearly, each relation of this type would give a counter-example to the above theorem if the assumption gi+1/gi+1≠−1 were dropped. Our next goal is to state a normal form result for the groupoid   GM=Germ({exp(awg):g∈M,a∈C}).For this, given g∈M, and a∈C, we define the following germ:   Φa,g={p−1/gt−agp−g,ifg≠0sexp(a),ifg=0,where ta and sα are the translation and scalings germs, respectively; and the power map pr is defined as in Section 1.2. In other words, Φa,g∈GM is simply the germ obtained from the (multivalued) flow map exp(awg) by choosing the zeroth branch of the logarithm in the definition of the power maps x↦x−g and x↦x−1/g. The phenomena described in the previous remark leads us to define the following concept. We say that an additive sub-monoid M of C has no rational antipodal points if   M∩(MQ≤0)={0}.Using our Main Theorem, we shall prove the following: Theorem 1.12 (Normal form in GM) Suppose that Mhas no rational antipodal points. Then, each element of the groupoid GMis uniquely defined by a path  [Φa0,0,Φa1,g1,…,Φan,gn]for some n≥0, gi∈Mand ai∈Csuch that: gi∈M⧹{0}and ai∈C⧹{0}, for 1≤i≤n, Im(a0)∈]−πi,πi], gi≠gi+1, for 1≤i≤n−1. Remark 1.13 For sub-monoids M having antipodal points, it follows from Remark 1.11 that a normal form result as above would depend on precise characterization of the set of free points. This seems to be a very difficult problem. As a hint, we refer to Fig. 2, reproduced from [11]. It shows a numerically computed representation the set of free points in the plane Cλ, where λ=a2/2. Figure 2. View largeDownload slide The known free points are unshaded. Figure 2. View largeDownload slide The known free points are unshaded. 1.4. HNN-extensions in Homeo(R,+∞) Going in another direction, we can consider the analogous problem for the group Homeo(R,+∞) of germs at +∞ of real homeomorphisms defined in open intervals of the form {x:x>x0} and which go to infinity as x goes to infinity. The group operation being the composition. Consider the following subgroups of Homeo(R,+∞),   T={ta:x↦x+a,a∈R},S+={sα:x↦αx,α∈R>0}Exp={e:x↦exp(x),e−1:x↦ln(x)},where ln:R*→R is obviously taken as the real branch of the logarithm. Let Aff+=T⋊S+ denote the subgroup of real positive affine maps. As it is well known, the conjugation by the exponential map defines an isomorphism   θ:T⟶S+ta⟼etae−1=sexp(a)and we can consider the group Aff+⋆θ derived from (T,Aff+,θ) by HNN extension. We recall (see for example, [24, 1.4]) that given a group G with presentation G=⟨F∣R⟩ and an isomorphism θ:H→K between two subgroups H,K⊂G, the HNN extension derived from (H,G,θ) is a group G⋆θ with presentation   G⋆θ=⟨F,𝚔∣R,𝚔h𝚔−1=θ(h),∀h∈H⟩.The new generator 𝚔 is called stable letter. Consider now the subgroup GAff+,Exp of Homeo(R,+∞) generated by Aff+∪Exp. From the universal property of the HNN extensions, we know that there is a unique morphism   ϕ:Aff+⋆θ⟶GAff+,Expwhich is the identity when restricted to Aff+ and which maps the stable letter to the exponential map. We claim that GAff+,Exp contains no other relations besides the one expressing that exp conjugates T to S+. In other words, Theorem 1.14 ϕ:Aff+⋆θ→GAff+,Expis an isomorphism. Remark 1.15 Based on the above result, we can obtain a quite economic presentation for the group GAff+,Exp, namely   GAff+,Exp=⟨R,𝚔∣(𝚔a𝚔−1)b(𝚔a𝚔−1)−1=exp(a)b,∀a,b∈R⟩,where R is equipped with its usual additive group structure. For instance, the multiplicative structure of S+ is easily obtained by defining sexp(a)≔𝚔a𝚔−1. As another consequence, we obtain a large collection of (apparently new) free subgroups inside Homeo(R,+∞). Indeed, consider the family of subgroups {Tn}n∈Z*⊂Homeo(R,+∞) given by   T0=T,Tn=θn(T0)=enT0e−n,∀n∈Z*,where for n>0 (resp. n<0), en denotes the n-fold composition of e (resp. e−1). Notice that S+=T1. We define   An=θn(Aff+)=Tn⋊Tn+1,∀n∈Z,where Tn+1 acts on Tn by conjugation (exactly as S+ acts on T). Corollary 1.16 The subgroup of GAff+,Expgenerated by ⋃n∈ZAnis isomorphic to the infinite free amalgamated product given by the following diagram:  where the north-east and north-west arrows are, respectively, the identity inclusions and the monomorphism S+=θ(T). In [12], Glass attributes to Higman the following question: Do T and Pow+={x↦xr:r∈R>0}generate their free product? The above corollary allows us to answer this question affirmatively. Indeed, as A1=S+⋊Pow+, the above diagram shows that the subgroup GAff+,Pow+ of Homeo(R,+∞) generated by T∪S+∪Pow+ has the presentation   GAff+,Pow+=(T⋊S+)⋆S+(S+⋊Pow+),where the amalgam is obviously made over S+. 1.5. Transseries and a finitary version of Lemma 1 We follow the notation from [9]. Let T=R[[[x]]] be the real ordered field of well-based transseries and P⊂T be the subset large positive transseries. Then, P is a group under the composition operation and there is a injective homomorphism   T:GAff+,Exp→Pwhich associates to each element g∈GAff+,Exp its transseries at infinity. Indeed, each germ in GAff+,Exp defines element in the Hardy field H(Ran,exp) (see for example, [26]), and therefore this homomorphism is a direct consequence of the embedding of H(Ran,exp) into T (see [26, Corollary 3.12]). In this subsection, we shall be concerned with the following property (see for example, [6, 13]): Definition 1.17 Given an element ϕ∈P and a subgroup H⊂P, we shall say that H and ϕ are immiscible if the subgroup generated by H∪ϕHϕ−1 is isomorphic to the free product H⋆H. For each integer k≥1, let Gk⊂P denote the subgroup real formal series at +∞ which are tangent to identity to order k, that is, the group of transseries of the form   x↦x+∑j≥k−1bjx−j,withbj∈R.The following problem is stated in [7] (see also [6, Section 1.4]): Immiscibility problem. Prove that G2and ϕ are immiscible in the following cases:   ϕ:x↦exp(x),ϕ:x↦x+ln(x),orϕ:x↦xλ,where λ∈R>0⧹Q>0. Remarks 1.18 (1) The immiscibility problem naturally appears in the study of the Poincaré first return map in the vicinity of an elementary polycycle. Such study is an essential ingredient in the proofs of the Finiteness theorem of limit cycles for polynomial vector fields in the plane (see in [6, 13]). According to the strategy sketched in [7, 8], one expects that a positive answer to the immiscibility problem would allow us to significantly simplify these proofs. (2) The immiscibility problem has an obvious negative answer if G2 is replaced by G1. Indeed, given an arbitrary non-identity element f∈G1 and a scalar a∈R*, consider the series   g=sexp(a)fsexp(−a)which is also an element of G1. Then, using the identity eta=sexp(a)e one can rewrite   g=etae−1fet−ae−1.Since the translation ta is an element of G1, one obtains the following relation in the subgroup generated by G1∪eG1e−1:   etae−1fet−ae−1=sexp(a)fsexp(−a),which shows that this subgroup is not isomorphic to the free product G1⋆G1. Notice that an element f∈Gk can be written as the limit of a (Krull convergent) sequence {fn}n≥k⊂P given by   f0=1,fn=T(exp(anw−n))fn−1,wherew−n=x−n(x∂∂x)with constants an∈R uniquely determined by f and the flow maps exp(anw−n) being given by Example 1.9. This motivates us to consider the subgroup Gk,finite⊂Gk of those elements f which can be expressed as finite words, namely   f=T(exp(a1w−k1)⋯exp(anw−kn))for some n≥0, ai∈R and ki∈Z≥k. Notice that each Gk,finite is indeed defined by an analytic function in a neighborhood of infinity and lies in the image of the morphism T considered above. It also lies in the Hardy field H(Ran,exp) (cf. [26, Section 3.11]). In order to formulate our next result, let Λ be the subset of all non-identity elements g∈GAff+,Exp of the form   g=g1⋯gnfor some n≥1 and gi∈{e,l}∪{pr:r∈R>0⧹Q>0}. As a consequence of the previous theorem and the Normal form Theorem, we obtain the following finitary immiscibility property: Theorem 1.19 Let g be an arbitrary element in Λ. Then, G2,finiteand ϕ=T(g)are immiscible. Remarks 1.20 (1) Notice that the result includes the case where g is given by towers of exponentials and powers, such as   g:x→eexr,r∈R>0⧹Q>0.However, it does not include the so-called inverse log-Lambert map,   L:x↦x+ln(x)which is a solution of the differential equation   (x1+xddx)L=1.The map L plays an important role in proof of the finiteness of limit cycles. Indeed, it constitutes one of the building blocks in the construction of the Dulac transition map near a hyperbolic saddle or a saddle-node. We believe that it is possible to adapt our proof to include this function in the statement of the above theorem. (2) The passage from G2,finite to G2 in the immiscibility problem seems to be outside the reach of the tools developed in this paper. A possible strategy of proof could consist in appropriately identifying G2 to some subset of ends in the Bass–Serre tree defined by the HNN-extension Aff+⋆θ. 2. Formal theory in GPSL(2,C),Exp In this section, we will start our proof of the Main Theorem. As a first step, we recall some basic universal constructions in groupoid theory, following closely [4]. 2.1. Free product and quotient of groupoids Let G and H be groupoids, and let j1:G→K, j2:H→K be morphisms of groupoids. We say that these morphisms present K as the free product of G and H if the following universal property is satisfied: if g:G→L and h:H→L are morphisms of groupoids which agree on Obj(G)∩Obj(H), then there is a unique morphism k:K→L such that kj1=g, kj2=h. Such free product always exists (see [4, Section 8.1]) and will be noted G∗H. If the groupoids G and H have no common morphism except the identity, the elements of G∗H are can be identified with the set of paths   [g1,g2,…,gn]which are either equal to [1] or where each gi belongs to either G or H, no gi is the identity, and gi, gi+1 do not belong to the same groupoid. We now recall the construction of the quotient of a groupoid by a set of relations. In a groupoid G, suppose given, for each object p, a set R(p) of elements of G(p) (the vertex group at p). The disjoint union R of the R(p) is called a set of relations in G. We define the normal closure N=N(R) of R as the following subgroupoid: Given an object x∈Obj(G), a consequence of R at x is either the identity at x or any morphism of the form   an−1ρnan⋯a1−1ρ1a1for which ai∈G(x,xi) and ρi, or ρi−1, is an element of R(xi). The set of all consequences at a point x, which we note N(x), is a subgroup of G(x) and the disjoint union N of all N(x) has the structure of a totally disconnected normal subgroupoid of G (see [4], Section 8.3), where by totally disconnected groupoid we mean a groupoid where each morphism has its source equal to its target. It can be shown that N is the smallest wide normal groupoid of G which contains R. Let G/N(R) be the quotient groupoid (see [4, Theorem 8.3.1]). The projection π:G→G/N(R) has the following universal property: for each morphism of groupoids f:G→H which annihilates R, there exists a unique morphism f′:G/N(R)→H such that f=f′π. 2.2. Product normal form in GPSL(2,C)∗ΓGExp The essence of our Normal Form theorem is to present GPSL(2,C),Exp as the quotient of a free product of groupoids by some explicit set of relations. For this, we consider the groupoids   GPSL(2,C)=Germ(PSL(2,C)),GExp=Germ({exp})and let ΓGExp denote the free groupoid on the graph of GExp, that is, the groupoid defined by the set of reduced paths on GExp (see Section 1.1). Let F=GPSL(2,C)∗ΓGExp be the free product of these groupoids. A first necessary step to obtain a normal form in F is to describe the normal forms in GPSL(2,C) and ΓGExp. We need two preparatory Lemmas: Lemma 2.1 Each element g∈PSL(2,C)can be written as one of the following expressions:   g=sαtawtborg=sαtbfor some uniquely determined constants α∈C*and a,b∈C. Moreover, if we consider the region Ω⊂Cgiven by  Ω={α:Re(α)>0}∪{α:Re(α)=0,Im(α)>0}(see Fig. 1, at the Introduction), the following holds: Each right coset of H0=T⋊{s−1}in PSL(2,C)contains a unique element of the form  g=sρwtb,org=sρfor some constants b∈Cand ρ∈Ω. Each right coset of H1=S⋊{w}in PSL(2,C)contains a unique element of the form  g=tawtb,g=tcw,org=tbfor some constants c∈C⧹{0}, b∈Cand a∈Ωsuch that b≠−1/a. Proof The first part of the lemma follows from the well-known presentation of PSL(2,C) (see for example, [14, XI]). In particular, we recall the following relation in PSL(2,C):   1a+1z=−1a2(−a+1z+1a),∀a∈C⧹{0},z∈C,or, equivalently,   wtaw=s−1/a2t−awt1/a.Now, in order to prove items (i) and (ii), it suffices to study the orbit of sαtb and sαtawtb under the left multiplication by H0 and H1, respectively. For instance, given sαtawtb∈PSL(2,C) such that a≠0, the above relation in PSL(2,C) allows us to write   sαtawtb≡(s−a2sαw)sαtawtb≡s−a2wtawtb≡t−awtb+1/a,where ≡ denotes the equivalence in H1⧹PSL(2,C). Therefore, the coset H1sαtawtb contains either an element of the form tawtb with a∈Ω and b≠−1/a or an element of the form tcw, with c≠0.□ To state the next result, we introduce the symbols   lk:z↦lnk(z),∀k∈Z,where we recall, lnk denotes the kth branch of the logarithm. Notice that l0=l. Lemma 2.2 Each element in ΓGExpis either the identity 1or a path of the form  [lk1,…,lkn,e,…,e︸s−times]for some positive integers n,s, not both zero, and integers k1,…,kn∈Zsuch that the rightmost germ lknand the leftmost germ ein the path are not mutually inverses. Proof Each germ g∈GExp is defined by a path   [g1,g2,…,gm],where each gi is either equal to e or to lk for some k∈Z. We transform this path to a reduced one by successively canceling out each two consecutive germs gi,gi+1 such that gigi+1=1. Recall now the following (unique) two relations in GExp (see the discussion at the Introduction),   (1)elk=1and(2)lke=1,ifs(e)∈Jk,for all k∈Z. Therefore, after performing all possible cancellations in the above path, we either obtain the identity path, or a path as above such that no germ lk has a germ e to its left; and furthermore, that no consecutive germs lk,e are mutually inverse. This is precisely a path of the form in the statement of the lemma.□ We now consider normal forms inside the free product groupoid F=GPSL(2,C)∗ΓGExp. Definition 2.3 A product normal form in F is a path of the form   g=[g0,h1,f1,g1,…,hn,fn,gn]for some n≥0, such that the following holds: The germ g0 lies in GPSL(2,C) (with possibly g0=1). For 1≤i≤n, hi is either equal to e or to lk, for some k∈Z. If hi=e then fi∈H0 and gi is given by item (i) of Lemma 2.1. If hi=lk then fi∈H1 and gi is given by item (ii) of Lemma 2.1. There are no subpaths [lk,1,1,e] or [e,1,1,lk] such that the germs lk and e are mutually inverse.We denote by PNF the set of all product normal forms. As a consequence of the definition of F and the previous two lemmas, we obtain the following. Proposition 2.4 Each morphism of Fcan be uniquely defined by an element of PNF. Proof By the definition of a free product, each non-identity element g∈F can be uniquely identified with a path   g=[g1,g2,…,gn]such that the following conditions hold: gi∈GPSL(2,C)∪ΓGExp, for i=1,…,n. No two consecutive morphisms gi, gi+1 belong to the same groupoid. No gi is the identity morphism.Given such a path, we can uniquely obtain a path in PNF. Indeed, proceeding from left to right, for i=1,…,n, we do the following: If gi∈ΓGExp, then we use Lemma 2.2 to write   gi=[lk1,…,lkn,e,…,e]and, in the expression of g, we replace gi by the subpath   [lk1,1,1,lk2,…,lkn,1,1,e,1,1,…,e]. If i≥1, gi belongs to GPSL(2,C) and gi−1 has an e as its last symbol, then we use Lemma 2.1 to write   gi=fg′,forsomef∈H0,andg′givenbyLemma2.1(i)and we replace gi by the subpath [f,g′] in the expression of g. If i≥1, gi belongs to GPSL(2,C) and gi−1 has an lk as its last symbol, then we use Lemma 2.1 to write   gi=fg′,forsomef∈H1,andg′givenbyLemma2.1(ii)and we replace gi by the subpath [f,g′] in the expression of g.This concludes the proof.□ Remark 2.5 Recall that the subgroup of translations by 2πiZ lies in the intersection GExp∩GPSL(2,C). Therefore, the normal forms in the free product groupoid GPSL(2,C)∗GExp are more subtle to describe than those in F. Let now NF be the set of normal form paths defined in Remark 1.2 of the Introduction. Clearly, there is a natural embedding of NF into PNF given by   [g0,h1,g1,…,hn,gn]∈NF⟶[g0,h1,1,g1,…,hn,1,gn]∈PNF.To simplify the notation, we will keep the symbol NF to denote the image of this embedding. 2.3. Quotienting GPSL(2,C)∗ΓGExp Now, we consider the following collection Rel of relations in F:   Rel{le=t−2πir,ifs(e)∈Jr.es−1=weeta=sexp(a)e,∀a∈C,where l=l0 is the 0th branch of the logarithm. Notice that, for simplicity, we have written these relations in the form of an equality of germs w=u, but this should be understood as saying that w composed with the inverse of u is a relation (in the sense of Section 2.1) at every point where the corresponding germs are defined. Let F/N(Rel) denote the quotient groupoid, as defined in the previous subsection, and let   π:F⟶F/N(Rel)be the canonical morphism. The following theorem will be proved in the next subsection. Theorem 2.6 Each element in the quotient F/N(Rel)is uniquely defined by a normal form in NF. We now observe that, by construction and the universal property of F, there is a uniquely defined groupoid epimorphism   ϕ:F→GPSL(2,C),Expwhich is induced by the inclusion morphisms GPSL(2,C)→GPSL(2,C),Exp and GExp→GPSL(2,C),Exp. Using the obvious relations between the exponential, the affine maps and the involution, we conclude that this morphism factors out through the canonical morphism π:F→F/N(Rel), that is, we have a commutative diagram   for a uniquely defined morphism φ:F/N(Rel)→GPSL(2,C),Exp. As an immediate consequence of this discussion and Theorem 2.6, we obtain: Corollary 2.7 The first statement of the Main Theorem is true. 2.4. Reduction to normal forms in F/N(Rel) This subsection is devoted to the proof of Theorem 2.6. For this, we briefly recall the basic concepts of reduction systems (see for example, [3]). An abstract reduction system is a pair (X,→) where the reduction → is a binary relation on the set X. Traditionally, we write x→y (or y←x) instead of (x,y)∈→. The binary relation →* is the reflexive transitive closure of →. In other words, x→*y if and only if there is x0,…,xn such that x=x0→x1→⋯→xn=y. The binary relation ↔* is the reflexive transitive symmetric closure of →. Equivalently, x↔*y if and only if there are z1,…,zn∈X such that   x↔z1↔z2⋯↔zn↔y,where ↔=←∪→. We also say that: x∈X is reducible if there is a y∈X such that x→y. x∈X is in normal form if it is not reducible. x∈X is a normal form of y∈X if y→*x and x is a normal form. x,y∈X are joinable if there is a z∈X such that x→*z←*y.A reduction system (X,→) is called terminating if there is no infinite descending chain x0→x1→⋯. In this case, each element x has at least one normal form. A reduction system (X,→) is called confluent if y1←*x→*y2 implies that the elements y1 and y2 are joinable. We say that (X,→) is Church–Rosser if x↔*y implies that x and y are joinable. These two properties are usually pictured by the following respective diagrams:   We shall use the following consequences of the definitions: if (X,→) is terminating and confluent, then every element has a unique normal form (see [3, Lemma 2.1.8]). The Church–Rosser and the confluent properties are equivalent (see [3, Theorem 2.1.5]).We are going to apply this formalism to the set X=PNF of product normal forms (see Definition 2.3). In order to simplify the notation, in the remainder of this subsection, we shall identify a path [f1,…,fn] with a word f1f2⋯fn in the letters f1,…,fn. We stress that the formal word f1f2⋯fn should not be confounded with the element of the groupoid GPSL(2,C),Exp defined by the corresponding path. The letter l0 will be written simply l. Moreover, the identity path [1] will be identified with the empty word ε. Thus, for instance, the path [e,1,1,e] will be written simply as ee. First of all, we introduce the following reduction rules (recall that both sides of the ⇒ relation should be seen as paths in F):   lk⇒t2πikl,∀k∈Z*el⇒ε,le⇒t−2πir,ifs(e)∈Jres−1⇒we,eta⇒sexp(a)e,∀a∈C,lw⇒s−1lifarg0(s(w))≠πlw⇒t2πis−1lifarg0(s(w))=πlsα⇒tbl,∀α∈C*,where, in this last rule, we define b∈C as follows:   b={ln0(α),if−π<arg0(α)+arg0(s(sα))≤πln−1(α),ifπ<arg0(α)+arg0(s(sα))≤2πln1(α),if−2π<arg0(α)+arg0(s(sα))≤−π.The reduction system (PNF,→) is now defined as follows: Given g,h∈PNF, we say that g→h if there exists some reduction rule u⇒v as above such that one can write   g=g′ug″,forsomeg′,g″∈PNF,and h∈PNF is the product normal form of the path g′vg″. Remark 2.8 Notice that the simple substitution g′ug″→g′vg″ would not map PNF into itself. For instance, if b=ln0(−2) then, applying the fifth substitution rule to g=etaetb, one would obtain etas−2e, which is not in PNF, since tas−2 should be decomposed in H0T0 as (tas−1)s2. Proposition 2.9 The reduction system (PNF,→)is terminating and confluent. Moreover, the set of normal forms of (PNF,→)is precisely the subset NF. Proof We claim that there can be no infinite sequence of reductions. To prove this, we define a well-order on PNF which will decrease after each reduction. First of all, recall that a germ f∈H0∪H1 is either the identity or can be uniquely expressed as follows: In H0: f=tas−1, f=ta or f=s−1, for some a∈C⧹{0}, In H1: f=sαw, f=sα or f=w, for some α∈C*⧹{1}.Accordingly, we define the h-length lh(f)∈{0,1,2} by   lh(f)=2iff∈{tas−1,sαw},lh(f)=1iff∈{ta,sα,s−1,w},and we put lh(f)=0 if f=1. Consider now a path g=g0h1f1g1⋯hnfngn in PNF. We define its h-length as the integer n-vector   lh(g)=(lh(fn),lh(fn−1),…,lh(f1))∈{0,1,2}n.We further define m(g) to be the total number of germs of type lk, for k∈Z*, and n(g) to be the total number of germs of type e or l in the expression of g. Finally, we define a total order in PNF by saying that g<g′ if   (m(g),n(g),lh(g))<lex(m(g′),n(g′),ln(g′)),where <lex is the lexicographical ordering in the set of positive integer vectors. By inspecting the rules in Rel, one sees that if g′→g then g<g′. Moreover, a path g∈PNF is not reducible if and only if the following holds: m(g)=0, lh(g)=(0,…,0), and g contains no subpath of the form el or le.According to Definition 1.1, this corresponds precisely to say that g∈NF. Thus, we have proved that (PNF,→) is terminating and that its set of normal forms is NF. In order to prove the confluence of the reduction system, we use Bendix–Knuth criteria as stated in [10, Lemma 6.2.4]. Thus, it suffices to consider all shortest paths for which at least two of the above reduction rules can be applied (that is, they overlap) and show that the paths obtained after applying these reductions are then joinable. For instance, one sees that   The computation for the other possible overlaps is straightforward but quite tedious. We omit this computation.□ We are now ready to prove Theorem 2.6: Proof of Theorem 2.6 We need to prove that each coset of F/N(Rel) contains exactly one element of NF. By the fact that (PNF,→) is terminating, we know that each coset of F/N(Rel) contains at least one element of NF. Now, the essential remark is that the equivalence relation ↔* on PNF defines precisely the cosets of the quotient groupoid F/N(Rel). Indeed, for each relation u=v in the list Rel given at Section 2.3, one sees that u↔*v. Reciprocally, for each reduction rule u⇒v, one sees that uv−1 belongs to N(Rel). Therefore, assume that there exist two elements g,g′ in NF such that π(g)=π(g′) (where π:F→F/N(Rel) is the quotient map). This is equivalent to say that g↔*g′. Since (PNF,→) is confluent, it is Church–Rosser. Therefore, g↔*g′ implies that g and g′ are joinable. But since both g and g′ are normal forms (and hence not reducible), we conclude that g=g′.□ Remark 2.10 (Word problem and decidability) One could ask if the reduction system (PNF,→) would allow us to algorithmically solve the word problem in F/N(Rel). Equivalently, one asks if, given an element g∈PNF, there exists an algorithm to decide if   g→*1.Notice that the reduction rules in (PNF,→) assume the existence of an oracle which, given a complex constant α∈C, will answer affirmatively or negatively to the question   Isα=0?Even assuming that the constants appearing in the initial path g are, say, rational numbers, this oracle will eventually need to test new constants which are exp–log expressions in these initial constants, such as   eee2log(3/4)+e−3ee10−eee2ln5−ln(ln(3/2)).The existence of an algorithm for the above oracle is strongly related to the decidability of (R,exp) and the known algorithms assume Schanuel's conjecture. On the positive side, using the results of [22, 25] (see also [17, Section 2.1]), one can prove the following: Assuming Schanuel’s Conjecture, the word problem is decidable for the groupoid  FQ/N(Rel),where FQdenotes the free product groupoid GPSL(2,Q)*ΓGexp. 3. From normal forms to field extensions Our present goal is to prove the second part of the Main Theorem. Many of the following constructions will be carried out for arbitrary normal forms, not necessarily satisfying the tameness property. We shall explicitly indicate the points where this assumption will be necessary. Given a point p∈P1(C), we denote by (M,∂) the differential field of meromorphic germs at p equipped with the usual derivation ∂=d/dz with respect to some arbitrary local coordinate z at p (with constant subfield Const(∂)=C). Given a normal form g∈NF, with source point p=s(g), our next goal is to construct a sequence of field extensions in M which will encode the necessary information to study the identity   φ(g)=id,where we recall that φ:NF→GPSL(2,C),Exp is the mapping which associates a germ in GPSL(2,C),Exp to each path in NF. 3.1. Algebraic paths and Cohen field In this subsection, we consider field extensions defined by algebraic paths. Let a∈NF be an algebraic path of length n≥0. Thus, we can uniquely write   a=θ0pα1θ1⋯pαnθn,where each exponent αi lies in Ω∩Q⧹{1}, θ0∈PSL(2,C), θi∈T1⧹{1}, for 1≤i≤n and θ1,…,θn−1 are not the identity. We consider the sequence of algebraic field extensions in M  En⊂En−1⊂⋯⊂E0=Einductively defined as follows. First, En=C(xn) is the field defined by the identity germ xn=φ(1). Then, for each i=0,…,n−1, we define   Ei=Ei+1(xi),where xi is a germ of solution of the algebraic equation   xiv=θ(xi+1)u,where we have written θ=θi+1 and αi+1=u/v for some co-prime integers u,v. Here, the branch of the vth-root is uniquely chosen accordingly to the source/target compatibility condition determined by a. We will say that the resulting field   E=C(x0,…,xn)is the Cohen field of a. In the seminal paper [5], Cohen has studied the Cohen field for algebraic normal forms of affine type. In what follows, we shall make essential use of the following immediate consequence of a result in [5]. Theorem 3.1 (cf. [5, Theorem 3.2]) Assume that a is an algebraic normal form of affine type and length n≥0, with associated Cohen field E=C(x0,…,xn). Then, we can write  E=C(x0,xn)that is, x1,…,xn−1are rational functions of x0and xn. Moreover, if a is not a rational path (Recall (see the Introduction) that an algebraic path is called rational if each power map in its basic decomposition has a exponent in Ω∩Z), then E is a strict algebraic extension of C(xn). Notice that the second statement of the theorem does not hold if a is not of affine type. Example 3.2 Consider the algebraic normal form   a=p1/2t−1/4p2θp2,where θ is a Möebius map such that θ(0)=−1/2 and θ(∞)=1/2. Then, we can write Ea=C(x,y,z), where x,y,z satisfy the relations   z2=−x2(x2+1)2andy=z2.Obviously, C(x,z) is not a strict algebraic extension of C(x). From now on, the elements y=xn and z=θ0(x0) (where θ0 is the Möebius part of a) will be called, respectively, the tail and the head elements of the Cohen field of a. Corollary 3.3 Assume that a is a non-identity algebraic normal form of affine type, with Cohen field E. Then, the head and tail elements z,y∈Ecannot satisfy the relation  yz=1. Proof We consider the following three possible cases: a is not a rational path. a is a rational path of length n≥1. a=θ is a Möebius map.In case (1), the result is an immediate consequence of Theorem 3.1. In case (2), we write the decomposition of a as   a=θ0pu1θ1⋯punθnfor some integers u1,…,un∈Ω∩Z. It follows that E is the function field of an algebraic curve C⊂(P1(C))n+1 which defined by the vanishing of the ideal n⊂C[X0,…,Xn] generated by the equations   xn−1=θn(xn)un,…,x0=θ1(x1)u1(after appropriately eliminating the denominators in the expression of the Möebius maps). In particular, the rational map y=xn defines a degree 1 unbranched covering C→P1(C), while the rational map z=θ(x0) defines a branched covering C→P1(C) of degree u=∣u1⋯un∣≥2. Therefore, y/z is a non-constant rational map on C. Finally, in the case (3), we have the identity x0=xn=y. Therefore, we can write   zy=θ(y)yand the right-hand side is non-constant because θ≠1. This concludes the proof.□ Remark 3.4 Example 1.4 shows that Corollary 3.3 is false if we drop the condition that a is of affine type. 3.2. The field associated to g∈NF We will now generalize the construction of the previous section to arbitrary normal forms. Assume that g∈NF has an algebro-transcendental decomposition   g=a0γ1a1⋯γmam,m≥0.Then, we inductively construct a chain of subfields in M,   Lm⊂Fm⊂⋯⊂L0⊂F0=F,as follows. First, we define Lm=C(ym) to be the field defined over C by the identity germ ym=φ(1). Then, we put: Fi=Li(zi), where zi∈M is the germ defined   zi=φ(aiγi+1ai+1⋯γmam),i=0,…,m. Li=Fi+1(yi), where yi∈M is the germ defined   yi=φ(γi+1ai+1⋯γmam),i=0,…,m−1.We will say that F=C(y0,z0,…,ym,zm) is the field associated to g. Notice that each field extension Fi+1⊂Li (described in item (2)) is obtained by adjoining to Fi+1 a (germ of) non-zero solution y to one of the following differential equations:   ∂(y)y=∂(z),∂(y)=∂(z)zor∂(y)y=α∂(z)z,where we have written write z=zi+1. These equations correspond respectively to the case where γi+1 is given by e, l or pα, for some α∈Ω⧹Q. On the other hand, each extension Li⊂Fi (described in item (1)) is algebraic, obtained by adjoining a germ of solution zi of a polynomial equation with coefficients in C(yi). More precisely, if we consider the Cohen field E associated to the maximal algebraic path ai (as defined in Section 3.1), and write E=C(x0,…,xn), then we can embed C(yi,zi) into E by setting   y=xn,andz=θ0(x0).In particular, the first part of Theorem 3.1 can be reformulated as follows: If g=a is an algebraic path of affine type, then the fields F and E coincide. Remark 3.5 It follows from the above discussion that the differential field (F,∂) associated to a normal form is a Liouvillian extension of (C(x),∂) (see [27, Section 1.5]). Furthermore, one always has   trdegCF≤m+1,where m=height(g).The second part of the Main Theorem will be a direct consequence of Corollary 3.3 and the following result. Theorem 3.6 Let g∈NFtamebe a tame normal form of height m≥1and associated field F=C(y0,z0,…,ym,zm). Then  trdegCF=m+1.Moreover, {y0,…,ym}forms a transcendence basis for F/C. 3.3. Three lemmas on twisted equations The proof of Theorem 3.6 uses induction on the height of a normal form and is essentially based on the three lemmas stated below, which treat special types of differential equation in (F,∂). For future reference, the equations (3.1), (3.2) and (3.3) below will be called, respectively, the first, second and third twisted equations. To fix the notation, we consider a tame normal form g∈NFtame of height m≥0 with algebro-transcendental decomposition   g=a0γ1a1⋯γmam,m≥0and associated differential field F=C(y0,z0,…,ym,zm) (equipped with the usual derivation ∂=d/dz induced from M). In the following statements, we use the following definition. Given a germ γ∈{e,l,pα:α∈Ω⧹Q}, the γ-augmention of g (or, shortly, the γ-augmented path) is the path   gaug=γa0γ1a1⋯γnanobtained by adjoining γ to the left of g (with the obvious compatibility condition that the source of the germ γ coincides with the target of the germ θ0). Note that gaug is not necessarily in normal form. We will say that gaug is a nice augmentation of g if gaug lies in NFtame and moreover the right-hand side of the above displayed equation is precisely the algebro-transcendental decomposition of gaug. Lemma 3.7 Assume {y0,…,ym}is a transcendence basis for F/Cand suppose that the e-augmented path   gaug=eθ0γ1θ1⋯γnθnis a nice augmentation of g. Let f∈Fbe a non-zero solution of the equation  (∂−μ∂(z0))f=0 (3.1)for some μ∈C. Then, necessarily μ=0and f∈C. Lemma 3.8 Assume that {y0,…,ym}is a transcendence basis for F/Cand suppose that the p-augmented path  gaug=pαθ0γ1θ1⋯γnθn,forsomeα∈Ω⧹{1}is a nice augmentation of g. Let f∈Fbe a non-zero solution of the equation  (∂−μ∂(z0)z0)f=0 (3.2)for some μ∈C⧹Q*. Then, μ=0and f∈C. Lemma 3.9 Assume that {y0,…,ym}is a transcendence basis for F/Cand suppose that the l-augmented path  gaug=lθ0γ1θ1⋯γnθnis a nice augmentation of g. Let f∈Fbe a non-zero solution of the equation  ∂f=c∂(z0)z0 (3.3)for some c∈C. Then, necessarily c=0and f∈C. We postpone the proofs of these lemmas to Section 3.7. 3.4. The proof of Theorem 3.6 Let us see how the previous results imply Theorem 3.6. Proof of Theorem 3.6 As we said above, the proof is by induction in m=height(g). Therefore, we start with the case m=0. By definition, g=a is an algebraic path of affine type, and the associated field K=C(y,z) has transcendency degree one over C. Now, given m≥0, we assume by induction that all tame normal forms of height ≤m satisfy the conclusions of the theorem. Let h∈NFtame be a normal form of height m+1. Then, we can write the decomposition   h=a0γ0gfor some algebraic path a0∈PSL(2,C), a germ γ0∈{e,l,pα:α∈Ω⧹Q}. Further,   g=a1γ2θ2⋯γn+1θn+1is a tame normal form of height m and the augmented path gaug=γ0g is a nice augmentation of g (see definition at Section 3.3). Let us denote by F and F′ the differential fields associated to g and h, and write   F=C(y1,z1,…,ym+1,zm+1),F′=C(y0,z0,…,ym+1,zm+1).Recall that, by construction, z0 is algebraic over C(y0). Therefore, to prove the induction step, it suffices to show that the element y0 is transcendental over F. To simplify the notation, from now on we will write y=y0 and z=z1. Let us assume for a contradiction, that y is algebraic over F. We choose a minimal polynomial f∈F[X] for y, say   f=f0+⋯+fd−1Xd−1+Xd,fk∈F,where we can assume that d≥1 and that f0 is non-zero. We discuss separately the cases where γ0=e (extension of exp type), γ0=l (extension of log type) and γ0=pα (extension of power type). Suppose that the extension F⊂F′ is of exp type. Recalling that y satisfies the equation ∂(y)=y∂(z), it follows from the equation ∂(f(y))=0 that the polynomial   p=∂(f0)+⋯+(∂(fd−1)+(d−1)∂(z)fd−1)Xd−1+d∂(z)Xdvanishes on y. As a consequence, the polynomial of degree at most d−1 given by q=p−d∂(z)f also vanishes y. By the minimality of f, this polynomial must vanish identically. This is equivalent to the collection of equations   (∂−(k−d)∂(z))fk=0,k=0,…,d−1.We claim that this implies d=0, which contradicts the definition of f. Indeed, assume for a contradiction that d≥1. Then, we are precisely in the hypothesis of Lemma 3.7, that is, each fk satisfies the first twisted equation with μ=(k−d). In particular, for k=0, one has   (∂−d∂(θ(x)))f0=0and the lemma implies that f0=0, which is absurd. Assume now that the extension F⊂F′ is of power type. Then, it follows from the relation ∂(y)=αy∂(z)/z that the polynomial   p=∂(f0)+⋯+(∂(fd−1)+α(d−1)∂(z)zfd−1)Xd−1+αd∂(z)zXdalso vanishes on y. Hence, by the minimality of d, the polynomial   q=p−αd∂(z)zfmust vanish identically. This corresponds to the collection of equations   (∂−α(d−k)∂(z)z)fk=0,k=0,…,d−1.We claim that this set of equations has no solution if d≥1. Indeed, in this case, each one of the above equations corresponds to a twisted equation as described in Lemma 3.8 with μ=α(d−k)∈C⧹Q. In particular, for k=0, we conclude from that lemma that f0=0, which is absurd. Finally, in case where the extension F⊂F′ is of log type, we have   ∂(y)=∂(z)z.Hence, the polynomial of degree at most d−1 given by   q=(∂(f0)+∂(z)zf1)+⋯+(∂(fd−1)+d∂(z)z)Xd−1also vanishes on y. By the minimality of d, this polynomial vanishes identically and this corresponds to say that the collection of equations   ∂fk=∂(z)z(k+1)fk+1,fork=0,…,d−1hold, where we put fd=1. Taking k=d−1, the rightmost equation is exactly the twisted equation from Lemma 3.9. From the lemma, it follows that d=0, which contradicts our assumption. This concludes the proof of the theorem.□ Remark 3.10 As some readers may have noticed, the above computations are very similar to the classical computations of the Picard–Vessiot extension K/k for the elementary linear differential equations   ∂(y)=a,or∂(y)=ay,witha∈kover a given differential field (k,∂) (see for example, [27, Examples 1.18 and 1.19]). In this simple setting, the computation of the differential Galois group of the extension K/k reduces to studying when these equations have no solution in the base field. 3.5. Resonance relations and Ax Theorem In this subsection, we recall some results about differential field extensions and differentials forms, following closely Wilkie's notes [29]. They are key ingredients in proof of the celebrated Ax's Theorem (cf. [2]). We observe that the results described here are completely independent of Theorem 3.6 and Lemmas 3.7, 3.8 and 3.9. In this subsection, k⊆K will denote arbitrary fields of characteristic zero, and k will be assumed to be algebraically closed. We will say that elements x1,…,xn∈K satisfy a power resonance relation (over k) if there exist integers r1,…,rn, not all zero, such that   ∏i=1nxiri∈k.Similarly, we will say that y1,…,yn∈K satisfy a linear resonance relation (over k) if there exist integers r1,…,rn, not all zero, such that   ∑i=1nriyi∈k.Since k is supposed algebraically closed, we can assume in both cases that the integers r1,…,rn are coprime. We will denote by Derk(K) the set of derivations δ:K→K whose constant subfield Const(δ) contains k. For each n∈N, Ωkn(K) denotes the K-vector space of alternating, K-linear n-forms on Derk(K) and   d:Ωkn(K)⟶Ωkn+1(K)is the total differential map. The space Ωk1(K) is the dual of Derk(K), and it is usually called the space of Kähler differentials of K over k. The space Ωk0(K) is identified to K. We say that a 1-form ω∈Ωk1(K) is closed (resp. exact) if dω=0 (resp. ω=du for some u∈K). Finally, given an intermediate field k⊂K0⊂K, we say that a form ω∈Ωk1(K) is defined over K0 if ω=∑aidbi, for some ai,bi∈K0. Lemma 3.11 Suppose that K0is a field such that k⊂K0⊂Kand tr.deg.k(K0)=nfor some n≥1. Let δ∈Derk(K)be a derivation such that Const(δ)=k, and suppose that  ω1,…,ωn∈Ωk1(K)are closed 1-forms defined over K0satisfying ωi(δ)=0, for i=1,…,n. Then ω1,…,ωnare linearly dependent over k. The above statement and its proof can be found, for instance, in Wilkie's notes [29, Theorem 2]. Lemma 3.12 Suppose that there exists non-zero x1,…,xm∈Kand elements e1,…,em∈knot all zero such that the differential form  ∑i=1meidxixiis exact. Then, x1,…,xmsatisfy a power resonance relation over k. Moreover, assume that there exists a derivation δ∈Derk(K)and y1,…,ym∈Ksuch that  Const(δ)=k,andδ(xi)xi=δ(yi)for each 1≤i≤m. Then y1,…,ymsatisfy a linear resonance relation over k. Proof The first statement is proved in [29]. For the second statement, it suffices to remark that if the monomial m=∏xiri belongs to k, then the linear form l=∑riyi satisfies   δ(l)=∑riδ(yi)=∑riδ(xi)xi=δ(m)m=0which implies that l∈k.□ 3.6. Resonances and transcendental equations We now consider an arbitrary normal form g∈NF of height m≥0, with algebro-transcendental decomposition,   g=a0γ1a1⋯γmam,m≥0and associated differential field (F,∂), where we write   F=C(y0,z0,…,ym,zm)as in Section 3.2. Further, for each i=0,…,m, we denote by Fi=C(yi,zi) the field associated to the maximal algebraic subpath ai. Proposition 3.13 Assume that a non-zero element f∈Fsatisfies one of the following three equations:   (1)∂(f)f=∂(z0),(2)∂(f)=∂(z0)z0or(3)∂(f)f=β∂(z0)z0for some β∈C⧹Q. Then, the elements  y0,z0,…,ym,zm∈Fsatisfy either a power resonant relation or a linear resonant relation over C. Proof Based on the algebraic-transcendental expansion of g written above, we consider the following subsets of {1,…,m}  Ie={j:γj=e},Il={j:γj=l},andIp={j:γj=pαj,forsomeαj∈Ω⧹Q},and define a collection of closed 1-forms ω1,…,ωm∈ΩC1(F) as follows:   ωj={dyj−1/yj−1−dzj,ifj∈Iedyj−1−dzj/zj,ifj∈Ildyj−1/yj−1−αjdzj/zj,ifj∈Ip,where, in this last case, αj denotes the exponent in the power map γj=pαj. Similarly, we define the closed 1-form ω0 as   ω0={df/f−dz0,iffsatisfies(1)df−dz0/z0,iffsatisfies(2)df/f−βdz0/z0,iffsatisfies(3).Then, by Lemma 3.11, since ωj(∂)=0 for all 0≤j≤m, and trdegC(F)≤m+1, there exist constants c0,…,cm∈C, not all zero, such that the relation   c0ω0+⋯+cmωm=0holds. We consider, first of all, the case where f satisfies equation (2). Then, by suitably regrouping the terms in the above relation, we obtain a 1-form   c0dz0z0+∑j∈Iecidyi−1yi−1+∑j∈Ilcjdzjzj+∑j∈Ipcj(dyj−1yj−1+αjdzjzj)which is exact. Therefore, we can apply Lemma 3.12 to conclude that there exists a monomial   M=z0v0∏j∈Ieyj−1uj−1∏j∈Ilzjvj∏j∈Ipyj−1uj−1zjvjwith integer exponents not all zero, which belongs to C. This proves the result. We consider now the case (1). Here, we conclude from the relation ∑cjωj=0 that the 1-form   c0dff+∑j∈Iecjdyi−1yi−1+∑j∈Ilcjdzjzj+∑j∈Ipcj(dyj−1yj−1−αjdzjzj)is exact. Hence, applying again Lemma 3.12, we show that there exists a monomial of the form   M=fw∏j∈Ieyj−1uj−1∏j∈Ilmzjvj∏j∈Ipyj−1uj−1zjvj∈C(for some integers w,vj,uj, not all zero) which belong to C. Notice that if w=0 we are done because this would give the desired relation. By the same reason, we would be done if c0=0. Hence, from now on, we can assume that w is non-zero and that c0=1. By computing the logarithmic derivative dM/M, we can write   dff=∑j∈Iempj−1dyj−1yj−1+∑j∈Ilmqjdzjzj+∑j∈Ipm(pj−1dyj−1yj−1+qjdzjzj)with pj=−uj/w and qj=−vj/w being rational numbers. We can now replace this expression for df/f in the relation ∑cjωj=0 and, again by suitably regrouping the terms, conclude that the 1-form   ∑j∈Ie(cj+pj−1)dyi−1yi−1+∑j∈Il(cj+qj)dzjzj+∑j∈Ip(cj+pj−1)dyj−1yj−1+(−cjαj+qj)dzjzjis exact. If at least one of the coefficients of this 1-form is non-zero, then we can apply again Lemma 3.12 in order to obtain a monomial satisfying the conditions in the statement. So, let us assume that all these coefficients vanish. Since αi∉Q for each i∈Ip, we conclude that   cj=pj−1=qj=0,forallj∈Ip.In particular, the monomial M has simply the form   M=fw∏j∈Ieyj−1uj−1∏j∈Ilmzjvj.If we consider the linear form   l=wz0+∑j∈Ieuj−1zj+∑j∈Ilvjyj−1it is easy to see that ∂(l)=∂(M)/M=0. Therefore, l∈Const(∂)=C. It remains to consider the case of equation (3). The treatment is similar to the previous case. Here, we obtain a 1-form   c0(dff−βdz0z0)+∑j∈Iecidyi−1yi−1+∑j∈Ilcjdzjzj+∑j∈Ipcj(dyj−1yj−1+αjdzjzj)which is exact, and hence there exists a monomial   M=fwz0v0∏j∈Ieyj−1uj−1∏j∈Ilmzjvj∏j∈Ipyj−1uj−1zjvj∈C(with exponents not all zero). Assuming that c0=1 and that w is non-zero (otherwise we are done), we can apply exactly the same reasoning as above to conclude that the 1-form   (q0−β)dz0z0+∑j∈Ie(cj+pj−1)dyi−1yi−1+∑j∈Il(cj+qj)dzjzj+∑j∈Ip(cj+pj−1)dyj−1yj−1+(−cjαj+qj)dzjzjis exact (where q0=−v0/w). Since β∉Q, the coefficient in front of dz0/z0 cannot vanish. Therefore, we can apply Lemma 3.12 in order to obtain a monomial which satisfies the desired relation. This concludes the proof.□ For later use, we need to establish a more precise statement about the existence of power/linear relations in the fields F0,…,Fm associated to the algebraic subpaths a0,…,am. Given a normal form g=a0γ1a1⋯γmam as in the beginning of the subsection and an equation for f∈F as in the statement of the previous proposition, we define the augmentation of g as the path   gaug=γ0a0γ1⋯γmamwhich is obtained by concatenating to g the symbol γ0=e (resp. l or pβ) if f satisfies equation (1) (resp. (2) or (3)). Further, given an index 0≤j≤m−1 and two symbols γ,γ′∈{e,l,p}, we will say that the algebraic subpath aj of gaug lies in a [γ,γ′]segment if   γj=γandγj+1=γ′. Corollary 3.14 Assume that {y0,…,ym}is a transcendence basis for F/C. Let f∈Fbe a nonzero solution of one of the equations (1), (2) or (3) from Proposition3.13. Then, there exists at least one index 0≤j≤m−1such that Either ajlies in a [e,l]segment and yj,zjsatisfy a linear resonance relation in Fj. Or ajlies in a [l,e], [l,p], [p,e]or [p,p]segment and yj,zjsatisfy a power resonance relation in Fj.In particular, if m=0then there is no non-zero element f∈Fsatisfying (1), (2) or (3). Proof The hypothesis imply that {dy0,…,dym} is a basis of ΩC1(F) and that the F-subspaces generated by ΩC1(F0),…,ΩC1(Fm) are F-linearly independent. Moreover, since each zj is algebraic over yj, the 1-form dzj lies in the one-dimensional F-subspace generated by dyj. From Proposition 3.13, we conclude that if a non-zero element f∈F satisfies (1), (2) or (3), then either there exists a monomial M=∏j=0mzjvjyjuj or a linear form l=∑j=0mvjzj+ujyj (with integers uj,vj not all zero) which belong to C. Taking the logarithmic derivative dM/M in the former case or the derivative dl in the later case, we obtain   ∑j=0mvjzjdzj+ujyjdyj=0,or∑j=0mvjdzj+ujdyj=0,respectively. Therefore, by the linear independency of dy0,…,dyn, either vjzjdzj+ujyjdyj=0 or vjdzj+ujdyj=0 for all 0≤j≤m. In the former case, we conclude that d(zvjyjuj)=0, while in the latter case d(vjzj+ujyj)=0. Now, to conclude the proof, it suffices to consider more carefully the expressions of m and l obtained in the proof of the previous proposition. For instance, we consider the case where   m=z0v0∏j∈Ieyj−1uj−1∏j∈Ilzjvj∏j∈Ipyj−1uj−1zjvj∈Cwhich, by the above argument, implies a collection of power resonance relations of the form d(zjvjyjuj)=0, for j=0,…,n. Notice that no relation of type d(zjvj)=0 (that is, with uj=0) or d(yjuj) (that is, with vj=0) can appear, since this would imply that zj or yj belong to C, contradicting the fact that both yj and zj are germs of invertible maps. Thus, there necessarily exists a monomial relation of the form zjvjyjuj∈C with exponents uj,vj both non-zero. But looking to the above expression for m, we conclude that this can only happen in the index j is such that j∈Il∪Ip and j+1∈Ip∪Ie. This is equivalent to say that aj lies in an [l,e], [l,p], [p,e] or [p,p] segment. The other cases can be treated in an analogous way.□ 3.7. Proofs of lemmas on twisted equations We now proceed to the proof of Lemmas 3.7, 3.8 and 3.9. We keep the notation introduced in Section 3.3. Proof of Lemma 3.7 Let us assume that μ≠0. By contradiction, we assume that there exists a non-zero f∈F such that   ∂(f)f=∂(z0).Writing the algebro-transcendental decomposition of the e-augmented path gaug as   gaug=ea0γ1a1⋯γmam,m≥0we let zj,yj denote the head and tail elements of the Cohen differential field Ej associated to the algebraic path aj, for j=0,…,m. Defining γ0=e, we can now apply Corollary 3.14 to conclude that there exists at least one index 0≤j≤m−1 such that Either γj=e, γj+1=l and yj,zj satisfy a linear resonance relation. Or γj∈{l,pα:α∈Ω⧹Q}, γj+1∈{e,pα:α∈Ω⧹Q} and yj,zj satisfy a power resonance relation.If m=0 we get our desired contradiction. If m≥1, we will deduce the contradiction using Corollary 3.3. For this, we treat cases (1) and (2) separately. To simplify the notation, we define   a=aj,y=yjandz=zjand write the expansion of the algebraic path a (of affine type) as   a=θ0p1θ1⋯pnθn,n≥0.In the case (1), y and z satisfy a relation of the form vz+uy=c, for some u,v∈Z* and c∈C. We consider then the modified algebraic path  a*=normalformreductionofs−v/ut−c/va.Explicitly, for a given as above, we can write   a*=θ0*p1θ1⋯pnθn,where the Möebius part of a* is given by θ0*=s−v/ut−c/vθ0. In particular, the assumption that a is an algebraic path of affine type implies that the same property holds for a*. Now, by the definition of a*, the head and tail elements z* and y* of the Cohen field E* associated to a* should satisfy the relation   y*z*=1.Hence, we will obtain the desired contradiction to Corollary 3.3 once we show that   a*≠1.To prove that this always holds, observe that a*=1 if and only if a=(s−v/ut−c/v)−1=tc/vs−u/v. Since a is a maximal algebraic subpath (lying in an [e,l] segment) of the augmented path gaug, this would contradict the hypothesis that gaug is a nice augmentation of g, as stated in Section 3.3. Indeed, if either c≠0 or −u/v∉Ω⧹{1}, then the subpath eal is certainly not in normal form. On the other hand, if c=0 and −u/v∈Ω⧹{1} then, according to our definition of algebro-transcendental decomposition, the corresponding subpath es−u/vl should instead be considered as a rational power map p−u/v. This concludes the proof of (1). Consider now the case (2). We write the corresponding power resonance relation as yuzv=c, for some u,v∈Z* and c∈C*. Since the algebraic path a lies in a [γ,η]-segment (with γ∈{l,p} and η∈{e,p}) and gaug is a nice augmentation of g, it follows from the definition of NF that its Möebius part θ0 necessarily lies in T1⧹{1}. We introduce now the modified algebraic path   a*=sc1/upv/ua,which is also a non-identity normal form by the discussion of the above paragraph. Similarly to the previous case, the head and tail elements z*,y* of the Cohen field E* associated to a* satisfy the relation y*z*=1 (up to a convenient choice of the branch of c1/u). Furthermore, a* is an algebraic normal form of affine type, and the above identity contradicts Corollary 3.3 when applied to a*. This concludes the proof of the lemma.□ Proofs of Lemmas 3.8 and 3.9 We follow exactly the same strategy of the previous proof. Namely, we assume for a contradiction that c≠0 and that there exists a non-zero element f∈F satisfying one of the following two equations:   ∂(f)f=μ∂(z0)z0or∂(f)=∂(z0)z0,where μ∈C⧹Q. Considering the algebraic transcendental decomposition of the augmented path gaug, the same alternatives (1) and (2) listed in the previous proof appear. By repeating the same reasoning, we obtain a contradiction.□ 4. Some consequences We proceed to prove the other results stated in the Introduction. Proof of Theorems 1.7, 1.10 and 1.12 We will only prove Theorem 1.7, since the other two results are immediate consequences. First of all, we remark that GAff,PowR is a subgroupoid of GPSL(2,C),Exp. Therefore, if we consider the free product groupoid F=GPSL(2,C)∗ΓGExp and the groupoid morphism   φ:F→GPSL(2,C),Expdefined in Section 2.3, then each germ lying in the GAff,PowR is the image of a (not necessarily unique) path in F. Further, we can assume that such path of the form   g=θ0pr1θ1⋯prnθn,n≥1,where each pri is a power map with exponent ri∈R and each θi is an affine map. Possibly making some simplifications, we can further assume that θ1,…,θn≠1 and that r1,…,rn≠1. As a consequence, g is a product normal form, that is, an element of the subset PNF⊂F given by Definition 2.3. Applying the reduction system (PNF,→) defined in Section 2.4, we can make the reduction   g→*g′,where g′ has the same form as g, but with the additional property that each affine map θ1,…,θnis a translation. The subset of paths in F satisfying these properties will be called normal forms of power-translation type, and noted NFPT. Notice that a path in NFPT is not necessarily an element of NF (see Definition 1.1), because the exponents r1,…,rn of the power maps do not necessarily lie in the region Ω described in Remark 1.2. However, the reduction from NFPT to NF can be easily obtained. Indeed, assuming that g∈NFPT is written as above, its normal form reduction g→*h gives the path   h=θ0wε1ps1θ1wε2ps2⋯wεnpnθn,where we define each pair (si,εi)∈Ω×{0,1} as follows:   (si,εi)={(ri,0)ifri∈Ω(−ri,1)if−ri∈Ω.We remark the following two facts: The normal form h lies in NFtame. If g,g′∈NFPT reduce to a same normal form h∈NFtame then necessarily g=g′.Indeed, the assumption R∩Q<0=∅ implies that a subpath of the form θiw appears in the expansion of h if and only if the power map pri+1 has an exponent in R⧹Q. Therefore, the algebro-transcendental decomposition of h can only contain maximal algebraic subpaths of affine type. This proves (1). The proof of (2) is immediate, since the original powers r1,…,rn∈R can be read out from the expression of the normal form. Based on these remarks, the result is now an immediate consequence of the second part of the Main Theorem.□ Proof of Theorems 1.14 and 1.19 We will only give the details of the proof of Theorem 1.14, since Theorem 1.19 is an immediate consequence of this result. Keeping the notation of Section 1.4, we want to prove that the homomorphism   ϕ:Aff*θ+→GAff+,Exp,is injective. Using Britton's normal form (see for example, [16, IV.2]), and the right transversals to T0,T1 to H0,H1 defined in Remark 1.2, it follows that we can (setwise) identify Aff*θ+ to a set BNF (so-called Britton normal forms) contained in the free product Aff+∗⟨𝚔i:i∈Z⟩ (where 𝚔 denotes the stable letter of the HNN-extension). By definition, each f∈BNF can be uniquely written as   f=θ0γ1θ1⋯γnθn,n≥0,where θ0,…,θn are affine maps, γi∈{𝚔,𝚔−1}, and If γi=𝚔 then θi∈S+. If γi=𝚔−1 then θi∈T. There are no subwords of the form 𝚔1𝚔−1 or 𝚔−11𝚔.The set BNF has a natural group structure which is inherited from the group structure of Aff*θ+. Using the above expansion for f∈BNF, we define maximal interval of existence If∈(R,+∞) of f as the largest open neighborhood of ∞ (of the form ]Af,+∞[ for some Af∈R) such that each one of the n+1truncations of the above normal form, namely   f[i]=θiγi+1θi+1⋯γnθnfori=0,…,nmaps under ϕ to a germ ϕ(f[i])∈GAff+,Exp which extends analytically to an invertible function defined on the interval If. To simplify the notation, we denote also by   ϕ(f):If→Rthe corresponding (uniquely determined) analytic function. Now, we consider a mapping ρ1:BNF→NFtame which sends an element f∈BNF to a tame normal form ρ1(f)∈NFtame. If f is written as above, this mapping is defined as follows: Each symbol θi is replaced by a corresponding germ of affine map. Each symbol 𝚔 (resp. 𝚔−1) is replaced by a germ of exponential (resp. principal branch of logarithm) map. The source point of the rightmost affine germ θn is chosen to be Af+1 (or 0 if Af=−∞).Notice that condition (3) uniquely determines the choice of all germs given in (1) and (2) due to the necessarily source/target compatibility conditions. Consequently, the mapping is well-defined by these conditions and, moreover, injective. Similarly, we consider the mapping   ρ2:BNF→GPSL(2,C),Expdefined as follows: given f∈BNF, we consider the analytic function ϕ(f):If→R and let ρ2(f)∈GPSL(2,C),Exp be the germ of ϕ(f) at the point Af+1 (or at 0 if Af=−∞). By construction, if φ:NF→GPSL(2,C),Exp denotes the mapping defined at the Main Theorem, the following diagram   is commutative. Now, we reason by contradiction assuming that there exists a non-identity Britton normal form f∈BNF lying in the kernel of ϕ. Then, it follows that ϕ(f):If→R is the identity map and, consequently, that ρ2(f) is the identity germ. On the other hand, ρ1(f) is a non-identity tame normal form and it follows from the Main theorem that φ◦ρ1(f) cannot be the identity germ. This is a contradiction.□ Funding This work has been partially supported by the ANR project STAAVF and by the CAPES/COFECUB project MA731-12. Acknowledgements I would like to thank Dominique Cerveau, Robert Roussarie, Robert Moussu, Jean-Jacques Risler, Jean-Philippe Rolin, Jean-Marie Lion, Bernard Teissier, Norbert A'Campo, Frank Loray, Étienne Ghys and Thomas Delzant for numerous enlightening discussions. References 1 R. K. Amayo and I. Stewart, Infinite-dimensional Lie Algebras , Noordhoff International Publishing, Leyden, 1974. Google Scholar CrossRef Search ADS   2 J. Ax, On Schanuel's conjectures, Ann. 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