Geometric presentations of Lie groups and their Dehn functions

Yves Cornulier; Romain Tessera

doi:10.1007/s10240-016-0087-3

Geometric presentations of Lie groups and their Dehn functions

Cornulier, Yves; Tessera, Romain 2016-12-20 00:00:00 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS by YVES CORNULIER and ROMAIN TESSERA ABSTRACT We study the Dehn function of connected Lie groups. We show that this function is always exponential or poly- nomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local ﬁelds, uses and extends Abels’ theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these groups. CONTENTS 1. Introduction ...................................................... 80 1.1. Dehn function of Lie groups ......................................... 80 1.2. Riemannian versus combinatorial Dehn function of Lie groups ...................... 82 1.3. Main results . ................................................. 83 1.4. The obstructions as “computable” invariants of the Lie algebra ...................... 88 1.5. Examples .................................................... 89 1.6. Description of the strategy ........................................... 93 1.7. Sketch of the proof of Theorem F ....................................... 94 1.8. Introduction to the Lie algebra chapters ................................... 97 1.9. Guidelines ................................................... 100 2. Dehn function: preliminaries and ﬁrst reductions .................................. 101 2.A. Asymptotic comparison ............................................ 101 2.B. Deﬁnition of Dehn function .......................................... 101 2.C. Dehn vs Riemannian Dehn .......................................... 103 2.D. Two combinatorial lemmas on the Dehn function .............................. 105 2.E. Gromov’s trick ................................................. 106 3. Geometry of Lie groups: metric reductions and exponential upper bounds . . . ................. 108 3.A. Reduction to triangulable groups ....................................... 108 3.B. At most exponential Dehn function ...................................... 108 3.C. Reduction to split triangulable groups .................................... 112 4. Gradings and tameness conditions .......................................... 114 4.A. Grading in a representation .......................................... 114 4.B. Grading in a standard solvable group .................................... 116 4.C. Tameness conditions in standard solvable groups .............................. 118 4.D. Cartan grading and weights .......................................... 121 5. Algebraic preliminaries about nilpotent groups and Lie algebras ......................... 124 5.A. Divisibility and Malcev’s theorem ....................................... 124 5.B. Powers and commutators in nilpotent groups ................................ 125 5.C. Lazard’s formulas ............................................... 129 6. Dehn function of tame and stably 2-tame groups .................................. 130 6.A. Tame groups .................................................. 130 6.B. Length in nilpotent groups .......................................... 134 6.C. Application of Gromov’s trick to standard solvable groups ......................... 138 6.D. Dehn function of stably 2-tame groups .................................... 139 6.E. Generalized tame groups ........................................... 140 7. Estimates of areas using algebraic presentations ................................... 143 7.A. Afﬁne norm on varieties over normed ﬁelds ................................. 143 7.B. Area of words of bounded combinatorial length . .............................. 146 8. Central extensions of graded Lie algebras ...................................... 153 8.A. Basic conventions ............................................... 153 The authors are supported by ANR Project GAMME (ANR-14-CE25-0004). DOI 10.1007/s10240-016-0087-3 80 YVES CORNULIER, ROMAIN TESSERA 8.B. The blow-up .................................................. 156 8.C. Homology and restriction of scalars ..................................... 159 8.D. Auxiliary descriptions of H (g) and Kill(g) ................................ 165 2 0 0 9. Abels’ multiamalgam ................................................. 173 9.A. 2-tameness . .................................................. 173 9.B. Lemmas related to 2-tameness ........................................ 173 9.C. Multiamalgams of Lie algebras ........................................ 175 9.D. Multiamalgams of groups ........................................... 179 10. Presentations of standard solvable groups and their Dehn function ........................ 182 10.A. Outline of the section . ............................................ 182 10.B. Abels’ multiamalgam ............................................. 184 10.C. Algebraic and geometric presentations of the multiamalgam ........................ 185 10.D. Presentation of the group in the 2-tame case . ................................ 187 10.E. Quadratic estimates and concluding step for standard solvable groups with zero Killing module .... 190 10.F. Area of welding relations ........................................... 191 10.G. Concluding step for standard solvable groups ................................ 196 10.H. Dehn function of generalized standard solvable groups ........................... 198 11. Central and hypercentral extensions and exponential Dehn function ....................... 198 11.A. Introduction of the section .......................................... 199 11.B. FC-Central extensions ............................................. 199 11.C. Hypercentral extensions ............................................ 201 11.D. Proof of Theorem 11.C.4 ........................................... 203 11.E. An example without central extensions .................................... 206 12. G not 2-tame ...................................................... 208 12.A. Combinatorial Stokes formula ........................................ 209 12.B. Loops in groups of SOL type ......................................... 210 12.C. Groups with the SOL obstruction . ..................................... 215 Acknowledgements ..................................................... 217 References ......................................................... 217 1. Introduction 1.1. Dehn function of Lie groups. — The object of study in this paper is the Dehn function of connected Lie groups. For a simply connected Lie group G endowed with a left-invariant Riemannian metric, this can be deﬁned as follows: the area of a loop γ is the inﬁmum of areas of ﬁlling discs, and the Dehn function δ (r) is the supremum of areas of loops of length at most r. The asymptotic behavior of δ (when r →+∞) actually does not depend on the choice of a left-invariant Riemannian metric. If G is an arbitrary connected Lie group and K a compact subgroup such that G/K is simply connected (e.g. K is a maximal compact subgroup, in which case G/K is diffeomorphic to a Euclidean space), we can endow G/K with a G-invariant Riemannian metric and thus deﬁne the Dehn function δ (r) in the same way; its asymptotic behavior depends G/K only on G, neither on K nor on the choice of the invariant Riemannian metric, and is called the Dehn function of G. Let us provide some classical illustrating examples. If for some maximal compact subgroup K, the space G/K has a negatively curved G-invariant Riemannian metric, then the Dehn function of G has exactly linear growth. Otherwise, G is not Gromov- hyperbolic, and by a very general argument due to Bowditch [Bo95](notspeciﬁc to Lie groups), the Dehn function is known to be at least quadratic. On the other hand, the Dehn function is at most quadratic whenever G/K can be endowed with a non-positively curved GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 81 invariant Riemannian metric, notably when G is reductive. It is worth emphasizing that many simply connected Lie groups G fail to have a non-positively curved homogeneous space G/K and nevertheless have a quadratic Dehn function. Characterizing Lie groups with a quadratic Dehn function is a very challenging problem, even in the setting of nilpotent Lie groups. Indeed, although connected nilpotent Lie groups have an at most polynomial Dehn function, there are examples with Dehn function of polynomial growth with arbitrary integer degree. Besides, the Dehn function of a connected Lie group is at most exponential, the prototypical example of a Lie group with an exponential Dehn function being the three-dimensional SOL group. A simple main consequence of the results we describe below is the following theo- rem. Theorem A. —Let G be a connected Lie group. Then the Dehn function of G is either exponential or polynomially bounded. Using the well-known fact that polycyclic groups are virtually cocompact lattices in connected Lie groups, we deduce Corollary B. — The Dehn function of a polycyclic group is either exponential or polynomially bounded. Let us mention that in both results, “polynomially bounded” cannot be improved to “of polynomial growth”, since S. Wenger [We11] has exhibited some simply connected 2 2 nilpotent Lie groups (with lattices) with a Dehn function satisfying n≺ δ(n) n log n. Our results are more precise than Theorem A: we characterize algebraically which ones have a polynomially bounded or exponential Dehn function. In order to state the re- sult, we describe below two “obstructions” implying exponential Dehn function; the ﬁrst being related to SOL, and the second to homology in degree 2; these two obstructions appeared in a related context in Abels’ seminal work on p-adic algebraic groups [Ab87]. We prove that each of these obstructions indeed implies that the Dehn function has an exponential growth, and conversely that if none of these obstructions is fulﬁlled, then the group has an at most polynomial Dehn function, proving in a large number of cases that the Dehn function is at most quadratic or cubic. These results can appear as unexpected. Indeed, it was suggested by Gromov [Gro93,5.A ] that the only obstruction should be related to SOL. This has been proved in several important cases [Gro93, Dru04, LP04] but turns out to be false in general. Our approach relies on the dynamical structure arising from the action of G on itself by conjugation. A crucial role is played by some naturally deﬁned subgroups that are contracted by suitable elements. The presence of these subgroups is a non-discrete feature, which is invisible in any cocompact lattice (when such lattices exist). In particular, it sounds unlikely that Corollary B can be proved directly with no reference to ambient 82 YVES CORNULIER, ROMAIN TESSERA Lie groups, and the obstructions themselves are not convenient to state directly in terms of the structure of those discrete polycyclic groups. The remainder of this introduction is organized as follows: in Section 1.2,wede- ﬁne a combinatorial Dehn function for compactly presented locally compact groups, and use it to state a version of Theorem A for algebraic p-adic groups. Then Section 1.3 is dedicated to our main results. The most difﬁcult part of the main theorem is the fact that in the absence of the two obstructions, the Dehn function is polynomially bounded. In Section 1.4, we describe a useful characterization of these obstructions in terms of Lie alge- bra gradings. We apply these results to various concrete examples in Section 1.5. We outline the general strategy in Section 1.6. We introduce an additional condition under which we prove that the Dehn function is quadratic (Theorem F). In Section 1.7, we sketch the proof of this theorem; this allows us emphasize the key ideas of our approach. A substan- tial part of our work is purely algebraic and possibly of interest for other purposes, we introduce it independently in Section 1.8. 1.2. Riemannian versus combinatorial Dehn function of Lie groups. — The previous ap- proaches consisted of either working with groups admitting a cocompact lattice and using combinatorial methods, or using the Riemannian deﬁnition. Our approach, initiated in [CT10], consists of extending the combinatorial language and methods to general locally compact compactly generated groups. In particular, Lie groups are treated as combina- torial objects, i.e. groups endowed with a compact generating set and the corresponding Cayley graph. The object of study is the combinatorial Dehn function, usually deﬁned for discrete groups, which turns out to be asymptotically equivalent to its Riemannian coun- terpart. This unifying approach allows to treat p-adic algebraic groups and connected Lie groups on the same footing. We now give the combinatorial deﬁnition of Dehn function (rechristening the above deﬁnition of Dehn function as Riemannian Dehn function); see Section 2.B for more details. Let G be a locally compact group, generated by a compact subset S. Let F be the free group over the (abstract) set S and F→ G the natural epimorphism, and K its kernel (its elements are called relations). We say that G is compactly presented if for some ,K is generated, as a normal subgroup of F , by the set K of elements with length at most −1 with respect to S, or equivalently if K is generated, as a group, by the union gK g g∈F of conjugates of K in F ; this does not depend on the choice of S; the subset K is called a set of relators. Assuming this, if x∈ K, the area of x is by deﬁnition the number area(x) de- −1 ﬁned as its length with respect to gK g . Finally, the Dehn function of G is deﬁned g∈F as δ(n)= sup area(x): x∈ K,|x|≤ n . In the discrete setting (S ﬁnite), this function takes ﬁnite values, and this remains true in the locally compact setting. If G is not compactly presented, a good convention is to set δ(n) =+∞ for all n. The Dehn function of a compactly presented group G depends on GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 83 the choices of S and , but its asymptotic behavior does not. In addition, for a simply connected Lie group, the combinatorial and Riemannian Dehn functions have the same asymptotic behavior, see Proposition 2.C.1. With this deﬁnition at hand, we can now state a version of Theorem A in a non- Archimedean setting. Theorem C. —Let G be an algebraic group over some p-adic ﬁeld. Then the Dehn function of G is at most cubic, or G is not compactly presented. Before providing more detailed statements, let us compare Theorems A and C.Itis helpful to have in mind a certain analogy between Archimedean and non-Archimedean groups, where exponential Dehn function corresponds to not compactly presented. On the other hand, a striking difference between these two theorems is the absence for p- adic groups of polynomial Dehn functions of arbitrary degree. The explanation of this fact can be summarized as follows. In the connected Lie group setting, Dehn functions of “high polynomial degree” witness to the presence of simply connected non-abelian nilpotent quotients, see Theorem 10.H.1 for a precise statement. By way of contrast, any totally disconnected, compactly generated locally compact nilpotent group is compact- by-discrete; and if a compactly generated locally compact group G is isomorphic to the -points of some p-adic algebraic group, then any nilpotent quotient of G is group of Q compact-by-abelian. 1.3. Main results. — We now turn to more comprehensive statements. Let us ﬁrst introduce the two main classes of groups we will be considering in the sequel. Deﬁnition 1.1. —A real triangulable group is a Lie group isomorphic to a closed connected group of real triangular matrices. Equivalently, it is a simply connected solvable group in which for every g, the adjoint operator Ad(g) has only real eigenvalues. It can be shown that every connected Lie group G is quasi-isometric to a real triangulable Lie group. Namely, there exist compactly generated Lie groups G , G , G 1 2 3 and maps G← G→ G← G , 1 2 3 where G is real triangulable, and each arrow is a proper continuous homomorphism with cocompact image and thus is a quasi-isometry, see Lemma 3.A.1. Let A be an abelian group and consider a representation of A on a K-vector space V, where K is a ﬁnite product of complete normed ﬁelds. Let V be the largest A-equivariant quotient of V on which A acts with only eigenvalues of modulus one. Deﬁnition 1.2. — A locally compact group is a standard solvable group if it is topologically isomorphic to a semidirect product U A so that 84 YVES CORNULIER, ROMAIN TESSERA (1) A is a compactly generated locally compact abelian group (2) U decomposes as a ﬁnite direct product U ,where each U is normalized by A and can i i be written as U= U (K ),where U is a unipotent group over some nondiscrete locally i i i i compact ﬁeld of characteristic zero K ; (3) (U/[U, U]) ={0}. For a group G satisfying (1) and (2), condition (3) implies that G is compactly gener- ated, and conversely if U is totally disconnected, the failure of condition (3) implies that G is not compactly generated. If G is a compactly generated p-adic group as in Theorem C, then it has a Zariski closed cocompact subgroup which is a standard solvable group (with a single i and K= Q ). Many real Lie groups have a closed cocompact standard solvable i p group; however, for instance, a simply connected nilpotent Lie group is not standard solv- able unless it is abelian. We now introduce a very special but important class of standard solvable groups. Deﬁnition 1.3. —A group of SOL type is group U A,where U= K× K ,where K , 1 2 1 ∗∗ K are nondiscrete locally compact ﬁelds of characteristic zero, and A⊂ K× K is a closed subgroup 1 2 ∗∗ of K× K containing, as a cocompact subgroup, the cyclic group generated by some element (t , t ) with 1 2 1 2 |t| > 1 >|t|. Note that this is a standard solvable group. We call it a non-Archimedean group 1 2 of SOL type if both K and K are non-Archimedean. 1 2 −1 Example 1.4. —If K= K= K and A is the set of pairs (t, t ), then G is the usual 1 2 k− group SOL(K). More generally, A is the set of pairs (t , t ) where (k,) is aﬁxedpairof positive integers, then this provides another group, which is unimodular if and only k= . Another example is (R× Q ) Z,where Z acts as the cyclic subgroup generated by (p, p) (note that|p| > 1 >|p| ); the latter contains the Baumslag-Solitar group Z[1/p] Z as R Q a cocompact lattice. Also, deﬁne, for λ> 0, the group SOL as the semidirect product R R where λ∗ 2 R is identiﬁed with the subgroup{(t, t ): t > 0} of (R ) .Notethat SOL has index 2 in SOL(R); there are obvious isomorphisms SOL SOL−1,and the SOL ,for λ≥ 1, λ λ λ are pairwise non-isomorphic. These are the only real triangulable groups of SOL type. Deﬁnition 1.5. — (SOL obstruction) A locally compact group has the SOL obstruction (resp. non-Archimedean SOL obstruction) if it admits a homomorphism with dense image to a group of SOL type (resp. non-Archimedean SOL type). That the SOL obstruction implies that the Dehn function grows at least exponen- tially is the contents of the forthcoming Theorems E.1 and E.2. In the positive direction, we start with following result, which generalizes [Gro93, 5.A ], [Dru04, Theorem 1.1 (2)], [LP04]and [CT10]. Theorem D. —Let G= U A be a standard solvable group. Suppose that every closed subgroup of G containing A (thus of the form V A with V a closed A-invariant subgroup of U)is GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 85 a standard solvable group and does not satisfy the SOL obstruction. Then G has an at most quadratic Dehn function. The main result of [CT10] is essentially the case when the normal subgroup U is abelian (but on the other hand works in arbitrary characteristic), which itself generalizes results of Gromov [Gro93], Dru¸ tu [Dru04], Leuzinger-Pittet [LP04]. Theorem D turns out to be a particular instance of the much more general Theo- rem F below, but is considerably easier: the material is the length estimates of the begin- ning of Section 6 and Gromov’s trick described in Section 2.E. A direct proof of Theo- rem D is given in Section 6.D. In [Gro93,5.B ], quoth Gromov, “We conclude our discussion on lower and upper bounds for ﬁlling area by a somewhat pessimistic note. The present methods lead to satisfactory results only in a few special cases even in the friendly geometric surroundings of solvable and nilpotent groups.” Indeed, for standard solvable groups without the SOL obstruction, it seems that Theorem D is the best result that can be gotten without bringing forward new ideas. The ﬁrst example of a standard solvable group without the SOL obstruction but not covered by Theorem D is Abels’ group A (K) (see the previous subsection). In this particular ex- ample, the authors obtain a quadratic upper bound for the Dehn function in [CT13]. In this case, the group is tractable enough to work with explicit matrices, but such a pedes- trian approach becomes hopelessly intricate in an arbitrary group as in Theorem E.4. In a more general context, we have to deal with groups without the SOL ob- struction but not satisfying the assumptions of Theorem D. In this context, we have to introduce the 2-homological obstruction. For this, we need to recall a fundamental no- tion introduced and studied by Guivarc’h [Gui80]and laterbyOsin[Os02]. Let G be a real triangulable group. Its exponential radical G is deﬁned as the intersection of its lower central series and actually consists of the exponentially distorted elements in G. Let g be its Lie algebra. In the case of a standard solvable group, the role of exponential radical is played by U itself (it can be checked to be equal to the derived subgroup of G, so is a characteristic subgroup). Deﬁnition 1.6 (2-homological obstruction). • The real triangulable group G is said to satisfy the 2-homological obstruction if ∞∞ H (g ) ={0}, or equivalently if the A-action on H (g ) has some nonzero invariant 2 0 2 vector. • The standard solvable group G= U A is said to satisfy the 2-homological obstruction if H (u) ={0}, that is to say, H (u ) ={0} for some j . If moreover j can be chosen so that 2 0 2 j 0 K is non-Archimedean, we call it the non-Archimedean 2-homological obstruction. In most cases, including standard solvable groups, the 2-homological obstructions can be characterized by the existence of suitable central extensions. For instance, if a real 86 YVES CORNULIER, ROMAIN TESSERA triangulable group G has a central extension G, also real triangulable, with nontrivial kernel Z such that Z⊂ (G) then it satisﬁes the 2-homological obstruction. The converse is true when G admits a semidirect decomposition G N, but nevertheless does not in general, see Section 11.E. We are now able to state our main theorem, which immediately entails Theo- rems A and C. Theorem E. —Let G be a real triangulable group, or a standard solvable group. • if G satisﬁes one of the two non-Archimedean (SOL or 2-homological) obstructions, then G is not compactly presented; • otherwise G is compactly presented and has an at most exponential Dehn function. Moreover, in this case – if G satisﬁes one of the two (SOL or 2-homological) obstructions, then G has an exponential Dehn function; – if G satisﬁes none of the obstructions, then it has a polynomially bounded Dehn function; in the case of a standard solvable group, the Dehn function is at most cubic. This result can be seen as both a generalization and a strengthening of the follow- ing seminal result of Abels [Ab87]. Theorem (Abels). — Let G be a standard solvable group over a p-adic ﬁeld. Then G is compactly presented if and only if it satisﬁes none of the non-Archimedean obstructions. Let us split Theorem E into several independent statements. The ﬁrst two provide lower bounds and the last two provide upper bounds on the Dehn function. Theorem E.1. —Let G be a standard solvable or real triangulable group. If G satisﬁes the SOL (resp. non-Archimedean SOL) obstruction, then G has an at least exponential Dehn function (resp. is not compactly presented). We provide a uniﬁed proof of both assertions of Theorem E.1 in Section 12. Note that the non-Archimedean case is essentially contained in the “only if ” (easier) part of Abels’ theo- rem above, itself inspired by previous work of Bieri-Strebel, notably [BiS78,Theorem A]. Part of the proof consists in estimating the size of loops in the groups of SOL type, where our proof is inspired by the original case of the real 3-dimensional SOL group, due to Thurston [ECHLPT92], which uses integration of a well-chosen differential form. Our method in Section 12 is based on a discretization of this argument, leading to both a simpliﬁcation and a generalization of the argument. Theorem E.2. —Let G be a standard solvable or real triangulable group. If G satisﬁes the 2-homological (resp. non-Archimedean 2-homological) obstruction, then G has an at least exponential Dehn function (resp. is not compactly presented). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 87 The case of standard solvable groups reduces, after a minor reduction, to a simple and classical central extension argument, see Section 11.B. The case of real triangulable groups is considerably more difﬁcult; in the absence of splitting of the exponential rad- ical, we construct an “exponentially distorted hypercentral extension”. This is done in Section 11. Theorem E.3. —Let G be a real triangulable group. Then G has an at most exponential Dehn function. ◦◦ Let G= U A be a standard solvable group and U the identity component in U.If G/U is compactly presented, then G is compactly presented with an at most exponential Dehn function. Since compact presentability is stable under taking extensions [Ab72], for an arbi- trary locally compact group G with a closed connected normal subgroup C, it is true that G is compactly presented if and only G/C is compactly presented. We do not know if this can be generalized to the statement that if G/C has Dehn function f (n), then G has Dehn function max(f (n), exp(n)).Theorem E.3, which follows from Theorem 3.B.1 and Corollary 3.B.6, contains two particular instances where the latter assertion holds, which are enough for our purposes. The ﬁrst instance, namely that every connected Lie group has an at most exponential Dehn function, was asserted by Gromov, with a sketch of proof [Gro93, Corollary 3.F ]. Example 1.7. —Fix n∈ Z with|n|≥ 2. Consider the group G= (R× Q ) Z, n n n where Q is the product of Q where p ranges over distinct primes divisors of n.Here n p ◦◦ R and G/U Q Z, which, as a hyperbolic group (it is an HNN ascending n n extension of the compact group Z ), has a linear Dehn function. So, by Theorem E.3,G n n has an at most exponential Dehn function. Since|n|≥ 2, it admits a prime factor p,so G admits the group of SOL type (R× Q ) Z as a quotient, and therefore G satisﬁes the p n n SOL obstruction and thus has an at least exponential Dehn function by Theorem E.1. We conclude that G has an exponential Dehn function. This provides a new proof that its lattice, the Baumslag-Solitar group −1 n BS(1, n)= t, x| txt= x , has an exponential Dehn function. This is actually true for arbitrary Baumslag-Solitar groups BS(m, n),|m| =|n|, for which the exponential upper bound was ﬁrst established in [ECHLPT92, Theorem 7.3.4 and Example 7.4.1] and independently in [BGSS92], and the exponential lower bound, attributed to Thurston, was obtained in [ECHLPT92, Example 7.4.1]. Let us provide a useful corollary of Theorems E.1 and E.3. Corollary E.3.a. —Let G= U A be a standard solvable group in which A has rank 1 (i.e., has a closed inﬁnite cyclic cocompact subgroup). Then exactly one of the following occurs 88 YVES CORNULIER, ROMAIN TESSERA • G satisﬁes the non-Archimedean SOL obstruction and thus is not compactly presented; • G satisﬁes the SOL obstruction but not the non-Archimedean one; it is compactly presented with an exponential Dehn function; • G does not satisfy the SOL obstruction; it has a linear Dehn function and is Gromov-hyperbolic. It is indeed an observation that if A has rank 1 and G does not satisfy the SOL obstruction, then some element of A acts on G as a compaction (see Deﬁnition 4.B.4) and it follows from [CCMT15] that G is Gromov-hyperbolic, or equivalently has a linear Dehn function. In this special case where A has rank 1, the 2-homological obstruction, which may hold or not hold, implies the SOL obstruction and is accordingly unnecessary to consider; see also Theorem D. Turning back to Theorem E, the fourth and most involved of all the steps is the following. Theorem E.4. —Let G be a standard solvable (resp. real triangulable) group not satisfying nei- ther the SOL nor the 2-homological obstructions. Then G has an at most cubic (resp. at most polynomial) Dehn function. The proof of Theorem E.4 for standard solvable groups is done in Section 6, relying on algebraic preliminaries, occupying Sections 8 and 9. We actually obtain, with some additional work, a similar statement for “generalized standard solvable groups”, where A is replaced by some nilpotent compactly generated group N (see Theorem 10.H.1). The case of real triangulable groups requires an additional step, namely a reduction to the case where the exponential radical is split, in which case the group is general- ized standard solvable. This reduction is performed in Section 3.C, and relies on results from [C11]. 1.4. The obstructions as “computable” invariants of the Lie algebra. — The obstructions were introduced above in a convenient way for expository reasons, but the natural frame- work to deal with them uses the language of graded Lie algebras, which we now de- scribe. Let G be either a standard solvable group U A or a real triangulable group with exponential radical also denoted, for convenience, by U. Let u be the Lie algebra of U; it is a Lie algebra (over a ﬁnite product of nondiscrete locally compact ﬁelds of characteristic zero). It is, in a natural way, a graded Lie algebra. In both cases, the grading takes values into a ﬁnite-dimensional real vector space, namely Hom(G/U, R). It is introduced in Section 4.B, relying on Theorem 4.A.2 (see Section 1.5 for a representative particular case.) In this setting, there are useful restatements (see Propositions 4.C.3 and 4.D.9)ofthe obstructions. We say that α∈ Hom(G/U, R) is a weight of G (or of U, when u is endowed with the grading) if u = 0 and is a principal weight of G if α is a weight of U/[U, U].The deﬁnitions imply that 0 is not a principal weight (although it can be a weight). We say GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 89 that two nonzero weights α, β are quasi-opposite if 0 ∈[α, β], i.e., β =−tα for some t > 0. We write U= U× U as the product of its Archimedean and non-Archimedean parts. a na Then we have the following restatements: • G satisﬁes the SOL obstruction⇔ U admits two quasi-opposite principal weights (Propositions 4.C.3 and 4.D.9); • G satisﬁes the non-Archimedean SOL obstruction⇔ U admits two quasi- na opposite principal weights (Proposition 4.C.3 applied to G/G ); • G satisﬁes the 2-homological obstruction⇔ H (u) ={0}; 2 0 • G satisﬁes the non-Archimedean 2-homological obstruction⇔ H (u ) ={0}; 2 na 0 • G fails to satisfy the assumption of Theorem D⇔ 0 belongs to the interval joining some pair of weights. Here, H (u) denotes the homology of the Lie algebra u; the grading on u in the real vector space Hom(G/U, R) canonically induces a grading of H (u) in the same space (see Section 8.A), and H (u) is its component in degree zero. 2 0 Another important module associated to u is Kill(u), the quotient of the second symmetric power u u by the submodule generated by elements of the form[x, y] z− x [y, z]; thus, in case of a single ﬁeld (that is, U= U in Deﬁnition 1.2), the invariant quadratic forms on u are elements in the dual of Kill(u). Theorem F. —Let G= U A be a standard solvable group not satisfying any of the SOL or 2-homological obstructions. Suppose in addition that Kill(u) ={0}. Then G has an at most quadratic Dehn function (thus exactly quadratic if A has rank at least two). Theorem F is proved in Section 10.E.Partofitisprovedalong with Theorem E.4 and involves the same difﬁculties, except that the welding relations do not appear. The condition Kill(u) ={0} corresponds to the existence of certain central exten- sions of G as a discrete group. Using asymptotic cones, it will be shown in a subsequent paper that it implies, in many cases (without the SOL and 2-homological obstructions), that the Dehn function grows strictly faster than a quadratic function. 1.5. Examples. — Let us give a few examples. All are standard solvable con- nected Lie groups G= U A so that the action of A on the Lie algebra u of U is R-diagonalizable. In this context, we call Hom(A, R) the weight space. The grading of the Lie algebra u in Hom(A, R) is given by −1 α(v) u= u∈ u |∀v∈ A,v uv= e u . When we write the set of weights, we use boldface for the set of principal weights. We underline the zero weight (or just denote to mark zero if zero is not a weight). 90 YVES CORNULIER, ROMAIN TESSERA 1.5.1. Groups of SOL type. — For a group of SOL type (Deﬁnition 1.3), the weight space is a line and the weights lie on opposite sides of zero: 1 2 By deﬁnition it satisﬁes the SOL obstruction. On the other hand, it satisﬁes the 2-homological obstruction only in a few special cases. For instance, (R× Q ) Z does p p not satisfy the 2-homological obstruction, and the real group SOL (see Example 1.4) satisﬁes the 2-homological obstruction only for λ= 1. 3 2 2 3 1.5.2. Gromov’s higher SOL groups. — For the group R R where R acts on R as the group of diagonal matrices with positive diagonal entries and determinant one (often called higher-dimensional SOL group, but not of SOL type nor even satisfying the SOL obstruction according to our conventions), the weight space is a plane in which the weights form a triangle whose center of gravity is zero: Since there are no opposite weights, we have (u⊗ u)= 0 and therefore H (u) 0 2 0 and Kill(u) (which are subquotients of (u⊗ u) ) are also zero. It was stated with a sketch 0 0 of proof by Gromov that this group has a quadratic Dehn function [Gro93,5.A ]. Dru¸ tu 3+ε obtained in [Dru98, Corollary 4.18] that it has a Dehn function n ,and then ob- tained a quadratic upper bound in [Dru04, Theorem 1.1], a result also obtained by Leuzinger and Pittet in [LP04]. The quadratic upper bound can also be viewed as an illustration of Theorem D. (The assumption that 0 is the center of gravity is unessential: the important fact is that 0 belongs to the convex hull of the three weights but does not lie in the segment joining any two weights.) 1.5.3. Abels’ﬁrstgroup.— The group G= A (K) consists of matrices of the form ⎛ ⎞ 1 u u u 12 13 14 ⎜ ⎟ 0 s u u 2 23 24× ⎜ ⎟ ; s∈ K , u∈ K. i ij 00 s u 3 34 0 001 Its weight conﬁguration is given by 13 24 12 34 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 91 This example is interesting because it does not satisfy the SOL obstruction but admits opposite weights. A computation shows that H (u)= 0and Kill(u)= 0 (see 2 0 0 Abels [Ab87, Example 5.7.1]). ¯¯ If G= U A is the quotient of G by its one-dimensional center, then G does not satisfy the SOL obstruction but satisﬁes H (u¯) = 0, i.e. satisﬁes the 2-homological 2 0 obstruction. This example was studied speciﬁcally by the authors in the paper [CT13], where it is proved that it has a quadratic Dehn function, which also follows from Theo- rem F. 1.5.4. Abels’ second group. — This group was introduced in [Ab87, Example 5.7.4]. Consider the group U A, where A R and U is the group corresponding to the quotient of the free 3-nilpotent Lie algebra generated by the 3-dimensional K-vector space of basis (X , X , X ) by the ideal generated by[X ,[X , Y]] for all i, j ∈{1, 2, 3}. 1 2 3 i i j Its weight structure is as follows 12 23 ∗∗ (The sign∗∗ indicates that the degree 0 subspace u is 2-dimensional.) A compu- tation (see Remark 8.D.3)shows that H (u)= 0and Kill(u) is 1-dimensional. 2 0 0 This example was introduced by Abels as typically difﬁcult because although H (u) vanishes, U is not the multiamalgam of its tame subgroups (see Section 1.7); as 2 0 we show in this paper, this is reﬂected in the fact that Kill(u) is nonzero. This results in a signiﬁcant additional difﬁculty in order to estimate the Dehn function, which is at most cubic by Theorem E.4. 1.5.5. Semidirect products with SL .— We consider the groups V(K) SL (K), 3 3 where V is one of the following three irreducible modules: V= V ,the standard 3- dimensional module; V= Sym (V ), the 6-dimensional second power of V and V , 20 10 10 11 the 8-dimensional adjoint representation. (The notation is borrowed from [FuH,Lec- ture 13], writing V instead of .) These groups are not solvable but have a cocompact ij ij K-triangulable subgroup, namely V(K) T (K),where T (K) is the group of lower tri- 3 3 angular matrices. It is a simple veriﬁcation that for an arbitrary nontrivial irreducible representation, the group V(K) T (K) admits exactly three principal weights, namely the two principal negative roots r , r of SL itself, and the highest weight of the rep- 21 32 3 resentation V , which can be written as aL− bL ,where±L , L , L are the minimal ab 1 3 1 2 3 92 YVES CORNULIER, ROMAIN TESSERA vectors of the weight lattice, arranged as in the following picture. −L L L 2 1 −L−L 1 2 r r 31 32 The three principal weights always form a triangle with zero contained in its interior. In particular, V(K) T (K) does not satisfy the SOL obstruction. Let us write the weight diagram for each of the three examples (we mark some other points in the weight lattice as· for the sake of readability). More speciﬁcally, for V , the weights conﬁguration looks like L L 2 1 .. . L r r 31 32 We see that there are no quasi-opposite weights at all. This is accordingly a case for which Theorem D applies directly. Thus K SL (K) has a quadratic Dehn function (it can be checked to also hold for K SL (K) for d≥ 3). For V , writing L= L+ L , the weights are as follows 20 ij i j 2L L 2L 2 12 1 .. r . L L 23 13 .. . r r 31 32 . 2L . Thus there are quasi-opposite weights but no opposite weights. Theorem 10.E.1 implies that V (K) SL (K) has a quadratic Dehn function. 20 3 For V , writing L= L− L , the weights are as follows 11 ij i j L L 23 13 .. . .. r , L∗∗ L 21 21 12 .. .. . r , L r , L 31 31 32 32 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 93 In this case, there are opposite weights, there is an invariant quadratic form in degree zero (akin to the Killing form), deﬁned by φ(r , L )= 1for all i < j and all other products ji ij being zero, so Kill(u) = 0. However, a simple computation shows that H (u)= 0. So 0 2 0 Theorem E.4 implies that sl (K) SL (K) has an at most cubic Dehn function. 3 3 1.6. Description of the strategy. — Let us now describe the main ideas that underlie the proof of Theorems E.4 and F. We then proceed to sketch, with more details, the proof of Theorem F. General picture. — A central idea, already essential in Gromov’s approach is to use non- positively curved subgroups (called tame subgroups in the sequel). It was previously used in Abels’ work on compact presentability of p-adic groups [Ab87]. Abels considers a certain abstract group, obtained by amalgamating the tame subgroups over their intersections (more details are given in 1.7). Our combinatorial approach to the Dehn function allows us to take advantage of the consideration of this “multiamalgam” G. One similarly deﬁnes a multiamalgam of the tame Lie subalgebras, denoted by gˆ . Strategy. — To simplify the discussion, let us assume that G is standard solvable over a single nondiscrete locally compact ﬁeld of characteristic zero K, satisfying none of the SOL and 2-homological assumptions. Roughly speaking, the strategy is as follows. Ideally, one would like to obtain G= G; this turns out to be true under the additional assump- tion that Kill(u) vanishes. In general we obtain that G is a central extension of G, and describe generators of the central kernel. Second, we need to be able to decompose any combinatorial loop into boundedly many loops corresponding to relations in the tame subgroups. It turns out that both steps are quite challenging. While Abels’ work provides substantial material to tackle the ﬁrst step, we had to introduce completely new ideas to solve the second one. The ﬁrst step: giving a compact presentation for G.— Under the assumption that the group G does not satisfy the SOL obstruction, we show, improving a theorem of Abels, that ˆˆˆ the multiamalgam is a central extension of G. Thus we have G= U A, where Uis a central extension of U, whose central kernel is centralized by A. At ﬁrst sight, it seems that the condition H (u)= 0 should be enough to ensure that G= G. However, it turns 2 0 out that in general, U is a “wild” central extension, in the sense that it does not carry any locally compact topology such that the projection onto U is continuous. This strange phenomenon is easier to describe at the level of the Lie algebras. There, we have that gˆ= uˆ a,where uˆ is a central extension in degree 0 of u, seen as Lie algebras over Q.Now, if in the last statement, we could replace Lie algebras over Q by Lie algebras over K, then clearly H (u)= 0 would imply that uˆ= u. This happens if and only if the natural 2 0 morphism H (u)→ H (u) is an isomorphism; we show in Section 8 that this happens 0 2 0 if and only if the module Kill(u) (see Section 1.4) vanishes. In fact, there are relatively 0 94 YVES CORNULIER, ROMAIN TESSERA simple examples, already pointed out by Abels where Kill(u) does not vanish (see the previous subsection). As a consequence, even when none of the obstructions hold, we need to complete the presentation of G with a family of so-called welding relations, which encode generators of the kernel of U→ U. At the Lie algebra level, these relations encode K-bilinearity of the Lie bracket. The second step: reduction to special relations. — The second main step is to reduce the esti- mation of area of arbitrary relations to that of relations of a special form (e.g., relations inside a tame subgroup). This idea amounts to Gromov and is instrumental in Young’s approach for nilpotent groups [Y13a]and for SL (Z) [Y13b]; it is also used in [CT10, d≥5 CT13]. The main difference in this paper is that we have to perform such an approach without going into explicit calculations (which would be extremely complicated for an arbitrary standard solvable group, since the unipotent group U is essentially arbitrary). Our trick to avoid calculations is to use a presentation of U that is stable under “ex- tensions of scalars”. Let us be more explicit, and write U= U(K),sothat U(A) makes sense for any commutative K-algebra A. We actually provide a presentation, based on Abels’ multiamalgam and welding relations, of U(A) for any K-algebra A. When apply- ing it to a suitable algebra A of functions of at most polynomial growth, we obtain area estimates (in the sketch of Section 1.7, we do this with the algebra A of sequences of el- ements of at most exponential growth). This is the core of our argument; it is performed in Section 10.E (in the setting of Theorem F) and in Section 10.G in general, relying on the general approach developed in Section 7.B. The presentation itself is established in Sections 8 and 9. Last step: computation of the area of special relations. — For a standard solvable group not satisfying the SOL and 2-homological obstructions, these relations are of two types: those that are contained (as loops) in a tame subgroup and thus have an at most quadratic area; and the more mysterious welding relations. We show that welding relations have an at most cubic area. When the welding relations are superﬂuous, namely when Kill(u) vanishes, Theorem 10.E.1 asserts that G then has an at most quadratic Dehn function. A study based on the asymptotic cone, in a paper in preparation by the authors, will show that conversely, in some cases where Kill(u) does not vanish, the Dehn function of G grows strictly faster than quadratic. We mention this to enhance the important role played by the welding relations in the geometry of these groups. We suspect that they might be relevant as well in the study of the Dehn function of nilpotent Lie groups. 1.7. Sketch of the proof of Theorem F.— In the following, we emphasize the main ideas, omitting some technical issues occurring in the detailed, rigorous proof completed in Section 10.E. To begin with, let us explain in detail the notion of multiamalgam. Let G be a group and (H ) a family of subgroups. The multiamalgam Gof the H is the quotient of i i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 95 the free product of the H by the obvious amalgamation relations (identifying H∩ H to i i j its image in both H and H for all i, j ). We say that G is the multiamalgam of the H if i j i the canonical homomorphism G→ G is an isomorphism. Basic example: suppose that G= G . Then G is the multiamalgam of the family of subgroups (G⊕ G ) . This is because G is the quotient of the free product by the i j i =j commutation relations, and each of these commutation relations lives within one pair of summands. Consider now a standard solvable group G= U A satisfying the hypotheses of Theorem F. We call tame subgroup of G a subgroup of the form V A, where V is a closed A-invariant subgroup such that some element of A acts on V as a contraction. Let (U A) be the family of maximal tame subgroups. There are ﬁnitely i 1≤i≤ν many. We now deﬁne a presentation of the multiamalgam G. Let S be a compact sym- metric neighborhood of 1 in U , T a compact symmetric generating subset of A with nonempty interior, and S= S T. Let R be the set of bounded length relators of the following four types: • bounded length relators of (A, T), • relators of length 3 inside S, −1−1 • relators of the form ts t s for s, s∈ S, t∈ T, −1 • (amalgamation relators) relators of the form s s when s∈ S , s∈ S have the i j same image in U∩ U . i j Fix k≥ 1and let ℘ ∈{1...,ν} be a k-tuple of indices. Let W (n) be the set of −1 group words in the free group F of the form t s t where t∈ F with|t |≤ n and S j j j T j j=1 j s∈ S .Let δ (n) be the supremum of areas (for the presentationS| R) of those null- j ℘ ℘ homotopic words in W (n). Gromov’s trick (see Section 2.E) says that there exists k and ℘ such that δ (n) 2 2 n implies that the Dehn function of G is n . Showing that Gromov’s trick can be performed requires some length estimates which take some work but we do not insist on this here. So we ﬁx k and are reduced to showing δ (n) n . For each n,weﬁxsome w(n)∈ W (n); we decompose it as −1 w(n)= t (n)s (n)t (n) . j j j i=1 Let us proceed to show that the area of w(n) is ﬁnite and quadratically bounded in terms −1 of n.For each n,deﬁne σ (n) as the element of U represented by t (n)s (n)t (n) .Then j ℘ j j j σ (n),viewedasanelement of thefreeproduct U ∗ ··· ∗ U , has a trivial image j 1 ν i=1 in U. exp Let K be the K-algebra of sequences in K with at most exponential growth. exp Viewing σ as a function of n, simple estimates show that σ∈ U (K ). j j ℘ j 96 YVES CORNULIER, ROMAIN TESSERA Using the universal property of the multiamalgam, there is a canonical well-deﬁned exp exp exp homomorphism from the multiamalgam U(K ) of the U (K ) to U(K ). It is at this point that we use in an essential way the assumptions of Theorem F, namely that G does not satisfy the SOL and 2-homological obstructions, and Kill(u) ={0}. These assump- tions, thanks to the preliminary algebraic work of Sections 8 and 9, imply that this homo- exp exp morphism U(K )→ U(K ) is an isomorphism (Corollary 9.D.4). For q, r ∈{1,...,ν}, exp denote by η the inclusion of U∩ U into U . So, the multiamalgam of the U (K ) qr q r q i is the quotient of the free product by the normal subgroup generated by the relators −1 exp exp η (x)η (x) ,when (q, r) ranges over pairs and x ranges over U (K )∩ U (K ). qr rq q r exp Therefore, there exists a ﬁnite family (μ (n)) of elements of U(K ),and m 1≤m≤p elements q , r ∈{1,...,ν},suchthat μ (n)∈ U∩ U ,elements (g (n)) in the free m m m q r m m m exp product of the U (K ) such that −1−1 σ= g η (μ )η (μ ) g . j m q r m r q m m m m m m j=1 m=1 Thus for every n we have, in U ∗ ··· ∗ U 1 ν −1−1 σ (n)= g (n)η (μ )(n)η (μ )(n) g (n) . j m q r m r q m m m m m m j=1 m=1 Denote by g (n) a choice of word of length≤ Cn representing g (n) in U A, m m q in the generators S∪ T; this canbedonebecause g (n) has exponential size, so can q m −cn cn be written as t st with s∈ S and t∈ T, and c independent of n.Wealsodeﬁne −cn cn−cn cn simultaneously η (μ )(n) and η (μ )(n) to be of the form t st and t s t ,with q r m r q m m m m m s∈ S , s∈ S having the same image in U∩ U . q r q r m m m m So the word −1 −1 −1 w (n)= w(n) g (n) η (μ )(n) η (μ )(n) g (n) . m q r m r q m m m m m m m=1 is null-homotopic in (U ∗· · · ∗ U ) A and has an at most linear length. 1 ν −1 Let us obtain a uniform bound on the area of η (μ )(n)η (μ )(n) for each q r m r q m m m m m m.Wehave −cn −1 cn −1 η (μ )(n)η (μ )(n)= t ss t; q r m r q m m m m m −1 since ss is a relator, its area is equal to 1. Thus the area of w(n)w (n) is≤ p (indepen- dently of n). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 97 This shows that we are reduced to proving that w (n) has an at most quadratic area. For convenience, we rewrite w (n)= h (n), m=1 where h is a word in T∪ S for some ν and ν -tuple ℘ , independent of n. Let us show m ℘ that the area of a word w= h that is null-homotopic in (U ∗ ··· ∗ U ) Ais at m 1 ν m=1 most quadratic with respect to the total length|h| (with a constant depending only on ℘ ). This is shown by induction on ν . Since w is null-homotopic, there exists either m such that h is null-homotopic, or there exists m such that ℘= ℘ . In the second case, m m+1 this reduces to the case of ν− 1. In the ﬁrst case, it also reduces to the case of ν− 1, at the cost of replacing h with 1, which has at most quadratic cost since U A has an m ℘ at most quadratic Dehn function. So w (n) has an at most quadratic area. It follows that w(n) has an at most quadratic area. Since this holds for every choice of w, this shows that δ (n) n . Thus G has an at most quadratic Dehn function. This completes the sketch of proof of Theorem F. The ﬁrst two conditions (absence of SOL and 2-homological obstructions) were necessary as otherwise, the Dehn function is at least exponential. On the other hand, in the proof of Theorem E.4 we also have to proceed when the Killing module is not assumed zero in degree 0. In this case, U can be described as a quotient of the multiamalgam by some welding relators,inawaythatis exp inherited to U(K ) (see 10.D). So, in the above proof, some further relations appear (in −1 addition to the η (μ )(n)η (μ )(n) ), and we establish (Section 10.F) that these fur- q r m r q m m m m m ther relations have at most cubic area with respect to some compact presentation (where the additional relators are some welding relations of bounded length). 1.8. Introduction to the Lie algebra chapters. — Although Sections 8 and 9 can be viewed as technical sections when primarily interested in the results about Dehn func- tions, they should also be considered as self-contained contributions to the theory of graded Lie algebras. Recall that graded Lie algebras form a very rich theory of its own interest, see for instance [Fu, Kac]. Let us therefore introduce these chapters indepen- dently. In this context, the Lie algebras are over a given commutative ring A,withno ﬁniteness assumption. This generality is essential in our applications, since we have to consider (ﬁnite-dimensional) Lie algebras over an inﬁnite product of ﬁelds. We actually consider Lie algebras graded in a given abelian group W , written additively. 1.8.1. Universal central extensions. — Recall that a Lie algebra is perfect if g =[g, g]. It is classical that every perfect Lie algebra admits a universal central extension. We pro- vide a graded version of this fact. Say that a graded Lie algebra is relatively perfect in degree zero if g ⊂[g, g] (in other words, H (g) ={0}). In Section 8.B,toany graded 0 1 0 Lie algebra g, we canonically associate another graded Lie algebra g˜ along with a graded 98 YVES CORNULIER, ROMAIN TESSERA Lie algebra homomorphism τ: g˜→ g, which has a central kernel, naturally isomorphic to the 0-component of the 2-homology module H (g) . 2 0 Theorem G (Theorem 8.B.4). — Let g be a graded Lie algebra. If g is relatively perfect in degree zero, then the morphism g˜→ g is a graded central extension with kernel in degree zero, and is universal among such central extensions. An important feature of this result is that it applies to graded Lie algebras that are far from perfect: indeed, in our case, the Lie algebras are even nilpotent. 1.8.2. Restriction of scalars. — In Section 8.C, we study the behavior of H (g) un- 2 0 der restriction of scalars. We therefore consider a homomorphism A→ B of commuta- tive rings. If g is a Lie algebra over B, then it is also a Lie algebra over A and therefore to A B avoid ambiguity we denote its 2-homology by H (g) and H (g) according to the choice 2 2 of the ground ring. There is a canonical surjective A-module homomorphism H (g)→ A,B H (g);wecallits kernelthe welding module and denote it by W (g).Itisagraded A- 2 2 A,B A B module, and W (g) is also the kernel of the induced map H (g)→ H (g) . 0 0 0 2 2 2 These considerations led us to introduce the Killing module.Let g be a Lie algebra ⊗3 over the commutative ring B. Consider the homomorphism T from g to the symmetric square g g,deﬁnedby T (u⊗ v⊗ w)= u [v, w]−[u,w] v (all tensor products are over B here). The Killing module is by deﬁnition the cokernel of T . The terminology is motivated by the observation that for every B-module m, Hom (Kill (g), m) is in natural bijection with the module of invariant B-bilinear forms g× g→ m.If g is graded, then Kill(g) is canonically graded as well. Theorem H. —Let Q⊂ K be ﬁelds of characteristic zero, such that K has inﬁnite transcendence degree over Q.Let g be a ﬁnite-dimensional graded Lie algebra over K, relatively perfect in degree zero (i.e., H (g) ={0}). Then the following are equivalent: 1 0 Q,K (i) W (g) ={0}; Q,R (ii) W (g⊗ R) ={0} for every commutative K-algebra R; K 0 (iii) Kill (g) ={0}. In particular, assuming moreover that H (g) ={0}, these are also equivalent to: 2 0 (iv) H (g) ={0}; (v) H (g⊗ R) ={0} for every commutative K-algebra R. K 0 The interest of such a result is that (iii) appears as a checkable criterion for the vanishing of a complicated and typically inﬁnite-dimensional object (the welding mod- ule). Let us point out that this result follows, in case g is deﬁned over Q, from the results GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 99 of Neeb and Wagemann [NW08]. In our application to Dehn functions (speciﬁcally, in the proof of Theorems 10.E.1 and 10.G.1), we make an essential use of the implication (iii)⇒(v), where Q= Q, K is a nondiscrete locally compact ﬁeld, and R is a certain al- gebra of functions on K. That (iii) implies the other properties actually does not rely on the speciﬁc hypotheses (restriction to ﬁelds, ﬁnite dimension), see Corollary 8.C.8.The converse, namely that the negation of (iii) implies the negation of the other properties, follows from Theorem 8.C.13. 1.8.3. Abels’ multiamalgam. — Section 9 is devoted to the study of Abels’ multia- malgam gˆ and to its connection with the universal central extension g˜→ g.Here, we again consider arbitrary Lie algebras over a commutative ring R, but we now assume that the abelian group W is a real vector space. Given a graded Lie algebra, a Lie subalge- bra is called tame if 0 does not belong to the convex hull of its weights. Abels’ multiamal- gam gˆ is the (graded) Lie algebra obtained by amalgamating all tame subalgebras of g along their intersections (see 9.C for details); it comes with a natural graded Lie algebra homomorphism gˆ→ g. Abels deﬁnes a 2-tame nilpotent graded Lie algebra to be such that 0 does not belong to the convex hull of any pair of principal weights (this condition is related to the condition that the SOL-obstruction is not satisﬁed). A more general notion of 2-tameness, for arbitrary W -graded Lie algebras, is introduced in 9.A. Although Abels works in a speciﬁc framework (p-adic ﬁelds, ﬁnite-dimensional nilpotent Lie algebras), his methods imply with minor changes the following result. Theorem (essentially due to Abels, see Theorem 9.C.2). — If g is 2-tame, then gˆ→ g is the universal central extension in degree 0. In other words, gˆ is canonically isomorphic to g˜ . This means that in this case, gˆ is an excellent approximation of g, the discrepancy being encoded by the central kernel H (g) . 2 0 We actually need the translation of this result in the group-theoretic setting, which involves signiﬁcant difﬁculties. Assume now that the ground ring R is a commutative al- gebra over the ﬁeld Q of rationals. Recall that the Baker-Campbell-Hausdorff formula deﬁnes an equivalence of categories between nilpotent Lie algebras over Q and uniquely divisible nilpotent groups. Then g is the Lie algebra of a certain uniquely divisible nilpo- tent group G, and we can deﬁne the multiamalgam G of its tame subgroups, i.e. those subgroups corresponding to tame subalgebras of g. Note that it does not follow from ab- ˆˆ stract nonsense that G is controlled in any way by g, because G is deﬁned in the category of groups and not of uniquely divisible nilpotent groups. In a technical tour de force, Abels [Ab87, §4.4] managed to prove that G is nilpotent and asked whether the exten- sion G→ G is central. The following theorem, which we need for our estimates of Dehn function, answers the latter question positively. Theorem I (Theorem 9.D.2). — Let g be a 2-tame nilpotent graded Lie algebra over a commuta- ˆˆ tive Q-algebra R.If g is 2-tame, then G is nilpotent and uniquely divisible, and G→ G corresponds to 100 YVES CORNULIER, ROMAIN TESSERA gˆ→ g under the equivalence of categories. In particular, the kernel of G→ G is central and canonically isomorphic to H (g) . 1.8.4. On the “computational” aspect. — Let g be a Lie algebra (over some com- mutative ring) graded in some abelian group. Theorems E and F involve the condi- tions H (g) ={0} and Kill(g) ={0}. Typically, in the cases of interest, g is a ﬁnite- 2 0 0 dimensional nilpotent Lie algebra over a ﬁeld, and these conditions mean the surjectivity of some explicit matrices. There are simple families of examples for which g has dimen- 3 2 sion n , while g= g has dimension n , so that most of the contribution to α =0 dim(g) is in degree zero. Hence, when computing H (g) and Kill(g) , the main con- 2 0 0 tribution comes from those terms involving g . It turns out that under mild assumptions fulﬁlled in both theorems (implied by the failure of the SOL obstruction), we can get rid of these terms. Namely, recalling that the homology module H (g) is the cokernel of the map d: 2 3 3 2 3 2 g→ g, we observe (see Section 8.D.1 for details) that d maps ( g )→ g ; 3 0 let H (g) be the cokernel of the latter module homomorphism; it is endowed with a canonical map H (g)→ H (g) . 0 2 0 ⊗3 2 Similarly, Kill(u) is the cokernel of the map T: g→ S g (the second symmetric ⊗3 2 power) mapping x⊗ y⊗ z to[x, y] z− x [y, z], which maps ((g ) ) into (S g ) ; 0 0 let Kill (g) be the cokernel of the latter; it admits a canonical module homomorphism into Kill(g) . The following theorem is contained in Theorem 8.D.2 for H and in Theo- rem 8.D.9 for the Killing module. Theorem J. —Let g be a nilpotent Lie algebra (over any commutative ring) graded in some real vector space. Suppose that for every α we have g=[g , g]. Then the natural module 0 β−β β/∈{0,α,−α} homomorphisms H (g)→ H (g) and Kill (g)→ Kill(g) are isomorphisms. 0 2 0 0 0 The assumption is satisﬁed if g is nilpotent and 2-tame, that is, g/[g, g] has no pair {α, β} of weights such that 0 belongs to the segment[α, β] [Ab87, Lemma 4.3.1(c)]. 1.9. Guidelines. • The reader interested in the proof of Theorem E.1 candirectlygotoSection 12, which is essentially independent. • The reader interested in the proof of Theorem E.2 can directly go to Sec- tion 11, and refer when necessary to the preliminaries of Sections 4, 5,and, more scarcely, Section 8. • The reader interested in the proof of the polynomial upper bounds will ﬁnd the ﬁrst important steps in Section 3 and the weight deﬁnitions in Section 4,and then can proceed to Sections 6, 7,and 10, and refer when necessary to all the preceding sections. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 101 • The reader interested speciﬁcally in the algebraic results, as described in Sec- tion 1.8, can directly go to Sections 8 and 9, and refer when necessary to the preliminary Section 5. Although being of preliminary nature here, Sections 2 (introducing the Dehn func- tion) and 4 (weights and tameness conditions, Cartan grading) can be read for their own interest. 2. Dehn function: preliminaries and ﬁrst reductions This section contains general preliminaries on the Dehn function. In particular, we state Gromov’s trick in Section 2.E, essentially following arguments from [CT10]. 2.A. Asymptotic comparison. Deﬁnition 2.A.1. —If f , g are real functions deﬁned on any set where∞ makes sense (e.g., in a locally compact group, it usually refers to convergence to the point at inﬁnity in the 1-point compactiﬁca- tion), we say that f is asymptotically bounded by g and write f g if there exists a constant C≥ 1 such that for all x close enough to∞ we have f (x)≤ Cg(x)+ C; if f g f we write f g and say that f and g have the same asymptotic behavior,orthe same -asymptotic behavior. Deﬁnition 2.A.2. —If f , g are non-decreasing non-negative real functions deﬁned on R or N, we say that f is bi-asymptotically bounded by g and write f g if there exists a constant C≥ 1 such that for all x f (x)≤ Cg(Cx)+ C; if f g f we write f≈ g, and say that f and g have the same bi-asymptotic behavior, or the same ≈-asymptotic behavior. Besides the setting, the essential difference between -asymptotic and≈-asymp- n n totic behavior is the constant at the level of the source set. For instance, 2≈ 3 but n n 3 . 2.B. Deﬁnition of Dehn function. — If G is a group and S any subset, we denote by |g| the (possibly inﬁnite) word length of g∈ G with respect to S; the length |·| takes S S ﬁnite values on the subgroup generated by S. If H is a group and R⊂ H, we deﬁne the area of an element w∈ H as the (pos- −1 sibly inﬁnite) word length of w with respect to the union hRh ;wedenoteitby h∈H area (w). It takes ﬁnite values on the normal subgroup generated by R. R 102 YVES CORNULIER, ROMAIN TESSERA Now let S be an abstract set and F thefreegroup over S, and π: F→ Ga S S surjective homomorphism. Its kernel is denoted by K; elements in K are called relations. If a subset R of K is given, the elements in the normal subgroup generated by R are called null-homotopic. Thus any null-homotopic word is a relation, and conversely, if R generates K as a normal subgroup (then R is called a system of relators), then relations are null-homotopic. By a common abuse of terminology, elements in a system of relators are called relators. We deﬁne the Dehn function δ (n)= maxn/2, sup area (w): w∈ K,|w|≤ n . S,π,R R S (The term n/2 is not serious, and only avoids some pathologies. Intuitively, it corresponds n−n to the idea that the area of a word s s , which is zero by deﬁnition, should account for at least n.) We like to think of δ as the Dehn function of G, but in this general setting, S,π,R this function as well as its asymptotic behavior can depend on the choice of S, π and R. It can also take inﬁnite values. Note that often S is a given subset of G and since π is obviously chosen as the unique homomorphism extending the identity of S, we write the Dehn function as δ . S,R Let now G be a compactly generated LC-group (LC-group means locally compact group), and S a compact generating symmetric subset whose interior contains 1. View S as an abstract set and consider the surjective homomorphism F→ G which is the identity on S, and K its kernel. Let K(d) be the intersection of K with the d -ball in (F , |·| ). We say that G is compactly presented if for some d≥ 0, the subset K(d) generates S S K as a normal subgroup. This does not depend on the choice of S (but the minimal value of d can depend). If so, the function δ takes ﬁnite values (see Lemma 2.B.2 below), S,K(d) and its≈-asymptotic behavior does not depend on the choice of S and d . Thelatteristhencalledthe Dehn function of G. If G is not compactly presented, we say by convention that the Dehn function is inﬁnite. For instance, when we say that a compactly generated LC-group G has a Dehn function f (n), we allow the possibility that G is not compactly presented, i.e. its Dehn function is (eventually) inﬁnite. Proposition 2.B.1. —The≈-behavior of the Dehn function is a quasi-isometry invariant of the compactly generated locally compact group. On the proof. — This follows from a more general statement for arbitrary connected graphs. In this case, the Dehn function can be deﬁned as soon that the graph is large-scale simply connected, i.e., can be made simply connected by adding 2-cells with a bounded number of edges, thus deﬁning a notion of area for arbitrary loops. The quasi-isometry invariance of the asymptotic behavior of the Dehn function for arbitrary connected graph is similar to the usual one showing that the Dehn function is a quasi-isometry invariant among ﬁnitely generated groups, see [Al91]or[BaMS93, Theorem 26]. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 103 Lemma 2.B.2. —Asabove,let G be an LC-group with a compact symmetric generating subset S,and K the kernel of the canonical homomorphism F→ G. Assume that, for some d≥ 4,the intersection K(d) of the d -ball in F with K generates K as a normal subgroup. Then δ (n) is S S,K(d) ﬁnite for all n. Proof.—Fix n.Denoteby S the product of n copies of S, with the product topology n n (rather than the n-ball in F ). Consider the product map p: S→ F ,and let (S ) be the S S 0 n n n inverse image of K(n) in S ;thus (S ) is a closed subset of S , hence is compact. By assumption, the area function area◦ p takes ﬁnitevalueson (S ) .Wehavetoshow K(d) 0 that it is bounded on (S ) ; thus it is enough to show that this function is locally bounded. n n Indeed, given s= (s ,..., s )∈ (S ) , there exists a neighborhood V of s in (S ) such that 1 n 0 0 −1 for all s= (s ,..., s )∈ Vand all1≤ k≤ n, we have the element (s ... s ) (s ... s ) 1 k 1 n 1 k represents an element t∈ S. Note that t= 1. k n Fix s∈ V, and deﬁne t ,..., t∈ Sas above. Set t= 1, and deﬁne, for 0≤ k≤ n 1 n 0 n+1 s= s ,..., s , t , s ,..., s∈ S (k) 1 k k k+1 n Then p(s )= p(s ), p(s )= p(s),and forall 0≤ k≤ n− 1 (0) (n) −1 −1 −1 −1 area p s s= area s ... s s t s t s ... s K(d) (k) K(d) k+1 k+1 (k−1) k+2 n k+1 k k+2 n −1 −1 = area s t s t≤ 1; K(d) k+1 k+1 k+1 k −1 it follows that area (p(s ) p(s))≤ n. Therefore K(d) area p s≤ area p(s)+ n,∀s∈ V, K(d) K(d) proving the local boundedness. 2.C. Dehn vs Riemannian Dehn. — Let X is a Riemannian manifold. If γ is a Lips- chitz loop in X, deﬁne the area of γ as the inﬁmum of areas of Lipschitz disc ﬁllings of γ . Deﬁne the ﬁlling function of X as the function mapping r to the supremum F(r) of areas of all Lipschitz disc ﬁllings of Lipschitz loops of length≤ r. Proposition 2.C.1. —Let G be a locally compact group with a proper cocompact isometric action on a simply connected Riemannian manifold X. Then G is compactly presented and the Dehn function of G satisﬁes δ(n)≈ max F(n), n . (The max(·, n) is essentially technical: unless X has dimension≤ 1or is compact, it can be shown that F(r) grows at least linearly.) The proof is given, for G discrete, by Bridson [Bri02, Section 5]. Here we only pro- vide a complete proof of the easier inequality F(n) δ(n), because the proof in [Bri02] 104 YVES CORNULIER, ROMAIN TESSERA makes a serious use of the assumption that G is ﬁnitely presented. For the converse in- equality δ(n) max(F(n), n), the (highly technical) proof given in [Bri02] uses general arguments of ﬁlling in Riemannian manifolds and a general cellulation lemma, and the remainder of the proof carries over our broader context, so we shall only provide a brief sketch of the converse. Lemma 2.C.2. —Let X be a simply connected Riemannian manifold with Isom(X) acting cocompactly on X. Then the Riemannian area of loops of bounded length is bounded. Proof.—Write G= Isom(X). By cocompactness, there exists r such that for every x∈ X, the exponential is (1/2, 2)-bilipschitz from the r -ball in T X to X. In particular, 0 x given a loop of length≤ r , it passes through some point x; its inverse image by the exponential at x has length≤ 2r and can be ﬁlled by a disc of area≤ π r in T X, and its 0 x image by the exponential is a ﬁlling of area≤ 4π r in X. −1 Now ﬁx a positive integer m≥ 6/r . LetDbe an m -metric lattice in X, that 0 0 −1 is, a maximal subset in X in which any two distinct points have distance≥ m .Note −1 that every point in X is at distance≤ m to some element in D. Also note that, by properness of X, the intersection of D with any bounded subset is ﬁnite. For any x, y∈ D −1 with d(x, y)≤ 3m , ﬁx a geodesic path S(x, y) between x and y.Consideracompact subset such that G= X. Consider a 1-Lipschitz loop f :[0, k]→ X and let us bound the Riemannian area of f by some constant depending only of k. By homogeneity, we can suppose that −1−1 f (0)∈ .For all 0≤ n≤ km ,let x be a point in D that is m -close to f (nm ).Fix 0 n 0 0 −1 geodesic paths joining x and f (nm ). So there is a homotopy from f to the concate- −1 nation of the S(x , x ) through km “squares” of perimeter at most 6m≤ r .Bythe n n+1 0 0 above, each of these km squares can be ﬁlled with area≤ π 4r . The remaining loop is a concatenation of k segments of the form S(x , x ) with x∈ D (with n taken modulo n n+1 n km ). For given k, there are only ﬁnitely many such loops, since all the points x are at 0 n −1 distance≤ k/2+ m to . Since X is simply connected, each of these loops has ﬁnite area. So the remaining loop has area≤ a for some a <∞. So we found a Lipschitz k k ﬁlling of the original loop, of area≤ a+ km π r . k 0 Partial proof of Proposition 2.C.1. — Fix a compact symmetric generating set S in G and a set R of relators. Fix x∈ X. Set r= sup d(x , sx ).If s∈ S, ﬁx a r-Lipschitz map 0 0 0 s∈S j :[0, 1]→ X mapping 0 to x and 1 to sx .If w= s ... s ,deﬁne j :[0, k]→ Xas s 0 0 1 k w follows: if 0≤ ≤ k− 1and 0≤ t≤ 1, j (+ t)= s ... s j (t). It is doubly deﬁned for w 1 s an integer, but both deﬁnitions coincide. So j is r-Lipschitz. If w represents 1 in G, then j (0)= j (k). w w If w is a word in the letters in S and represents the identity, let A(w) be the area of the loop j . w GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 105 By Lemma 2.C.2,A(w) is bounded when w is bounded. Also, it is clear that −1 A(gwg )= A(w) forall groupwords g. This shows that there exists a constant C > 0, namely C= sup A(r),suchthat A(w)≤ Carea(w) for some constant C. r∈R For some r , every point in X is at distance≤ r of a point in Gx .Considera 0 0 0 loop of length k in X, given by a 1-Lipschitz function u :[0, k]→ X. For every n (modulo k), let g x be a point in Gx with d(g x , u(n))≤ r .Wehave d(g x , g x )≤ 2r+ 1. n 0 0 n 0 0 n 0 n+1 0 0 −1 N By properness, there exists N (depending only on r )suchthat g g∈ S .If σ is a 0 n+1 n −1 word of length N representing g g ,and σ= σ ...σ ,thenwepassfrom u to j n+1 0 k−1 σ by a homotopy consisting of k squares of perimeter≤ 4r+ 2. By Lemma 2.C.2,there is a bound M on the Riemannian area of such squares. So the Riemannian area of u is bounded by kM+ A(j )≤ kM+ Carea(σ )≤ kM+ δ (Nk). This shows that 0 σ 0 0 S,R F(k)≤ kM+ δ (Nk). 0 S,R For the (more involved) converse inequality, we only give the following sketch: let ρ> 0 be such that each point in X is at distance <ρ/8toGx , and assume in addition that ρ> ρ= ,where κ is an upper bound on the sectional curvature of X. Let S be 2 κ the set of elements in G such that d(gx , x )≤ ρ and R the set of words in F ,oflength 0 0 S at most 12 and representing 1 in G. Then the proof in [Bri02, §5.2] shows thatS| R is a presentation of G with Dehn function δ(n) bounded above by 4λ (F(ρn)+ ρn+ 1), where λ= 1/ min(4 κ/π, α(r,κ)) and α(r,κ) is the area of a disc of radius r in the standard plane or sphere of constant curvature κ . 2.D. Two combinatorial lemmas on the Dehn function. — This subsection contains sev- eral general lemmas about the Dehn function, which will be used at some precise parts of the paper. 2.D.1. Free products. — Recall that a function R→ R is superadditive if it satisﬁes ≥0 f (x+ y)≥ f (x)+ f (y) for all x, y. For instance, the function r → r is superadditive for every α≥ 1. Lemma 2.D.1. — Let f be a superadditive function. Let (G ) be a ﬁnite family of (abstract) groups, each with a presentationS| R, with Dehn function≤ f . Then the free product H of the G i i i has Dehn function δ≤ f with respect to the presentation S| R. i i Proof.—Let w= s ... s be a null-homotopic word with n≥ 1. Because H is a free 1 n product, there exists i and 1≤ j≤ j+ k− 1≤ n such that every letter s for j≤ ≤ j+ k− 1 is in S and s ... s represents the identity in G . So, with the corresponding cost, i j j+k−1 i which is≤ f (k), we can simplify w to the null-homotopic word s ... s s ... s .Thus 1 j−1 k n δ(n)≤ f (k)+ δ(n− k) (with 1≤ k≤ n). Using the property that f is superadditive, we can thus prove by induction on n that δ(n)≤ f (n) for all n. (This argument is used in [GS99] for ﬁnitely generated groups.) 106 YVES CORNULIER, ROMAIN TESSERA 2.D.2. Conjugating elements. Lemma 2.D.2. —LetS| R be a group presentation, and r a bound on the length of the words in R. Then for every null-homotopic w∈ F , with length n and area α, we can write, in F , S S −1±1 w= g r g with r∈ R ∪{1} and g∈ F , with the additional condition|g|≤ n+ rα. i i i i S i S i=1 i Proof. — We start with the following claim: consider a connected polygonal planar complex, with n vertices on the boundary (including multiplicities); suppose that the num- ber of polygons is α and each has at most r edges. Fix a base-vertex. Then the distance in the one-skeleton of the base-vertex to any other vertex is≤ n+ rα. Indeed, pick an injective path: it meets at most n boundary vertices. Other vertices belong to some face, but each face can be met at most r times. So the claim is proved. Now a van Kampen diagram for a null-homotopic word of size n and area α with relators of length≤ r satisﬁes these assumptions, the distance from the identity to some vertex corresponds to the length to the conjugating element that comes into play. Thus the g can be chosen with|g|≤ n+ rα. i i S 2.E. Gromov’s trick. Deﬁnition 2.E.1. —Let F be the free group over an abstract set S,let G be an arbitrary group, and let π: F→ G be a surjective homomorphism. We call (linear) combing of (G,π) (or, informally, of G if π is implicit), a subset F⊂ F such that 1∈ F and some constant C≥ 1,we n Cn have the property that π(S)⊂ π(S∩ F) for all n≥ 0. Example 2.E.2. — Let T be any generating subset of a ﬁnitely generated abelian group A. Suppose that{t ,..., t }⊂ T is also a generating subset. Then the set of words { t} where (m ) ranges over Z , is a combing of F→ A. If every element of T is j T j=1 j equal or inverse to some t , the constant C can be taken equal to 1. The following is, up to minor changes explained below, established in [CT10, Proposition 4.3]: Theorem 2.E.3 (Gromov’s trick). — Let f: R→ R be a function such that for some >0 >0 α> 1, the function r → f (r)/r is non-decreasing. Let S be an abstract set, let G be an arbitrary group, and let π: F→ G be a surjective homomorphism. Let R⊂ F be a subset contained in Ker(π). Consider a combing F⊂ F . S S Assume that for all n, the area with respect to R of any w∈ Ker(π) of the form w= w w w 1 2 3 with max|w |≤ n with w∈ F is≤ f (n).ThenS| R is a presentation of G (i.e., R generates i i i Ker(π) as a normal subgroup) and for some constants C , C > 0, the Dehn function of G with respect to R is≤ C f (C n). Remark 2.E.4. — The statement in [CT10, Proposition 4.3] is awkward because it purportedly considers a locally compact group without specifying a presentation and GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 107 gives a conclusion on its Dehn function as a function (and not an asymptotic type of func- tion). Actually the local compactness assumption is irrelevant and the correct statement is the one given here, the proof given in [CT10] applying without modiﬁcation. The setting is just that of a group presentation; it is even not necessary to assume that the relators have bounded length. Second, the assumption was for words w of length≤ n, while here we assume that all w have length≤ n; this is more natural and handy, and requires no change in the proof. Third, [CT10, Proposition 4.3] is stated with the function n → n for α> 1, but it relies on [CT10, Lemma 4.1], which allows much more general functions. At the time [CT10] it was not obvious to the author how to state a general result without a compli- cated technical assumption, but it turns out that the non-decreasing assumption of f is very general: we are not even aware of any non-hyperbolic compactly presented locally compact group whose Dehn function is not≈-equivalent to a function f such that, say, 3/2 f (r)/r is non-decreasing. Proof of Theorem 2.E.3. — We ﬁrst claim that for any k≥ 3, any n and any w= w ...w∈ Ker(π) with w∈ F and sup|w |≤ n,wehave area (w)≤ f (n)= 1 k i i R k kf (Ckn/2). Indeed, for every i the length of π(w ...w ) is≤ n/2, so we can ﬁnd u∈ F 1 i i with π(u )= π(w ...w ) and|u |≤ Cn/2, with u= u= 1. Then i 1 i i 0 k k k −1 w= w= u w u; i i−1 i i=1 i=1 −1−1 since π(u w u )= 1, by assumption we have area (u w u )≤ f (Ckn/2). i−1 i R i−1 i i i Then, denoting δ= δ ,by[CT10, Lemma 4.2], we have, for all n≥ k, S,R δ(n)≤ kδ 2(C+ 1)n/k+ f C! n/k" ,∀n≥ k (where we have f (C! n/k") rather than f (Ck! n/k") because we consider here sup|w| k k i rather than|w|). So, using that! n/k"≤ 2n/k,wehave δ(n)≤ kδ 2(C+ 1)n/k+ kf C n ,∀n≥ k. If we deﬁne u (r)= kf (C r), then we claim that zu (y)/u (yz) tends to zero uni- k k k formly in y≥ 1, k≥ 1, when z →∞. Indeed, writing f (r)= r v(r) with v non- decreasing, we have 2 α 2 2 zu (y) z(C y) v(C y) v(C y) 1 ==≤ . 2 α 2 α−1 2 α−1 u (yz) (C yz) v(C yz) z v(C yz) z This uniform convergence to zero allows to use [CT10, Lemma 4.1], which yields that for some r ,some k, and some constant c > 0, we have f (r)≤ cu (r) for all r≥ r .Thus 0 k 0 δ(r)≤ ckf (C r) for all r≥ r . 0 108 YVES CORNULIER, ROMAIN TESSERA (Here the value of the constant C= ck is hidden behind the estimates from the proof [CT10, Lemma 4.1] but the constant C= C is explicit, C being given in Deﬁni- tion 2.E.1.) 3. Geometry of Lie groups: metric reductions and exponential upper bounds In the study of upper bounds for the Dehn function of Lie groups, there are two im- portant reduction steps, described in this section. The ﬁrst (Section 3.A) allows to reduce the problem to triangulable groups, with no cost on the Dehn function. The second (Sec- tion 3.C) passes to a generalized standard solvable group, with a polynomial cost on the Dehn function. In between (Section 3.B), we describe a general retraction argument in Lie groups, proving in particular that connected Lie groups have an at most exponential Dehn function. 3.A. Reduction to triangulable groups. — The following lemma is essentially borrowed from [C08]. Lemma 3.A.1. — For every connected Lie group G, there exists a series of proper homomor- phisms with cocompact images G← G→ G← G , with G triangulable. In particular, G is 1 2 3 3 3 quasi-isometric to G and the Dehn functions of G and G are≈-equivalent. Proof. — Let N be the nilradical of G. By [C08, Lemma 6.7], there exists a closed cocompact solvable subgroup G of G containing N, and a cocompact embedding G⊂ 1 1 H with H a connected solvable Lie group, such that H is generated by G and its center 2 2 2 1 Z(H ). In particular, every normal subgroup of G is normal in H . Let W be the largest 2 1 2 compact normal subgroup of H and deﬁne L= H /W, so L is a connected solvable 2 2 2 2 Lie group whose derived subgroup has a simply connected closure. By [C08, Lemma 2.4], there are cocompact embeddings L⊂ G⊃ G ,with G and G connected Lie 2 2 3 2 3 groups, and G triangulable. The latter statement follows from the fact that continuous proper homomorphisms with cocompact images are quasi-isometries, along with Proposition 2.B.1, which says that the≈-behavior of the Dehn function is a quasi-isometry invariant. 3.B. At most exponential Dehn function. Theorem 3.B.1 (Gromov). — Every connected Lie group has an at most exponential Dehn function. As usual, Gromov gives a rough sketch of proof [Gro93, Corollary 3.F ], but we are not aware of a complete written proof. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 109 The basic idea is that the law can be described using polynomials and exponentials, but this is not enough because iterated exponentials would be an issue. A workable lemma is the following. n n Recall that an exponential polynomial R→ R is a real-valued function each of whose n coordinates is a polynomial in the coordinates x ,..., x and in exponentials 1 n λ x j i e for suitable complex scalar numbers λ . Lemma 3.B.2. — Every simply connected solvable Lie group is isomorphic to a Lie group described as (R× R ,∗) with the law of the form (u ,v )∗ (u ,v )= P(v ,v , u , u ), v+ v; 1 1 2 2 1 2 1 2 1 2 −1 (u,v)= T(v, u),−v , where P is an exponential polynomial depending polynomially on the variable (u , u ) (that is, not 1 2 involving exponentials in the coordinates of u and u ), and where T is an exponential polynomial 1 2 depending polynomially on the variable u. For instance, the law of SOL can be described as v−λv 2 2 (x , y ,v )∗ (x , y ,v )= e x+ x , e y+ y ,v+ v 1 1 1 2 2 2 1 2 1 2 1 2 (here (, m)= (2, 1)). Proof. — Let N be the derived subgroup of G and H a Cartan subgroup; both are a simply connected nilpotent Lie group. By [Bou, Chap. 7, §1, 2, 3] (see also Proposi- tion 4.D.4), we have NH= G. Consider a supplement subspace of the Lie algebra of M= N∩ H in the Lie algebra of H, and deﬁne V as its exponential (it is usually not a subgroup). Set m= dim(V) and = dim(N). We identify V and N to R and R through the exponential and a choice of bases of log(V) and log(N). The product map M× V→ H is an algebraic isomorphism of varieties, which yields a decomposition H= M× V, for which the law is given by polynomials; for the moment we just need to use the fact that the product (0,v)(0,v ) can be described as (Q(v, v ), v+ v ) for some polynomial Q on 2k variables. The adjoint action of h on n is given by some Lie algebra homomorphism φ: h→ End(n). If h∈ Hand n∈ N, we have −1 hnh= exp Ad(h) log(n)= exp exp φ log(h) log(n) . Beware that in this expression, the ﬁrst exp is the exponential of the Lie algebra N, with inverse log (both being polynomials), while the second is the operator exponentiation on n. The operator exp(φ(log(h))) is a polynomial in the entries and exponentials of 110 YVES CORNULIER, ROMAIN TESSERA the coefﬁcients of log(h) (the exponentials appear because φ(log(h)) is not necessary nilpotent), possibly with complex coefﬁcients. Hence after identiﬁcation of H and N to real vector spaces using the Lie algebras, −1 we can write hnh= R(h, n), where R is a polynomial in the coefﬁcients of h and n,and in some (possibly complex) exponentials of the coefﬁcients of h. Also, describe the product of 3 elements in N by a polynomial: nn n= S(n, n , n ). The product map yields a decomposition G= VN, and for v, v∈ V, n, n∈ N, we −1 have (nv)(n v )= nvn v v(vv ), so in coordinates, we have −1 (n,v) n ,v= S n,vn v , Q v, v ,v+ v = S n, R v, n , Q v, v ,v+ v . −1−1−1−1 That the inverse map has the given form is similar, since (nv)= (v n v)v . Lemma 3.B.3. —If G is a simply connected solvable Lie group with a left-invariant Rieman- nian metric, there is an exponentially Lipschitz strong deformation retraction of G to the trivial subgroup, i.e. a map F: G ×[0, 1]→ G such that for all g, F(g, 0)= gand F(g, 1)= 1, and such that for some constant C,if B(n) is the n-ball in G then F is exp(Cn)-Lipschitz in restriction to B(n) ×[0, 1]. Proof. — We identify G with R× R and write its law as −1 (u ,v )∗ (u ,v )= u+ u , P(u , u ,v ,v ); (u,v)= (−u,−v), 1 1 2 2 1 2 1 2 1 2 with P as in Lemma 3.B.2. Deﬁne, for ((u,v), t,τ)∈ G ×[0, 1] , s (u,v), t,τ= (tu,τv). We need an upper bound on the differential of s at a given point (u ,v , t ,τ ). 0 0 0 0 Let L denote the left translation by g,given by h → g∗ h. Then −1 L◦ s L (u,v), t,τ (u ,v ) 0 0 (t u ,τ v ) 0 0 0 0 = T(τ v , t u ),−τ v∗ tP(v ,v, u , u), τ v+ τv 0 0 0 0 0 0 0 0 0 = P−τ v ,τv+ τv, T(τ v , t u ), tP(v ,v, u , u) , 0 0 0 0 0 0 0 0 0 −τ v+ τv+ τv 0 0 0 We can write it as Q(τ v ,τv ,τv,v ,v; t , t, u , u),where Q= (Q , Q ),Q is an expo- 0 0 0 0 0 0 1 2 1 nential polynomial depending polynomially on the last four variables and Q is a poly- nomial. It follows that each partial derivative of Q with respect to the variables (u,v, t,τ) is an exponential polynomial R in the same variables, polynomial on the last four GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 111 variables (with coefﬁcients depending only the group law). Each such partial deriva- tive, when (u,v, t,τ)= (0, 0, t ,τ ), is equal to R(τ v ,τ v , 0,v , 0; t , t , u , 0), which 0 0 0 0 0 0 0 0 0 0 canberewrittenas R (τ v ,v; t , u ),with R an exponential polynomial that is poly- 0 0 0 0 0 nomial in the last two variables. Hence, keeping in mind that τ , t belong to[0, 1], 0 0 the partial derivatives of Q at (0, 0, t ,τ ) at any (0, 0, t ,τ ) are bounded above by 1 0 0 0 0 c$v$ c 2 0 3 c e (1 +$u$) ,where c , c are positive constants depending only on the group law 1 0 1 2 and the choice of left-invariant Riemannian metric μ. Therefore, the differential of s at any (u ,v , t ,τ ), for the left-invariant Rieman- 0 0 0 0 nian metric μ, has the same bound. If (u ,v )∈ B(2n) (the 2n-ball around 1 in G), then$v $≤ n (up to rescaling μ) 0 0 0 c n 2 and$u $≤ e (for some ﬁxed constant c ). So, for every (u ,v , t ,τ )∈ B(2n) ×[0, 1] , 0 4 0 0 0 0 Cn the differential of s at (u ,v , t ,τ ) is bounded by e , for some positive constant C only 0 0 0 0 depending on (G,μ). In particular, since any two points in B(n) can be joined by a 2 Cn geodesic within B(2n), we deduce that the restriction of s to B(n) ×[0, 1] is e -Lipschitz. The function (g, t) → s(g, t, t) is the desired retraction. (We used an extra variable τ by anticipation, in order to reuse the argument in the proof of Proposition 3.B.5.) Proof of Theorem 3.B.1. — By Lemma 3.A.1, we can restrict to the case of a simply connected solvable (actually triangulable) group G. Given a loop of size n in G based at the unit element, Lemma 3.B.3 provides a Lipschitz homotopy with exponential area to the trivial loop. Remark 3.B.4. — Using Guivarc’h’s estimates on the word length in simply con- nected solvable Lie groups [Gui73, Gui80], we see that there exists a constant C such that if B(n) is the n-ball in G, then F(B(n),[0, 1]) is contained in the ball B(C n) (here F is the function constructed in the proof of Lemma 3.B.3). Thus in particular, F provides a ﬁlling of every loop of linear size, with exponential area and inside a ball of linear size. In particular, any virtually connected Lie group (and cocompact lattice therein) has a linear isodiametric function. d d Let G= (U× U ) Z be a standard solvable group. The group G= U Z a na 1 a can be embedded as a closed cocompact subgroup into a virtually connected Lie group G with maximal compact subgroup K. Consider a left-invariant Riemannian metric on the connected manifold G /K. The composite map G→ G /K is a G-equivariant 2 1 2 quasi-isometric injective embedding; endow G with the induced metric. Endow U Z 1 na with a word metric with respect to a compact generating subset, and endow G with a metric induced by the natural quasi-isometric embedding G→ (U Z )× G . na 2 Proposition 3.B.5. —Let G be a standard solvable group of the form (U× U ) Z , with a na the above metric. Then there is an exponentially Lipschitz homotopy between the identity map of G and its natural projection π to U Z . Namely, for some constant C,there is amap na d d σ: (U× U ) Z ×[0, 1]→ (U× U ) Z , a na a na 112 YVES CORNULIER, ROMAIN TESSERA such that for all g∈ G, σ(g, 0)= gand σ(g, 1)= π(g),and σ(g, t)= gif g∈ U Z ,and σ na Cn is e -Lipschitz in restriction to B(n) ×[0, 1]. Proof. — As in the deﬁnition of standard solvable group, write G= U Z and U= U× U . Since U is a simply connected nilpotent Lie group, we can identify it to a na a its Lie algebra thorough the exponential map. Deﬁne, for (v, w, u)∈ U× U Z and a na t ∈[0, 1], σ(v,w, u, t)= (tv, w, u). By the computation in the proof of Lemma 3.B.3, Cn σ is e -Lipschitz in restriction to B(n) ×[0, 1]; the presence of w does not affect this computation. Corollary 3.B.6. — Under the assumptions of the proposition, if G/U U A is com- na pactly presented with at most exponential Dehn function, then G has an at most exponential Dehn function. Proof. — Given a loop γ of size n in G, the retraction of Proposition 3.B.5 inter- polates between γ and its projection γ to G/G . The interpolation has an at most ex- ponential area because the retraction is exponentially Lipschitz; γ has linear length and hence has at most exponential area by the assumption. So γ has an at most exponential area. 3.C. Reduction to split triangulable groups. Proposition 3.C.1 ([C11]). — Let G be a triangulable real group and E= G its exponential ˘˘ radical. There exists a triangulable group G= E V and a homeomorphism φ: G→ G,sothat, denoting by d and d left-invariant word distances on G and G G˘ • φ restricts to the identity E→ E, • E is the exponential radical of G • V is isomorphic to the simply connected nilpotent Lie group G/E, • the map φ quasi-preserves the length: for some constant C > 0, −1 C|g|≤|φ(g)|≤ C|g|;∀g∈ G and is logarithmically bilipschitz −1 D(|g |+|h|) d (g, h)≤ d (g, h)≤ D(|g |+|h|)d (g, h);∀g, h∈ G, G G where C > 0 is a constant and where D is an increasing function satisfying D(n)≤ C log(n) for large n. Corollary 3.C.2. —Set{H, L}={G, G}. Suppose that the Dehn function δ of L satisﬁes δ (n) n . Then for any ε> 0, the Dehn function δ of H satisﬁes α+ε 2α+ε α δ (n) log(n) δ n log(n) log(n) n . H L GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 113 Proof. — Suppose, more precisely, that every loop of length n in L can be ﬁlled with area δ(n) in a ball of radius s(n);note that s can be chosen to be asymptotically equal to δ (by Lemma 2.D.2). −1 Start with a combinatorial loop γ of length n in H. It maps (by φ or φ )toa “loop” in L, in the Cn-ball, in which every pair of consecutive vertices are at distance ≤ Clog(n). Join those pairs by geodesic segments and ﬁll the resulting loop γ of length ≤ Cn log(n) by a disc consisting of δ(Cn log(n)) triangles of bounded radius (say,≤ C), inside the s(Cn log(n))-ball. Map this ﬁlling back to H. We obtain a “loop” γ consisting of Cn log(n) points, each two consecutive being at distance≤ Clog(s(Cn log(n))),with a ﬁlling by δ(Cn log(n)) triangles of diameter at most C log(s(Cn log(n))). Interpolate γ by geodesic segments, so as to obtain a genuine loop γ .So γ is ﬁlled by γ and Cn log(n) “small” loops of size ≤ Clog s Cn log(n)+ 1≤ Clog C Cn log(n)+ 1≤ C log(n). The loop γ itself is ﬁlled by δ(Cn log(n)) triangles of diameter at most Clog(s(Cn log(n))) and thus of size≤ 3C log(n). cn We know that H has its Dehn function bounded above by C e . So each of these small loops has area≤ C exp(3c(C log(n)))= C n .Wededucethat γ can be ﬁlled by 2 1 c 1+c+max(1,α) Cn log(n)+ δ Cn log(n) C n n triangles of bounded diameter. We deduce that H has a Dehn function of polynomial growth, albeit with an out- rageous degree, the constants c being out of control. Anyway, this provides a proof that Hhas a Dehn function≤ C n for some q, and we now repeat the above argument with this additional information. The small loops of size C log(n) therefore have area≤ C (C log(n)) and the 1 3 1 δ(Cn log(n)) triangles ﬁlling γ can now be ﬁlled by≤ C (3C log(s(Cn log(n)))) triangles of bounded diameter. We deduce this time that γ can be ﬁlled by at most q q Cn log(n)C C log(n)+ 3CC log s Cn log(n) δ Cn log(n) 3 1 3 ≈ log s n log(n) δ n log(n) triangles of bounded diameter. We have 2 2α log s(n log n)≤ log s n log n log(n), so we deduce that the Dehn function of γ is log(n) δ n log(n) . Since δ(n) n , the previous reasoning can be held with q of the form α+ ε for any ε> 0. This proves the desired result. 114 YVES CORNULIER, ROMAIN TESSERA Remark 3.C.3. — A variant of the proof of Corollary 3.C.2 shows that if the Dehn function of G is exponential, then the Dehn function of G is exp(n/ log(n) ), but is not strong enough to show that the Dehn function of G is exponential, nor even exp(n/ log(n) ) for small α≥ 0. 4. Gradings and tameness conditions This preliminary section describes basics on the weight decomposition of standard solvable groups and real triangulable groups. In Section 4.A, we state a general weight decomposition theorem for ﬁnite- dimensional representations of nilpotent groups over complete normed ﬁelds. We use it to introduce weights in standard solvable groups in Section 4.B, where we provide some use- ful characterizations. We introduce the tameness conditions in Section 4.C, which allow to reinterpret the SOL obstructions in a more conceptual way. In Section 4.D,wedeal with the Cartan grading in real triangulable groups, which is also needed in Section 11. 4.A. Grading in a representation. Lemma 4.A.1. —Let H⊂ G be an inclusion of ﬁnite index between nilpotent groups. Then any homomorphism f: H→ R has a unique extension f: G→ R. k k Proof.—If g∈ Gand g∈ H, the element f (g )/k does not depend on k,wedeﬁne ˜˜ it as f (g); note that this is the only possible choice for f and already proves uniqueness. To prove the existence, we need to check that f is a homomorphism. Since checking ˜˜˜ f (xy)= f (x)+ f (y) only involves two elements, we can suppose that G and H are ﬁnitely generated. We can also suppose that they are torsion-free, as the problem is not modiﬁed if we mod out by the ﬁnite torsion subgroups. So G and H have the same rational Mal- cev closure, and every homomorphism H→ R extends to the rational Malcev closure. ˜˜ Necessarily, the extension is equal to f in restriction to G, so f is a homomorphism. (Note that R could be replaced in the lemma by any torsion-free divisible nilpotent group.) Recall that for every complete normed ﬁeld K, the norm on K extends to every ﬁnite extension ﬁeld in a unique way [DwGS, Theorem 5.1, p. 17]. Theorem 4.A.2. —Let K be a non-discrete complete normed ﬁeld. Let N be a nilpotent topolog- ical group and V a ﬁnite-dimensional vector space with a continuous linear N-action ρ: N→ GL(V). Then there is a canonical decomposition V= V , α∈Hom(N,R) GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 115 where, for α∈ Hom(N, R), the subspace V is the sum of characteristic subspaces associated to α(ω) irreducible polynomials whose roots have modulus e for all ω∈ N;moreoverwehave, for α∈ Hom(N, R) n 1/n α(ω) V ={0}∪ v∈ V {0}:∀ω∈ N, lim$ρ(ω)· v$= e n→+∞ n 1/n α(ω) = v∈ V :∀ω∈ N, lim$ρ(ω)· v$≤ e n→+∞ (Notethatwedonot assume that ρ(N) has a Zariski-connected closure.) Proof. — Let us begin with the case when K is algebraically closed. Let N be the Zariski closure of ρ(N); decompose its identity component N= D× U into diagonaliz- able and unipotent parts. Consider the corresponding projections d and u into D and U. 0−1 0 Deﬁne N= ρ (N ); it is an open subgroup of ﬁnite index in N. Let D be the (ordinary) closure of the projection d(ρ(N )). We can decompose, with respect to D, the space V into weight subspaces: V= V ,where V ={v∈ V :∀d∈ D, d· v= γ γ γ∈Hom(D,K ) γ(d)v}.Write (log|γ|)(v)= log(|γ(v)|),so log|γ |∈ Hom(D, R).For δ∈ Hom(D, R), deﬁne V= V ,sothat V= V .If δ∈ Hom(D, R),then δ γ δ {γ: log|γ|=δ} δ∈Hom(D,R) δ◦ d◦ ρ∈ Hom(N , R). So by Lemma 4.A.1, it uniquely extends to a homomorphism δ: N→ R, which is continuous because its restriction δ◦ d◦ ρ to N is continuous. Note that δ → δ is obviously injective. If v∈ V {0} and ω∈ N , write it as a sum v= v where 0 = v∈ V , δ γ γ γ γ∈I where I is a non-empty ﬁnite subset of Hom(D, K ), actually consisting of elements γ for which log|γ |= δ. Then, changing the norm if necessary so that the norm is the supremum norm with respect to the norms on the V (which does not affect the limits because of the exponent 1/n), we have n n n 1/n 1/n $ρ(ω) v$= sup$u ρ(ω) d ρ(ω) v$ γ∈I 1/n =|γ d ρ(ω)| sup$u ρ(ω) v$ γ∈I 1/n = exp δ(ω) sup$u ρ(ω) v$; γ∈I the spectral radii of both u(ρ(ω)) and its inverse being equal to 1 and v = 0, we deduce n 1/n n 1/n 0 that lim$u(ρ(ω)) v$= 1. It follows that lim$ρ(ω) v$= exp(δ(ω)) for all ω∈ N k 0 n 1/n and v∈ V {0}.If ω∈ N, there exists k such that ω∈ N . Writing f (n) =$ρ(ω) v$ , δ ω we therefore have 1/k 1/k k ˆˆ lim f (kn)= lim f k(n)= exp δ ω= exp δ(ω); ω ω n→∞ n→∞ on the other hand a simple veriﬁcation shows that lim f (n+ 1)/f (n)= 1, and it n→∞ follows that lim f (n)= exp(δ(ω)). n→∞ ω 116 YVES CORNULIER, ROMAIN TESSERA ±1 It follows in particular that the spectral radius of ρ(ω) on V is exp(±δ(ω)) and since the δ are distinct, it follows that V is the sum of common characteristic subspaces associated to eigenvalues of modulus exp(δ(ω)) for all ω. Conversely, suppose that v∈ V and that there exists α∈ Hom(N, R) such that for n 1/n α(ω) all ω∈ Nwe have lim$ρ(ω)· v$≤ e , and let us check that v∈ V . Observe n→+∞ n−1 n 1/n that lim ($ρ(ω)· v $$ρ(ω )· v$)≤ 1. Write v= v with v∈ V and suppose δ δ δ δ∈J n→+∞ ˆˆ by contradiction that J contains two distinct elements δ ,δ .So δ = δ and there exist 1 2 1 2 ˆˆ ω∈ Nsuch that δ (ω) > δ (ω).Then 0 1 2 n 1/n n−1 1≥ lim$ρ(ω )· v $$ρ ω· v$ n→+∞ n 1/n−1 1/n ≥ lim$ρ(ω )· v$$ρ ω· v$ 0 δ δ 1 0 2 n→+∞ ˆˆ = exp δ (ω )− δ (ω ) > 1, 1 0 2 0 a contradiction; thus v∈ V . This concludes the proof in the algebraically closed case. Now let K be arbitrary. The above decomposition can be done in an algebraic clo- sure of K, and is deﬁned on a ﬁnite extension L of K,so W= V⊗ L= W , K α α∈Hom(N,R) where W satisﬁes all the characterizations. Consider a K-linear projection π of L onto K;itextends to a K-linear projection π= id⊗ π of W= V⊗ L onto V, V K K which commutes with the action of ρ(N). Clearly V= π (W ). Since K is a com- plete normed ﬁeld, π is Lipschitz. It then follows from the deﬁnition that π maps W into itself (where we naturally consider the inclusion V⊂ W): this follows from the characterization of those elements of W as those v∈ Wsuch that for all ω∈ N n 1/n α(ω) we have lim$ρ(ω)· v$≤ e . It follows that V= V ,where V= π (W )= α α α n→+∞ (W∩ V). So the proof is complete. Note that the proof, as a byproduct, characterizes the elements v of V α∈Hom(N,R) n−n 1/n as those in V for which, for all ω∈ N, we have lim ($ρ(ω)· v $$ρ(ω)· v$)≤ 1 n→+∞ (which is actually a limit, and equal to 1, if v = 0). 4.B. Grading in a standard solvable group. — Let G= U A be a standard solvable group in the sense of Deﬁnition 1.2. It will be convenient to consider the product ring K= K ; it is endowed with the supremum norm, and view U as U(K).Wethuscall j=1 G a standard solvable group over K. In a ﬁrst reading, the reader can assume there is a single ﬁeld K= K . Since K is a ﬁnite product of ﬁelds, a ﬁnite length K-module is the same as a direct sum V= V ,where each V is a ﬁnite-dimensional K -vector space. The length of V j j j as a K-module, is equal to dim V . K j Let u be the Lie algebra of U .So u= u is a Lie algebra over K and the j j j exponential map, which is truncated by nilpotency, is a homeomorphism u→ U. This GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 117 conjugates the action of D on U to a linear action on u, preserving the Lie algebra structure; for convenience we denote it as an action by conjugation. We endow u with the action of A, and with the grading in Hom(A, R), as intro- duced in Theorem 4.A.2.Thus u itself is graded by u= u . The ﬁnite-dimensional α j,α vector space W= Hom(A, R) is called the weight space. This is a Lie algebra grading: [u , u ]⊂ u ,∀α, β∈ Hom(A, R). α β α+β Note that since A is a compactly generated locally compact abelian group, it is d d 1 2 isomorphic to R× Z× K for some integers d , d , and K a compact abelian group. In 1 2 particular, if d= d+ d , then the weight space W= Hom(A, R) is a d -dimensional real 1 2 vector space. If we split u as the direct sum u= u⊕ u of its Archimedean and non- a na Archimedean parts, the weights of u and u , respectively, are called Archimedean weights a na and non-Archimedean weights. Example 4.B.1. — Assume that the action of A on u is diagonalizable. If K= R and the diagonal entries are positive, we have −1 α(v) (u )= x∈ u :∀v∈ A,v xv= e x j α j and if K= Q and the diagonal entries are powers of p,wehave j p −1−α(v)/ log(p) (u )= x∈ u :∀v∈ A,v xv= p x . j α j Example 4.B.2 (Weights in groups of SOL type). —Let G= (K× K ) Abe a group 1 2 of SOL type as in Deﬁnition 1.3, where A contains as a cocompact subgroup the cyclic subgroup generated by some element (t , t ) with|t| > 1 >|t|. Then the weight space 1 2 1 2 is a one-dimensional real vector space, and with a suitable normalization, the weights are α= log(|t|)> 0and α= log(|t|)< 0, and U= K . It is useful to think of the 1 1 2 2 α i weight α with multiplicity q , namely the dimension of K over the closure of Q in K i i i i (which is isomorphic to R or Q for some p). In particular, G is unimodular if and only if q α+ q α= 0. 1 1 2 2 For instance, if G= (R× Q ) Z,then u= R× Q , u= R ×{0}, u= p p p log(p)− log(p) {0}× Q . For an arbitrary standard solvable group, we deﬁne the set of weights W= α: u ={0}⊂ Hom(A, R). u α It is ﬁnite. Weights of the abelianization u/[u, u] are called principal weights of u. Lemma 4.B.3. — Every weight of u is a sum of≥ 1 principal weights; 0 is not a principal weight. 118 YVES CORNULIER, ROMAIN TESSERA Proof. — The ﬁrst statement is a generality about nilpotent graded Lie algebras. If P is the set of principal weights and v= u ,then u= v +[u, u], i.e., v gener- α∈P ates u modulo the derived subalgebra. A general fact about nilpotent Lie algebras (see Lemma 6.B.3 for a reﬁnement of this) then implies that v generates u. The ﬁrst assertion follows. The condition that 0 is not a principal weight is a restatement of Deﬁni- tion 1.2(3). Deﬁnition 4.B.4. —If U is a locally compact group and v a topological automorphism, v is called a compaction if there exists a compact subset ⊂ U that is a vacuum subset for v, in the sense that for every compact subset K⊂ U there exists n≥ 0 such that v (K)⊂ .Ifevery neighborhood of 1 is a vacuum subset, we say that v is a contraction. Proposition 4.B.5. —Let U A be a standard solvable group. Equivalences: (i) some element of A acts as a compaction of U; (ii) some element of A acts as a contraction of U; (iii) 0 is not in the convex hull in W of the set of weights; (iv) 0 is not in the convex hull in W of the set of principal weights. Proof. — Automorphisms of U are conjugate, through the exponential, to linear au- tomorphisms; in particular, contractions and compactions coincide, being characterized by the condition that all eigenvalues have modulus < 1. Thus (i)⇔(ii). By Lemma 4.B.3, weights are sums of principal weights, and thus (iii)⇔(iv). If v∈ A acts as a contraction of U, then α(v) < 0 for every weight α.Thus α → α(v) is a linear form on W , which is positive on all weights. Thus (ii)⇒(iii). Conversely, let L be the set of linear forms of W= Hom(A, R) such that (α) > 0 for every weight α∈ W . Suppose that 0 is not in the convex hull of W , or equivalently u u ∗∗ that L =∅. Since the image of A in the bidual W linearly spans W , it has a cocompact closure in bidual W ; since L is a nonempty open convex cone, it follows that it has a non-empty intersection with the image of A in the bidual W . Thus there exists v∈ A such that α(v) > 0 for every weight α. So (iii)⇒(ii) holds. 4.C. Tameness conditions in standard solvable groups. — Motivated by Proposition 4.B.5, we introduce the following terminology: Deﬁnition 4.C.1. — We say that the standard solvable group G= U A (or the graded Lie algebra u)is • tame if 0 is not in the convex hull of the set of weights; • 2-tame if 0 is not in the segment joining any pair of principal weights; • stably 2-tame if 0 is not in the segment joining any pair of weights. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 119 To motivate the adjective “stably”, observe that u is stably 2-tame if and only if every graded subalgebra of u is 2-tame. The importance of 2-tameness, ﬁrst brought forward by Abels, will appear gradu- ally in the paper, starting with Proposition 4.C.3. Remark 4.C.2. — Clearly tame⇒ stably 2-tame⇒ 2-tame; the converse implications do not hold in general; however they hold when W is 1- dimensional, i.e. when A has a discrete cocompact inﬁnite cyclic subgroup. Also, when u is abelian, then 2-tame and stably 2-tame are obviously equivalent. The following characterization of 2-tameness will be needed to show in Sec- tion 4.C.3 that if G is not 2-tame then it has an at least exponential Dehn function. Proposition 4.C.3. —Let G= U A be a standard solvable group. Then G is not 2-tame if and only if there exists a group V E of type SOL (see Deﬁnition 1.3) and a homomorphism f into V E whose image contains V and is dense. Proof. — In general, let V E be a standard solvable group, and consider a contin- uous homomorphism f: U A→ V E with dense image (assuming that the closure of the image is cocompact would be enough). Since U and V are the derived subgroups of thetwo groups,wehave f (U)⊂ V. Since f (U)⊂ V, passing to the quotients, f induces a continuous homomorphism A→ E with dense image, which induces an injective linear map f: Hom(E, R)→ Hom(A, R). Denote by f the Lie algebra map it induces u→ v. If we consider the weight decomposition u= u , we see that either f (u ) is zero, or α has the form α α α∈Hom(A,R) f (β) and f (u )⊂ v . In particular, f (u) is contained in the sum of v ,where β ranges α β β over elements in Hom(E, R) such that f (β) is a weight of U A. In particular, if we assume that f (U)= V, we deduce that f maps weights to weights. Assume now that V E is a group of type SOL; since E is abelian, f factors through a homomorphism U/[U, U] A→ V E. Recall from Example 4.B.2 that for a group of SOL type, the set of weights consists of a quasi-opposite pair, i.e. a pair of nonzero elements such that the segment joining them contains zero. Since f is injective R-linear and maps weights to weights, we deduce that U/[U, U] A admits a quasi-opposite pair of weights. Conversely, suppose that U A, where U/[U, U] has two nonzero weights α, β with β= tα for some t < 0. To construct the map f , ﬁrst mod out by[U, U], so we are reduced to the case when U is abelian. Modding out if necessary by all other nonzero weights, we can suppose that u= u⊕ u .Write u= u ,and modout all u except one, so that u is a Lie α β α j,α j,α α j 120 YVES CORNULIER, ROMAIN TESSERA algebra over a single ﬁeld K ; again modding out by a maximal invariant subspace and possibly replacing K with a ﬁnite extension, we can assume that u is 1-dimensional over j α K , with a scalar action. Similarly we can assume that u is 1-dimensional over K ,with j β j ∗∗ a scalar action. We can mod out by the kernel of the homomorphism A→ K× K ; j j let B be the closure of its image. Since α and β are proportional, E contains a cyclic cocompact subgroup. The resulting group V E, where V= K× K ,isoftypeSOL j j and the resulting homomorphism U A→ V E has a dense image containing V. Remark 4.C.4. — The homomorphism to a group of type SOL cannot always be 2 2 chosen to have a closed image. For instance, let Z act on R by −m−n m n (m, n)· (x, y, z)= 2 3 x, 2 3 y . 2 2 Let G= R Z be the corresponding standard solvable group. Then every nontrivial normal subgroup of G contains either R ×{0} or{0}× R; thus every proper quotient of G is tame and it follows that no quotient of G is of SOL type. Let G= U A is a standard solvable group in the sense of Deﬁnition 1.2.The Lie algebra u of U admits a grading in the weight space W= Hom(A, R), introduced in Section 4.B. Deﬁne the set of weights of u as the ﬁnite subset W ={α∈ W: u ={0}}. u α We say that a subset of W is conic if it is of the form W∩ C, with C an open u u convex cone not containing 0. Let C be the set of conic subsets of W .If C∈ C,deﬁne u= u; C α α∈C this is a graded Lie subalgebra of u; clearly it is nilpotent. Let U be the closed subgroup of U corresponding to u under the exponential map and G= U A. In particular, if C C C v∈ A, deﬁne H(v) ={α∈ W: α(v) > 0}. Deﬁnition 4.C.5. —The G are called the essential tame subgroups of G.The G H(v) C are called the standard tame subgroups of G. Note that each essential tame subgroup is standard tame, and there ﬁnitely many standard tame subgroups; moreover every standard tame subgroup is a ﬁnite intersection of essential tame subgroups. Remark 4.C.6. — These notions can be developed in a broader context, avoiding references to gradings and Lie algebras. Let G be a locally compact group with a ﬁxed semidirect product decomposition G= U A. A tame subgroup of G (relative to the given semidirect decomposition) is deﬁned as a subgroup of the form V A, which is tame, i.e., in which some element of A acts on V as a compaction (in the sense of Deﬁnition 4.B.4). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 121 For v∈ A, if we deﬁne U as the contraction subgroup −n n (4.C.7) x∈ U: lim v xv= 1 , n→+∞ then v acts as a compaction on U ([CCMT15, Prop. 6.17]) and therefore U Ais v v tame. We call this a essential tame subgroup.Then standard tame subgroups aredeﬁnedasﬁ- nite intersections of essential tame subgroups; this can be shown to match the previous deﬁnition in the setting of standard solvable groups. 4.D. Cartan grading and weights. — Here we introduce the notion of Cartan grading, especially in the context of real triangulable Lie algebras; this will especially be needed in Section 11. Triangulable Lie groups are not standard solvable in general and although the treatment is analogous to that in Section 4.B, we have to face some speciﬁc difﬁculties. All Lie algebras in this Section 4.D are ﬁnite-dimensional over a ﬁxed ﬁeld K of characteristic zero. ∞ k Deﬁnition 4.D.1. —If g is a Lie algebra, let g= g be the intersection of its lower k≥1 central series, so that g/g is the largest nilpotent quotient of g. Deﬁnition 4.D.2. —If G is a triangulable Lie group with Lie algebra g, deﬁne its exponen- ∞∞ tial radical G as the intersection of its lower central series (so that its Lie algebra is equal to g ). We need to recall the notion of Cartan grading of a Lie algebra, which is used in Section 11.D and Section 11.E.Let n is a nilpotent Lie algebra; denote by n the space of homomorphisms from n to K (that is, the linear dual of n/[n, n]). Let v be an n-module (ﬁnite-dimensional) with structural map ρ: n→ gl(v).If α∈ n , deﬁne the characteristic subspace v= v∈ v :∀g∈ n, ρ(g)− α(g) v= 0 . k≥1 The subspaces v generate their direct sum; we say that v is K-triangulable if v= v ; α α α∈n this is automatic if K is algebraically closed. If v is a K-triangulable n-module, the above decomposition is called the natural grading (in n )of v as an n-module. If v= v ,wecall v a nilpotent n-module. Deﬁnition 4.D.3 ([Bou]). — A Cartan subalgebra of g is a nilpotent subalgebra that is equal to its normalizer. We use the following proposition, proved in [Bou, Chap. 7, §1, 2, 3] (see Deﬁni- tion 4.D.1 for the meaning of g ). 122 YVES CORNULIER, ROMAIN TESSERA Proposition 4.D.4. — Every Lie algebra admits a Cartan subalgebra. If n is a Cartan subal- gebra, then g= n+ g and n contains the hypercenter of g (the union of the ascending central series). If g is solvable, any two Cartan subalgebras of g are conjugate by some elementary automorphism exp(ad(x)) with x∈ g (here ad(x) is a nilpotent endomorphism so its exponential makes sense). Let us provide a simple way to recognize a Cartan subalgebra. Proposition 4.D.5. —Let g be a Lie algebra graded in some abelian group W (see Section 8.A if necessary). Assume that g is nilpotent and that for every α = 0, we have[g , g ]= g . Then g is 0 0 α α 0 a Cartan subalgebra in g. Proof. — We have to show that the normalizer h of g in g is g itself. This is a 0 0 graded subalgebra; hence we have to show that h= 0for all α = 0. Since[g , h] is α 0 α contained in both g and g , we deduce that[g , h ]= 0. 0 α 0 α To show that h= g , it is enough to do it in the case K is algebraically closed. For α = 0, we can decompose the module g with respect to the action of the nilpotent Lie algebra g as a sum of characteristic subspaces V , for homomorphisms λ: g→ K. 0 λ 0 Then V= 0 since the contrary would contradict[g , g ]= g . On the other hand, since 0 0 α α h is contained in the centralizer of g , it is contained in V . We deduce that it is zero. α 0 0 Example 4.D.6. —For λ∈ K, let g(λ) be the 4-dimensional Lie algebra with basis (u, x, y, z) and nonzero brackets[x, y]= z,[u, x]= x,[u, y]= λy,[u, z]= (1+ λ)z.Then a Cartan subalgebra is given as follows: • if λ/ ∈{−1, 0}, it can be taken to be the 1-dimensional subalgebra generated by u; • if λ =−1, as the abelian subalgebra generated by u, z; • if λ= 0, as the abelian subalgebra generated by u, y. Indeed, we can consider the grading in Z for which u has weight 0, x has weight 1, y has weight λ,and z has weight 1+ λ. Then it satisﬁes the condition of Proposition 4.D.5. Other illustrating examples of Cartan subalgebras in triangulable Lie algebras can be found in [C08, Ex. 4.1 and 4.2]. We now turn to a partial converse for Proposition 4.D.5. Let g be a Lie algebra and n a Cartan subalgebra. For the adjoint representation, g is an n-module. Assume that g is K-triangulable as an n-module (e.g., this holds if K is algebraically closed). The corresponding natural grading is called the Cartan grading of g (relative to the Cartan subalgebra n); moreover the Cartan grading determines n,namely n= g .Wecall weights the set of α such that g = 0. The Cartan grading is a Lie algebra 0 α grading, i.e. satisﬁes[g , g ]⊂ g for all α, β∈ n . In addition it satisﬁes[g , g ]= g α β α+β 0 α α for all α = 0; hence it fulﬁlls the conditions of Proposition 4.D.5. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 123 ∞∞ We have n+ g= g,and g ⊂[g, g], so the projection n→ g/[g, g] is surjective, ∨∨ inducing an injection (g/[g, g])⊂ n . Now assume that g is K-triangulable. Then all the weights of the Cartan grading lie inside the subspace (g/[g, g]) of n; the advantage is that this space does not depend on n, allowing to refer to a weight α without reference to a the choice of a Cartan subalgebra (although the weight space g still depends on this choice). In view of Proposition 4.D.4, any two Cartan gradings of g are conjugate. In particular, the set of weights, viewed as a ﬁnite subset of (g/[g, g]) , does not depend on the Cartan grading. Actually, the subspace linearly spanned by weights is exactly (g/r) ,where r ⊃[g, g] is the nilpotent radical of g.The principal weights of g are by deﬁnition the weights of the Lie algebra ∞∞∞∞∞ g /[g , g] (that is, the nonzero weights of the Lie algebra g/[g , g]); note that every nonzero weight is a sum of principal weights, as a consequence of the following lemma, which relies on the (basic) results of Section 8.A.2. Lemma 4.D.7. —Deﬁne g= g .Then g is the subalgebra generated by g . α =0 Proof. — It is clear from the deﬁnition that[n, g ]= g for all α = 0 and therefore α α ∞∞ g⊂ g . Conversely, since g= n is nilpotent, Lemma 8.A.5 implies that g is contained in the subalgebra generated by g . If m is a g-module, deﬁne m ={x∈ m :∀g∈ g, gx= 0}. Lemma 4.D.8. —Let g be a K-triangulable Lie algebra with a Cartan grading, and i≥ 1. ∞ g∞ Then H (g ) ={0} if and only if H (g ) ={0}. i i 0 ∞ g∞ Proof. — By deﬁnition of Cartan grading, we have H (g )⊂ H (g ) ; this pro- i i 0 vides one implication. ∞∞∞ Note, that H (g ) is a trivial g -module, hence can be viewed as a (g/g )- module, and for any α,H (g ) is a nonzero g -submodule; since it is stable under both i α 0 ∞∞∞∞ g and g ; since g= g+ g we deduce that each H (g ) is a g-submodule of H (g ) 0 0 i α i ∞ g∞ g and H (g )= H (g ) . i i ⊗i On the other hand, it follows from the deﬁnition of Cartan grading that (g ) is a nilpotent g -module; hence its subquotient H (g ) is also a nilpotent g -module 0 i 0 0 and hence is a nilpotent g-module by the previous remark. Hence if H (g ) = 0, then i 0 H (g ) is nonzero as well. Now consider a real triangulable Lie algebra g, or the corresponding real triangu- lable Lie group. We call weights of g the weights of graded Lie subalgebra g endowed with the Cartan grading from g (note that since g= g+ g , this is precisely the set of weights of the graded Lie algebra g, except possibly the 0 weight). We deﬁne a real triangulable Lie algebra g, or the corresponding real triangulable Lie group G, to be tame, 2-tame,or stably 2-tame exactly in the same fashion as in Deﬁni- tion 4.C.1. In the case G is also a standard solvable group U D, we necessarily have 124 YVES CORNULIER, ROMAIN TESSERA u= g and its grading is also the Cartan grading; in particular whether G is tame (resp. 2-tame, stably 2-tame) does not depend on whether G is viewed as a real triangulable Lie group or a standard solvable Lie group. For α> 0, deﬁne SOL as the semidirect product R R, where the action is t−αt given by t· (x, y)= (e x, e y). Note that apart from the obvious isomorphisms SOL SOL , they are pairwise non-isomorphic. 1/α In a way analogous to Proposition 4.C.3,wehave Proposition 4.D.9. —Let G be a real triangulable group. Then G is not 2-tame if and only if it admits SOL as a quotient for some α> 0. Proof. — The “if ” part can be proved in the same lines as the corresponding state- ment of Proposition 4.C.3 and is left to the reader. ∞∞ Conversely, assume that G is not 2-tame. Clearly, G/[G , G] is not 2-tame, so we ∞∞ can assume that G is abelian. Endow g with a Cartan grading; then since g is abelian, ∞∞∞ 0isnot aweightof g and hence g= g g .Let G= G G be the corresponding 0 0 decomposition of G. In the same way as in the proof of Proposition 4.C.3, we can pass to a quotient that is still not 2-tame, for which the new G is 2-dimensional. Since the action of G on G is given by two proportional weights, its kernel has codimension 1 in N; in particular this kernel is normal in G; we see that the quotient is necessarily isomorphic to SOL for some α> 0. 5. Algebraic preliminaries about nilpotent groups and Lie algebras 5.A. Divisibility and Malcev’s theorem. — Recall that a group G is divisible (resp. uniquely divisible)iffor every n≥ 1, the power map G→ G mapping x to x ,issurjec- tive (resp. bijective). The following lemma is very standard. Lemma 5.A.1. — Every torsion-free divisible nilpotent group is uniquely divisible. k k Proof. — We have to check that x= y⇒ x= y holds in any torsion-free nilpotent k k group. Assume that x= y and embed the ﬁnitely generated torsion-free nilpotent group =x, y into the group of upper unipotent matrices over the reals. Since the latter is uniquely divisible (the k-th extraction of root being deﬁned by some explicit polynomial), we get the result. Let N be the category of nilpotent Lie algebras over Q (of possibly inﬁnite di- mension) with Lie algebras homomorphisms, and N the category of nilpotent groups with group homomorphisms, and N its subcategory consisting of uniquely divisible (i.e. divisible and torsion-free, by Lemma 5.A.1) groups. If g∈ N , consider the law on g Q g deﬁned by the Campbell-Baker-Hausdorff formula. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 125 Theorem 5.A.2 (Malcev [Mal49a, St70]). — For any nilpotent Lie algebra g over Q, (g, ) is agroup andif f: g→ h is a Lie algebra homomorphism, then f is also a group homomorphism (g, )→ (h, ). In other words, g → (g, ),f → f is a functor from N to N .Moreover, g k g Q this functor induces an equivalence of categories N→ N . Q Q The contents of the last statement is that • any group homomorphism (g, )→ (h, ) is a Lie algebra homomorphism; g h • any uniquely divisible nilpotent group (G,•) has a unique Q-Lie algebra struc- ture g= (G,+,[·,·]) such that =•. Recall that an element in a group is divisible if it admits n-roots for all n≥ 1. Lemma 5.A.3 (see Lemma 3 in [Ho77]). — In a nilpotent group, divisible elements form a subgroup. Lemma 5.A.4 (see Theorem 14.5 in [Ba60]). — Let G be nilpotent and uniquely divisible, n n with lower central series (G ). Then G/G is torsion-free (hence uniquely divisible) for all n. The following lemma is needed in the proof of Theorem 9.D.2. Lemma 5.A.5. — In the category of s-nilpotent groups, any free product of uniquely divisible groups is uniquely divisible. Proof. — First, the free groups in this category are torsion-free: to see this, it is enough to consider the case of a free group of ﬁnite rank in this category; such a group is of the form F/F with F a free group; it is indeed torsion free for all i: this is a result about the lower central series of a non-abelian free group and is due to Magnus [Mag35] (see also [Se, IV.6.2]). Let G , G be torsion-free uniquely divisible s-nilpotent groups and G∗ G their 1 2 1 s 2 s-nilpotent free product, which is divisible by Lemma 5.A.3.Denoteby (N ) the lower s+1 central series of G∗ G .Let g be a non-trivial element in G∗ G= (G∗ G )/N 1 2 1 s 2 1 2 and let us show that g is not torsion in this group. By [Mal49b], a free product of torsion- free nilpotent groups is residually torsion-free nilpotent, and therefore there exists t≥ s+ 1 such that the image of g is not torsion in (G∗ G )/N . Applying Lemma 5.A.4 to 1 2 t s+1 (G∗ G )/N , we see that (G∗ G )/N is torsion-free, so since g is non-trivial, it is not 1 2 1 2 torsion. So G∗ G is uniquely divisible by Lemma 5.A.1. 1 s 2 5.B. Powers and commutators in nilpotent groups. — This subsection includes several lemmas, which will be used at various places of the paper. Denote by ((·,·)) group commutators, namely −1−1 ((x, y))= x y xy, 126 YVES CORNULIER, ROMAIN TESSERA and n-fold group commutators (5.B.1) ((x ,..., x ))= ((x ,((x ,..., x )))). 1 n 1 2 n Deﬁne similarly n-fold Lie algebra brackets. When n≥ 2 is not speciﬁed, we just call them iterated group commutators or Lie algebra brackets. 1 i+1 i If G is a group, its lower central series is deﬁned by G= Gand G= ((G, G )) (the group generated by commutators ((x, y)) when (x, y) ranges over G× G ). The group s+1 Gis s-nilpotent if G ={1}. In particular, 0-nilpotent means trivial, 1-nilpotent means abelian, and more generally, s-nilpotent means that (s+ 1)-fold group commutators van- ish in G. Similarly, if g is a Lie algebra, its lower central series is deﬁned in the same way (and does not depend on the ground commutative ring), and s-nilpotency has the same meaning. The following lemma will be used in the proof of Lemma 6.E.4. Lemma 5.B.2. —Let N be an s-nilpotent group, and let i be an integer. Then there exists an integer m= m(i, s) such that for all x, y∈ N, we can write ((x, y ))= w ...w , 1 m ±1±1 where each w (1≤ j≤ m) is an iterated commutator (or its inverse) whose letters are x or y , that ±1±1 is, w= ((t ,..., t )) for some k≥ 2 and t ∈{x , y}. j j,1 j,k j j,i Example 5.B.3. — In a 3-nilpotent group N, we have, for all x, y∈ Nand i∈ Z i i−i(i−1)/2 ((x, y ))= ((x, y)) ((y, x, y)) . Proof of Lemma 5.B.2. — We shall prove the lemma by induction on s. The statement is obvious for s= 0 (i.e. when N is the trivial group), so let us suppose s≥ 1. Applying the induction hypothesis modulo the s-th term of the descending series of N, one can write ((x, y ))= wz,where w has the form w ...w where m= m(i, s− 1),and where z lies in 1 m the s-th term of the lower central series of the subgroup generated by x and y, which will ±1±1 i be denoted by H. Since the word length according to S ={x , y} of both ((x, y )) and w is bounded by a function of i and s, this is also the case for z. Now H is generated by the set of iterated commutators T ={((x , x ..., x ))| x ,..., x∈ S}. Therefore, z can be 1 2 s 1 s ±1 written as a word in T , whose length only depends on s and i. So the lemma follows. The following lemma will be used in the proof of Lemma 6.B.3. Lemma 5.B.4. — In any uniquely divisible s-nilpotent group, if x= exp v i i ((x ,..., x ))= exp[v ,v ,...,v]. 1 s 1 2 s GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 127 Proof. — (Refer to (5.B.1) for the conventions deﬁning iterated Lie brackets or group commutators.) Use the convenient convention to identify the group and the Lie algebra through the exponential and with this convention, the lemma simply states that ((x ,..., x )) =[x , x ,..., x ]∀x ,..., x∈ G. 1 s 1 2 s 1 s We prove the result by induction on s;itistrivial for s= 1 (abelian groups); assume it holds for s− 1. By induction, ((x ,..., x )) =[x ,..., x ]+ O(s),where O(s) means some 2 s 2 s combination of s-fold Lie algebra brackets. It follows from the Baker-Campbell-Hausdorff formula that if[x, y] is central then ((x, y)) =[x, y]. We can apply this to x= x and y= ((x ,..., x )). This yields 1 2 s ((x ,((x ,..., x ))))= x ,[x ,..., x ]+ O(s) =[x ,..., x]. 1 2 s 1 2 s 1 s The following proposition is the key new feature in the proof of Theorem 9.D.2. Proposition 5.B.5. —Let x, y be elements of a uniquely divisible nilpotent group G.Let N r−r be the normal subgroup generated by the elements of the form x y , where r ranges over Q. Then N is divisible. Equivalently, G/N is torsion-free. We need the following lemma. Lemma 5.B.6. —Let x, y be elements of a uniquely divisible nilpotent group G,and n an −1 1/n 1/k−1/k integer. Then (xy ) is contained in the normal subgroup generated by{x y: k∈ Z}. Proof.—Let G be s-nilpotent. By [BrG11, Lemma 5.1] (see however Re- mark 5.B.7), there exists a sequence of rational numbers a , b ,..., a , b such that in 1 1 k k any uniquely divisible s-nilpotent group H and any u,v∈ H, we have 1/n −1 1/n−1/n a b i i uv= u v u v . i=1 In particular, picking (H, u,v)= (R, 1, 2), we see that a= b= 0. Therefore, if i i a b 1/n−1/n i i d is a common denominator to a ,..., b and n, u v as well as u v belong to 1 k i=1 1/d−1/d−1 1/n the normal subgroup generated by u v ,hence (uv ) as well. In particular, this applies to (H, u,v)= (G, x, y). Remark 5.B.7. — In the above proof, we used [BrG11, Lemma 5.1] to be concise. However, this is not very natural, because the latter is proved using the Hall-Petrescu formula; the problem is that in this formula, exponents are put outside the commutators. The proof of the Hall-Petrescu formula can easily be modiﬁed to prove by induction on the degree of nilpotency a similar formula with exponents inside the commutators. In 128 YVES CORNULIER, ROMAIN TESSERA [BrG11], in order to shorten the argument (as we also do), instead of processing this in- duction, they work with a much simpler induction based on the Hall-Petrescu formula; this is very unnatural, because if we do not allow ourselves to use the Hall-Petrescu for- mula, to go through the latter is very roundabout; moreover the exponents a , b obtained k k in [BrG11]dependon s, while in a direct induction, we pass from the s-nilpotent case to the (s+ 1)-nilpotent case by multiplying on the right by some suitable iterated commu- tator of powers. r−r ρ Proof of Proposition 5.B.5. — By Lemma 5.B.6, N contains elements (x y ) for any r−r ρ ρ∈ Q. So N is generated as a normal subgroup by the divisible subgroups N ={(x y ): ρ∈ Q}, and therefore N is divisible by Lemma 5.A.3. The following lemma is used in the proof of Lemma 9.B.1. Lemma 5.B.8. —Let g be a Lie algebra over the commutative ring R.Let m be a generating [1][i][1][i−1] R-submodule of g.Deﬁne g= m, and by induction the submodule g =[g , g] for i≥ 2 (namely, the submodule generated by the brackets of the given form). Let (g ) be the lower central series i[m] of g. Then for all i we have g= g . m≥i Proof.—Letuscheck that [i][j][i+j] (5.B.9) g , g⊂ g∀i, j≥ 1. Note that (5.B.9) holds when either i= 1or j= 1. We prove (5.B.9) in general by induction on k= i+ j≥ 2, the case k= 2 being already settled. So suppose that k≥ 3 and that the result is proved for all lesser k. We argue again by induction, on min(i, j), the case min(i, j)= 1 being settled. Let us suppose that i, j≥ 2and i+ j= k and let us check that (5.B.9) holds. We can suppose that j≤ i. By the Jacobi identity and then the induction hypothesis [i][j][i][1][j−1] [g , g ]= g , g , g [1][i][j−1][j−1][1][i] ⊂ g , g , g+ g , g , g [1][i+j−1][j−1][1+i] ⊂ g , g+ g , g [i+j][j−1][1+i] ⊂ g+ g , g Since j≤ i,wehave min(j− 1, i+ 1)< min(i, j), the induction hypothesis yields [j−1][1+i][i+j] [g , g ]⊂ g and (5.B.9)isproved. It follows from (5.B.9) that, deﬁning for all i≥ 1 {i}[j] g= g , j≥i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 129 {i} the graded submodule g is actually a Lie subalgebra of g. Let us check by induction on i{i}[1]{1} 1 i≥ 1that g= g . Since g generates g, it follows that g= g= g . Now, suppose i≥ 2 and the equality holds for i− 1. Then g is, in our notation, the Lie subalgebra generated by i−1{i−1}[j] g, g= g, g= g, g j≥i−1 [j][j+1]{i} = g, g= g= g; j≥i−1 j≥i−1 {i} i{i} since g is a Lie subalgebra we deduce that g= g . 5.C. Lazard’s formulas. — Lazard’s formulas (which will be used in Section 10.D and Section 10.F) are, in a sense, inversion formulas for the Baker-Campbell-Hausdorff formula: roughly speaking, they express the Lie algebras laws in terms of the group law; actually it involves taking roots and we ﬁnd it convenient to state them by “canceling denominators” as below. Here, F denotes the free group on 2 generators). Theorem 5.C.1 (Lazard [La54]). — For every s≥ 1, there exist group words A , B∈ F s s 2 and positive integers q , q , such that for every simply connected s-nilpotent Lie group G with Lie algebra g,and all x, y∈ G, writing X= log(x)∈ g, Y= log(y)∈ g, we have log A (x, y)= q (X+ Y) and log B (x, y)= q[X, Y] s s s Here A stands for Add, and B for Bracket. Remark 5.C.2. — We see that we have the formal equality of group words A (x, 1)= x . Indeed, A (x, 1) has the form x for some ∈ Z, and by evaluation in s s R we obtain = q . Similarly, we have the formal equality B (x, 1)= 1. s s The usual Lie ring axioms are reﬂected in the following proposition: Proposition 5.C.3. — In every s-nilpotent group G, abbreviating A= A , B= B ,q= q , s s s we have identities∀x, y, z∈ G, −1 A(x, y)= A(y, x); B(x, y)= B(y, x); B A(x, y), z= A B(x, z), B(y, z); q q A A(x, y), z= A x , A(y, z); k k k B x , y= B x, y= B(x, y)∀k∈ Z. 130 YVES CORNULIER, ROMAIN TESSERA Proof. — If G is a simply connected nilpotent Lie group, this follows from the cor- responding identities in the Lie algebra: commutativity of addition, anti-commutativity of the bracket, distributivity, associativity of addition; in the last equality if follows from the fact that log(x )= k log(x). Therefore this holds in every subgroup of such a group G, and in particular in any ﬁnitely generated free s-nilpotent group, and therefore in any s-nilpotent group, by substitution. 6. Dehn function of tame and stably 2-tame groups In Section 6.A, we introduce the class of tame groups, showing in particular that their Dehn function is at most quadratic. In Section 6.C, we formulate and prove a ver- sion of Gromov’s trick for standard solvable groups involving its tame subgroups. This result relies on some lengths estimates on nilpotent algebraic groups over local ﬁelds, which are obtained in Section 6.B.ThenSection 6.D is dedicated to the proof of Theo- rem D from the introduction (which is reformulated in terms of 2-tameness). Finally we extend this result to generalized 2-tame groups in Section 6.E. 6.A. Tame groups. — We introduce here a class of locally compact groups, called tame groups, including tame standard solvable groups from Deﬁnition 4.C.1. Tame groups are, for our purposes, the best-behaved. Using a “large-scale Lip- schitz homotopy”, these groups are shown to be compactly presented with an at most quadratic Dehn function and a have simple presentation (Theorem 6.A.4 and Corol- lary 6.A.7). While standard solvable groups are “usually” not tame, their geometrical study will be based on the family of their tame subgroups. Deﬁnition 6.A.1. —Wecalla tame group any locally compact group with a semidirect product decomposition G= U A,where A is a compactly generated abelian group, such that some element of A acts as a compaction of U (in the sense of Deﬁnition 4.B.4), in which case we call U A a tame decomposition. Recall that this means that there exist v∈ A and a compact subset (called vac- n−n uum subset) of U such that v v= U uniformly on compact subsets, in the n≥0 n−n sense that for every compact subset K of U there exists n such that K⊂ v v .If −n n we apply this to itself, so that v v⊂ ,wesee,byreplacing with the larger −n n−n n−1 v v= v v , that we can suppose v v⊂ , in which case we call 0≤k≤n−1 k≥0 a stable vacuum subset. Remark 6.A.2. —If G= U A is a tame decomposition, and if W is the largest compact subgroup in A, then W admits a direct factor A in A (isomorphic to R× Z for some k,≥ 0), so that G= UW A ;if x∈ A actsasacompaction on U thenit GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 131 also acts as a compaction on UW, and so does the projection of x on A (because the set of compactions of a given locally compact group is stable by multiplication by inner automorphisms [CCMT15, Lemma 6.16]). This shows that assuming that A= R× Z is not really a restriction. We are going to prove that tame groups are compactly presented with an at most quadratic Dehn function; in order to make quantitative statements, we use the following language. Let v∈ A be an element acting (by conjugation on the right) as a compaction of U. Let m be an integer. Let S be a compact symmetric subset of U with unit. Let T be a compact symmetric generating subset of A with unit. Deﬁnition 6.A.3. — We say that (m, S , T) is adapted to U A and v if v∈ T,the −1 subset S is a stable vacuum subset for the right conjugation u → v uv by v, and there exist non- −k 2 k −1 negative integers k,≥ 1 with k+ ≤ msuch that v S v⊂ S and (v w) S (v w)⊂ S U U U for all w∈ T. There always exists such an adapted triple; more precisely, for every symmetric generating subset T of A containing{1,v} and every symmetric stable vacuum subset S⊂ Uof v, (m, S , T) is adapted for all m large enough. Also, if S is a stable vacuum U U U ±k} k subset for some v∈ A and T is arbitrary, then (2, S , T ∪{v ) is adapted to v for large enough k. Theorem 6.A.4. —Let G be a tame group with a tame decomposition U A, with an element v acting as a compaction of U. Denote by π the canonical projection G→ A. Let (m, S , T) be adapted to U A and v in the above sense. Then S= S∪ T is a compact U U generating subset of G,and G being endowed with the corresponding word metric, the function γ: G× N→ G; (g, n) → gv is 1-Lipschitz in each variable and satisﬁes: γ(g, 0)= g, and d(γ (g, n), v π(g))≤ 1 whenever n≥ m|g|. The function γ should be understood as “locally” a large-scale homotopy between the identity of G and its projection onto A. This is only “local” since the time needed to reach A depends on the size of the element. See however Remark 6.A.10. −1−1 Proof of Theorem 6.A.4. — Since v S v⊂ S ,wehave v Sv⊂ S. So, for n≥ 0, U U −n n n the automorphism g → v gv is 1-Lipschitz, and since the left multiplication by v is an isometry, we deduce that γ(·, n) is 1-Lipschitz. Also, assuming that v∈ T, it is immediate that γ(g,·) is 1-Lipschitz. 132 YVES CORNULIER, ROMAIN TESSERA It remains to prove the last statement. Consider an element in U, of size at most n. We can write it as g= s u i i i=1 with s∈ Tand u∈ S . Deﬁning t= s ... s , we deduce i i U i i n −1 g= t t u t . n i i i=1 So t= π(g) and −n n−n−1 n v gv= π(g) v t u t v . i i i=1 −1 −n n By deﬁnition of , the element v= v t u t v belongs to S .So i i i U −n n v gv= π(g) v . i=1 −k 2 k−kj n kj Since v (S )v⊂ S , setting j =! log (n)",wehave v (S )v⊂ S .Thuswe U U U 2 U obtain −n−k! log (n)" n+k! log (n)" 2 2 v gv∈ π(g)S , mn mn since for all n we have! log (n)"≤ n and k+ ≤ n,weobtain d(gv ,v π(g))≤ 1. A ﬁrst consequence is an efﬁcient way of writing elements in a tame group: Corollary 6.A.5. — Under the assumptions of Theorem 6.A.4,for every x∈ G of with|x|= mn−mn n, we can write x= π(x)v sv and s∈ S . mn−1 Proof.—Deﬁne s= (v π(x)) γ(x, mn).Bythe theorem, s∈ S , while x= mn−mn π(x)v sv . The corollary allows us to introduce the following deﬁnition, which will be used in the sequel. Let G= U A betameand v∈ A act as a compaction on U. Let (m, S , T) be adapted to U Aand v in the sense of Deﬁnition 6.A.3,and S= S∪ T. By Corol- −mn mn lary 6.A.5,if x∈ Uand n =|x| , the element s= v xv belongs to S . S U Deﬁnition 6.A.6. —For x∈ U with|x|= nas above, we deﬁne x∈ F as the word S S mn−mn v sv , which has length 2mn+ 1 and represents x. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 133 We now write the consequence of Theorem 6.A.4, that tame groups have a “nice” presentation and a good control on the Dehn function. Corollary 6.A.7. — Under the assumptions of Theorem 6.A.4, G admits a presentation with generating set S in which the relators are the following (1) All relators of the form s s s for s , s , s∈ S ,whenever s s s= 1 in U; 1 2 3 1 2 3 U 1 2 3 −1−1 (2) all relators of the form ws w s whenever w∈ T,s , s∈ A and ws w s= 1 in G; 1 2 1 2 1 2 (3) a ﬁnite set R of deﬁning relators of A (with respect to the generating subset T) including all commutation relators. The Dehn function of this presentation is bounded above by mn+ δ (n),where δ is the Dehn T,R T,R 3 3 function of the presentationT| R of the abelian group A. Corollary 6.A.8. —If G is a tame locally compact group, then it has an at most quadratic Dehn function. This follows since A has an at most quadratic Dehn function. Remark 6.A.9. — Since G has a 1-Lipschitz retraction onto A (namely the canon- ical projection), we deduce that the Dehn function of G is exactly quadratic when A has rank at least 2. On the other hand, if A has rank one (i.e., contains Z as a cocompact lattice), then G is hyperbolic and thus its Dehn function is linear [CCMT15]. Proof of Corollary 6.A.7. — Start from a combinatorial loop γ= (u ) of length n. 0 i i This is a combinatorial loop in the Cayley graph of (G, S), i.e., d(u , u )≤ 1. Since γ: i i+1 n j (g, n) → gv is 1-Lipschitz by Theorem 6.A.4,each γ= (u v ) is also a loop of length n. j i i Deﬁne μ as the loop (v π(u )) in A. We know that for j≥ mn,wehave d(μ (i), γ (i))≤ 1 j i i j j for all i. So the “homotopy” consists in γ γ ··· γ μ 0 1 mn mn and is ﬁnished by a homotopy from μ to a trivial loop inside A. So we have to describe mn each step of this homotopy. To go from γ to γ ,weuse n squares, each with vertices of the form j j+1 j j j+1 j+1 u v , u v , u v , u v i i+1 i+1 i −1 for i= 1,... n (modulo n). To describe this square, we discuss whether u u belongs to i+1 S or T: in the ﬁrst case this is a relator of the form (2) and in the second case it is a relator of the form (3). To go from γ to μ ,weuse n squares with vertices mn mn mn mn mn mn u v , u v ,π(u )v ,π(u )v . i i+1 i+1 i 134 YVES CORNULIER, ROMAIN TESSERA −1−1 We discuss again: if u u∈ T, this is a relator of the form (3). If u u∈ S , actually i+1 i+1 U i i π(u )= π(u ) and this square actually degenerates to a triangle of the form (1). i i+1 Finally, the loop μ ,oflength n can be homotoped within A to a constant loop mn with area≤ δ (n). T,R Remark 6.A.10 (Asymptotic cones of tame groups are contractible). —Let X beamet- ric space and ω a nonprincipal ultraﬁlter on the set of positive integers. If (x ), (y ) n n are sequences in X, deﬁne d ((x ), (y ))= lim d(x , y )/n ∈[0,∞].The asymptotic cone ω n n ω n n (Cone (X), d ) is deﬁned as the metric space consisting of those sequences (x ) with ω ω n d ((x ), (x )) <∞, modulo identiﬁcation of (x ) and (y ) whenever d ((x ), (y ))= 0. ω n 0 n n ω n n A straightforward corollary of Theorem 6.A.4 is that if G is a tame locally compact group, then all its asymptotic cones are contractible. Indeed, the large-scale Lipschitz mapping γ induces a Lipschitz map γ˜: Cone (G)× R→ Cone (G) ω≥0 ω (x ), t → γ x ,tn n n such that γ(˜ x, 0)= x and γ(˜ x, t)∈ Cone (A) if t≥ C|x|. Deﬁning h(x, t) =˜γ(x, t|x|), then h is continuous, h(x, 0)= x and h(x, C)∈ Cone (A) for all x∈ Cone (G).Inother ω ω words, h is a homotopy between the identity of Cone (G) and a map whose image is contained in Cone (A). Since Cone (A) is (bilipschitz) homeomorphic to a Euclidean ω ω space, this shows that Cone (G) is contractible. 6.B. Length in nilpotent groups. — As in Deﬁnition 1.2,let G= U Abe a standard solvable group. We ﬁx a decomposition U= U ,where U= U (K ),where K is (j) (j) (j) j j j=1 some nondiscrete locally compact ﬁeld. We write K= K . We say that a subgroup j=1 VofUis K-closed if it is a direct product V ,where V= V∩ U is Zariski-closed (j) (j) (j) j=1 in U . (j) The condition that V is K-closed is equivalent to the requirement that log(V) is a Lie K-subalgebra of the Lie K-algebra u of U. When K is R or Q for some prime p,itis just equivalent to the condition that V is closed (for the ordinary topology) and divisible. We ﬁx on G an admissible ﬁnite family U ,..., U of tame subgroups of U, where 1 ν by admissible we mean it fulﬁlls all the following conditions: (1) each U is K-closed in U (2) U is generated by U . i=1 (3) each U is normalized by A. For instance, we can choose them to be the standard tame subgroups (there are ﬁnitely many, see Deﬁnition 4.C.5), but in the proof of Theorem 6.D.1 we will make another choice. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 135 We can ﬁnd a compact symmetric subset T⊂ A containing 1 and compact sym- metric subsets S of U containing 1, such that for every i,S is a stable vacuum subset for i i i some element of T. Deﬁne S= T∪ S . We assume that S= S∩ U for all i (we can i i i always replace S with S∩ U so that it holds). i i Notation 6.B.1. —For r ∈[−∞,+∞] and n∈ N,let T⊂ F be the n-ball of F and S T (r) −1 max(0,r) deﬁne S as the set elements of F of the form tst , where s∈ S and t∈ T .Let π be the S i canonical projection F→ G. Theorem 6.B.2. — Fix a standard solvable group G, an admissible family of tame subgroups and generating subset as above. There exist constants λ> 0, μ ,μ > 1, a positive integer κ and a 1 2 κ -tuple (q ,..., q ) of elements in{1,...,ν} satisfying 1 κ • for every small enough norm $·$ on u, denoting by U[r] the exponential of the r-ball in (u, $·$), we have, for all n, the inclusions n n n n n n n 2n S⊂ U μ T , U μ⊂ S , U μ T⊂ S . 2 1 1 n (λn) (λn) n • the n-ball S of G is contained in the projection π(S ... S T ) (which is itself contained q q 1 κ in the (κλ+ 1)n-ball). We ﬁrst need a general lemma. Lemma 6.B.3. — Fix an integer s≥ 1.Let K= K be a ﬁnite product of nondiscrete locally compact ﬁelds of characteristic zero (or of characteristic p > s). Let u be a s-nilpotent ﬁnite length Lie algebra over K (ﬁnite length just means that under the canonical decomposition u= u , each u is ﬁnite-dimensional over K ) and ﬁx a submultiplicative norm $·$ on u.Let U be the corresponding nilpotent topological group; if x∈ U, write$x$=$ log(x)$. Let (u ) be K-subalgebras of u; suppose that the u generate u as a K-subalgebra modulo i 1≤i≤c i [u, u].Let U= exp(u )⊂ U be the corresponding subgroups. i i Then there exists an integer κ and a constant K such that every element x∈ U can be written x ... x with x∈ u ,and sup$x $≤ Kmax($x$, 1). 1 κ k i k i k Proof. — We argue by induction on s.If s= 1, the assumption is that u is abelian and generated by the u . So there exist subspaces h⊂ u such that u= h ,soif x∈ u i i i i and p is the projection to h ,then x= p (x), and if K= sup$p$ (operator norm) i i i 0 i i=1 then$p (x)$≤ K$x$. i 0 Suppose now s > 1 and that the result is proved for s− 1. Denoting by (u ) the lower central series, we choose the norm on u/u to be the quotient norm. We use the in- duction hypothesis modulo U , so that there exist d and K≥ 1such that every x∈ Ucan be written as y ... y z,with$y $≤ K max($x$, 1) and z∈ U . By the Baker-Campbell- 1 d i Hausdorff formula, there exists a constant C > 0 (depending on s and d )suchthatfor all 136 YVES CORNULIER, ROMAIN TESSERA x ,... x in u,wehave 1 d+1 $x ··· x $≤ Csup max$x$ ,$x$ . 1 d+1 i i −1−1 Since z= y ... y x, we deduce d 1 ! " s s $z$≤ Cmax sup max$y$ ,$y$ , max$x$ ,$x$ i i s s s s s s ≤ Cmax max K , K$x$ , max$x$ ,$x$= C K max$x$ , 1 . s s Consider the K-multilinear map (u/[u, u])→ u given by the s-fold bracket. Since its image generates u as a K-submodule and since the u generate u modulo[u, u],wecan ﬁnd a ﬁnite sequence (i ) and a ﬁxed ﬁnite family (κ ) with κ∈ u such j 1≤j≤j jk 1≤j≤j ,1≤k≤s jk i 0 0 j that, setting ζ =[κ ,...,κ] we have u= Kζ (we can normalize so that$ζ $= 1). j j1 js j j Denote C= sup$κ$.Let K be the supremum of the norm of projections onto the jk j,k submodules Kζ . Then we can write z= λ ζ with λ∈ K, satisfying sup|λ |≤ K$z$. j j j j If L is a normed ﬁeld, denote β(L)= inf{|x|:|x|∈ L,|x| > 1}.Deﬁne α= sup β(K )≥ 1 (it only depends on K). 1/s Hence for any element λ∈ K, there exist μ ,...,μ∈ K with sup|μ |≤|λ| α , 1 s k 0 s s 1/s such that λ= μ . We apply this to write λ= μ ,with|μ |≤|λ| α . k j jk jk j 0 k=1 k=1 Now identify the Lie algebra and the group through the exponential map. By Lemma 5.B.4,wehave λ ζ= ((μ κ ,...,μ κ )). j j j1 j1 js js We have 1/s 1/s 1/s 1/s 1/s |μ κ |≤ α C K$z$≤ α C CK max K$x$, K$x$ jk jk 0 0 and x= y ... y ((μ κ ,...,μ κ )), 1 d j1 j1 js js which is a bounded number of terms in U ,eachwithnorm 1/s ≤ max A$x$, B$x$ , 1/s 1/s where A= K max(α C (CK ) , 1) and B= α C (CK K ) . 0 0 Proof of Theorem 6.B.2. — We can identify U with its Lie algebra u through the exponential map. Use an embedding of U into matrices over K to deﬁne a matrix norm $·$ on U, which is submutiplicative. Since both the embedding of U into its image and GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 137 its inverse are polynomial maps, there is a polynomial control of the two norms $·$ and $·$ with respect to each other, say$u$≤ max(A$u$ , B) and$u$≤ max(A$u$ , B ), with A, A≥ 1, B, B≥ 0. n n n Let us now prove the existence of μ > 1 such that the inclusion S⊂ U[μ]T n n holds for all n. Any element x in S can be rewritten as a product ua,where a∈ T and −1 n u is a product of at most n elements, each of which has the form tat ,where t∈ T and a∈ U∩ S. If C≥ 1 is an upper bound for$a$ when a∈ U∩ Sand λ≥ 2 is an upper 1 1 bound for the Lipschitz constant of elements of T viewed as operators on u,weobtain −1 n−1 k nk $tat $≤ C λ .Hence$tat$≤ max(A C λ , B ). Since $·$ is submultiplicative, it 1 1 1 k nk k k n follows that$u$≤ max(nA C λ , n). Deﬁning λ= 2A C λ ,weobtain$u$≤ λ for 1 1 1 1 2 k kk nkk k all n. In turn, we deduce$x$≤ max(An A C c , An , B). Thus there exists μ and n 2 0 n n such that for all n≥ n and all x= ua∈ S we have$u$≤ μ ; enlarging μ if necessary 0 2 n n n we can ensure n= 0. Hence S⊂ U[μ]T holds for all n. The U generate U modulo[U, U] (because u ⊂[u, u] and u is contained in one i 0 α of the u for every nonzero weight α). Hence we can apply Lemma 6.B.3: there exists c≥ 1and a κ -tuple (q ,..., q ) ∈{1,...,ν} such that every x∈ U can be written as 3 1 κ x with x∈ U with sup$x $≤ e max(1,$x$). Then there exists c such that for q 4 =1 every i and every y∈ U , there exists t∈ Tand m∈ N such that m≤ c max(1, log$y$) i 4 (c max(1,log$y$)) m−m and t yt∈ S .Inother words, forevery i and y∈ U ,wehave y∈ S . i i In particular, setting c= c c 5 3 4 (c max(1,log$x$)) (c[c+max(0,log$x$)]) (c max(1,log$x$)) 4 4 3 5 x∈ S⊂ S⊂ S; q q q thus if we set c= c c we have 5 3 4 (c max(1,log$x$)) x∈ S Hence, for every r > 0we have r (c max(1,r)) κ c max(1,r) 5 5 U e⊂ S⊂ S; =1 −1 in particular, if C= κ c and μ= e , n max(n,C) U μ⊂ S n n n−C n−C n Thus U[μ ]⊂ S for all n≥ C. In particular, U[μ ]⊂ S⊂ S for all n≥ 0. Multi- −C n−C n plying the norm by μ redeﬁnes U[μ] to be U[μ] and hence with this new norm n n we have U[μ ]⊂ S for all n. For the second inclusion, let us go back to the inclusion of U[e], which yields (for all n≥ 0), with λ= c log μ > 0 5 2 n (max(c ,λn)) U μ⊂ S; 2 q =1 as in the previous case, multiplying the norm allows to assume c= 0. 5 138 YVES CORNULIER, ROMAIN TESSERA 6.C. Application of Gromov’s trick to standard solvable groups. — As above, let U ,..., U 1 ν be an admissible family of tame subgroups in U. If c is a positive integer and ℘ is a c-tuple of elements in{1,...,ν}, we can consider the product U= U . ℘ ℘ =1 Then we can deﬁne, for r ≤∞, a set of null-homotopic words (r) U (r)= w= w ...w| π(w)= 1,∀, w∈ S⊂ F . ℘ 1 c S If R is a subset of the kernel of F→ G consisting of words of bounded length, deﬁne δ (n)= sup area (w) ∈[0,∞]. ℘,S,R S,R w∈U (n) (The area can a priori take inﬁnite values, since we do not assume that R normally generates the kernel.) The following theorem is an essential reduction in the study of the Dehn function of standard solvable groups. Theorem 6.C.1. —Let G= U A be a standard solvable group. Fix an admissible family U ,..., U of tame subgroups and generating subsets as in Section 6.B;alsoﬁxasubset R of the kernel 1 ν of π: F→ G. Let f be a function such that r → f (r)/r is non-decreasing for some α> 1.Suppose that for every c and every c-tuple ℘ ∈{1,...,ν} we have δ (n) f (n). Then G is compactly ℘,S,R presented byS| R and has Dehn function f (n). Remark 6.C.2. — The proof actually even shows that the following holds, G, S and R being ﬁxed as well α> 1: There exists c and a c-tuple ℘ satisfying: if for some f we have δ (n) f (n),then G is compactly presented byS| R and has Dehn function f (n). ℘,S,R However, we will only use the result in the slightly weaker form given in Theorem 6.C.1. (λn) (λn) n Proof.—Denote W= S ... S T ,with λ, k as given by Theorem 6.B.2 (we use q q 1 κ the notation of 6.B.1). Hence π(W ) contains the n-ball in G; note that W is contained n n in the[(2κλ+ 1)n+ κ]-ball of F , which itself is contained in the κ n-ball, where κ= 2κλ+ κ+ 1. A simple observation is that (λn) (λn) (λn+n) (λn+n) (λn+2n) (λn+2n) 3n W W W⊂ S ... S S ... S S ... S T; n n n q q q q q q κ κ κ 1 1 1 in particular, setting λ= λ+ 2, and deﬁning ℘= q for 1≤ ≤ κ , p ∈{0, 1, 2},we +pκ have (λ n) (λ n) W W W∩ Ker(π)⊂ S ... S∩ Ker(π)= U λ n . n n n ℘ ℘ ℘ 1 3κ Thus the assumption implies that words in W W W∩ Ker(π) have an area n n n f (n).ByTheorem 2.E.3, it follows thatS| R is a presentation of G and has a Dehn function f (n). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 139 6.D. Dehn function of stably 2-tame groups. — The following is a reformulation of Theorem D from the introduction. Theorem 6.D.1. —Let G= U A be a stably 2-tame standard solvable group (see Def- inition 4.C.1). Then G has a linear or quadratic Dehn function (linear precisely when A has rank one). Remark 6.D.2. —Theorem 6.D.1 is actually a corollary of Theorem 10.E.1,be- causefor astably2-tamegroup,wehave H (u)= 0and Kill(u)= 0 (because both are 2 0 0 subquotients of (u⊗ u) which is itself trivial if u is stably 2-tame). The point is that The- orem 10.E.1 is considerably more difﬁcult, since it relies on the work of Section 7.B and the algebraic work of Section 8 and Section 9. Proof of Theorem 6.D.1. — If A has rank one, then G is tame and thus hyperbolic, see Remark 6.A.9, and thus has a linear Dehn function. Otherwise, A being a retract of G, the Dehn function has a quadratic lower bound. Now let us prove the quadratic upper bound. We prove the quadratic upper bound by arguing by induction on the length of u as a K-module. If ≤ 2, then G is tame, hence has an at most quadratic Dehn function. Assume now that ≥ 2 and the result is proved for lesser . In order to apply Theorem 6.C.1, we ﬁrst need to choose a suitable admissible family of tame subgroups. Let P be the (ﬁnite) set of principal weights, that is, the set of weights of u/[u, u]. Given a principal weight α;deﬁne u= u ; this is a graded [α] tα t>0 Lie algebra (beware that u need not be a Lie subalgebra). Let α ,...,α be representa- α 1 μ tives of the principal weights modulo positive collinearity; write U= exp(u ).Then [α][α] i i we choose U ,..., U as admissible family of tame subgroups, and we let S ,... S [α][α] 1 ν 1 μ be ﬁnite generating subsets of the U ’s satisfying the conditions of Section 6.B.Welet [α] (n) (n) (β n) β≥ 1 be an integer such that for all n≥ 1, S S⊂ S (the existence of β follows 1 1 1 from the second part of Theorem 6.B.2 applied to U A, for the admissible fam- [α] ily{U}). [α] Let v be the sum of[u, u] and all u when β ranges over all weights not positively collinear to α ;then v is a graded ideal. Note that u= u+ v.Deﬁne V= exp(v). 1[α] Let V ,..., V be the standard tame subgroups of V and denote by acompact 1 ν i subset of V (with all the previous requirements we made for S ). i i Fix k≥ 1, ℘ ∈{1,...,μ} . For positive integers n ,..., n ,consideranelement 1 k (n ) (n ) 1 k w in F belonging to S ... S , we do the following: if there are any two consecutive ℘ ℘ 1 k (n) (m) (β max(m,n)) occurrences of S , we glue them using the inclusion S S⊂ S in G. The cost 1 1 1 of this is at most quadratic since it lies in the tame subgroup U A. If not, we con- [α] sider i maximal such that ℘= 1. If i= 1or i does not exist, say we are done. Otherwise (n ) (n ) (n ) j (n ) (n ) j j (n ) i i i j= ℘ = 1. We ﬁrst observe that S S is contained in S S ((S , S )) (inclusion i−1 j 1 j 1 j 1 140 YVES CORNULIER, ROMAIN TESSERA in F ). Then α and α belong to a single standard tame subgroup, and hence the com- S 1 j mutator belongs to a single standard tame subgroup of V A. So the commutator can be rewritten in V A, thanks to the second part of Theorem 6.B.2,asanelement of (c(n+n )) (c(n+n )) 1 2 1 2 ... ,where c and κ only depend on G. We note that doing this, we q q 1 κ make some generators in appear, but they stay at the right of any occurrence of S . i 1 Therefore we can reiterate some bounded number of times (say≤ k ) until we obtain a (c n) word w w with|w |≤ c n (n= n ), w∈ S ,and w awordin S∪ ∪ T. In 1 2 2 i 1 2 i i 1 i≥2 i particular, assuming that w represents the trivial element in G, w represents an element of V. There exists a standard tame subgroup of U containing U ; then its intersection [α] with V is a standard tame subgroup, say V .So w can be replaced with some element w 1 1 −1 c n of with quadratic cost since the word w w lies in a tame subgroup of G. So from 1 1 1 w we passed with quadratic cost to a word in V of length≤ C|w|. Since by induction V has an at most quadratic Dehn function, we deduce that w has an at most quadratic area with respect to n. Since this works for every ℘ (with constants possibly depending on ℘ ), we can apply Theorem 6.C.1 to conclude that G has an at most quadratic Dehn function. 6.E. Generalized tame groups. Deﬁnition 6.E.1. — A locally compact group is generalized tame if it has a semidirect product decomposition U N where some element c of N acts on U as a compaction, and N is nilpotent and compactly generated. Thus, the assumption that N is abelian in tame groups is relaxed to nilpotent. If G= U N is a tame generalized standard solvable group it is tempting to believe that, in a way analogous to Section 6.A, there is a large-scale Lipschitz deformation retraction of G onto N. However, the proof only carries over when the element c of N acting as a compaction of N can be chosen to be central in N. Unfortunately, this can not always be assumed and, in order to get an upper bound on the Dehn function, we need a more complicated approach. Theorem 6.E.2 (The generalized tame case). — Consider a generalized standard solvable group G= U N such that there exists c∈ N acting on U as a compaction. Let δ be the Dehn function of N, and let f be a function such that δ fand r → f (r)/r is non-decreasing for some α> 1. Then the Dehn function δ of G satisﬁes δ δ f . In particular, if f can be chosen to be G N G ≈-equivalent to the Dehn function of N, then δ≈ δ . G N Note that in all examples we are aware of, we can indeed choose f to be≈- equivalent to δ ,suchthat r → f (r)/r is non-decreasing (unless N admits Z as a lattice, in which case G is tame and has a linear Dehn function). In general, we can always ﬁx α−α α> 1 and consider the function r → f (r)= r sup s δ (s). 1≤s≤r GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 141 Remark 6.E.3. —Theorem 6.E.2 has some similarity with a theorem of Varopoulos [Var00, Main theorem, p. 57] concerning connected Lie groups. Namely, for a simply connected Lie group of the form U NwithU, N simply connected nilpotent Lie groups such that N contains an element acting as a contraction on N, he proves that there exists a “polynomially Lipschitz” homotopy from the identity of G to its projection on N. To prove the theorem, we need the following lemma, whose proof is much more complicated than we could expect at ﬁrst sight (see Remark 6.E.6). Lemma 6.E.4. — Relations of the form s t, where s and t belong to S and where w is a word of length nin S have area δ (n). N N Proof. — For the sake of readability, we ﬁrst consider the (easier) case when N is 2-nilpotent. Denote j= in,where i is an integer to be determined latter in the proof, but that will only depend on G and S (hence is to be considered as bounded). Up to conjugating by a power of c, it is enough to evaluate the area of the relation j j wc c s t . Since conjugation by c is a contraction (see Deﬁnition 4.B.4), it turns out that j j c−1 c t= u∈ S . It is straightforward to check that the relation t u has area j.Hence we wc are left to consider the relation s u. n−1−n n Denoting y= ((w, c ))= w c wc ,wehave j j j j i (6.E.5) wc= c w((w, c ))= c y . j−i Moreover the area of the relation ((w, c ))y is controlled by the Dehn function of N, so we are reduced to compute the area of wy s u. Denote N the Zariski closure of the range of N in Aut(U). The algebraic group N a a decomposes as a direct product AV where A (resp. V) is semisimple (resp. unipotent). Let us write c= c c and w= w w according to this decomposition. Endow the Lie algebra d v d v n n of U with some norm. The crucial observation is that y= ((w, c ))= ((w , c )). It follows that the matrix norm of y (acting on the Lie algebra of U) is at most Cn for some C, D depending only on G and S. Let K > 1 be a constant such that the matrix norm of every subword of wy is at n i k most K .Let z be a preﬁx of wy :itisofthe form ry ,where r is a subword of wy,and D n k≤ i. The matrix norm of z is therefore at most Cin K . Since c acts as a contraction, one can choose i be such that the matrix norm of c −2 j D−n is less than K .Hence thematrixnormof c z is less than Cin K which is bounded by j i some function of i, C, D and K. It follows that for any preﬁx r of c wy has bounded matrix j i norm. Let a for q= 1, 2,... be the sequence of letters of the word c wy ,and let z= q q 142 YVES CORNULIER, ROMAIN TESSERA z−1 a ... a . It follows that the elements s are bounded in U. Now let t be words in S of 1 q U bounded length representing the elements s . We conclude by reducing successively the relations (t ) t whose area are bounded. This solves the case where N is 2-nilpotent. q−1 q i i If N is not assumed to be 2-nilpotent, then the relation ((a, b ))= ((a, b)) does not j n i hold anymore. Therefore we cannot simply replace ((w, c )) by ((w, c )) ,aswedid above. We shall use instead a more complicated formula, namely the one given by Lemma 5.B.2. According to that lemma, one can write ((w, c )) as a product of m iterated commutators ±1±n (or their inverses) in the letters w and c . The rest of the proof is then identical to the i n i 2-nilpotent case, replacing in the previous proof, the power of commutators y= ((w, c )) by this product of (iterated) commutators. Remark 6.E.6. — Lemma 6.E.4 is not as trite as it may look at ﬁrst sight, and fails when N is not nilpotent. For instance, consider the free group F =s, t, and the semidi- rect product U F ,where U R is written as N ={(u )} is written multiplicatively, 2 x x∈R −1 −1 2−n n−1 sxs= x for all x∈ Nand[t, N]= 1. Write m= s ts .Then m v m v can be shown n n 1 n 1 to have an exponential area. Lemma 6.E.7. — Consider a semidirect product of groups G= U N;let c beanelement −1−n n of N and a symmetric subset of U such that ⊂ c cand c c= U.Let T be a n≥0 −1−1 symmetric generating subset of N containing{1, c}, such that tt⊂ c c. Then S= ∪ T t∈T n−2n 2n n 5n+1 generates G and S⊂ (c c )T⊂ S . Proof. — It is immediate that S generates G. By an immediate induction, we have 2−k k n−! log n"! log n" 2 2 ⊂ c c . In particular, ⊂ c c . −1 Consider a word w of length n in S. Then it can be rewritten as ( t ω t )t, i i i=1 i where each t and t belong to T and each ω belong to . Thus, in G, we have i i n−n n n−n n n n S⊂ c c T= c c T −n−! log n" n+! log n" n−2n 2n n 5n+1 2 2 ⊂ c c T⊂ c c T⊂ S Proof of Theorem 6.E.2. — Since N is a Lipschitz retract of G, we have δ δ ;let N G us prove that δ f . We pick the 1-element family (U ),where U , as an admissible family of sub- 1 1 groups; we let S be a stable vacuum subset for some element c∈ T acting as a com- paction on U. We ﬁrst bound (assuming c∈ T) the area of relations of length≤ n,ofthe form −n n i i w= ( c ω c )w with ω∈ and w a word in T (necessarily n≤ n). Since w is i i i i=1 a relation in N, with cost≤ f (n), we can replace w with the trivial word. Then after n n−n n−n n−n n−n i i i i conjugation by c , we are reduced to the word c ω c . The element c ω c i i i=1 n−n n−n i i represents an element s∈ , and by Lemma 6.E.4, the cost to pass from c ω c to s i i i is f (n).Hence theareaof w is f (n). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 143 We can apply Lemma 6.E.7 (if necessary, replace c with a large power c of it and ±k replace T with T ∪{c} for its hypotheses to be fulﬁlled. Thanks to the length estimates of Lemma 6.E.7, we can use Gromov’s trick (Theorem 2.E.3) to conclude that the Dehn function of G is f . We also need a generalization of Theorem 6.B.2. We deﬁne an admissible family of subgroups as in 6.B (A being replaced with N), and let T be a compact symmetric gen- erating subset of N. Since the proof of the following theorem is an immediate adaptation of that of Theorem 6.B.2, we omit the proof. Theorem 6.E.8. — Fix a generalized standard solvable group G, an admissible family of tame subgroups and generating subset as above. There exists a constant λ> 0, a positive integer κ and a κ -tuple (q ,..., q ) of elements in{1,...,ν} such that for every n, the n-ball of G is contained in the 1 κ (λn) (λn) n projection π(S ... S T ). q q 1 κ 7. Estimates of areas using algebraic presentations In this section, we provide a new method allowing to obtain upper bounds on the Dehn function of groups with “enough algebraic structure”. The basic idea is to consider a group G given by an “algebraic presentation”, that is, it is isomorphic to the quotient of a free product U (K) of algebraic groups U ∗ i i i=1 over some locally compact ﬁeld K (or, more generally, a ﬁnite product of such ﬁelds), by the subgroup generated by ﬁnitely many “algebraic families of relators”. An algebraic family of relators is an algebraic subvariety of some U × ··· × U ,where (i ,..., i ) is a i i 1 ﬁxed sequence of integers from 1, .., ν , which is interpreted as a set of words of length of the free product. In order to get estimates, we need to suppose that G itself is of the form G(K), and assume that the above presentation holds in a very strong sense. Namely, we need to assume that for every commutative K-algebra A,the group G(A) is the quotient of U (A) by the relators (evaluated in A). ∗ i i=1 Our main result consists in controlling the size and word length of a set of relations that belong to a same algebraic family. More precisely, consider some algebraic family M of relations: namely a subvariety of some U ×· · · × U such that for every K-algebra A, j j 1 k the A-points of this variety are mapped to the neutral element of G . Then there exists an integer N such that every element x= (x ,..., x ) of M can be written as a word of length 1 k at most N in U (A) consisting of a product of conjugates of relators. Moreover the ∗ i i=1 norms of the letters of this word are polynomially controlled by the norms of the x ’s. 7.A. Afﬁne norm on varieties over normed ﬁelds. 7.A.1. Norms. — In this paragraph, let (K, |·|) be a normed ﬁeld. The afﬁne d d space A (K)= K is endowed with the sup norm $·$. 144 YVES CORNULIER, ROMAIN TESSERA Let X be an afﬁne K-variety with basepoint x (a ﬁxed element of X(K)). For every d and every pointed closed K-embedding φ: X→ A (pointed means mapping x to 0), we can deﬁne a “length function” on X(K) by (x) =$φ(x)$. The following proposition asserts that up to polynomial distortion, this length is unique. Proposition 7.A.1. —Let φ, ψ be two choices of pointed embeddings. Then for some positive constants C, c > 0, (x)≤ Cmax (x) , 1 ,∀x∈ X(K) ψ φ d d Proof. — If the embeddings are into A and A ,withsay d≤ d , using that the d d trivial embedding A (K)⊂ A is isometric, we can enlarge d to assume that d= d .Now −1 consider the isomorphism η= ψ◦ φ: φ(X)→ ψ(X).Then η extends to a regular map d d d η: A→ A : indeed, ﬁrst view η as a regular map φ(X)→ A , which is the same as the data of d regular maps φ(X)→ A and remind that regular maps on φ(X) are by deﬁnition restrictions of regular maps of the afﬁne space. Then, viewing A embedded 2d d into A as the left factor A ×{0},wecan extend η to a K-automorphism 2d 2d η˜: A→ A (x, y) → η (x)+ y, x Now if c is the total degree of η˜ , there exist a positive constant C such that for all u∈ 2d A (K) we have $˜η(u)$≤ Cmax$u$ , 1; therefore for all x∈ X(K) we have (x) =$ψ(x)$=$η◦ φ(x)$≤ Cmax$φ(x)$, 1 = Cmax (x) , 1 . Note that the analogue for distances is not true: for instance if we take the disjoint union of two lines K ×{0, 1}, setting φ(x, t)= (x, t) (two parallel lines) and ψ(x, t)= (x, tx ) (a line and a parabola), then considering the sequence of pairs of points (n, 0) and (n, 1) we see that$ψ(x)− ψ(y)$ cannot be bounded in terms of$φ(x)− φ(y)$. Example 7.A.2. — Assume that K has characteristic zero. Let U be a unipotent group over K. Given an embedding ψ of U as a Zariski-closed subgroup of SL ,and ﬁxing a norm on the algebra of matrices, we obtain two norms on U: the one given by this embedding, and the one given by the embedding ψ of the Lie algebra u into sl , pulled back to U through the exponential. Each of these two norms on U is polynomially bounded by the other, by Proposition 7.A.1. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 145 Remark 7.A.3. — We can deﬁne the logarithmic length on X(K) associated to φ as L (x)= log 1+ (x) . φ φ As a consequence of Proposition 7.A.1, for any two choices ψ and φ,wehave L L , ψ φ i.e. there exist positive constants c≥ 1and C≥ 0such that −1 c L (x)− C≤ L (x)≤ cL (x)+ C,∀x∈ X(K). φ ψ φ 7.A.2. Ring of polynomial growth functions. — Let K be a normed ﬁeld. Endow K with the supremum norm and consider the algebra P= P (K) of functions from Y× R Y Y≥0 to K that have most polynomial growth with respect to the real variable, uniformly in Y, that is, satisfying ∃c,α > 0, c∈ R,∀t≥ 0,∀y∈ Y,$f (y, t)$≤ ct+ c . Lemma 7.A.4. —Let Y be a set. Then for any K-afﬁne variety X, there are canonical bijections ∼∼ Y Y X K→ X(K) , X P (K)→ P X(K) Y Y Proof. — The ﬁrst equality is formal: if A= K[X], then, Hom being in the category of K-algebras, we have Y Y Y X K= Hom K[X], K= Hom K[X], K= X(K) . Given a K-closed embedding of X into the d -dimensional afﬁne space, it is just given by d Y d Y the function A (K )→ A (K) (f ,..., f ) → y → f (y),..., f (y) . 1 d 1 d Y Y d d Then (X(K ))= X(K) and (A (P (K)))= P (A (K)), and hence Y Y Y d P X(K)= X(K)∩ P A (K) Y Y Y d = X K∩ A P (K)= X P (K) . Y Y 7.A.3. Remark. — All the results from this Section 7.A immediately to the case of X(K) when K is a ﬁnite product of normed ﬁelds (endowed with the sup norm). The only difference lies in the language: K-afﬁne variety should be replaced with: afﬁne scheme of ﬁnite type over K;if K= K , this is just the data of one K -afﬁne variety over K for j j j j=1 each j . 146 YVES CORNULIER, ROMAIN TESSERA 7.B. Area of words of bounded combinatorial length. 7.B.1. The setting: generators (*). — By K we mean a ﬁnite product of normed ﬁelds (in a ﬁrst reading, we can assume it is a single normed ﬁeld). We recall the reader familiar with varieties but not schemes that it is not much here: over a single ﬁeld, “scheme of ﬁnite type” can be thought of as “variety” and “afﬁne group scheme of ﬁnite type” as “linear algebraic group”. Over a ﬁnite product K of ﬁelds, the datum of a scheme is just the datum of one scheme over each of the ﬁelds K . Nevertheless, we stick to the word “scheme” because it is the only rigorous setting in which the assertions we state are precise. Let U ,..., U be afﬁne K-group schemes of ﬁnite type. Write U= U (K). 1 ν i i We need to introduce some “norm function” on each U . For this, as explained in Section 7.A, we ﬁx a closed K-embedding of U into SL , so that the norm, written$u$, i q of any element u of U makes sense (using the operator norm on q× q matrices, where K is endowed with the sup norm). 7.B.2. The setting: relators (**). — We are going to introduce some functors X from the category of commutative (associative unital) K-algebras to the category of sets, de- noted by A → X[A], we use brackets rather than parentheses to emphasize that these objects are possibly not representable by a scheme. We need to consider words of some given length whose letters belong to the disjoint union of the U . To do so, we ﬁx integers 1≤ ω ,...,ω≤ ν . Hence we can consider i 1|ω| |ω| the product U= U .Thus ω ω =1 |ω| U (A)= U (A) ω ω =1 is an obvious way to parameterize the set of words u ... u with u∈ U (A) for all . ω ω ω ω 1|ω| We are going to consider closed subschemes of U , which can then be used to parameterize sets of words. Formally, this is done as follows. Let H[A] be the free product U (A). There is an obvious product map ∗ i i=1 π: U (A)→ H[A],(u ,..., u ) → u ... u . ω 1|ω| 1|ω| Now ﬁx ﬁnitely many closed subschemes R ,..., R⊂ U (they will play the role 1 ξ ω of algebraically parameterized relators). For convenience, write R[A]= R (A) ∪ ··· ∪ R (A) for any K-algebra A (if K were a ﬁeld, it would be representable by a closed subscheme and we could then assume ξ= 1; anyway this is not an issue); clearly the π together deﬁne a map π: R[A]→ H[A]. Deﬁne Q[A] as the quotient of H[A] by the normal subgroup generated by π (R[A]). Informally, Q is a group generated by algebraic generators and algebraic sets of relators. A priori, Q is not representable by a group scheme over K (for instance, if R is empty, Q[A] is the free product H[A]). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 147 7.B.3. A family of relations (***). — Now given an integer c≥ 0and a c-tuple ℘= (℘ ,...,℘ ) of integers in{1,...,ν}, so we have the product 1 c U (A)= U (A) ℘ ℘ i=1 and we deﬁne L[A]= (f ,..., f )∈ U (A): f ... f≡ 1in Q[A] . ℘ 1 c ℘ 1 c (It should rather be denoted L[A]) but since R is ﬁxed we omit it in the notation.) In R,℘ other words, denoting by N[A] the normal subgroup of H[A] generated by R[A],we −1 have L[A]= (π ) (N[A]). We cannot a priori represent L as a K-closed subscheme of U . We obtain results ℘ ℘ in case it turns out to be a closed subscheme, or more generally when it contains a K- closed subscheme. Namely, let M⊂ U be a K-closed subscheme and assume that it is contained in L , in the sense that for every (reduced) commutative K-algebra A we have M(A)⊂ L[A] (equality as subsets of U (A)). Also assume that the unit element of ℘ ℘ U (K) belongs to M(K) (this is no restriction since it belongs to L[K]). ℘ ℘ Deﬁne V[A] as the union of all U (A) in H[A]. The following lemma roughly says that any element x= (x ,..., x )∈ M(K) can 1 c be written as a product of boundedly many conjugates of relators whose coefﬁcients are controlled by polynomials in the size of x. Lemma 7.B.1. —Fix U ,..., U as in (*), ω and R as in (**). Given any ℘ and any 1 ν family M of relations as in (***), there exist positive integers m,μ,α satisfying the following: for all x= (x ,..., x )∈ M(K), there exist 1 c • elements ρ (x)∈ U (K), 1≤ k≤ m, 1≤ ≤|ω|, such that setting k ω |ω| ρ˜ (x)= ρ (x),...,ρ (x)∈ U (K), k k1 k|ω| ω =1 we have ρ˜ (x)∈ R[K] for every k; • elements h (x) in V[K], 1≤ k≤ m, 1≤ ≤ μ, satisfying the inequalities $h (x)$,$ρ (x)$≤ (2 +$x$)− 2,∀k,; k k and such that, setting |ω| μ ρ (x)= π ρ˜ (x)= ρ (x), h (x)= h (x), k K k k k k =1 =1 148 YVES CORNULIER, ROMAIN TESSERA we have the equality in H[K]: −1 (7.B.2) x ... x= h (x)ρ (x)h (x) . 1 c k k k k=1 Remark 7.B.3. —Thatfor every x we can write an equality as in (7.B.2) follows directly from the fact that M(K)⊂ L[K], but this gives such an equality with m,μ de- pending on x, and with an efﬁcient upper bound on the norm of relators and conjugating elements. Lemma 7.B.1 is a uniform version of this fact, and relies on the stronger in- clusion M(A)⊂ L[A] for some well-chosen K-algebra A. While a suitable choice of exp A would be the algebra K (as in the sketch of proof of Section 1.7), we ﬁnd more convenient to work with the algebra P (K) introduced in Section 7.A.2. The difference is unessential, and especially allows us to avoid to argue by contradiction as we did in Section 1.7. Proof of Lemma 7.B.1. — Let us ﬁx an abstract set Y; we will assume that its cardinal is at least equal to that of K. The ring P (K) (Section 7.A.2) can be described as the set of functions Y× R→ K satisfying ≥0 ∃α> 0,∀t≥ 0,∀y∈ Y,$f (y, t)$≤ (2+ t)− 2. For any afﬁne K-scheme X of ﬁnite type (with a given embedding in an afﬁne space), X(K) inherits of a norm and the set P (X(K)) is well-deﬁned. There is an obvious inclusion L[P (K)]⊂ P[L (K)]. It is not necessarily an equality (see Example 7.B.8); ℘ Y Y ℘ however the equality M(P (K))= P (M(K)) holds (see Lemma 7.A.4). Y Y For x∈ U (K),recallthat$x$= max$x$. Now assume that Y is inﬁnite of ℘ 1≤i≤c i cardinal at least equal to that of K. We claim that there exists a function f= (f ,..., f ): (Y× R )→ M(K) 1 c≥0 satisfying (a)$f (y, t)$≤ t for all (y, t)∈ Y× R ; ≥0 (b) for every x= (x ,..., x )∈ M(K), there exists y∈ Ysuch that f (y,$x$)= x. 1 c To construct such an f , consider an arbitrary injective map x → y(x) from U (K) to Y (it exists because of the cardinality assumption on Y); for each x∈ M(K),deﬁne f (y(x),$x$)= x and deﬁne f (y, t)= 1for every (y, t) not of the form (y(x),$x$),where 1 here denotes the unit element (1,..., 1) of U (K), which belongs to M(K) by assump- tion. By construction, f takes values in M(K).By(a), f∈ P (M(K)). Hence f∈ M(P (K))= M(P), where write P= P (K) as a shorthand. So, by the Y Y inclusion M(P)⊂ L (P),wehave f∈ L (P).Hence π (f )= f ... f∈ H[P] belongs to ℘ ℘ 1 c the normal subgroup generated by π (R[P]). This means that there exist P GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 149 • integers m,μ; • ρ∈ U (P),1≤ k≤ m,1≤ ≤|ω|, such that setting ρ˜= (ρ ,...,ρ )∈ k i k k1 k|ω| U (P),wehave ρ˜∈ R[P] for every k; ω k • elements h in V[P],1≤ k≤ m,1≤ ≤ μ, such that, setting |ω| μ ρ= π (ρ˜ )= ρ , h= h , k P k k k k =1 =1 we have, in H[P], the equality −1 (7.B.4) f ... f= h ρ h . 1 c k k k=1 Hence for all y∈ Yand t∈ R , we have the equality in H[K] ≥0 −1 f (y, t)... f (y, t)= h (y, t)ρ (y, t)h (y, t) , 1 c k k k k=1 μ μ where ρ (y, t)= ρ (y, t) and h (y, t)= h (y, t). Since each of the elements k k k k =1 =1 ρ , h belongs to some U (P), there exists α> 0such that for all k, and all y, t,wehave k k i α α $ρ (y, t)$≤ (2+ t)− 2,$h (y, t)$≤ (2+ t)− 2. k k Hence for all x= (x ,..., x )∈ M(K), choosing one y= y(x) satisfying (b) and 1 c t =$x$, and using the shorthands ρ (x)= ρ y(x),$x$ , h (x)= h y(x),$x$ , k k k k μ μ ρ (x)= ρ y(x),$x$= ρ (x), h (x)= h y(x),$x$= h (x), k k k k k k =1 =1 we precisely obtain that the norms of both ρ (x) and h (x) is bounded above by (2+ k k $x$)− 2 and that the equality (7.B.2) holds in H[K]. 7.B.4. Generating subsets and presentations (****). — Keep the previous setting (with U , R given). Assume that we have, in addition, a group H, subgroups H ,..., H gener- i 1 ν ating H, and homomorphisms ψ: U→ H (in the examples we have in mind, all ψ are i i i i given as inclusions). We assume that the resulting homomorphism H[K]→ His trivial on R[K], or equivalently factors through Q[K]. LetS| be a presentation of H . We suppose that for each i and for every i i i x∈ U , the word length of ψ (x) with respect to the generating subset S of H is i i i i log(1 +$x$).For x∈ U , ﬁx a representing word x in S of ψ (x),ofsize log(1 +$x$). i i i 150 YVES CORNULIER, ROMAIN TESSERA Assume in addition the following: • all presentationsS| have Dehn function bounded above by some superad- i i ditive function δ ; • there is a subset R⊂ R[K] and a function δ such that for every r= (r ,..., r )∈ R[K],the area of r ... r with respect to S| ∪ R 1|ω| 1|ω| i i i i |ω| is ﬁnite and≤ δ (max|r| ). 2 S =1 i 7.B.5. The fundamental theorem. Theorem 7.B.5. — In the above setting (*), (**), (****), and given a family of relations as in (***), there exist constants s, s , m > 0 such that for every x= (x ,..., x )∈ M(K), denoting 1 c n= max|x| , the area of x ... x with respect to S| ∪ R is≤ δ (sn)+ mδ (s n). i i S 1 c i i 1 2 i i i Proof.—Fix x as above. As in Lemma 7.B.1, write the equality in H[K] −1 (7.B.6) x ... x= h (x)ρ (x)h (x) 1 c k k k k=1 (with all the additional features of the statement of the lemma). Deﬁning[x]= log(2 +$x$) and similarly[u]= log(2 +$u$) for any i and u∈ U (K), the upper bounds on$h (x)$ and$ρ (x)$ provided by the lemma can be i k k rewritten as h (x) , ρ (x)≤ α[x]. k k |ω| μ Deﬁne S= S .Write ρˆ (x)= ρ (x) and h (x)= h (x).Deﬁne the i k k k k =1 =1 word in the free group over S −1−1 ˆˆ w= (x ... x ) h (x)ρˆ (x)h (x) 1 c k k k k=1 Hence, by (7.B.6), the image of w in H[K] is trivial. Denoting by |·| the word length in H[K] with respect to S. By assumption, there −1 exist γ,ζ≥ 1such that for every i and u∈ U (K),wehave ζ[u]≤|u|≤ γ[u];in i S particular|u|≤ γ[u]. Also deﬁne n= max|x| .Hence i S i=1 −1−1 n≥ ζ max[x ]≥ ζ[x] and c m|ω| μ |w|≤|x |+|ρ (x)|+ 2|h (x)| i k k i=1 k=1 =1 =1 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 151 c m|ω| μ ≤ γ[x ]+ ρ (x)+ 2 h (x) i k k i=1 k=1 =1 =1 ≤ γ c+ mα(|ω|+ 2μ)[x]≤ sn, where s= ζγ (c+ mα(|ω|+ 2μ)). By Lemma 2.D.1 (Dehn function of free products of presentations), the area of w with respect to S| is≤ δ (sn). i i 1 We have, by deﬁnition of w: −1−1 ˆˆ x ... x= h (x)ρˆ (x)h (x) w; 1 c k k k k=1 the area of ρˆ (x) with respect to the presentation P = S| ∪ R is k i i ! " ! " |ω||ω| ≤ δ max|ρ (x)|≤ δ max γ ρ (x)≤ δ γα[x]≤ δ (γ αζ n) 2 k S 2 k 2 2 =1 =1 −1 ˆˆ so the area of h(x)ρˆ (x)h (x) with respect to the presentation P is k k k=1 ≤ mδ (γ αζ n). Therefore the area of x ... x with respect to P is 1 c ≤ δ (sn)+ mδ (γ αζ n). 1 2 7.B.6. Restatement of the theorem involving a group word. — Let F be the free group on the generators t ,..., t . We say that w∈ F is essential if all t appear in the reduced form 1 c c i of w (possibly with a negative exponent). We call the word t ... t , which is essential, the 1 c tautological word; denote it by τ . Let w be an essential word in F ,oflength c (necessarily c≥ c if w is essential). Let ℘ be a c -tuple of elements of{1,...,ν}. −1 Write w as a reduced word, namely w= w ...w with w ∈{t , t , 1≤ j≤ c};if 1 c i j ±1 w= t , we write W(i)= j and w[i]=±1. Let us deﬁne a c-tuple ℘= w• ℘ of elements in{1,...,ν}:for 1≤ i≤ c, write −1 ℘= j if w ∈{t , t}. For instance τ• ℘= ℘ ; another less trivial example is i i j −2−1 w(t , t , t )= t t t t ,℘= (5, 6, 3)⇒ w•℘= (6, 5, 5, 3, 6). 1 2 3 2 3 1 2 Deﬁne (w• U )(A) as w[k] w[] −1 (u ,..., u )∈ U (A) :∀j∈ 1,..., c ,∀k,∈ W{j} , u= u . 1 c ℘ Thus w• U is a closed subscheme of U . In the above example, we have ℘ ℘ −1 (w• U )(A)= u , u , u , u , u| (u , u , u )∈ (U× U× U )(A) . ℘ 6 5 5 3 3 5 6 3 5 6 6 152 YVES CORNULIER, ROMAIN TESSERA When w is essential, we have an isomorphism φ of schemes U → w• U ,de- w ℘ ℘ ﬁned on U (A) by w[1] w[c] φ (u ,..., u )= u ,..., u w 1 c W(1) W(c) Note that φ is the identity. If M is a closed subscheme of U ,wedeﬁne w• M= φ (M); τ ℘ w this is a closed subscheme of w• U . The following theorem seems to be a generalization of Theorem 7.B.5 (namely when w= t ... t ), but is actually a simple consequence. It will be useful to refer to it. 1 c Let ℘ be a c-tuple of elements in{1,...,ν}.If w∈ F ,denoteby L[A] the set of (x ,..., x )∈ U (A) such that w(x ,..., x ) equals 1 in Q[A]. 1 c ℘ 1 c Theorem 7.B.7. — In the above setting (*), (**), (****), let c be a positive integer and ℘ a c -tuple of elements of{1,...,ν}.Let M be a closed subscheme of U and let w be an essential word in F , such that M (A)⊂ L[A] for all A. Then there exist constants s, s , m > 0 such that for every x= (x ,..., x )∈ M (K),de- 1 c noting n= max|x| , the area of w(x ,..., x ) with respect to S| ∪ R is≤ i i S 1 c i i δ (sn)+ mδ (s n). 1 2 Proof.—Deﬁne ℘= w• ℘ and M= w• M as above. Thus π (M(A)) is precisely the set elements of the form w(x ,..., x )∈ H[A] with 1 c (x ,..., x )∈ M (A). In particular, M(A)⊂ L[A].Hence M satisﬁes the assumption of 1 c (***) and Theorem 7.B.5 can be applied. Y Y Example 7.B.8. — Here is an example in which the inclusion L[K ]⊂ L[K] is ℘ ℘ strict. We choose K= R, ν= 1, and U (A)= A for every A (so U is the 1-dimensional 1 1 unipotent group). We choose R= R to be the singleton{1} (formally, R(A) ={1}⊂ A). Hence Q(A) is equal to A/Z1 ,and for ℘ ={1},wehave L[A]= Z1 . In particular, A ℘ A Y Y the inclusion L[K ]⊂ L[K] is proper as soon as Y contains two distinct elements. ℘ ℘ Anyway this example is not really serious because assuming that R is a subscheme con- Y Y taining 0 (which could easily be arranged), we would have L[K ]= L[K] for every ℘ ℘ ﬁnite Y, and the serious issue is when Y is inﬁnite. Indeed, now deﬁning R= R as equal to{0, 1} (namely R= Spec(R[t]/(t− t))), we have R(A) equal to the set of idempotents in A, and hence L[A] is the additive subgroup of A generated by idempotents. In par- Y Y ticular, L[R] is equal to the set of bounded functions Y→ Z, while L[R] is equal to ℘ ℘ the set of all functions Y→ Z, which differs from the former if Y is inﬁnite. The inclu- sion L[P (R)]⊂ P (L (R)) is also proper, since then L[P (R)] is equal to the set of ℘ Y Y ℘ ℘ Y bounded functions Y→ Z, while P (L (R)) is equal to the set of all functions Y→ Z of Y ℘ at most polynomial growth, uniformly in Y. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 153 8. Central extensions of graded Lie algebras This section contains results on central extensions of graded Lie algebras, which will be needed in Section 9.Let Q⊂ K be ﬁelds of characteristic zero (for instance, Q= Q and K is a nondiscrete locally compact ﬁeld). To any graded Lie algebra, we associate a central extension in degree zero, which we call its “blow-up”, whose study will be needed in Section 9. We are then led to following problem: given a Lie algebra g over K, we need to compare the homologies H (g) and H (g) of g viewed as a Lie algebra over K and as a Lie algebra over Q by restriction of scalars. When g is deﬁned over Q, i.e. g= K⊗ l, this problem has been tackled in several papers [KL82, NW08]. Here most of the work is carried out over an arbitrary commutative ring; this generality will be needed as we need to apply the results over suitable rings of functions. 8.A. Basic conventions. — The following conventions will be used throughout this chapter. The letter R denotes an arbitrary commutative ring (associative with unit). Un- less explicitly stated, modules, Lie algebras are over the ring R and are not assumed to be ﬁnitely generated. The reader is advised not to read this part linearly but rather refer to it when necessary. Gradings. — We ﬁx an abelian group W , called the weight space. By graded mod- ule, we mean an R-module V endowed with a grading,namelyan R-module decomposi- tion as a direct sum V= V . α∈W Elements of V are called homogeneous elements of weight α.An R-module homomor- phism f: V→ W between graded R-modules is graded if f (V )⊂ W for all α.If V is α α a graded module and V is a subspace, it is a graded submodule if it is generated by homo- geneous elements, in which case it is naturally graded and so is the quotient V/V .By the weights of V we generally mean the subset W⊂ W consisting of α∈ W such that V ={0}.Weuse thenotation V= V . α =0 By graded Lie algebra we mean a Lie algebra g endowed with an R-module grading g= g such that[g , g ]⊂ g for all α, β∈ W . α α β α+β Tensor products. — If V, W are modules, the tensor product V⊗ W= V⊗ Wis deﬁned in the usual way. The symmetric product V V is obtained by modding out by the R-linear span of all v⊗ w− w⊗ v and the exterior product V∧ V is obtained by 154 YVES CORNULIER, ROMAIN TESSERA modding out by the R-linear span of all v⊗ w+ w⊗ v (or equivalently all v⊗ v if 2 is invertible in R). More generally the nth exterior product V ∧· · · ∧ V is obtained by modding out the nth tensor product V⊗ ...⊗ Vby all tensors v ⊗· · · ⊗ v+ w ⊗ ··· ⊗ 1 n 1 w , whenever for some 1≤ i = j≤ n,wehave w= v , w= v and w= v for all k = i, j . n i j j i k k If W , W are submodules of V, we will sometimes denote by W∧ W (resp. W 1 2 1 2 1 W ) the image of W⊗ W in V∧ V (resp. V V). In case W= W= W, the latter map 2 1 2 1 2 factors through a module homomorphism W∧ W→ V∧ V (resp. W W→ V V), and this convention is consistent when this homomorphism is injective, for instance when W is a direct factor of V. If V, W are graded then V⊗ W is also graded by (V⊗ W)= V⊗ W . α β γ {(β,γ ):β+γ=α} When V= W, we see that V∧Vand V V are quotients of V⊗ V by graded submodules and are therefore naturally graded; for instance if W has no 2-torsion (V∧ V)= (V∧ V )⊕ V⊗ V . 0 0 0 α−α α∈(W−{0})/± Homology of Lie algebras. — Let g be a Lie algebra (always over the commutative ring R). We consider the complex of R-modules d d d 4 3 2 1 ··· g∧ g∧ g∧ g −→ g∧ g∧ g −→ g∧ g −→ g −→ 0 given by d (x , x ) =−[x , x] 2 1 2 1 2 d (x , x , x )= x ∧[x , x ]+ x ∧[x , x ]+ x ∧[x , x] 3 1 2 3 1 2 3 2 3 1 3 1 2 and more generally the boundary map i+j d (x ,..., x )= (−1)[x , x ]∧ x ∧ ··· ∧# x ∧· · · ∧# x ∧ ··· ∧ x; n 1 n i j 1 i j n 1≤i≤j≤n and deﬁne the second homology group H (g)= Z (g)/B (g),where Z (g)= Ker(d ) is 2 2 2 2 2 the set of 2-cycles and B (g)= Im(d ) is the set of 2-boundaries. (We will focus on d 2 3 i for i≤ 3 although the map d will play a minor computational role in the sequel. This is of course part of the more general deﬁnition of the nth homology module H (g)= Ker(d )/Im(d ), which we will not consider.) If A→ R is a homomorphism of n n n+1 commutative rings, then g is a Lie A-algebra by restriction of scalars, and its 2-homology as a Lie A-algebra is denoted by H (g).If g is a graded Lie algebra, then the maps d are graded as well, so H (g) is naturally a graded R-module. 2 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 155 Iterated brackets. — In a Lie algebra, we deﬁne the n-fold bracket as the usual bracket for n= 2 and by induction for n≥ 3as [x ,..., x ]= x ,[x ,..., x] . 1 n 1 2 n Central series and nilpotency. — Deﬁne the lower central series of the Lie algebra g by 1 i+1 i s+1 g= g and g =[g, g] for i≥ 1. We say that g is s-nilpotent if g ={0}. 8.A.1. The Hopf bracket. — Consider a central extension of Lie algebras 0→ z→ g→ h→ 0. Since z is central, the bracket g∧ g→ g factors through an R-module homomorphism B: h∧ h→ g, called the Hopf bracket. It is unique for the property that B(p(x)∧ p(y))= [x, y] for all x, y∈ g (uniqueness immediately follows from surjectivity of g∧ g→ h∧ h). Lemma 8.A.1. —For all x, y, z, t∈ h we have B([x, y]∧[z, t]) =[B(x∧ y), B(z∧ t)]. Proof. — Observe that if x¯, y¯ are lifts of x and y then B(x∧ y) =[x ¯, y¯],and that [¯ x, y¯] is a lift of[x, y]. In view of this, observe that both terms are equal to[[x¯, y¯], [¯ z, t]]. 8.A.2. 1-tameness. Deﬁnition 8.A.2. — We say that a Lie algebra g is 1-tame if it is generated by g= g . α =0 Lemma 8.A.3. —Let g be a graded Lie algebra. Then g is 1-tame if and only if we have g=[g , g],where β ranges over nonzero weights. 0 β−β Proof. — One direction is trivial. Conversely, if g is 1-tame, then g is generated as an abelian group by elements of the form x =[x ,..., x] with k≥ 2and x homogeneous 1 k i of nonzero weight. So x =[x , y] with y =[x ,..., x ]∈ g and x∈ g . 1 2 k 1 Lemma 8.A.4. —Let g be a graded Lie algebra. Then the ideal generated by g coincides with the Lie subalgebra generated by g and in particular is 1-tame. Proof.—Let h be the subalgebra generated by g ; it is enough to check that h is an ideal, and it is thus enough to check that[g , h]⊂ h.Set h= g and h =[g , h], 0 1 d d−1 so that h= h . It is therefore enough to check that[g , h ]⊂ h for all d≥ 1. This d 0 d d d≥1 is done by induction. The case d= 1is clear. If d≥ 2, x∈ g , y∈ g , z∈ h , then using 0 d−1 the induction hypothesis x,[y, z]= y,[x, z]−[y, x], z ∈[g , h ]⊂ h d−1 d and we are done. 156 YVES CORNULIER, ROMAIN TESSERA We use this to obtain the following result, which will be used in Section 11. Lemma 8.A.5. —Let g be a graded Lie algebra with lower central series (g ), and assume s+1 that g is s-nilpotent. Then g is contained in the subalgebra generated by g . In particular, if g is 0 0 ∞ i∞ nilpotent and g= g , then g is contained in the subalgebra generated by g . s+1 Proof. — By Lemma 8.A.4, it is enough to check that g is contained in the ideal s+1 s+1 j generated by g . It is sufﬁcient to show that g∩ g⊂ j.Eachelement x of g∩ g 0 0 can be written as a sum of nonzero (s+ 1)-fold brackets of homogeneous elements. Each of those brackets involves at least one element of nonzero degree, since otherwise all its entries would be contained in g and it would vanish. So x belongs to the ideal generated by g . 8.B. The blow-up. Deﬁnition 8.B.1. —Let g be an arbitrary graded Lie R-algebra (the grading being in the abelian group W ). Deﬁne the blow-up graded algebra g˜ as follows. As a graded R-module, g˜= g α α for all α = 0,and g˜= (g∧ g) /d (g∧ g∧ g) . 0 0 3 0 Deﬁne a graded R-module homomorphism τ: g˜→ g by τ(x)= xif x∈ g˜ and τ(x∧ y)= [x, y] if x∧ y∈ g˜ (which of course factors through 2-boundaries). Let us deﬁne the Lie algebra structure[·,·] on g˜ . Suppose that x∈ g˜ , y∈ g˜ . α β • if α+ β = 0,deﬁne[x, y] =[τ(x), τ (y)]; • if α+ β= 0,deﬁne[x, y]= τ(x)∧ τ(y). Lemma 8.B.2. — With the above bracket, g˜ is a Lie algebra and τ is a Lie algebra homomor- phism, whose kernel is central and naturally isomorphic to H (g) . Its image is the ideal g +[g, g] 2 0 of g. Proof.—Letusﬁrstcheck that τ is a homomorphism (of non-associative algebras). Let x, y∈ g˜ have weight α and β . In each case, we apply the deﬁnition of[·,·] and then of τ .If α+ β = 0 τ[x, y]= τ τ(x), τ (y)= τ(x), τ (y); if α+ β= 0then τ[x, y]= τ τ(x)∧ τ(y)= τ(x), τ (y) . By linearity, we deduce that τ is a homomorphism. Since τ is an isomorphism for α = 0 and τ =−d , the kernel of τ is equal by deﬁnition to H (g) . Moreover, by deﬁnition 0 2 2 0 the bracket[x, y] only depends on τ(x)⊗ τ(y), and it immediately follows that Ker(τ ) is central in g˜ . GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 157 Let us check that the bracket is a Lie algebra bracket; the antisymmetry being clear, we have to check the Jacobi identity. Take x∈ g˜ , y∈ g˜ , z∈ g˜ . From the deﬁnition α β γ above, we obtain • if α+ β+ γ = 0,[x,[y, z]] =[τ(x),[τ(y), τ (z)]]; • if α+ β+ γ= 0,[x,[y, z]]= τ(x) ∧[τ(y), τ (z)] (the careful veriﬁcation involves discussing on whether or not β+ γ is zero). Therefore, the Jacobi identity for (x, y, z) immediately follows from that of g in the ﬁrst case, and from the fact we killed 2-boundaries in the second case. We have τ(g˜ )= g and τ(g˜ ) =[g, g] . Therefore the image of τ is equal to 0 0 g +[g, g]. The latter is an ideal, since it contains[g, g]. Deﬁnition 8.B.3. — We say that a graded Lie algebra g is relatively perfect in degree zero if it satisﬁes one of the following (obviously) equivalent deﬁnitions (a) 0isnot aweightof g/[g, g]; (b) g ⊂[g, g]; (c) g= g +[g, g]; (d) g˜→ g is surjective (in view of Lemma 8.B.2). (e) g is generated by g +[g , g]; 0 0 Note that if g is 1-tame, then it is relatively perfect in degree zero, but the converse is not true, as shows the example of a nontrivial perfect Lie algebra with grading concen- trated in degree zero. The interest of this notion is that it is satisﬁed by a wealth of graded Lie algebras that are very far from perfect (e.g., nilpotent). Theorem 8.B.4. —Let g be a graded Lie algebra. If g is relatively perfect in degree zero (e.g., g is 1-tame), then the blow-up g˜→ g is a graded central extension with kernel in degree zero, and is universal among such central extensions. That is, for every surjective graded Lie algebra homomorphism h→ g with central kernel z= z , there exists a unique graded Lie algebra homomorphism φ: g˜→ h so that the composite map g˜→ h→ g coincides with the natural projection. Proof. — By Lemma 8.B.2, g˜→ g is a central extension with kernel in degree zero. Denote by B: g∧ g→ h the Hopf bracket associated to h→ g (see Section 8.A). Let us show uniqueness in the universal property. Clearly, φ is determined on g . ˜˜˜˜ So we have to check that φ is also determined on[g, g]. Observe that the map g∧ g→ h, x∧ y → φ([x, y]) =[φ(x), φ(y)] factors through a map w: g∧g→ h,so w(τ(x)∧τ(y))= [φ(x), φ(y)] for all x, y∈ g˜ . Since p◦ φ= τ and since h is generated by the image of φ and by its central ideal Ker(p), we deduce that for all x, y∈ h,wehave w(p(x)∧ p(y)) =[x, y]. By the uniqueness property of the Hopf bracket (see Section 8.A), we deduce that w= B . So for all x, y∈ g˜ , φ([x, y]) is uniquely determined as B (τ (x)∧ τ(y)). h 158 YVES CORNULIER, ROMAIN TESSERA −1 Now to prove the existence, deﬁne φ: g˜→ h to be p on g˜ ,and φ(x∧ y)= B (x∧ y) if x∧ y∈ g˜ .Itisclear that φ is a graded module homomorphism and that h 0 p◦ φ= τ . Let us show that φ is a Lie algebra homomorphism, i.e. that the graded module homomorphism σ: g˜∧ g˜→ h, (x∧ y) → φ([x, y] ) −[φ(x), φ(y)] vanishes. Since p◦ φ is a homomorphism and p is bijective, we have (p◦ σ)= 0, so it is enough to check that σ vanishes in degree 0. If x and y have nonzero opposite weights, noting that p◦ φ is the identity on g , −1−1 φ[x, y]= φ(x∧ y)= B (x∧ y)= p (x), p (y)= φ(x), φ(y) . If x∧ y and z∧ t belong to g˜= (g∧ g) , then, using Lemma 8.A.1,wehave 0 0 φ[x∧ y, z∧ t]= φ[x, y]∧[z, t]= B[x, y]∧[z, t] = B (x∧ y), B (z∧ t)= φ(x∧ y), φ(z∧ t) . h h By linearity, we deduce that σ= 0 and therefore φ is a Lie algebra homomorphism. ˜˜ Corollary 8.B.5. —If g is relatively perfect in degree zero then g= g. Proof. — Observe that g˜→ g has kernel z concentrated in degree zero, so it immediately follows that[z, g ]= 0. We need to show that z is central in g˜ . Since ˜˜ g =[g, g]+ g , it is enough to show that[z,[g˜, g˜]]={0}. By the Jacobi identity, ˜˜˜˜˜ [z,[g˜, g˜]]⊂[g˜,[z, g˜]]. Now since g˜→ g has a central kernel,[z, g˜] is contained in the ˜˜˜˜˜ kernel of g˜→ g˜ , which is central in g˜.So[g˜,[z, g˜]]={0}.Thus z is central in g˜.The universal property of g˜ then implies that g˜→ g˜ is an isomorphism. Remark 8.B.6. — There is an immediate generalization of this blow-up construc- tion, where we replace{0} by an arbitrary subset X⊂ W , deﬁning g˜ to be equal to g if α α 2 3 α/∈ X and to ( g/d ( g)) if α∈ X . Then immediate adaptations of Lemma 8.B.2, 3 α Theorem 8.B.4 and its corollary hold, with essentially no change in the proofs. (Deﬁni- tion 8.B.3 is valid replacing g with g= g and g with g ,exceptthatwe 0 X α WX α∈X have to remove (the analogue of) Condition (e) of Deﬁnition 8.B.3. Nevertheless, under the assumption that no weight in X is the sum of a weight X and of a weight in W X , Condition (e) is still equivalent; in particular this works if X is a subgroup of W and in particular for X ={0}. Otherwise, the reader can ﬁnd examples with gradings in{0, 1} and X ={1}, where the ﬁrst four conditions hold but not (e).) Lemma 8.B.7. —Let g be a graded Lie algebra and g its blow-up. If g is 1-tame, then so is g. Proof. — Suppose that g is 1-tame. Then by linearity, it is enough to check that for every x∈ g and u,v of nonzero opposite weights, the element x ∧[u,v] belongs to 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 159 [g˜ , g˜]. This is the case since modulo 2-boundaries, this element is equal to u ∧[x,v]+ v ∧[u, x]∈ (g∧ g ) . Lemma 8.B.8. —Let g be ﬁnitely many graded Lie algebras (all graded in the same abelian group) and g˜ their blow-up. If g= g satisﬁes the assumption that g/[g, g] has no i i opposite weights, then the natural homomorphism g→ g˜ is an isomorphism. Equivalently, i i H ( g )→ H (g ) is an isomorphism. 2 i 0 2 i 0 Proof.—Foreach i, there are homomorphisms g→ g→ g , whose composi- i j i tion is the identity, and hence H (g )→ H ( g )→ H (g ) , whose composition is the 2 i 0 2 j 0 2 i 0 identity again. So we obtain homomorphisms ! " H (g )→ H g→ H (g ) , 2 i 0 2 i 2 i 0 whose composition is the identity. To ﬁnish the proof, we have to check that H (g )→ 2 i 0 H ( g ) is surjective, or equivalently that in Z ( g ) , every element x is the sum of 2 i 0 2 i 0 an element in Z (g ) and a boundary. Now observe that 2 i 0 ! " Z g= Z (g )⊕ (g∧ g ) . 2 i 2 i 0 i j 0 i<j So we have to prove that for i = j , g∧ g consists of boundaries. Given an element x∧ y i j (x∈ g , y∈ g homogeneous), the assumption on g/[g, g] implies that, for instance, y is a i j sum[z ,w] of commutators. Projecting if necessary into g , we can suppose that all k k j z and w belong to g .So k k j x∧ y= x∧[z ,w ]= d (x∧ z∧ w ). k k 3 k k k k 8.C. Homology and restriction of scalars. — We now deal with two commutative rings A, B coming with a ring homomorphism A→ B (we avoid using R as the previous re- sults will be used both with R= A and R= B). If g is a graded Lie algebra over B,itcan then be viewed as a graded Lie algebra over A by restriction of scalars. This affects the deﬁnition of the blow-up. There is an obvious surjective graded Lie algebra homomor- A B phism g˜→ g˜ . The purpose of this part is to describe the kernel of this homomorphism A B (or equivalently of the homomorphism H (g)→ H (g) ), and to characterize, under 0 0 2 2 suitable assumptions, when it is an isomorphism. Our main object of study is the following kernel. A,B Deﬁnition 8.C.1. —If g is a Lie algebra over B,wedeﬁne the welding module W (g) A B as the kernel of the natural homomorphism H (g)→ H (g), or equivalently of the homomorphism 2 2 A,B A B (g∧ g)/B (g)→ (g∧ g)/B (g).If g is graded, it is a graded module as well, and W (g) A B 0 2 2 2 A B then also coincides with the kernel of g˜→ g˜ . 160 YVES CORNULIER, ROMAIN TESSERA The following module will also play an important role. Deﬁnition 8.C.2. —If g is a Lie algebra over B, deﬁne its Killing module Kill(g) (or Kill (g) if the base ring need be speciﬁed) as the cokernel of the homomorphism T: g⊗ g⊗ g→ g g u⊗ v⊗ w → u [v, w]+ v [u,w]. If g is graded, it is graded as well. Note that we can also write T (u⊗ w⊗ v) =[u,v] w− u [v, w],and thus we see that the set of B-linear homomorphisms from Kill(g) to any B-module M is naturally identiﬁed to the set of the so-called invariant bilinear maps g× g→ M. Lemma 8.C.3. —Let v be a B-module. Then the kernel of the natural surjective A-module homomorphism v⊗ v→ v⊗ v is generated, as an abelian group, by elements A B (8.C.4) λx⊗ y− x⊗ λy A A with λ∈ B,x, y∈ v. The same holds with⊗ replaced by or∧. Proof.—Endow v⊗ v with a structure of a B-module, using the structure of B-module of the left-hand v,namely λ(x⊗ y)= (λx⊗ y) if λ∈ B, x, y∈ v. Let W be the subgroup generated by elements of the form (8.C.4); it is clearly an A-submodule, and is actually a B-submodule as well. The natural A-module surjective homomorphism φ: (v⊗ v)/W→ v⊗ v is B-linear. To show it is a bijection, we A B observe that by the universal property of v⊗ v,wehavea B-module homomorphism ψ: v⊗ v→ (v⊗ v)/W mapping x⊗ y to x⊗ y modulo W. Clearly, ψ and φ are B A B A inverse to each other. Let us deal with∧,the case of being similar. The group v∧ v is deﬁned as the quotient of v⊗ v by symmetric tensors (i.e. by the subgroup generated by elements of the form x⊗ y+ y⊗ x), and the group v∧ v is deﬁned the quotient of v⊗ v by B B symmetric tensors. By the case of⊗, this means that v∧ v is the quotient of v⊗ v B A by the subgroup generated by symmetric tensors and elements (8.C.4). This implies that v∧ v is the quotient of v∧ v by the subgroup generated by elements (8.C.4) (with⊗ B A replaced by∧) Proposition 8.C.5. — For any Lie algebra g over B,the A-module homomorphism A,B : B⊗ g⊗ g→ W (g) A A λ⊗ x⊗ y → λx∧ y− x∧ λy A,B is surjective. If g is graded and is 1-tame, then : B⊗ (g⊗ g)→ W (g) is surjective in 0 A A 0 0 restriction to B⊗ (g⊗ g ) . 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 161 Proof.—Thegroup (g∧ g)/B (g) is deﬁned as the quotient of g∧ g by 2- A A boundaries, and (g∧ g)/B (g) is the quotient of g∧ g by 2-boundaries, or equivalently, B B by Lemma 8.C.3, is the quotient of g∧ g by 2-boundaries and elements of the form (8.C.6) λx∧ y− x∧ λy. A A A B It follows that the kernel of (g∧ g)/B (g)→ (g∧ g)/B (g) is generated by elements A B 2 2 of the form (8.C.6), proving the surjectivity of . For the additional statement, deﬁne W= (B⊗ (g⊗ g )) .Wehavetoshow that any element λx∧ y− x∧ λy as in (8.C.6)with x, y of zero weight belongs to W .By A A linearity and Lemma 8.A.3, we can suppose that y =[z,w] with z,w of nonzero opposite weight. So, modulo boundaries, λx ∧[w, z]− x∧ λ[w, z]=−w ∧[z,λx]− z ∧[λx,w]− x∧ λ[w, z] =−w∧ λ[z, x]+ λw ∧[z, x], which belongs to W . Proposition 8.C.7. —Let g be a Lie algebra over B. Then the map A,B : B⊗ g⊗ g→ W (g) A A of Proposition 8.C.5 factors through the natural projection B⊗ g⊗ g→ B⊗ Kill (g); A A A moreover in restriction to B⊗ g⊗[g, g], it factors through B⊗ Kill (g). In particular, if g is A A A graded and relatively perfect in degree 0 (e.g., 1-tame), then factors through B⊗ Kill (g). 0 A Proof. — All⊗,∧, are meant over A. λ λ λ Write (x⊗ y)= (λ⊗ x⊗ y). It is immediate that (y⊗ x)= (x⊗ y) (even before modding out by 2-boundaries), and thus factors through g g.Let us now check that factors through Kill (g). Modulo 2-boundaries: x [y, z]− y [z, x]= λx ∧[y, z]− x∧ λ[y, z] − λy ∧[z, x]+ y∧ λ[z, x] =−z ∧[λx, y]+ z ∧[x,λy]= 0. λ λ For the last statement, by Lemma 8.C.3, we have to show that (μx [y, z])= (x μ[y, z]) for all x, y, z∈ g and μ∈ B. Indeed, using the latter vanishing we get λ λ λ μx [y, z]= y [z,μx]= y [μz, x] = x [y,μz] . 162 YVES CORNULIER, ROMAIN TESSERA In turn, we obtain, as an immediate consequence: Corollary 8.C.8. —Let g be a graded Lie algebra over B; assume that g is relatively perfect in A,B degree 0. If Kill (g) ={0} then W (g) ={0}. 0 0 Proof. — By Propositions 8.C.5 and 8.C.7, induces a surjection B⊗ 0 A A,B Kill(g)→ W (g) . Under the additional assumptions that A= Q is a ﬁeld of characteristic = 2and g is deﬁned over Q, i.e. has the form l⊗ B for some Lie algebra l over Q, Corol- lary 8.C.8 easy follows from [NW08, Theorem 3.4]. We are essentially concerned with ﬁnite-dimensional Lie algebras g over a ﬁeld K of characteristic zero (K playing the role of B)and A= Q, but nevertheless in general we cannot assume that g be deﬁned over Q. We will also use the more speciﬁc application. Corollary 8.C.9. — Assume that A= Q and B= K= K is a ﬁnite product of locally j=1 compact ﬁelds K , each isomorphic to R or some Q .Let B be the closed unit ball in K . Assume that j p K j g is ﬁnite-dimensional over K, that is, g= g with g ﬁnite-dimensional over K . Suppose that g j j j is 1-tame and g/[g, g] has no opposite weights. For every j and weight α,let V be a neighborhood j,α Q,K K of 0 in g= (g ) . Then the welding module W (g)= Ker(H (g)→ H (g) ) is generated j,α j α 2 2 2 0 0 by elements of the form (λ⊗ x⊗ y) with λ∈ B and x∈ V ,y∈ V ,where α ranges over K j,α j,−α nonzero weights and j= 1,...,τ . A,B Proof. — By Proposition 8.C.5,W (g) is generated by elements of the form (λ⊗ x⊗ y ),with λ∈ K and x , y∈ g are homogeneous of nonzero opposite weight. By linearity and Lemma 8.B.8, these elements for which λ , x , y∈ K× g× g j j,α j,−α j α =0 are enough. Given such an element (λ , x , y )∈ K× g× g , using that K= Q+ j j,α j,−α j B , write λ= ε+ λ with ε∈ Q and λ∈ B . Also, since g= QV , write x= μx K K j,±α j,±α and y= γ y with μ, γ∈ Q, x∈ V , y∈ V . Clearly, by the deﬁnition of ,wehave j,α j,−α (ε⊗ x⊗ y )= 0, so ⊗ x⊗ y= (λ⊗ μx⊗ γ y)= μγ (λ⊗ x⊗ y), and (λ⊗ x⊗ y) is of the required form. 8.C.1. Construction of 2-cycles. — We now turn to a partial converse to Corol- lary 8.C.8. Deﬁne HC (B) (or HC (B) if A is implicit) as the quotient of B∧ B by the A- 1 A submodule generated by elements of the form uv∧ w+ vw∧ u+ wu∧ v. This is the ﬁrst GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 163 cyclic homology group of B (which is usually deﬁned in another manner; we refer to [NW08] for the canonical isomorphism between the two). Lemma 8.C.10. — Assume that K is a ﬁeld of characteristic zero and Q a subﬁeld. Suppose that K contains an element t that is transcendental over Q. Assume that either K has characteristic zero, Q Q −1 or K⊂ Q((t)). Then HC (K) ={0}.Moreprecisely,the imageof t∧ t in HC (K) is nonzero. 1 1 Proof. — We denote W by the Q-linear subspace of K∧ K generated by elements of the form uv∧ w+ vw∧ u+ wu∧ v. Let us begin with the case of Q((t)). Deﬁne a Q-bilinear map F: Q((t))→ Qby ! " i j (8.C.11)F x t , y t= kx y . i j k−k k∈Z −1 Observe that the latter sum is ﬁnitely supported. (In Q[t, t], the above map appears in the deﬁnition of the deﬁning 2-cocycle of afﬁne Lie algebras, see [Fu].) We see that F is −1 alternating by a straightforward computation, and that F(t, t )= 1. Setting f (x∧ y)= F(x, y),if x= x t , etc., we have f (xy∧ z)= k(xy) z k−k k∈Z = k x y z i j−k k∈Z i+j=k =− kx y z; i j k i+j+k=0 thus f (xy∧ z+ yz∧ x+ zx∧ y) =− (k+ i+ j)x y z= 0. i j k i+j+k=0 −1 Hence f factors through a Q-linear map from HC (Q((t)))→ Q mapping t∧ t to 1. The proof is thus complete if K⊂ Q((t)). −1 Now assume that K has characteristic zero; let us show that t∧ t has a nontrivial image in HC (K). Since K has characteristic zero, the above deﬁnition (8.C.11) imme- −∞ 1/n−1 diately extends to the ﬁeld Q((t ))= Q((t )) of Puiseux series and hence t∧ t n>0 −∞ has a nontrivial image in HC (Q((t ))) for every ﬁeld Q of characteristic zero. To prove the general result, let (u ) be a transcendence basis of K over Q(t). j j∈J Replacing Q by Q(t: j∈ J) if necessary, we can suppose that K is algebraic over Q(t). −∞ ˆˆ Let Q be an algebraic closure of Q. By the Newton-Puiseux Theorem, Q((t )) is al- gebraically closed. By the Steinitz Theorem, there exists a Q(t)-embedding of K into Q Q −∞ L= Q((t )). This induces a Q-linear homomorphism HC (K)→ HC (L) mapping 1 1 164 YVES CORNULIER, ROMAIN TESSERA Q Q −1−1 the class of t∧ t in HC (K) to the class of t∧ t in HC (L); the latter is mapped in 1 1 Q Q −1−1 turn to the class t∧ t in HC (L), which is nonzero. So the class of t∧ t in HC (K) 1 1 is nonzero. Theorem 8.C.12. —Let g be a Lie algebra over B, which is deﬁned over A, i.e. g B⊗ g A A for some Lie algebra g over A. Consider the homomorphism A A A ϕ: (g∧ g)/B (g)→ M= HC (B)⊗ Kill (g ) A A 2 1 (λ⊗ x)∧ (μ⊗ y) → (λ∧ μ)⊗ (x y). A,B Then ϕ is well-deﬁned and surjective, and ϕ(W (g))= 2M. In particular, if 2 is invertible in A, A,B g is agradedLie algebraand M= HC (B)⊗ Kill(g ) ={0}, then W (g) ={0}. A 0 1 A 0 0 The above map ϕ was considered in [NW08], for similar motivations. Assum- ing that A is a ﬁeld of characteristic zero, the methods in [NW08]can providea more precise description (as the cokernel of an explicit homomorphism) of the kernel A,B A B W (g)= Ker(H (g)→ H (g)). Since we do not need this description and in order 2 2 not to introduce further notation, we do not include it. Proof of Theorem 8.C.12.—Letusﬁrstview ϕ as deﬁned on g∧ g. The surjectivity is A,B trivial. By Proposition 8.C.5 (with grading concentrated in degree 0), we see that W (g) is generated by elements of the form λx∧ μy− μx∧ λy with x, y∈ g (we omit the⊗ signs, which can here be thought of as scalar multiplication); the image by ϕ of such an element is 2(λ∧ μ)(x y), which belongs to 2M. Conversely, since 2M is generated by A,B elements of the form 2(λ∧ μ)(x y), we deduce that ϕ(W (g))= 2M. Let us check that ϕ vanishes on 2-boundaries (so that it is well-deﬁned on g∧ g modulo 2-boundaries): ϕ tx ∧[uy,vz]= (t∧ uv)⊗ x [y, z]; ϕ uy ∧[vz, tx]= (u∧ vt)⊗ y [z, x]= (u∧ vt)⊗ x [y, z]; ϕ vz ∧[tx, uy]= (v∧ tu)⊗ z [x, y]= (v∧ tu)⊗ x [y, z] and since t∧ uv+ u∧ vt+ v∧ tu= 0inHC (A), the sum of these three terms is zero. The last statement clearly follows. 8.C.2. The characterization. — Using all results established in the preceding para- graphs, we obtain Theorem 8.C.13. —Let g be a ﬁnite-dimensional graded Lie algebra over a ﬁeld K of char- acteristic zero, assume that g/[g, g] has no opposite weights. Let Q be a subﬁeld of K, so that K has inﬁnite transcendence degree over Q. We have equivalences GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 165 Q,K Q • W (g) ={0} (i.e., H (g)→ H (g) is an isomorphism); 0 0 0 2 2 2 • Kill (g) ={0}. Corollary 8.C.14. — Under the same assumptions, we have equivalences • H (g) ={0} (i.e. the blow-up g˜→ g is an isomorphism); • H (g)= Kill (g) ={0}. 0 0 The interest is that in both the theorem and the corollary, the ﬁrst condition is a problem of linear algebra in inﬁnite dimension, while the second is linear algebra in ﬁnite dimension (not involving Q) and is therefore directly computable in terms of the structure constants of g. Proof of Theorem 8.C.13. — Suppose that Kill (g)= 0. By Corollary 8.C.8,the induced homomorphism H (g)→ H (g) is bijective. 0 2 0 Conversely, suppose that Kill (g) = 0. Since g is ﬁnite-dimensional over K, there exists a subﬁeld L⊂ K, ﬁnitely generated over Q, such that g is deﬁned over L, i.e. we can K L L write g= g⊗ K. Obviously, Kill (g)= Kill (g )⊗ K, so Kill (g ) = 0. Let (x y) L L L L L L be the representative of a nonzero element in Kill (g ).Let λ be an element of K, tran- −1 scendental over L. By Lemma 8.C.10, the element λ∧ λ has a nontrivial image in HC (K).ByTheorem 8.C.12 (applied with (A, B)= (L, K)), we deduce that −1−1 c= λx∧ λ y− λ x∧ λy L L L is not a 2-boundary, i.e. is nonzero in H (g) . In particular the element c (written as c 0 Q L K L with∧ instead of∧ )isnonzero in H (g) since its image in H (g) is c , while its image Q L 0 L 2 2 c in g∧ g and hence in H (g) is obviously zero. K K 0 8.D. Auxiliary descriptions of H (g) and Kill(g) .— We now prove the result in- 2 0 0 cluding Theorem J as a particular case. In this subsection, all Lie algebras are over a ﬁxed commutative ring R. Deﬁnition 8.D.1. —Let g be a graded Lie algebra. We say that g is doubly 1-tame if for every α we have g=[g , g]. 0 β−β β/∈{0,α,−α} In view of Lemma 8.A.3, doubly 1-tame implies 1-tame, and the reader can easily ﬁnd counterexamples to the converse. This deﬁnition will be motivated in Section 9, because it is a consequence of 2-tameness (Lemma 9.B.1(1)), which will be introduced therein. The purpose of this subsection is to provide descriptions of H (g) and Kill(g) . 2 0 0 166 YVES CORNULIER, ROMAIN TESSERA 8.D.1. The tame 2-homology module. — Let us begin with the trivial observation that if α+ β+ γ= 0and α, β, γ = 0, then α+ β, β+ γ,γ+ α = 0. It follows that d maps (g∧ g∧ g ) into g∧ g .Deﬁne the tame 2-homology module H (g)= Ker(d )∩ (g∧ g ) /d (g∧ g∧ g ) . 0 2 0 3 0 We are going to prove the following result. Theorem 8.D.2. —Let g be a graded Lie algebra. The natural homomorphism H (g)→ H (g) induced by the inclusion (g∧ g )→ (g∧ g) is surjective if g is 1-tame, and is an 2 0 0 0 isomorphism if g is doubly 1-tame. Remark 8.D.3. — In Abels’ second group (Section 1.5.4), (g∧ g) and (g∧ g∧ g) 0 0 have dimension 4 and 5, while (g∧ g ) and (g∧ g∧ g ) have dimension 3 and 2. 0 0 Thus, we see that the computation of H (g) is in practice easier than the computation of H (g) . 2 0 Lemma 8.D.4. —Let g be any graded Lie algebra. If g is 1-tame, then (1) g∧ g⊂ Im(d )+ (g∧ g ); 0 0 3 0 (2) g∧ g∧ g⊂ Im(d )+ g∧ (g∧ g ); 0 0 0 4 0 0 Proof. — Observe that (1) is a restatement of Lemma 8.B.7. The second assertion is similar: if u,v have nonzero opposite weights and x, y∈ g , then, modulo Im(d ), the element x∧ y ∧[u,v] is equal to y∧ u ∧[v, x]−[x, y]∧ u∧ v+ x∧ v ∧[u, y]− y∧ v ∧[u, x]− x∧ u ∧[v, y], which belongs to g∧ (g∧ g ) . 0 0 Proposition 8.D.5. — Consider the following R-module homomorphism : g⊗ g⊗ g⊗ g→ (g∧ g )/d (g∧ g∧ g ) u⊗ v⊗ x⊗ y → x∧ y,[u,v]− y∧ x,[u,v] . If g is doubly 1-tame, then there exists an R-module homomorphism : g⊗ g→ (g∧ g ) /d (g∧ g∧ g ) 0 0 0 3 0 such that whenever α, β are non-collinear weights, we have [x, y]⊗[u,v]= (u⊗v⊗ x⊗ y),∀x∈ g , y∈ g , u∈ g ,v∈ g . α−α β−β Moreover, is antisymmetric, i.e. factors through g∧ g . 0 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 167 Proof. — Suppose that α, β are non-collinear weights and that x∈ g , y∈ g , α−α u∈ g , v∈ g .Wehave d◦ d (x∧ y∧ u∧ v)= 0. If we write this down in (g∧ β−β 3 4 g ) /d (g∧ g∧ g ) , four out of six terms vanish and we get 0 3 0 d[x, y]∧ u∧ v+ d[u,v]∧ x∧ y= 0, 3 3 which expands as (8.D.6) (u⊗ v⊗ x⊗ y)+ (x⊗ y⊗ u⊗ v)= 0. Deﬁne, for w∈ g , (w)= x ∧[y,w]− y ∧[x,w]. The mapping 0 x,y (x, y) → ∈ Hom g ,(g∧ g ) /d (g∧ g∧ g ) x,y 0 0 3 0 is bilinear and in particular extends to a homomorphism σ: (g⊗ g )→ Hom g ,(g∧ g ) /d (g∧ g∧ g ) . 0 0 0 3 0 Since g is doubly 1-tame, any w∈ g can be written as[u ,v] with u∈ g , 0 i i i β v∈ g , β ∈{ / 0,±α}, so, using (8.D.6) i−β i (w)= (u⊗ v⊗ x⊗ y) x,y i i =− (x⊗ y⊗ u⊗ v )= [x, y] . i i u ,v i i i i This shows that σ(x⊗ y) only depends on[x, y], i.e. we can write σ(x⊗ y)= σ ([x, y]). Deﬁne, for z,w∈ g (z⊗ w)= σ (z)(w). By construction, whenever z =[x, y] and w =[u,v],with x∈ g , y∈ g , u∈ g , v∈ g α−α β−β and α, β are not collinear, we have [x, y]⊗[u,v]= (u⊗ v⊗ x⊗ y); from (8.D.6) we see in particular that is antisymmetric. Proof of Theorem 8.D.2.—If g is 1-tame, the surjectivity immediately follows from Lemma 8.D.4(1). Now to show the injectivity of the map of the theorem, suppose that c∈ (g∧ g ) is a 2-boundary and let us show that c belongs to d (g∧ g∧ g ) .Inviewof 0 3 0 Lemma 8.D.4(2), we already know that c belongs to d (g∧ g∧ g ) ,and letuswork 3 0 168 YVES CORNULIER, ROMAIN TESSERA again modulo d (g∧ g∧ g ) , so that we can suppose that c belongs to d (g∧ (g∧ 3 0 3 0 g ) ), and we wish to check that c= 0. Since g is doubly 1-tame, we can write c= d[u ,v ]∧ x∧ y 3 i i i i with x∈ g , y∈ g , u∈ g , v∈ g , α ,β nonzero and α =±β .Write w =[u ,v]. i α i−α i β i−β i i i i i i i i i i i Then c= w ∧[x , y]+ x ∧[y ,w ]+ y ∧[w , x] , i i i i i i i i i i i the ﬁrst term belongs to g∧ g and the second to (g∧ g ) /d (g∧ g∧ g ) ; since c 0 0 0 3 0 is assumed to lie in (g∧ g ) we deduce that (8.D.7) w ∧[x , y ]= 0 i i i in g∧ g . Therefore 0 0 c= x ∧[y ,w ]+ y ∧[w , x ]= (u⊗ v⊗ x⊗ y ). i i i i i i i i i i i i Now using Proposition 8.D.5 we get c= w ∧[x , y]= w ∧[x , y]= 0 i i i i i i i i again by (8.D.7). 8.D.2. The tame Killing module. Deﬁnition 8.D.8. —Let g be a graded Lie algebra over R. Consider the R-module homomor- phism T: (g g)⊗ g→ g g R R R u v⊗ w → u [v, w]+ v [u,w]. By deﬁnition, Kill(g) is the cokernel of T .Wedeﬁne Kill (g) as the cokernel of the restriction of T to (g g )⊗ g→ (g g ) . Note that T satisﬁes the identities, for all x, y, z T (x y⊗ z)+ T (y z⊗ x)+ T (z x⊗ y)= 0. There is an canonical homomorphism Kill (g)→ Kill(g) . 0 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 169 Theorem 8.D.9. —Let g be a graded Lie algebra. If g is 1-tame then the homomorphism Kill (g)→ Kill(g) is surjective; if g is doubly 1-tame then it is an isomorphism. 0 0 Lemma 8.D.10. —Let g be an arbitrary Lie algebra. Then we have the identity T w, x,[y, z]= T[x, z],w, y− T[x, y],w, z − T[w, y], x, z+ T[w, z], x, y; Proof. — Use the four equalities T[x, z],w, y= w [x, z], y +[x, z] [w, y], T[y, x],w, z= w [y, x], z +[y, x] [w, z], T[w, z], x, y= x [w, z], y +[w, z] [x, y], T[y,w], x, z= x [y,w], z +[y,w] [x, z]; the sum of the four right-hand terms is, by the Jacobi identity and cancellation of ([·,·] [·,·])-terms, equal to −w [z, y], x− x [z, y],w= T w, x,[z, y] . Lemma 8.D.11. —Let g be an arbitrary graded Lie algebra. • Let α, α ,β,β be nonzero weights, with α+ β, α+ β ,α+ β, α+ β = 0,and (x, x , y, y )∈ g× g × g× g .Then, modulo T (g g⊗ g ), we have α α β β T x, x , y, y= 0. (1) and T x, x , y, y= T y, y , x, x . (2) • Let α ,β,γ,γ be weights with β, γ, γ = 0, α ∈{ / −γ,−γ}.For all w∈ g , x∈ 0 0 0 α g , y∈ g , y∈ g we have β γ γ T w , x, y, y= T x, y ,w , y− T[x, y],w , y . (3) 0 0 0 • Let α, α ,β,β be nonzero weights with α+ α ,β+ β = 0.For all x∈ g , x∈ g , y∈ α α g , y∈ g we have β β T x, y, x , y= T x, y , y, x− T y, x , x, y . (4) Proof. — These are immediate from the formula given by Lemma 8.D.10 applied to (x, x , y, y ), resp. (x, y, x , y ), resp. (w , x, y, y ), resp. (x, y, x , y ). 0 170 YVES CORNULIER, ROMAIN TESSERA Lemma 8.D.12. —Let g be a doubly 1-tame graded Lie algebra. • Let α, β, γ be weights with α, β = 0 and α+ β+ γ= 0,and (x, y, z)∈ g× g× g . α β γ Then, modulo T (g g⊗ g ), we have T (x, y, z)= 0. (1) • Let α, α ,β,β be weights, with α, β = 0 and α+ α+ β+ β= 0,and (x, x , y, y )∈ g× g × g× g . Then, modulo T (g g⊗ g ), we have α α β β T x, y , y, x= T y, x , x, y . (2) Proof. — Let us check the ﬁrst assertion. If γ = 0 this is trivial, so assume γ= 0 (so α =−β ). Since g is doubly 1-tame, we can write z=[u ,v] with u ,v of opposite i i i i weights, not equal to±α. Then Lemma 8.D.11(1) applies. Let us check the second assertion. We begin with two particular cases • α+ β = 0. Then α+ β = 0and by (1), modulo T (g g⊗ g ),both T ([x, y], y, x ) and T ([y, x], x, y ) are zero. • α+ β= 0, α+ α = 0. Then α ,β are nonzero, so we obtain the assertion by applying Lemma 8.D.11(4)and then (1) of the current lemma. Finally, the only remaining case is when (α,β,α ,β ) = (α, α,−α,−α), as we assume now. To tackle this last case, let us use this to prove ﬁrst the following: if γ is a nonzero weight, w∈ g , z∈ g ,z∈ g ,then T (w , z, z )+ T (w , z , z)= 0. Indeed, since 0 0 γ−γ 0 0 g is doubly 1-tame, this reduces by linearity to w =[u, u] with u∈ g , u∈ g and 0 δ−δ δ/ ∈{0,±γ}. So, using twice (2) in one of the cases already proved, we obtain T w , z, z= T u, u , z, z = T z, z , u, u =−T z , z , u, u =−T u, u , z , z =−T w , z , z Now suppose that (α,β,α ,β )= (α, α,−α,−α). Then, using the antisymmetry property above and again using one last time one already known case of (2), we obtain T x, y , y, x =−T x, y , x , y =−T x , y , x, y = T y, x , x, y . GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 171 Lemma 8.D.13. —Let g be a 1-tame graded Lie algebra. Then (1) (g g)= T (g g⊗ g)+ (g g ) ; 0 0 0 (2) T (g g⊗ g )= T (g g⊗ g) . 0 0 Proof. — Suppose that x, y∈ g . To show that x y belongs to the right-hand term in (1), it sufﬁces by linearity to deal with the case when y =[u,v] with u,v homogeneous of nonzero opposite weight. Then x [u,v]= T (x, u,v)− u [x,v], which is the sum of an element in ((g g)⊗ g) and an element in (g g ) .So(1) is 0 0 proved. Let us prove (2). By linearity, it is enough to prove that any element T (x, y,[u,v]), where x, y have weight zero and u,v have nonzero opposite weight, belongs to T (g⊗ g⊗ g ) : the formula in Lemma 8.D.10 expresses T (x, y,[u,v]) as a sum of four terms in T (g⊗ g⊗ g ) . Proposition 8.D.14. —Let g be a doubly 1-tame graded Lie algebra. Then T (g g⊗ g)∩ (g g )⊂ T (g g⊗ g ) . 0 0 Proof.—Fix u,v∈ n , n (β = 0) and consider the R-module homomorphism β−β : g→ M= (g g) /T (g g⊗ g ) u,v 0 0 0 w → u [v, w]. The mapping (u,v) → ∈ Hom (g , M) is bilinear. Therefore it extends to a map- u,v R 0 ping s → deﬁned for all s∈ (g⊗ g ) . s 0 If s= x⊗ y∈ g∧ g, we writes=[x , y].Now,for s, s∈ (g⊗ g ) ,deﬁne i i i i 0 ˆˆ (s⊗ s )= (s)∈ M. In other words, (u⊗ v)⊗ (x⊗ y)= u v,[x, y] . We have (u⊗ v)⊗ (x⊗ y)= u v,[x, y] =−T[x, y], u,v +[x, y] [u,v] and similarly (x⊗ y)⊗ (u⊗ v) =−T[u,v], x, y +[u,v] [x, y], Given a map deﬁned on a tensor product such as , we freely view it as a multilinear map when it is convenient. 172 YVES CORNULIER, ROMAIN TESSERA so ˆˆ (u⊗ v)⊗ (x⊗ y)− (x⊗ y)⊗ (u⊗ v) = T[u,v], x, y− T[x, y], u,v= 0 ˆˆˆ by Lemma 8.D.12(2). Thus, is symmetric and we can write (s⊗ s )= (s s ).Note ˆˆ that (trivially) (s s )= 0 whenever s is a 2-cycle (i.e.s = 0), so by the symmetry factors through a map : g g→ Msuch that 0 0 s s= s s for all s, s∈ (g∧ g ) . In other words, we can write u v,[x, y]= [u,v] [x, y] . Now consider some element in T (g g⊗ g)∩ (g g ) . By Lemma 8.D.13,it can be taken in T (g g⊗ g ). We write it as τ= T (x , y , z ), i i i with x , y , z homogeneous and y , z of nonzero weight. Since we work modulo T (g i i i i i g⊗ g ) , we can suppose x is of weight zero for all i, and we have to prove that τ= 0. 0 i So τ= x [y , z]+ y [x , z] , i i i i i i i i the ﬁrst term belongs to g∧ g and the second to the quotient Kill (g) of (g∧ g ) 0 0 0 0 by T (g g⊗ g ) ,so (8.D.15) x [y , z ]= 0. i i i Now, writing x=[u ,v] with u ,v of nonzero opposite weight i ij ij ij ij τ= y [x , z] i i i = y z ,[u ,v] i i ij ij i,j = [y , z] [u ,v] i i ij ij i,j = [y , z] x= [y , z] x=0by(8.D.15). i i i i i i i i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 173 Proof of Theorem 8.D.9. — The ﬁrst statement follows from Lemma 8.D.13(1) and the second from Proposition 8.D.14. 9. Abels’ multiamalgam 9.A. 2-tameness. — In this section, we deal with real-graded Lie algebras, that is, Lie algebras graded in a real vector space W.AsinSection 8, Lie algebras are, unless explicitly speciﬁed, over the ground commutative ring R. Let g be a real-graded Lie algebra. We say that P⊂ W is g-principal if g is gen- erated, as a Lie algebra, by g= g (note that this only depends on the structure P α α∈P of Lie ring, not on the ground ring R). We say that P (or (g, P))is k-tame if when- ever α ,...,α∈ P , there exists an R-linear form on W such that (α )> 0for all 1 k i i= 1,..., k.Notethat P is 1-tame if and only if 0∈ / P and is 2-tame if and only if for all α, β∈ P we have 0 ∈[ / α, β].Notethat k-tame trivially implies (k− 1)-tame. We say that the graded Lie algebra g is k-tame if there exists a g-principal k-tame subset. Note that for k= 1 this is compatible with the deﬁnition in Section 8.A.2. Example 9.A.1. —Asusual,wewrite W ={α: g ={0}}. g α • g= sl with its standard Cartan grading, W ={α: 1≤ i = j≤ 3}∪{0} (with 3 g ij α= e− e , (e ) denoting the canonical basis of R ); then{α ,α ,α} and ij i j i 12 23 31 {α ,α ,α} are g-principal and 2-tame. 21 13 32 • If P is the set of weights of the graded Lie algebra g/[g, g],thenany g-principal set contains P ; conversely if g is nilpotent then P itself is g-principal. Thus if g 1 1 is nilpotent, then g is k-tame if and only if 0 is not in the convex hull of k weights of g/[g, g]. 9.B. Lemmas related to 2-tameness. — This subsection gathers a few technical lemmas needed in the study of the multiamalgam in Section 9.C and Section 9.D.The reader can skip it in a ﬁrst reading. The following lemma was proved by Abels under more speciﬁc hypotheses (g nilpo- tent and ﬁnite-dimensional over a p-adic ﬁeld). As usual, by[g , g] we mean the module β γ generated by such brackets. In a real vector space, we writeα, β for the segment joining α and β (so as to avoid any confusion with Lie brackets). Lemma 9.B.1. —Let g be a real-graded Lie algebra and P⊂ W a g-principal subset. Suppose that (g, P) is 2-tame. Then (1) for any ω∈ W , we have g=[g , g], with β ranging over W− Rω; 0 β−β This paper will not deal with k-tameness for k≥ 3 but this notion is relevant to the study of higher-dimensional isoperimetry problems. 174 YVES CORNULIER, ROMAIN TESSERA (2) if R α∩ P =∅, then g=[g , g], with (β, γ ) ranging over pairs in W− Rα + α β γ such that β+ γ= α. Lemma 9.B.1 is a consequence of the more technical Lemma 9.B.2 below (with i= 1). Actually, the proof of Lemma 9.B.1 is based on an induction which makes use of the full statement of Lemma 9.B.2. Besides, while Lemma 9.B.1 is enough for our pur- poses in the study of the multiamalgam of Lie algebras in Section 9.C, the statements in Lemma 9.B.2 involving the lower central series are needed when studying multiamalgams of nilpotent groups in Section 9.D. Lemma 9.B.2. — Under the assumptions of Lemma 9.B.1,let (g ) be the lower central series i i i i i of g and g= g∩ g (note that g thus means (g ) and can be distinct from (g ) ). Then α 0 0 α 0 i k (1) for any ω∈ W , we have g=[g , g], with β ranging over W Rω; 0 j+k=i β β−β i k (2) if R α∩ P =∅, then g=[g , g], with (β, γ ) ranging over pairs in α j+k=i β γ W Rα such that β+ γ= α; i k (3) if α/∈ P , then g=[g , g], with (β, γ ) ranging over pairs in W such that α β γ j+k=i β+ γ= α and 0 does not belong to the segmentβ, γ. [1][i][1][i−1] Proof.—Deﬁne g= g , and by induction the submodule g =[g , g] α∈P [i][i] for i≥ 2; note that this depends on the choice of P.Deﬁne g= g∩ g .Notethateach [i] g is a graded submodule of g. Let us prove by induction on i≥ 1 the following statement: in the three cases, [j] [i][k] (9.B.3) g⊂ g , g α β γ j,k≥1, j+k=i β+γ=α... where in each case (β, γ ) satisﬁes the additional requirements of (1), (2), or (3) (we encode this in the notation ). β+γ=α... Since in all cases, α/∈ P,the case i= 1 is an empty (tautological) statement. Sup- [i] pose that i≥ 2 and the inclusions (9.B.3) are proved up to i− 1. Consider x∈ g .By [1] [i−1] deﬁnition, x is a sum of elements of the form[y, z] with y∈ g , z∈ g with β+ γ= α. β γ If β and γ are linearly independent over R, the additional conditions are satisﬁed and we are done. Otherwise, since β∈ P,wehave β = 0 and we can write γ= rβ for some r∈ R. There are three cases to consider: • r > 0. Then we are in Case (3), and 0∈ /β, γ, so the additional condition in (3) holds. • r= 0. Then β= α = 0 is a principal weight, which is consistent with none of Cases (1), (2) or (3). • r < 0. Then since β∈ P,wehave R γ∩ P =∅, so we can apply the induction [j] hypothesis of Case (2) to z; by linearity, this reduces to z =[u,v] with u∈ g , δ GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 175 [k] v∈ g , j+ k= i− 1, δ+ ε= rβ,and δ, ε not collinear to β.Notethat α= β+ δ+ ε. By the Jacobi identity, [j][k+1][j+1] [k] (9.B.4) x= y,[u,v]∈ g , g+ g , g . α−ε δ α−δ ε Note that none of δ, α− δ, ε, α− ε belongs to Rα.Hence we aredonein both Cases (2) or (3). In case (1), if ω∈ Rβ , then since δ and ε are not collinear to ω, we see that (9.B.4) satisﬁes the additional conditions of (1). However, the case ω/∈ Rβ is trivial since then the writing x =[y, z] itself already satisﬁes the additional condition of (1). At this point, (9.B.3) is proved. By Lemma 5.B.8,for all i≥ 1, we have g= [] i[] g . In particular g= g and we have ≥i α ≥i α j j [] k k g⊂ g , g⊂ g , g , α β γ β γ j+k= β+γ=α... j+k=i β+γ=α... so i k g⊂ g , g , α β γ j+k=i β+γ=α... k i and since the inclusion[g , g ]⊂ g is clear, we get the desired equality β γ α i k g= g , g . α γ j+k=i β+γ=α... 9.C. Multiamalgams of Lie algebras. 9.C.1. The deﬁnition. — By convex cone in a real vector space, we mean any subset stable under addition and positive scalar multiplication (such a subset is necessarily con- vex). Let C be the set of convex cones of W not containing 0. Let g be a real-graded Lie algebra over the ring R.If C∈ C,deﬁne g= g; C α α∈C this is a graded Lie subalgebra of g (if W is ﬁnite, g is nilpotent). Denote by x →¯ x the g C inclusion of g into g. Deﬁnition 9.C.1. —Deﬁne gˆ= gˆ as the multiamalgam (or colimit) of all g ,where C ranges over C . 176 YVES CORNULIER, ROMAIN TESSERA This is by deﬁnition an initial object in the category of Lie algebras h endowed with compatible homomorphisms g→ h. It can be realized as the quotient of the Lie R-algebra free product of all g ,bythe ideal generated by elements x− y,where x, y range over elements in g , g such that x¯ =¯ y C D and C, D range over C . Note that among those relators, we can restrict to homogeneous x, y as the other ones immediately follow. Since the free product as well as the ideal are ˆˆ graded, g is a graded Lie algebra; in particular, g is also the multiamalgam of the g in the category of Lie algebras graded in W . The inclusions g→ g induce a natural homomorphism gˆ→ g. 9.C.2. Link with the blow-up. — Let g be the blow-up introduced in Section 8.B.For every C∈ C , the structural homomorphism g˜→ g is an isomorphism, and therefore C C we obtain compatible homomorphisms g→ g˜⊂ g, inducing, by the universal property, C C a natural graded homomorphism gˆ→ g˜ . R R Theorem 9.C.2. —If g is 1-tame, then the natural Lie algebra homomorphism κ: gˆ→ g˜ is surjective, and if g is 2-tame, κ is an isomorphism. In particular, if g is 2-tame then the kernel of R R ˆˆ g→ g is central in g and is canonically isomorphic (as an R-module) to H (g) . To prove the second statement, we need the following lemma. Lemma 9.C.3 (Abels). — If g is 2-tame, then gˆ→ g has central kernel, concentrated in degree zero. Proof of Theorem 9.C.2 from Lemma 9.C.3.—If g is 1-tame, then so is g˜ by Lemma 8.B.7, and the surjectivity of κ follows, proving the ﬁrst assertion. If g is 2-tame, then by Lemma 9.C.3, gˆ→ g has central kernel concentrated in degree zero, so by the universal property of the blow-up (Theorem 8.B.4), there is a section s of the natural map κ . By uniqueness in the universal properties, s◦ κ and κ◦ s are both identity and we are done. The last statement is then an immediate consequence of Lemma 8.B.2. Proof of Lemma 9.C.3.—If α is a nonzero weight of g, then there exists C∈ C such that α∈ C. By the amalgamation relations, the composite graded homomorphism g→ g→ gˆ does not depend on the choice of C. We thus call it i and call its image m . α C α α Let us ﬁrst check that whenever α, β are non-zero non-opposite weights, then we have the following inclusion in gˆ (9.C.4)[m , m ]⊂ m . α β α+β We begin with the observation that if γ,δ are nonzero weights with δ/∈ R γ ,then (9.C.5) i[g , g] =[m , m]. γ+δ γ δ γ δ GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 177 Indeed, in the above deﬁnition of i, we can choose C to contain both γ and δ.Inpar- ticular if α/∈ R β,then(9.C.4) is clear. Let us prove (9.C.4) assuming that α∈ R β.By −− 2-tameness, we can suppose that R β∩ P =∅, so by Lemma 9.B.1(2), g⊂[g , g], + β γ δ where (γ, δ) ranges over the set Q(β) of pairs of weights such that γ+ δ= β and γ,δ are not in Rβ(= Rα). So, applying (9.C.5), we obtain m⊂[m , m]. Therefore β γ δ (γ,δ)∈Q(β) [m , m ]⊂ m ,[m , m] . α β α γ δ (γ,δ)∈Q(β) Note that γ and α+ δ are not collinear, since otherwise we would infer that α =−β . If we ﬁx such (γ, δ)∈ Q(β), we get, by the Jacobi identity and using that α, γ, δ are pairwise non-collinear [m ,[m , m]]⊂ m ,[m , m]+ m ,[m , m] α γ δ γ δ α δ α γ ⊂[m , m ]+[m , m] γ δ+α δ α+γ ⊂ m= m , γ+δ+α α+β and ﬁnally[m , m ]⊂ m .So(9.C.4)isproved. α β α+β Now deﬁne v as the submodule of gˆ (9.C.6) v=[m , m]. 0 γ−γ γ =0 We are going to check that for every α = 0 (9.C.7)[m , v ]⊂ m α 0 α and (9.C.8)[v , v ]⊂ v . 0 0 0 Before proving (9.C.7), let us ﬁrst check that for every nonzero α,if X (α) is the set of nonzero weights not collinear to α then (9.C.9)[m , m ]⊂[m , m]. α−α β−β β∈X (α) Indeed, by 2-tameness, we can suppose that R (−α)∩P =∅, and apply Lemma 9.B.1(2), so g⊂[g , g]; −α γ δ (γ,δ)∈Q(−α) 178 YVES CORNULIER, ROMAIN TESSERA by (9.C.5) we deduce m⊂[m , m]; −α γ δ (γ,δ)∈Q(−α) so [m , m ]⊂ m ,[m , m] . α−α α γ δ (γ,δ)∈Q(−α) If (γ, δ)∈ Q(−α), we have, by the Jacobi identity and then (9.C.4) [m ,[m , m]]⊂ m ,[m , m]+ m ,[m , m] α γ δ γ δ α δ α γ ⊂[m , m ]+[m , m] γ δ+α δ α+γ =[m , m ]+[m , m]; γ−γ δ−δ we thus deduce (9.C.9). We can now prove (9.C.7), namely if α, γ = 0, then[m ,[m , m]]⊂ m .By α γ−γ α (9.C.9) we can assume that α and γ are not collinear, in which case (9.C.7) follows from (9.C.4) by an immediate application of Jacobi’s identity. To prove (9.C.8), we need to prove that v is a subalgebra, or equivalently that for every α = 0, we have v ,[m , m]⊂ v . 0 α−α 0 Indeed, using the Jacobi identity and then (9.C.7) [v ,[m , m]]⊂ m ,[m , v]+ m ,[v , m] 0 α−α α−α 0−α 0 α ⊂[m , m ]+[m , m ]⊂ v . α−α−α α 0 so (9.C.8)isproved. We can now conclude the proof of the lemma. By the previous claims (9.C.4), (9.C.7), and (9.C.8), if v is deﬁned as in (9.C.6) the submodule ( m )⊕ v is a Lie 0 α 0 α =0 subalgebra of gˆ . Since the m for α = 0 generate gˆ by deﬁnition, this proves that this Lie subalgebra is all of gˆ . Therefore, if φ: gˆ→ g is the natural map, we see that φ is the natural isomor- phism m→ g .Thus Ker(φ) is contained in gˆ .If z∈ Ker(φ) and x∈ m for some α α 0 α α = 0, then φ[z, x]= φ(z), φ(x)= 0,φ(x)= 0, −1 so[z, x]= φ (φ([z, x]))= 0. Thus z centralizes m for all α; since these generate gˆ,we deduce that z is central. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 179 9.C.3. On the non-2-tame case. — There is a partial converse to Theorem 9.C.2:if R is a ﬁeld and g/[g, g] is not 2-tame, then gˆ has a surjective homomorphism onto a free Lie algebra on two generators. In particular, if g is nilpotent (as in all our applica- tions), g˜ is nilpotent as well but gˆ is not. The argument is straightforward: the assumption implies that there is a graded surjective homomorphism g→ h,where h is the abelian 2-dimensional algebras with weights α, β with β∈ R α, inducing a surjective homo- morphism gˆ→ h. Now it follows from the deﬁnition that h is thefreeproductofthe 1-dimensional Lie algebras h and h and hence is not nilpotent. α β 9.D. Multiamalgams of groups. — In this part, g is a nilpotent real-graded Lie algebra over a commutative Q-algebra R of characteristic zero (although the results would work in characteristic p > s+ 1, where s is the nilpotency length of the Lie algebra involved). For C∈ C ,let G and G be the groups associated to g and g by Malcev’s equiv- g C C alence of categories between nilpotent Lie algebras over Q and uniquely divisible nilpo- tent groups, described in Theorem 5.A.2. Let the embedding G→ G corresponding to g→ g be written as g →¯ g.Let G be the corresponding amalgam, namely the group −1 generated by the free product of G for C∈ C , modded out by the relators xy when- C g ever x¯ =¯ y. The following result was proved by Abels [Ab87, Cor. 4.4.14] assuming that R= Q and that g is ﬁnite-dimensional. This is one of the most delicate points in [Ab87]. We provide a sketch of the (highly technical) proof, in order to indicate how his proof works over our general hypotheses. Theorem 9.D.1 (Abels). — Suppose that g is 2-tame and s-nilpotent. Then G is (s+ 1)- nilpotent. For the purpose of Section 6, we need a stronger result. Abels asked [Ab87, 4.7.3] whether G→ G is always a central extension. This is answered in the positive by the following theorem. Theorem 9.D.2. — Under the same hypotheses, the nilpotent group G is uniquely divisible, and its Lie algebra is gˆ in the natural way. In particular, the homomorphism G→ G has a central kernel, naturally isomorphic to H (g) as a Q-linear space. Proof. — Our proof is based on Theorem 9.D.1 and some generalities about nilpo- tent groups, which are gathered in Section 5. Since by Theorem 9.D.1, G is nilpotent, and since it is generated by divisible sub- groups, it is divisible (see Lemma 5.A.3). Therefore by the (standard) Lemma 5.A.1,to ˆˆ check that G is uniquely divisible, it is enough to check that it is torsion-free. Since G is known to be (s+ 1)-nilpotent, we see that G is the multiamalgam (=colimit) of the G within the category K of (s+ 1)-nilpotent groups. Therefore, G is the quotient of the free C 180 YVES CORNULIER, ROMAIN TESSERA product W in K of the G by amalgamations relations. By Lemma 5.A.5, W is a uniquely −1 divisible torsion-free nilpotent group. Since, for x, y∈ W, whenever xy is an amalgama- r−r tion relation, x y is an amalgamation relation as well for all r∈ Q,Proposition 5.B.5 applies to show that the normal subgroup N generated by amalgamation relations is di- visible. Therefore G/N is torsion-free, hence uniquely divisible. It follows that G is also the multiamalgam of the G in the category K of uniquely C 0 divisible (s+ 1)-nilpotent groups. By Malcev’s Theorem 5.A.2, this category is equivalent to the category of (s+ 1)-nilpotent Lie algebras over Q. Therefore, if H is the group associated to gˆ and H→ G are the homomorphisms associated to g→ gˆ,then H and C C the family of homomorphisms G→ H satisfy the universal property of multiamalgam in the category K , and this gives rise to a canonical isomorphism G→ H. In particular, ˆˆ by Theorem 9.C.2, the kernel W of G→ Gis central in G, and isomorphic to H (g) . Corollary 9.D.3. — Under the same hypotheses, if moreover H (g) ={0} then the central kernel of G→ G is generated, as an abelian group, by elements of the form −1 exp[λx, y] exp[x,λy] ,λ∈ R, x, y∈ g . If R= R is a ﬁnite product of Q-algebras (so that g= g canonically), then those elements j j j=1 j −1 exp([λx, y]) exp([x,λy]) with λ∈ R and x, y∈ g are enough. j j Proof. — The additional assumption and Theorem 9.D.2 imply that the kernel of Q,R G→ G is naturally isomorphic, as a Q-linear space, to the kernel W (g) of the natural map H (g)→ H (g) . 0 0 2 2 Let N be the normal subgroup of G generated by elements of the form −1 exp[λx, y] exp[x,λy] , x, y∈ g ,λ∈ R; clearly N is contained in the kernel W of G→ G. By Proposition 5.B.5, N is divisible, i.e., N is a Q-linear subspace of W. Therefore G/N is uniquely divisible and its Lie algebra can be identiﬁed with gˆ/N. By deﬁnition of N, in G/Nwe haveexp([λx, y])= exp([x,λy]) for all x, y∈ g . So in the Lie algebra of G/N, which is equal to gˆ/N we have[λx, y]=[x,λy]. Since by Proposition 8.C.5, the subgroup W is generated by elements of the form exp([λx, y]−[x,λy]) when x, y range over G and λ ranges over R, it follows that N= W. It remains to prove the last statement. By Lemma 8.B.8,H (g) can be identiﬁed 2 0 with the product H (g ) .Inparticular, by Proposition 8.C.5, it is generated by ele- 2 j 0 ments of the form[λx, y]−[x,λy] when x, y∈ g , λ∈ R ,and j= 1,...,τ . Then, we j,C j can conclude by a straightforward adaptation of the above proof. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 181 R R Corollary 9.D.4. — Under the same hypotheses, if moreover H (g)= Kill (g) ={0}, 0 0 then G→ G is an isomorphism. Proof. — As observed in the proof of Corollary 9.D.3, since H (g)= 0, the kernel Q,R of G→ G is isomorphic to W (g) . Since Kill (g)= 0, Corollary 8.C.8 implies that 0 0 Q,R W (g)= 0. Proof of Theorem 9.D.1 (sketched). — We only sketch the proof; the proof in [Ab87] (with more restricted hypotheses) is 8 pages long. Here the hypotheses are more general since Abels works with p-adic groups. The theorem is stated here assuming that R is a Q- algebra, but more generally, it works under the assumption that R is a Z[1/s!]-module, and in particular when it is a vector space over a ﬁeld of characteristic p > s. Fix a 2-tame g-principal subset P.Let S denote the set of all half-lines R w >0 i i i (w = 0) in W . Consider the lower central series (G ).Let M be the image of G∩ G 1 i i ˆˆ in Gand M= M .Let A be the normal subgroup of G generated by all M ,where C C C ranges over S . Abels also introduces more complicated subgroups. Let L denote the set of lines of W (i.e. its projective space). Each line L∈ L contains exactly two half lines: L= S ∪{0}∪ S . 1 2 Deﬁne the following subgroups of G M= M =M∪ M [L] S S 1 2 [L] and, by induction % & i i i k M= M∪ M∪ M , M , [L] [L] S S[L] 1 2 j+k=i where ((·,·)) denote the subgroup generated by group commutators. Abels proves the following lemma [Ab87, 4.4.11]: if L∈ L and C is a open cone in W such that L+ C⊂ C, then j j+k (9.D.5) M , M⊂ M . C[L] C j+k The interest is that M is a complicated object, while the right-hand term M is a [L] reasonable one. Abels obtains [Ab87, Prop. 4.4.13] the following result, which can appear as a group version of Lemma 9.B.2(1): for any L∈ L and any i,the ith term of the lower i i i central series G is generated, as a subgroup and modulo A ,bythe M where L ranges [L] over L −{L}, or, in symbols, % & i i i (9.D.6) G= M A . [L] L =L 0 182 YVES CORNULIER, ROMAIN TESSERA It is important to mention here that all three items of Lemma 9.B.2 are needed in the proof of (9.D.6) (encapsulated in the proof of [Ab87, Prop. 4.4.7]). This is also the step where the Baker-Campbell-Hausdorff formula is used to convey properties from the Lie algebra g to G. To conclude the proof, suppose that G is s-nilpotent. We wish to prove that s+1 ˆˆˆ ((G, G )) ={1}. Since G is easily checked to be generated by M for S ranging over S , it is enough to check, for each S∈ S s+1 (9.D.7) M , G ={1}. s+1 s+1 Now since G is s-nilpotent, A ={1}, so we can forget “modulo A ”in(9.D.6) (with i= s+ 1), so (9.D.7) follows if we can prove s+1 M , M ={1} [L] for all L∈ L with maybe one exception L ; namely, we choose L to be the line generated 0 0 by S. Then S+ Lis an open cone, (9.D.5) applies and we have s+1 s+1 s+2 M , M⊂ M , M⊂ M ={1}. S S+L [L][L] S+L Remark 9.D.8. — As we observed, Theorem 9.D.1 works when R is only assumed to be a Z[1/s!]-module. It follows that Theorem 9.D.2 works when R is assumed to be a Z[1/(s+ 1)!]-module, and in particular when it is a vector space over a ﬁeld of characteristic p > s+ 1. 10. Presentations of standard solvable groups and their Dehn function 10.A. Outline of the section. — This section, which is the most important of the pa- per, contains the conclusions of the proofs of Theorems E.4 and F and relies on essentially all the preceding sections. Let G= U A be a standard solvable group. In Section 10.B we consider the multiamalgam G of (standard) tame subgroups (which are ﬁnitely many) along their in- tersections (regardless of any topology), as originally introduced by Abels. This group admits a canonical homomorphism onto G. Under the assumption that G is “2-tame” (which means that G does not satisfy the SOL obstruction, which is one of the hypotheses of Theorem E), we use the algebraic results of Section 9 to check in Section 10.D that the kernel of the canonical homomorphism G→ G is central in G. Moreover, assuming that G does not satisfy the 2-homological obstruction, the work of Section 10.B provides an explicit generating family of the central kernel, called welding relations. This provides us with a ﬁrst “algebraic” presentation of G, deﬁned as follows. Let U A,..., U A be the standard tame subgroups. We consider the presentation P 1 ν GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 183 whose set of generators is the disjoint union A U , i=1 and whose set of relators are the following • all tame relations, namely relations of length 3 deﬁning the group law in each of the U A; • all amalgamation relations of length 2, namely if some u∈ U∩ U , then in this i j disjoint union it corresponds to two generators u∈ U and u∈ U ,and the i i j j −1 relator is then u u ; • all “algebraic” welding relations, introduced in Section 10.B (where “algebraic” means that these are words with letters in the alphabet U ). Next we obtain a compact presentation P of G, informally by replacing elements of the various U with efﬁcient representative words, and restricting to generators and relators in a large enough compact subset. It is given by a set of generators S= T S , i=1 where T is in bijection with some compact generating subset of A and S is in bijection with some compact subset of U so that S∪ T generates U A. From Section 6.C,we i i i have a way to associate to each i and each u∈ U ,aword u in the generators S T. i i Replacing each u with u is a way to pass from any word in the previous generators to a word in these new generators, which we call its geometric realization. The reader should keep in mind that even when we consider algebraic words of bounded length, their geometric realization can be unbounded since the length of u with respect to S is unbounded when u varies. The relators of P are: • deﬁning relators of each U A of bounded length; −1 • amalgamation relations of length 2, of the form s s for s∈ S and s∈ S rep- j i i j j resenting the same element in U∩ U ; i j • geometric realizations of algebraic welding relators whose length is bounded by some suitable number. To obtain an upper bound of the Dehn function of P , we proceed in 3 steps (1) we ﬁrst bound the area of a special class of relations, namely those geometric realizations of those algebraic relators; namely for the geometric realization of tame and amalgamation relations we obtain a quadratic upper bound (with respect to the word length with respect to S), and for the geometric realization of welding relations we obtain in Section 10.F a cubic upper bound. 184 YVES CORNULIER, ROMAIN TESSERA (2) the general method developed in Section 7.B allows to use these estimates so as to bound a more general class of relations, the so-called “relations of bounded combinatorial length”, that is, when n is ﬁxed, we consider all geometric real- izations of algebraic relations of length≤ n , and for such relations we show an asymptotic cubic upper bound (the constant depending on n ). (3) We conclude thanks to Gromov’s trick, which ensures that it is enough to con- sider such relations of bounded combinatorial length, using that the efﬁcient words u indeed have a bounded combinatorial length. The conclusions of these steps are written down in Section 10.E in the absence of welding relators, thus proving Theorem F, and in Section 10.G otherwise, ending the proof of Theorem E.4. Let us pinpoint that Section 10.E also provides information in the case where there are welding relators, in order to get quadratic estimates on the area of relations that follow only from the other types of relators. This is crucial in Section 10.F, where the cubic estimate is obtained by using n times a homotopy of area n . 10.B. Abels’ multiamalgam. — Let G be any group and consider a family (G ) of subgroups. We deﬁne the multiamalgam (or colimit) to be an initial object in the cate- gory of groups H endowed with homomorphisms H→ Gand G→ H, such that all composite homomorphisms G→ H→ G are equal to the inclusion. Such an object is deﬁned up to unique isomorphism commuting with all homomorphisms. It can be explic- itly constructed as follows: consider the free product H of all G ,denoteby κ: G→ H i i i −1 the inclusion and mod out by the normal subgroup generated by the κ (x)κ (x) when- i j ever x∈ G∩ G .Ifthe family (G ) is stable under ﬁnite intersections, it is enough to mod i j i −1 out by the elements of the form κ (x)κ (x) whenever x∈ G and G⊂ G . Clearly, the i j i i j multiamalgam does not change if we replace the family (G ) by a larger family (G ) such that each G is contained in some G ; in particular it is no restriction to assume that the family is closed under intersections. Also, the image of G→ G is obviously equal to the subgroup generated by the G . Remark 10.B.1. — Our (and, originally, Abels’) motivation in introducing the mul- tiamalgam is to obtain a presentation of G. Therefore, in this point of view, the ideal case is when G→ G is an isomorphism. In many interesting cases, which will be studied in the sequel, G→ G is a central extension. On the other hand, if the G have pairwise trivial intersection, the multiamalgam of the G is merely the free product of the G , and this generally means that the kernel of i i G→ G is “large”. Example 10.B.2. — Consider a group presentation G =S| R in which every relator involves at most two generators. If we consider the family of subgroups gener- ated by 2 elements of S, the homomorphism G→ G is an isomorphism. Instances of GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 185 such presentations are presentations of free abelian groups, Coxeter presentations, Artin presentations. Lemma 10.B.3. — In general, let G be any group, (G ) any family of subgroups. Suppose that each G has a presentationS| R and that S∩ S generates G∩ G for all i, j. Then the i i i i j i j multiamalgam of all G has a presentation with generators S , relators R and, for all (i, j) the i i i relators of size two identifying an element of S and of S whenever they are actually equal (if (G ) is i j i closed under ﬁnite intersections, those (i, j) such that G⊂ G are enough). i j Proof. — This is formal. Following a fundamental idea of Abels, we introduce the following deﬁnition. Deﬁnition 10.B.4. —Let G be a locally compact group G with a semidirect product decompo- sition G= U A (as in Section 4.C). Consider the family (G= U A) of its tame subgroups. i i 1≤i≤ν Let U be the multiamalgam of the U along their intersections; it inherits a natural action of A,and we ˆˆ deﬁne G= U A. Remark 10.B.5. —Byaneasyveriﬁcation, G is the multiamalgam of the G 10.C. Algebraic and geometric presentations of the multiamalgam. — Let G= U Abe a standard solvable group and (G ) the family of its standard tame subgroups, G= i 1≤i≤ν i U A. Consider the free product H= U . There is a canonical surjective homomor- ∗i phism p: H→ U. Besides, there are canonical homomorphisms j: U→ H, so that p◦ j i i i is the inclusion U⊂ U. Recall that u is deﬁned as equal to u for some subset C(i) of the weight space, i C(i) stable under addition and not containing 0. Lemma 10.C.1 (Algebraic presentation of the multiamalgam). — The multiamalgam U is, −1 through the canonical map p, the quotient of H by the normal subgroup generated by pairs j (x)j (x) i i where (i, i ) ranges over pairs such that C(i)⊂ C(i ) and x ranges over U . Our goal is to translate this presentation into a compact presentation of G. Let ¯¯¯ S be disjoint copies of the S . Deﬁne the disjoint union S= S Tand F the free i i i S ¯¯ group over S; note that there is a canonical surjection S→ S= S T. Denote by ζ i i ¯¯ the canonical bijection S→ S . There is a canonical surjection ζ: S→ Sgiven as the i i identity on T and as ζ on S . i i There is a canonical surjective homomorphism π¯: F→ H A, S 186 YVES CORNULIER, ROMAIN TESSERA and by composition, p ◦¯ π is a canonical surjection F→ U. If ι is the inclusion of S into i i F ,then p ◦¯ π◦ ι is the inclusion of S into U. i i Let us introduce some important sets of elements in the kernel of the map F¯→ U. We ﬁx a presentation of A over T, including all commutation relators between generators. (See Section 2.B for basic conventions about the meaning of relations and relators.) ¯¯¯ • R= R ,where R consists of all elements in Ker(π)∩ F .We tame tame,i tame,i TS call these tame relations. 1 1 1 ¯¯¯¯ • R= R ,where R consists of all elements in R that are relators tame,i tame i tame,i tame,i in the presentation of G given in Corollary 6.A.7,aswellasthe relators in R i tame,i of length two; we call these tame relators. −1 • R consists of the elements of the form σ (w) σ (w),where w ranges over amalg i i F ,and where (i, i ) ranges over pairs such that C(i)⊂ C(i ),and σ: F→ ST i S∪T i i F is the unique homomorphism mapping every s∈ S to ζ (s) and every t∈ T i i to itself. We call these amalgamation relations; • R consists of those amalgamation relations for which w∈ S . These are amalg words of length at most two. We call these amalgamation relators. Note that the quotient of F by R is just the free group F .Wedeﬁne R and S S tame amalg 1 1 ¯¯ R as the images of R and R in F . tame tame tame S Proposition 10.C.2 (Geometric presentation of the multiamalgam). — The multiamalgam U is, 1 1 through the canonical map p ◦¯ π , the quotient of F¯ by the normal subgroup generated by R∪ R , tame amalg and is also the quotient of F by the normal subgroup generated by R . tame Proof. — The quotient of F by the (normal subgroup generated by) tame relations is obviously isomorphic to ( U ) A. Since by Corollary 6.A.7,for each i, R is ∗i i tame,i contained in the normal subgroup generated by R , it follows that ( U ) Ais also ∗i i tame,i the quotient of F by the tame relators. Denote by J the inclusion of U into U . By Lemma 10.B.3, the quotient of i i∗ i −1 U by the relations J (u) J (u) for (i, i ) ranging over pairs such that C(i)⊂ C(i ),is ∗ i i i ˆˆ the multiamalgam U. It follows that the quotient of ( U ) Aby the R is U, and ∗ i amalg since obviously R is normally generated by R , we deduce that the quotient of amalg amalg ( U ) Aby the R is also U. ∗i amalg 1 1 ˆ¯ Hence U is naturally the quotient of F¯ by R∪ R . The second assertion tame amalg follows. Proposition 10.C.3 (Quadratic ﬁlling of tame and amalgamation relations). — For the presen- tation 1 1 ¯¯ S| R∪ R tame amalg GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 187 of G, the tame relations have an at most quadratic area with respect to their length, and the amalgamation −1 relations σ (w) σ (w) have an area bounded above by the length of w. In the presentation i i S| R tame of G, the tame relations have an at most quadratic area with respect to their length. Proof. — The tame relations have an at most quadratic area by Corollary 6.A.7. −1 Let us consider an amalgamation relation. It has the form σ (w) σ (w) with w i i of length n; then with cost≤ n we can replace all letters ζ (s) for s∈ S with ζ (s);the i i i −1 resulting word is then equal to σ (w) σ (w)= 1. i i The second assertion immediately follows. Proposition 10.C.2 is a ﬁrst step towards a compact presentation of G. 10.D. Presentation of the group in the 2-tame case. — The following two theorems, which use the notion of 2-tameness introduced in Deﬁnition 4.C.1, are established in Section 9 (Theorem 9.D.2 and Corollary 9.D.3), relying on Section 8. In the follow- ing statements, we identify U with its image in U. Also, recall that the decomposition K= K induces a decomposition u= u ,and u= u . j (j) i i(j) j=1 j j Theorem 10.D.1 (Presentation of a 2-tame group, weak form). — Assume that G= U A is a 2-tame standard solvable group. Then the homomorphism G→ G has a central kernel. If moreover the degree zero component of the second homology group H (u) vanishes, the kernel of G→ G is 2 0 generated by elements of the form −1 exp[λx, y] exp[x,λy] , 1≤ j≤ d, x, y∈ u ,λ∈ K ,. i(j) j We pinpoint the striking fact that H (u) ={0} does not imply that G→ Gis an 2 0 isomorphism. This was pointed out by Abels [Ab87, 5.7.4], and relies on the fact that the space H (u) of 2-homology of u,where u is viewed as a (huge) Lie algebra over the rationals, can be larger than H (u) . On the other hand, the fact that G→ Ghas 2 0 a central kernel is a new result, even in Abels’ framework (u ﬁnite-dimensional nilpotent Lie algebra over Q ); Abels however proved that Uis (s+ 1)-nilpotent if U is s-nilpotent and this is a major step in the proof. Actually, in order to use the results of Section 7.B, we need a (stronger) stable form of Theorem 10.D.1, namely holding over an arbitrary commutative K-algebra. We can view U as the group of K-points of U,where U is an afﬁne algebraic group over the ring K= K . Thus, for every commutative K-algebra A, U(A) is the group associated to the nilpotent Lie Q-algebra u⊗ A. Similarly, U is deﬁned so that U (A)⊂ U(A) is K i i the exponential of u⊗ A,and we deﬁne U[A] as the corresponding multiamalgam of i K the U (A). i 188 YVES CORNULIER, ROMAIN TESSERA Theorem 10.D.2 (Presentation of a 2-tame group, strong (stable) form). — Assume that G= U A is a 2-tame standard solvable group. Then for every K-algebra A, the homomorphism U[A]→ U(A) has a central kernel, which remains central in U[A] A. If moreover the zero degree component of the second homology group H (u) vanishes, the kernel of U[A]→ U(A) is generated by elements 2 0 of the form −1 exp[λx, y] exp[x,λy] , x, y∈ U (A), λ∈ A , j= 1..., d. i(j) j Note that we could state the theorem without decomposing along the decomposi- tion K= K , but we really need this statement when we estimate the area of welding relations in Section 10.F. Note that Theorem 10.D.1 is equivalent to the case A= K of Theorem 10.D.2, ˆˆ given the trivial observation that the kernels of G→ Gand U→ U coincide. Theorem 10.D.1 involves the Lie algebra bracket; in order to translate it into a purely group-theoretic setting, we need to use Lazard’s formulas from Section 5.C. We can now restate the second statement of Theorem 10.D.1 with no reference to the Lie algebra in the conclusion (except taking real powers): Theorem 10.D.3. —Let G= U A be a 2-tame standard solvable group such that H (u) ={0}. Choose s so that U is s-nilpotent. Then the (central) kernel of G→ G is generated by 2 0 elements of the form −1 λ λ B x , y B x, y , x, y∈ U ,λ∈ K , j= 1..., d, s+1 s+1 i(j) j where x denotes exp(λ log(x)). Again we need a stable form of the latter result. Theorem 10.D.4. —Let G= U A be a 2-tame standard solvable group such that H (u) ={0}. Choose s so that U is s-nilpotent. Then for every commutative K-algebra A, denoting 2 0 A= A⊗ K (so that A= A canonically), the (central) kernel of U[A]→ U(A) is generated j K j j by elements of the form −1 λ λ B x , y B x, y , x, y∈ U (A), λ∈ A , j= 1..., d, s+1 s+1 i(j) j where x denotes exp(λ log(x)). We can view the elements −1 λ λ B x , y B x, y s+1 s+1 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 189 as elements of the free product H= U ;now x, y range over the disjoint union x, y∈ ∗i i U . We call these welding relations in the free product H. i(j) By substitution, Theorem 10.D.3 gives rises to the set of relations in F ( ) −1 λ λ (10.D.5)R= B x , y B x, y: x, y∈ U ,λ∈ K , j= 1..., d . weld s+1 s+1 i(j) j in the free group F . We call these welding relations. We deﬁne the set R of welding relators weld as those welding relations for which$x$ ,$y$ ,|λ|≤ 1, where $·$ is the prescribed Lie algebra norm. It follows from Corollary 8.C.9 that G has a presentation with relators those of G (given in Lemma 10.B.3) along with welding relators. At this point, this already reproves Abels’ result. Corollary 10.D.6 (Compact presentation of G). — Let G be a standard solvable group. Assume that G is 2-tame and H (u ) ={0}.Then: 2 na 0 (a) (Abels) G is compactly presented; (b) if H (u) ={0}, a compact presentation of G is given by 2 0 1 1 1 ¯¯ (10.D.7) S| R∪ R∪ R . tame amalg weld (c) if H (u) ={0} and Kill(u) ={0}, a compact presentation of G is given by 2 0 0 1 1 ¯¯ S| R∪ R . tame amalg Proof. — First assume that H (u) ={0}. The above remarks show that if π is the 2 0 ˆˆ natural projection F→ G, then π (R ) generates normally the kernel of G→ G. weld Therefore, by Proposition 10.C.2, G admits the compact presentation (10.D.7). This proves (b). To obtain (a), It remains to prove that G is compactly presented only assuming H (u ) ={0},but then G/G is compactly presented by the previous case, and it follows 2 na 0 that G is compactly presented. 1 1 Finally (c) is by combining Proposition 10.C.2, which says thatS| R∪ R tame amalg is a presentation of G (for an arbitrary standard solvable group), and Corollary 9.D.4, ˆˆ which says that the natural projection U→ U is an isomorphism (and hence G→ Gas well). Remark 10.D.8. — Let us pinpoint that this does not coincide, at this point, with Abels’ proof. Abels did not prove that the kernel of G→ G is generated by welding relators. Instead (assuming that G is totally disconnected), he considered the multiamal- ˙˙˙ gamated product Uof all U and a compact open subgroup of U, and G= U A, i 190 YVES CORNULIER, ROMAIN TESSERA ˆ˙ˆ˙ assuming that is generated by the intersections ∩ U ,sothat U→ Uand G→ G are surjective. Deﬁning S= U∩ and using S= S∪ T as a generating subset of G (recall i i i that T is a compact generating subset of A), we see that is boundedly generated by S ˙ˆ (for instance, by the Baire category theorem). It follows that G is the quotient of Gby a normal subgroup normally generated by generators of bounded length. Hence Gis boundedly presented by S. There is a natural projection G→ G. Since welding relators are killed by the amal- gamation with , we know that the natural projection G→ G is an isomorphism. Not having the presentation by welding relators, Abels used instead topological arguments [Ab87, 5.4, 5.6.1] to reach the conclusion that G→ G is indeed an isomorphism; thus G is compactly presented. This approach, with the use of a compact open subgroup is, however, “unstable”, in the sense that it does not yield a presentation of U(A) Awhen A is an arbitrary commutative K-algebra, and the presentation with welding relators will be needed in the sequel in a crucial way when we obtain an upper bound on the Dehn function. 10.E. Quadratic estimates and concluding step for standard solvable groups with zero Killing module. — Let us call the rank of A the unique d such that A admits a copy of Z as a cocompact lattice. Theorem 10.E.1. —Let G= U A be a standard solvable group. If G is 2-tame and H (u) and Kill(u) both vanish, then the Dehn function of G is quadratic (or linear in case A has 2 0 0 rank≤ 1). If A has rank 0 then G is compact; if A has rank 1, then 2-tame implies tame, and in that case, the Dehn function is linear (see Remark 6.A.9). Otherwise, if A has rank at least 2, the Dehn function is at least quadratic. So the issue is to obtain a quadratic upper bound. Fix c and a c-tuple (℘ ,...,℘ ) of elements in{1,...,ν}.Recallthat U (r)⊂ F¯ 1 c ℘ (r) denotes the set of null-homotopic words of the form w ...w ,with w∈ S .Theo- 1 c rem 6.C.1 reduces the proof of Theorem 10.E.1 to proving that, for each given ℘,words in U (r) have an at most quadratic area with respect to r. We now proceed to prove this. We also prove a statement holding for more speciﬁc relations but without the vanishing assumptions, which will be used in the sequel. Theorem 10.E.2. —Let G= U A be a 2-tame standard solvable group. Let U ,..., U 1 ν be its standard tame subgroups. Then (1) Assume that H (u) and Kill(u) both vanish. Fix an integer c and any c-tuple 2 0 0 (℘ ,...,℘ ) of elements in{1,...,ν}. Then for r≥ 0, the area of words in U (r)⊂ F 1 c ℘ S is quadratically bounded in terms of r. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 191 (s+1) (2) Let s be such that U is s-nilpotent (that is, U ={1}). Then for every group word (s+2) w(x ,..., x ) that belongs to the (s+ 2)-th term F of the lower central series of the 1 c free group F , and for any u ,..., u in U , the relation w(u ,..., u ) in G has an at c 1 c i 1 c most quadratic area with respect to its total length (the constant not depending on c). Proof.—Theorem 10.E.1 makes use of the results of Section 7.B. With the notation of Section 7.B,for any K-algebra A we have H[A]= U (A). ∗ i i=1 Let us encode the amalgamation relations. For any 1≤ i = j≤ ν , write U (A)= ij U (A)∩ U (A),and let s be the inclusion of U (A) into U (A). Deﬁne a closed subscheme i j ij ij i R of U by deﬁning ij i i=1 R (A)= (u ,..., u ): u= 1∀k ∈{ / i, j}, u , u∈ U (A), and u u= 1 . ij 1 ν k i j ij i j Deﬁne R[A]= R (A). Thus, taking the quotient of H[A] by π (R[A]) means amal- ij A i<j gamating the U along their intersections; we thus denote this quotient as U[A] (in Sec- 1 1 tion 7.B it was denoted as Q[A]). Deﬁne R= R∪ R . tame amalg Using this, we now prove (1) and (2) separately. We begin with (1). Since u is 2-tame and H (u) and Kill(u) vanish, for every 2 0 0 A K commutative K-algebra A,wehave H (u⊗ A)= H (u)⊗ A ={0},and similarly K 0 0 K 2 2 Kill (u⊗ A) ={0}. Thus by Corollary 9.D.4, U[A]→ U(A) is an isomorphism for K 0 every A. It follows that L (deﬁned in Section 7.B.3) is a closed subscheme of U ,sowe ℘ ℘ can apply Theorem 7.B.5 with M= L . It gives an upper bound of the areas of elements in M(K) in terms of upper bounds on the Dehn functions of the U and of the areas of the relations of the form u ... u with (u ,..., u )∈ R[K]. In this case, the U have at most 1 ν 1 ν i quadratic Dehn function by Corollary 6.A.8, and the amalgamation relations have an at most linear area by Proposition 10.C.3. Thus by Theorem 7.B.5,elementsof M(K) have an at most quadratic area with respect to their length and R. To prove of (2), we argue as follows. By Theorem 9.D.1, U[A] is (s+ 1)-nilpotent for every A; thus, deﬁning M= U ,wehave M (A)⊂ L (A) (with the notation of The- orem 7.B.7. By Proposition 10.C.3, the tame and amalgamation relations have an at most 1 1 ¯¯ quadratic area with respect to the presentationS| R∪ R. We are thus in position tame amalg to apply Theorem 7.B.7, which yields the desired result. 10.F. Area of welding relations. — We now bound the area of welding relations, mak- ing use of the quadratic bound on the area of general nilpotency relations from Sec- tion 10.E. Theorem 10.F.1. — If the standard solvable group G= U A is 2-tame, then welding relations in G have an at most cubic area. More precisely, there exists a constant K such that for all j , all x, y∈ U , and all λ∈ K , i(j) j the welding relation (10.D.5) has area at most Kn , with n= log(1 +$x$+$y$+|λ|). 192 YVES CORNULIER, ROMAIN TESSERA Let us assume that the subgroup U, in Theorem 10.F.1,is s-nilpotent. In all this subsection, we write A= A ,B= B , q= q .Theorem 10.E.2(2) applies to the s+1 s+1 s+1 group words corresponding to equalities of Proposition 5.C.3, providing Theorem 10.F.2. —If G is 2-tame and U is s-nilpotent, for all x∈ U ,y∈ U and C C 1 2 z∈ U , the relations −1 A(x, y)= A(y, x); B(x, y)= B(y, x) B A(x, y), z= A B(x, z), B(y, z); q q A A(x, y), z= A x , A(y, z) k k k B x , y= B x, y= B(x, y) 1 1 ˆ¯¯ have an at most quadratic area in G for the presentationS| R∪ R. tame amalg −1 (By “the relation w= w has an at most quadratic area, we mean the relation w w has an at most quadratic area.) Note that in the last case, the quadratic upper bound is of the form c n ,where c may depend on k. k k Although we only use it as a lemma in the course of proving Theorem 10.F.1,we state Theorem 10.F.2 as a theorem, because it provides a geometric information which is not encoded in the Dehn function, namely a quadratic area for certain types of loops. We now proceed to prove Theorem 10.F.1. Using (with quadratic cost) the “bilin- earity” of B (or restricting scalars from the beginning), we can suppose that K is equal to R or Q (although the forthcoming argument can be adapted with unessential modiﬁ- cations to their ﬁnite extensions). If K= Q ,deﬁne π= p.If K= R,deﬁne π= 2. We j p j j j consider the following ﬁnite subset of Q: ( ) n−1 1 i (n)= λ= ε π: ε ∈{−π+ 1,...,π− 1}⊂ Q⊂ K . i i j j j j j i=0 Lemma 10.F.3. —For every κ> 0, there exists K > 0 such that for all n∈ N, all j , all x, y∈ U with log(1 +$x$+$y$)≤ n and all λ∈ (κ n), the welding relation i(j) −1 λ λ B x , y B x, y has area≤ Kn . Proof. — We can work for a given j , so we write π= π .Write κ n−1 λ= ε π ε ∈{−π+ 1,...,π− 1} . i i i=0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 193 If we set κ n−1 k−i λ= ε π , i k k=i we have λ= λ, λ= 0, and, for all i 0 κ n λ= πλ+ ε i i+1 i −1 i Set z= q y and σ= ε π (so σ= 0); consider the word i i−1 j=1 = A B λ x, π z , B(x, σ z) i i i−1 Here, for readability, we write λx (etc.) instead of x . This is natural since x can be identiﬁed to its Lie algebra logarithm. For i= 0, σ= 0 so, since 1= 1 and, formally, i−1 B(x, 1)= 1and A(x, 1)= x (see Remark 5.C.2), we have = B(λx, z) . For i= κ n, λ= 0and σ= λ so this is (using that formally B(1, y)= 1and n i−1 A(1, y)= y ) = B(x, λz) . κ n Let us show that we can pass from to with quadratic cost. In the following i i+1 computation, each means one operation with quadratic cost, i.e., with cost≤ K n for some constant K only depending on the group presentation. The tag on the right explains why this quadratic operation is valid, namely: • (1) means both the homotopy between loops lies in one tame subgroup. • (2) means the operation follows from Theorem 10.F.2;tobespeciﬁc: – (2) for distr left B A(x, y), z= A B(x, z), B(y, z) and similarly (2) on the right distr right – (2) for an equality of the type k k k B x, y= B x , y= B(x, y) , with k an integer satisfying|k|≤ max(q,π). q q – (2) for the identity A(A(x, y), z )= A(x , A(y, z)). assoc • Or a tag referring to a previous computation, written in brackets. 194 YVES CORNULIER, ROMAIN TESSERA −1−1 (10.F.4) λ x= q πλ q x+ ε q x i i+1 i −1−1 πλ q x+ ε q x (1) i+1 i −1−1 πλ q x+ ε q x (1) i+1 i −1−1 A πλ q x, ε q x (1) i+1 i So by substitution we obtain i−1−1 i (10.F.5) B λ x, π z B A πλ q x, ε q x , π z[10.F.4] i i+1 i −1 i−1 i A B πλ q x, π z , B ε q x, π z (2) i+1 i distr left −1 i−1 i A B πλ q x, π z , B q x, ε π z (2) i+1 i Q Independently we have −1 (10.F.6) B(x, σ z) B q x , σ z (1) i−1 i−1 −1 B q x, σ z (2) i−1 Q Again by substitution, this yields (10.F.7) = A B λ x, π z , B(x, σ z) i i i−1 −1 i−1 i−1 A A B πλ q x, π z , B q x, ε π z , B q x, σ z i+1 i i−1 [10.F.5,10.F.6] −1 i−1 i−1 A B πλ q x, π z , A B q x, ε π z , B q x, σ z i+1 i i−1 (2) assoc −1 i−1 i (10.F.8) A B πλ q x, π z , B q x, A ε π z, σ z i+1 i i−1 (2) distr right By similar arguments (2) q Q −1 i i+1 B πλ q x, π z B λ x, π z , i+1 i+1 and −1 i−1 i B q x, A ε π z, σ z B q x, q ε π z+ σ z (1) i i−1 i i−1 B x, ε π z+ σ z (2) i i−1 Q = B(x, σ z) i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 195 so substituting from (10.F.8)weget i+1 A B λ x, π z , B(x, σ z)= i i+1 i i+1 in quadratic cost, say≤ K n (noting that each has length≤ K n forsomeﬁxedcon- 1 i 2 stant K ). Note that the constant K only depends on the group presentation, because 2 1 the above estimates use a quadratic ﬁlling only ﬁnitely many times, each among ﬁnitely many types (note that we used (2) only for k in a bounded interval, only depending on K and s). It follows that we can pass from to with cost≤ K κ n .Onthe other 0 κ n 1 hand, by substitution of type (2) , we can pass with quadratic cost from B(λx, y) to −1 q−1 q B(λx, q y)= and from = B(x, λq y) to B(x, λy). So the proof of the lemma is 0 κ n complete. Conclusion of the proof of Theorem 10.F.1.—Letnow (n) be the set of quotients λ /λ with λ ,λ∈ (n), λ = 0. Lemma 10.F.3 immediately extends to the case when 2 2 λ∈ . It follows from the deﬁnition that (κ n) contains all elements of the form λ= j j κ n ε π ,with ε ∈{−π ,...,π}. i i j j i=−κ n j 2−2κ n 2κ n 2 κ n If K= R, π (κ n) contains all integers between−π and π .Thus (κ n) j j j j j −κ n κ n κ n 2 contains a set which is|π|-dense in the ball of radius|π|.If K= Q , π (κ n) j p 2κ n κ n contains a|π|-dense subset of Z .Thus (κ n) contains a|π|-dense subset of the −κ n−1 2 ball of radius|π|. In both cases, deﬁning = max(|π|,|π| ),weobtainthat (κ n) j j j −κ n κ n contains a -dense subset of the ball of radius in K . j j κ n 2n We now ﬁx j and write = , π= π .Wepick κ= 2/ log(),sothat = e . j j We assume that n≥ log(1/|q|),where|q| is the norm of q in K (if K= R this is an empty j j −κ n−1 n condition). It follows that |q| e≤ 1. Now ﬁx λ∈ K with|λ|≤ e . We need to prove that we can pass from B(λx, y) to q q B(x, λy) with cubic cost; clearly it is enough to pass from B(λx, y) to B(x, λy) with cubic cost. n κ n 2−κ n Since|λ|≤ e≤ , we can write λ= μ+ qε with μ∈ (κ n) and|qε|≤ . −κ n−1−n So|ε|≤ |q|≤ e . Also, assume that n≥ log(). So we can ﬁnd an integer k with n/ log()≤ k≤ ±k 2n/ log(). Thus, if we deﬁne η= π , with the choice of sign so that|η| > 1; we have n 2n n−1 e ≤|η|≤ e≤ e|ε| . Using a computation as in (10.F.4), we obtain, with quadratic cost B(λx, y)= B (μ+ qε)x, y −1 B A μq x, εx , y [10.F.4] −1 A B μq x, y , B(εx, y) (2) distr left 196 YVES CORNULIER, ROMAIN TESSERA and similarly, with quadratic cost. −1 B(x, λy) A B x, μq y , B(x, εy) By thepreviouscase, with cubiccostwehave −1−1 B μq x, y B x, μq y So it remains to check that with cubic cost we have (10.F.9)B(εx, y) B(x, εy). 2 n If η is the element introduced above, observe that η∈ (κ n) and e ≤|η|≤ n−1 e|ε| . We have, with cubic cost −1−1 (10.F.10)B(εx, y) B ηεx, η y; B η x, ηεy B(x, εy). −1−n−1−1 Since max(|ηε|,|η| )≤ e , it follows that all four elements ηεx, η y, η x, εy have norm at most one, and it follows that we can perform −1−1 (10.F.11)B ηεx, η y B η x, ηεy by application of a single welding relator. So (10.F.9) follows from (10.F.10)and (10.F.11). 10.G. Concluding step for standard solvable groups. Theorem 10.G.1. —Let G be a standard solvable group. If G is 2-tame and H (u)= 0 2 0 then its Dehn function is at most cubic. Exactly as in the proof of Theorem 10.E.1,Theorem 6.C.1, along with the cubic bound on the area of welding relators provided by Theorem 10.F.1, allows to reduce the proof of Theorem 10.G.1 to the following: Theorem 10.G.2. —Let G= U A be a 2-tame standard solvable group with H (u)= 0; 2 0 let U ,..., U be its standard tame subgroups; ﬁx a positive function n → g(n) such that n → g(n)/n 1 ν is eventually non-decreasing. Assume that the welding relations in G of length n have an area g(n). Fix c and a c-tuple (℘ ,...,℘ ) of elements in{1,...,ν}. Then words in U (n) have an 1 c ℘ 1 1 1 ¯¯ area, for the presentationS| R∪ R∪ R, asymptotically bounded above by g(n). tame amalg weld (We state the theorem with an arbitrary function g(n), because we do not know if the cubic bound is optimal; this might depend on G even assuming Kill(u) = 0.) Proof.—Theproof follows thesamelinesasTheorem 10.E.2(1); let us highlight the differences. We need to include welding relations in the deﬁnition of R: namely, deﬁning GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 197 R as in the proof of Theorem 10.E.2, we will deﬁne R[A]= R[A]∪ R (A),where ij weld ij R is deﬁned as follows. weld Let s be such that U is s-nilpotent. Deﬁne w∈ F as equal to w(x, x , y, y )= −1 B (x, y)B (x , y ) ,where B is a word as given by Theorem 5.C.1. s+1 s+1 s+1 We wish to deﬁne (recall that K= K ) j=1 R[A]= W (A), weld ii (j) 1≤i,i≤ν,1≤j≤d and we have to deﬁne W ⊂ U . ii (j) i =1 Deﬁne, for V, V unipotent K-groups, the following subscheme M= M of V× V,V V× V× V deﬁned by M (A)= v ,v ,v ,v∈ V× V× V× V (A): 1 2 1 2 ∃λ∈ A: v= v ,v= v . 2 2 1 Here v means exp(λ log(v)). That it is a subscheme can be seen on the Lie algebra, where it corresponds to the 4-tuples (v ,v ,v ,v ) such that there exists λ such that 1 2 1 2 v= λv and v= λv . Then writing in coordinates, this is equivalent to the condition 1 2 2 1 that • v v − v v = v v− v v= 0for all i, i , 2i 1i 1i 2i 2i 1i 1i 2i • v v= v v for all i; 2i 1i 2i 1i hence it is indeed a subscheme. Using the notation of Section 7.B.6,wehave −1 λ λ w• M (A)= B x , y B x, y| x∈ V(A), y∈ V (A), λ∈ K . s+1 s+1 V,V Hence if we deﬁne W = w• M ,wehave. ii (j) U ,U i(j) i (j) −1 λ λ W (A)= B x , y B x, y| x∈ U (A), y∈ U (A), λ∈ K . ii (j) s+1 s+1 i(j) i (j) j Now the deﬁnition of R is complete, and by Theorem 10.D.4, the quotient of H[A] by the normal subgroup generated by R[A] is canonically U(A), for every commutative K-algebra A. Hence we can, as in the proof of Theorem 10.E.2(1), deﬁne M as equal to L . Fix c and a c-tuple ℘ of elements in{1,...,ν}.ByTheorem 10.D.4, denoting 1 1 1 R= R∪ R∪ R ,wehave R⊂ R[K];byTheorem 10.D.6(b), the kernel of the tame amalg weld quotient map from F to G is generated as a normal subgroup by R. By assumption (and Proposition 10.C.3), the words deﬁned by the relations in R[K] have area g(r) with respect to their length r. Hence by Theorem 7.B.5,the area of words x ... x with 1 c x∈ U is g(r),where r= max|x|. i ℘ i i Since this holds for every given ℘,byTheorem 6.C.1 (which encapsulates Gro- mov’s trick), we deduce that the Dehn function of G is g(n). 198 YVES CORNULIER, ROMAIN TESSERA 10.H. Dehn function of generalized standard solvable groups. — We deﬁne a generalized standard solvable group as a locally compact group of the form U N, where the deﬁni- tion is exactly as for standard solvable groups (Deﬁnition 1.2), except that N is supposed to be nilpotent instead of abelian. Recall from Section 6.E that such a group is generalized tame if some element c of N acts on U as a compaction. Clearly split triangulable Lie groups, i.e. of the form G= G N, are special cases of generalized standard solvable groups. Theorem 10.H.1. —Let G be a generalized standard solvable group not satisfying any of the (SOL or 2-homological) obstructions. Suppose that the Dehn function of N is bounded above by some function f such that r → f (r)/r is non-decreasing for some α> 1. Then the Dehn function of G satisﬁes δ (n) nf (n).Ifmoreover Kill(u) ={0}, then G 0 δ (n) f (n). Note that in all examples we are aware of, the function f can be chosen to be equivalent to the Dehn function of N. The proof follows similar steps as in the case of standard solvable groups, so we only sketch it. The ﬁrst step is an upper bound for the Dehn function of generalized tame groups, which was obtained in Section 6.E. Sketch of proof of Theorem 10.H.1.—Theorem 6.E.2 implies that the generalized standard tame subgroups of G (i.e., the U N, where U are the standard tame subgroups i i of U) have their Dehn function f (n). The remainder of the proof follows the same steps as Theorems 10.E.1 and 10.G.1, allowing a reduction to giving estimates on the area of amalgamation and welding relations of some given combinatorial length. The amalgamation relations have a linearly bounded area with respect to their length, by the same trivial argument as in Proposition 10.C.3.Incase Kill(u) ={0}, this provides δ 0 N as an upper bound for the Dehn function. In the context of generalized standard groups, the analogue of Theorem 10.E.2 holds (with the same proof using Gromov’s trick and the results of Section 7.B), with an estimate f (n) instead of n . Accordingly, the same holds for its corollary, The- orem 10.F.2. The remainder of the proof of the generalized tame analogue of Theo- rem 10.F.1 works in the same way: we perform n times a homotopy of area f (n) instead of n , which yields an area nf (n) for welding relators. The concluding step is exactly as in Section 10.G and yields a Dehn function nf (n). 11. Central and hypercentral extensions and exponential Dehn function In this section, unless explicitly speciﬁed, all Lie algebras are ﬁnite-dimensional over a ﬁeld K of characteristic zero. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 199 11.A. Introduction of the section. — The purpose of this section is to prove the neg- ative statements of the introduction in presence of 2-homological obstructions (Theo- rem E.2), gathered in the following theorem. Theorem 11.A.1. (1) If G is a standard solvable group (see Deﬁnition 1.2) satisfying the non-Archimedean 2-homological obstruction, then G is not compactly presented; (2) if G is a standard solvable group satisfying the 2-homological obstruction, then it has an least exponential Dehn function (possibly inﬁnite); (3) if G is a real triangulable group with the 2-homological obstruction, then it has an at least exponential Dehn function. (1) and (2) are proved, by an elementary argument relying on central extensions, in Section 11.B. (3) is much more involved. The reason is that the exponential radical g is not necessarily split in g and the non-vanishing of H (g) does not necessarily yield a 2 0 central extension of g in degree zero (an explicit counterexample is given in Section 11.E). We then need some signiﬁcant amount of work to show that it provides, anyway, a hyper- central extension. 11.B. FC-Central extensions. — We use the following classical deﬁnition, which is a slight weakening of the notion of central extension. Deﬁnition 11.B.1. — Consider an extension (11.B.2)1→ Z→ G→ G→ 1. We say that it is an FC-central extension if i(Z) is FC-central in G, in the sense that every compact subset of i(Z) is contained in a compact subset of G that is invariant under conjugation. This widely used terminology is (lamely) borrowed from the discrete case, in which it stands for “Finite Conjugacy (class)”. In the following, the reader can assume, in a ﬁrst reading, that the FC-central extensions (deﬁned below) are central. The greater generality allows to consider, for in- stance, the case when Z is a nondiscrete locally compact ﬁeld on which the action by conjugation is given by multiplication by elements of modulus 1. Consider now an FC-central extension as in (11.B.2), and assume that G generated −1 by a compact subset S (symmetric with 1); Fix k, set W= Z∩ S ,and W= gW g , k k * k g∈G which is a compact subset of Z by assumption. The following easy lemma, is partly a restatement of [BaMS93, Lemma 5] (which deals with ﬁnitely generated groups and assumes Z is central). 200 YVES CORNULIER, ROMAIN TESSERA Lemma 11.B.3. — Keep the notation of Deﬁnition 11.B.1.Let * γ be any path in the Cayley graph of G with respect to S, joining 1 to an element z of Z.Let γ be the image of * γ in the Cayley graph of G (with respect to the image of S). If * γ can be ﬁlled by m (≤ k)-gons, then z∈ W . Proof.—If * γ can be ﬁlled by m (≤ k)-gons, then z can be written (in the free group −1 ˜ * * over S, hence in G) as z= h r h ,where r , h∈ G, r∈ Ker(G→ G)= Z having i i i i i i=1 −1 ˇˇ length≤ k with respect to S, i.e. r∈ W .Thus h r h∈ W ;hence z∈ W . i k i i k k In Deﬁnition 11.B.1, the group Z may or not be compactly generated; if it is the case, let U a compact generating set of Z, and deﬁne the distortion of Z in G as d (n)= max n, sup{|g|: g∈ Z,|g|≤ n}; G,Z U S if Z is not compactly generated set d (n) =+∞. Note that this function actually de- G,Z pends on S and U as well, but its∼-equivalence class only depends on (G, Z). Proposition 11.B.4. — Given a FC-central extension as in Deﬁnition 11.B.1,if G is compactly presented, then Z is compactly generated and its Dehn function satisﬁes δ (n)0 d (n). G G,Z Proof. — By Lemma 11.B.3, if G is presented by S and relators of length≤ k,then Z is generated by W , which is compact. If U is a compact generating set for Z, then W⊂ U for some .Write d(n)= d (n) (relative to S and U) and δ(n)= δ (n).Consider g∈ Zwith|g|≤ n and|g|= G,Z G S U d(n). Taking * γ to be a path of length≤ n in G joining 1 and g as in Lemma 11.B.3,we obtain that the loop γ in G has length≤ n and area m, and Lemma 11.B.3 implies that d(n) =|g|≤ m.So δ(n)≥ d(n)/. Proof of (1) and (2) in Theorem 11.A.1. — In the setting of (1), we assume that G= U A satisﬁes the non-Archimedean 2-homological obstruction, so that for some j , K is non-Archimedean and the condition Z= H (u ) = 0 means that the action of A on U 2 j 0 j can be lifted to an action on a certain FC-central extension 1→ Z→ U→ U→ 1, j j ˜˜ so that i(Z) is contained and FC-central in[U , U];here u is the blow-up of the graded j j Lie algebra u,asdeﬁnedinSection 8.B,and Z H (u ). Thus it yields an FC-central 2 j extension (11.B.5)1→ Z→ U A→ U A→ 1, ˜˜˜ where U= U× U ;notethat U A is compactly generated. By Proposition 11.B.4, j j j =j it follows that U A is not compactly presented. In the setting of (2), the proof is similar, with K being Archimedean; there is a dif- ference however: in (11.B.5), Z need not be FC-central in U A, because of the possible GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 201 real unipotent part of the A-action. Note that i(Z) is central in U. We then consider an A-irreducible quotient Z= Z/Z of Z and consider the FC-central extension 1→ Z→ U A→ U A→ 1. Since the real group U is globally exponentially distorted (in the sense that all elements of exponential size in U have linear size in U A), it follows that i(Z ) is exponentially dis- j j torted as well and by Proposition 11.B.4, it follows that U A has an at least exponential Dehn function. 11.C. Hypercentral extensions. — To prove Theorem 11.A.1(3), the natural approach seems to start with a real triangulable group G with H (g ) = 0 and ﬁnd a central 2 0 extension of G with exponentially distorted center. If the exponential radical of G is split, i.e. if G= G A for some nilpotent group A, the existence of such a central extension follows by a simple argument similar to that in the proof of Theorem 11.A.1(2). Unfortunately, in general such a central extension does not exist; although there are no simple counterexamples, we construct one in Section 11.E. Nevertheless, in order to prove Theorem 11.A.1(3), the geometric part of the ar- gument is the following variant of Proposition 11.B.4. Proposition 11.C.1. —Let G be a connected triangulable Lie group and G its exponential radical (see Deﬁnition 4.D.2). Suppose that there exists an extension of connected triangulable Lie groups 1 −→ N −→ H −→ G −→ 1 with N hypercentral in H (i.e. the ascending central series of H covers N) and with N∩ H ={1}. Then G has an (at least) exponential Dehn function. Proof. — We ﬁx a compact generating set S in H, and its image S in G. Let h∈ N be an element of linear size n in H and exponential size e in N. Pick a path of size n joining 1 to h in H, i.e. represent h by a word γ= x ... x in H with x∈ S. Push this n 1 n i path forward to G to get a loop of size n in G; let a be itsarea. So in thefreegroup over S, we have −1 γ= g r g n nj nj nj j=1 where r is a relation of G (i.e. represents the identity in G) of bounded size. By a standard argument using van Kampen diagrams (see Lemma 2.D.2), we can choose the size of g nj to be at most≤ C(a+ n), where C is a positive constant only depending on (G, S ).Push this forward to H to get −1 h= g r g , n nj nj nj j=1 202 YVES CORNULIER, ROMAIN TESSERA where r here is a bounded element of N, and g has length≤ C(a+ n) in H. Since the nj nj n −1 action of H on N by conjugation is unipotent, we deduce that the size in N of g r g is nj nj nj polynomially bounded with respect to a+ n,say (a+ n) (uniformly in j ). Therefore n n d+1 h has size (a+ n) . Since (h ) has exponential growth in N, we deduce that (a ) also n n n n grows exponentially. Let us emphasize that at this point, the proof of Theorem 11.A.1(3) is not yet complete. Indeed, given a real triangulable group G with H (g ) = 0, we need to check 2 0 that we can apply Proposition 11.C.1. This is the contents of the following theorem. If V is a G-module, V denotes the set of G-ﬁxed points. Also, see Deﬁnition 4.D.1 for the deﬁnition of g . ∞ G Theorem 11.C.2. —Let G be a triangulable Lie group. Suppose that H (g ) ={0}.Then there exists an extension of connected triangulable Lie groups 1 −→ N −→ H −→ G −→ 1 with N hypercentral in H and with N∩ H ={1}. The theorem will easily follow from the analogous (more general) result about solvable Lie algebras. By epimorphism of Lie algebras we mean a surjective homomorphism, and we de- note it by a two-headed arrow h g.Suchanepimorphismis k-hypercentral if its kernel z is contained in the kth term h of the ascending central series of h;when k= 1, that is, when z is central in h, it is simply called a central epimorphism. Also, if m is a g-module, we write m ={m∈ m :∀g∈ g, gm= 0}. Deﬁnition 11.C.3. — We say that a hypercentral epimorphism h g between Lie algebras ∞∞ has polynomial distortion if the induced epimorphism h g is bijective; otherwise we say it has non-polynomial distortion. The terminology is motivated by the fact that if G, H are triangulable Lie groups and H G is a hypercentral epimorphism, then its kernel is polynomially distorted in H if and only if the corresponding Lie algebra hypercentral epimorphism has polynomial distortion (in the above sense), and otherwise the kernel is exponentially distorted in H. ∞ g Theorem 11.C.4. —Let g be a K-triangulable Lie algebra. Assume that H (g ) ={0}. Then there exists a hypercentral epimorphism h g with non-polynomial distortion. ∞ g∞ The condition H (g ) ={0} can be interpreted as H (g ) ={0},where g is endowed 2 2 0 with a Cartan grading (see Section 4.D, especially Lemma 4.D.8). Theorem 11.C.4 will be proved in Section 11.D. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 203 Proof of Theorem 11.C.2 from Theorem 11.C.4.—Theorem 11.C.4 provides a hyper- central epimorphism h g, with kernel denoted by z, and with non-polynomial distor- tion, i.e. h∩ z ={0}. Since z is hypercentral, the action of g on z is nilpotent, hence triangulable, and since moreover g and z are triangulable, we deduce that h is triangu- lable. Let H→ G be the corresponding surjective homomorphism of triangulable Lie groups and Z its kernel, which is hypercentral. Then H∩ Z ={1}, because the Lie algebra counterpart holds. So the theorem is proved. 11.D. Proof of Theorem 11.C.4. Lemma 11.D.1. —Let g h l be epimorphisms of Lie algebras such that the composite homomorphism is a hypercentral epimorphism. If g l has polynomial distortion then so does h l. Proof. — This is trivial. Lemma 11.D.2. —Let g be a Lie algebra and (T ) be a one-parameter group of unipotent t∈K automorphisms of g.Let h g be a central epimorphism. Then there exists a Lie algebra k with an epimorphism ρ: k h such that the composite homomorphism k g is a central epimorphism, and such that the action of (T ) lifts to a unipotent action on k. Remark 11.D.3. — The conclusion of Lemma 11.D.2 cannot be simpliﬁed by the requirement that k= h, as we can see, for instance, by taking h to be the direct product of the 3-dimensional Heisenberg algebra and a 1-dimensional algebra and a suitable 1- parameter subgroup of unipotent automorphisms of g= h/[h, h]. Proof of Lemma 11.D.2. — Set z= Ker(h g) and denote the Hopf bracket (see Section 8.A.1)by [·,·]: g∧ g→ h. Pick a linear projection π: h→ z and deﬁne b(x∧ y)= π([x, y] ). This gives a linear identiﬁcation of h with g⊕ z, for which the law is given as x , z,x , z=[x , x], b(x∧ x ) 1 1 2 2 1 2 1 2 (in this proof, we write pairsx, z rather than (x, z) for the sake of readability). Observe that T naturally acts on g∧g, preserving Z (g) and B (g).If c∈ g∧g, deﬁne the function 2 2 α: K→ z u → b T c . If d is the dimension of g, let W denote the space of K-polynomial mappings of degree < 2d from K to z; the dimension of W is 2d dim(z).Now t → T is a polynomial 204 YVES CORNULIER, ROMAIN TESSERA of degree < d valued in the space of endomorphisms of g,soisalsopolynomialofdegree < 2d valued in the space of endomorphisms of g∧ g.So α is a polynomial of degree < 2d,from K to z. If c∈ g∧ g is a boundary then α= 0. So α deﬁnes a central extension k= g⊕ W (as a vector space) of g with kernel W, with law x ,ζ,x ,ζ=[x , x],α ,x ,ζ,x ,ζ ∈ g⊕ W. 1 1 2 2 1 2 x∧x 1 1 2 2 1 2 From now on, since elements of W are functions, it will be convenient to write elements of k asx,ζ(u),where u is thought of as an indeterminate. For t∈ K, the automorphism T lifts to an automorphism of k given by t t T x,ζ(u)= T x,ζ(u+ t) . This is obviously a one-parameter subgroup of linear automorphisms; let us check that these are Lie algebra automorphisms (in the computation, for readability we write the brackets as[·;·], with semicolons instead of commas). t t t t T x ,ζ (u); T x ,ζ (u)= T x ,ζ (u+ t); T x ,ζ (u+ t) 1 1 2 2 1 1 2 2 t t t t = T x; T x ,α (u) 1 2 T x∧T x 1 2 t t u t t = T x; T x , b T T x∧ T x 1 2 1 2 t t+u t+u = T[x; x], b T x∧ T x 1 2 1 2 = T[x; x],α (t+ u) 1 2 x∧x 1 2 = T[x; x],α (u) 1 2 x∧x 1 2 = T x ,ζ (u); x ,ζ (u) , 1 1 2 2 so these are Lie algebra automorphisms. Now the mapping k→ h ρ: x,ζ(u) → x,ζ(0) is clearly a surjective linear map; it is also a Lie algebra homomorphism: indeed ρ x ,ζ (u); ρ x ,ζ (u)= x ,ζ (0); x ,ζ (0) 1 1 2 2 1 1 2 2 =[x; x], b(x∧ x ) 1 2 1 2 =[x; x],α (0) 1 2 x∧x 1 2 = ρ[x; x],α (u) 1 2 x∧x 1 2 = ρ x ,ζ (u); x ,ζ (u) 1 1 2 2 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 205 Lemma 11.D.4. — The statement of Lemma 11.D.2 holds true if we replace, in both the hypotheses and the conclusion, central by k-hypercentral. Proof.—Thecase k= 1 was done in Lemma 11.D.2. Decompose h g as h h g,with h h central (with kernel z)and h g (k− 1)-hypercentral. By induction 1 1 1 hypothesis, there exists k with k h such that the composite epimorphism k g is (k− 1)-hypercentral and such that (T ) lifts to k. Consider the ﬁbered product h× k of the two epimorphisms h h and k h , so that the two lines in the diagram below are 1 1 central extensions and both squares commute. 0 z h h 0 0 z h× k k 0 Applying Lemma 11.D.2 again to h× k k,weobtain m h× l so that the h h 1 1 composite epimorphism m k is central and so that (T ) lifts to m. So the composite map m h is the desired homomorphism. We say that a Lie algebra g is spread if it can be written as g= n s where n is nilpotent, s is reductive and acts reductively on n.Itis spreadable if there exists such a decomposition. When g is solvable, s is abelian and a Cartan subalgebra of g is given by the centralizer h= C (s)= C (s)× s. In particular, the s-characteristic decomposition of g n g coincides with the h-characteristic decomposition, and the associated Cartan gradings are the same. This remark is useful when we have to deal with a homomorphism n s→ n s which is the identity on s: indeed such a homomorphism is graded for the Cartan gradings. Proof of Theorem 11.C.4. — We ﬁrst prove the result when g is spread, so g= u d. Let k be the dimension of u/g . We argue by induction on k. ∞∨ Suppose that k= 0. We have a grading of g , valued in d .Considerthe blow-up construction (Lemma 8.B.2): it gives a graded Lie algebra g and a graded surjective map ∞∞ g g with central kernel concentrated in degree zero and isomorphic to H (g ) . 2 0 206 YVES CORNULIER, ROMAIN TESSERA This grading, valued in d , deﬁnes a natural action of d on g and the epimorphism ∞∞ ∞∞ + + g d g d is central. By construction, the kernel H (g ) is contained in (g 2 0 d) so this (hyper)central epimorphism has non-polynomial distortion. Now suppose that k≥ 1. Let n be a codimension 1 ideal of g containing g d. Since n contains g⊕ d, the intersection of n with u is a hyperplane in u .Sothere 0 0 exists a one-dimensional subspace l⊂ u ,suchthat g= n l. Note that the grading of g, ∨∞∞∞ valued in d ,extends that of n and n= g and in particular, H (n ) ={0}. 2 0 By induction hypothesis, there exists a hypercentral epimorphism h n,with non-polynomial distortion. By Lemma 11.D.4, there exists a hypercentral epimorphism ad(l) m h (with kernel z) so that the action of e on n lifts to a unipotent action on m. This corresponds to a nilpotent action of l on m.Let z be the intersection of the ith term of the ascending central series of m with z,so z= z for some .Oneach z /z , the action of l i+1 i is nilpotent and the adjoint action of m is trivial. So the action of m l on each z /z , i+1 i hence on z, is nilpotent. That is, z is hypercentral in m l (this is where the argument would fail with “hypercentral” replaced by “central”). So m l g is the desired hy- percentral epimorphism: by Lemma 11.D.1, m n has non-polynomial distortion and therefore so does m l g. Now the result is proved when g is spread. In general, ﬁx a faithful linear represen- tation g→ gl ,and let h= u d be the splittable hull of g in gl (that is, the subalgebra n n generated by semisimple and nilpotent parts of elements of g for the additive Jordan de- composition, see [Bou, Chap. VII, §5]). If n is any Cartan subalgebra of h,then n= n∩g is a Cartan subalgebra of g [Bou,Chap. VII, §5,Ex. 8].Now n= (u∩ n)+ n ,but every n-weight of h vanishes on u∩n.Sothe n-grading of h extends the n -grading of g (in other ∞∞ words, the embedding g⊂ h is a graded map). In particular, since g= h , this equality is an isomorphism of graded algebras and we deduce that H (h ) ={0}.Soweobtain 2 0 ∞∞ a hypercentral extension m h, with non-polynomial distortion because g= h .By taking the inverse image of g in m, we obtain the desired hypercentral extension of g. 11.E. An example without central extensions. — We prove here that in the conclusion of Theorem 11.C.2, it is not always possible to replace, in the conclusion, hypercentral by central. We begin with the following useful general criterion. Proposition 11.E.1. —Let g be a ﬁnite-dimensional solvable Lie algebra with its Cartan grading. Then g has no central extension with non-polynomial distortion if and only if the image of (Ker(d )∩ (g∧ g )) in H (g) is zero (i.e. it is contained in Im(d )). 2 0 2 0 3 Proof. — Suppose that the image of the above map is nonzero. By the blow-up construction (Lemma 8.B.2), we obtain a central extension g˘ g with kernel H (g) 2 0 concentrated in degree zero. By the assumption, there exist x , y in g, of nonzero opposite i i weights±α ,suchthat x∧ y is a 2-cycle and is nonzero in H (g) . This means that in i i i 2 0 g˘ , the element z=[x , y] is a nonzero element of the central kernel H (g) .So z∈ g˘ i i 2 0 and the central epimorphism g˘ g does not have polynomial distortion. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 207 Conversely, suppose that there exists a central epimorphism g˘ g with non- polynomial distortion, with kernel z. Note that the Cartan grading lifts to g˘ , so that the kernel z is concentrated in degree zero. By assumption, z∩ g˘ contains a nonzero el- ement z. By Lemmas 8.A.3 and 8.A.4,wecan write,in g˘ , z=[x , y] with x , y of i i i i nonzero opposite weights. Thus in g, x∧ y is a nonzero element of H (g) . i i 2 0 Let G be the 15-dimensional R-group of 6× 6 upper triangular matrices of the form ⎛ ⎞ 1 x u u u u 12 13 14 15 16 ⎜ ⎟ 01 0 0 x x 25 26 ⎜ ⎟ ⎜ ⎟ 00 t u u u 3 34 35 36 ⎜ ⎟ (11.E.2) , ⎜ ⎟ 00 0 t u u 4 45 46 ⎜ ⎟ ⎝ ⎠ 00 0 0 1 x 0 0 0001 where t , t are nonzero. Its unipotent radical U consists of elements of the form (11.E.2) 3 4 ˜˜ with t= t= 1 and its exponential radical E consists of elements in U for which x= 3 4 12 x= x= x= 0. If D denotes the (two-dimensional) diagonal subgroup in G, the 25 26 56 ˜˜ quotient G/E is isomorphic to the direct product of D with a 4-dimensional unipotent group (corresponding to coefﬁcients x , x , x , x ). Note that the extension 1→ E→ 12 25 26 56 ˜˜˜ U→ U/E→ 1 is not split. Let Z the 2-dimensional subgroup of U consisting of matrices with all entries zero except u and x . Note that Z is hypercentral and has non-trivial intersection with the 16 26 exponential radical of G. ˜˜ Deﬁne G= G/Z; it is 13-dimensional. The weights of E= E/Z are arranged as follows (the principal weights are in boldface) 14 34 (11.E.3) 13 15 35 36 45 46 and the other basis elements of weight zero in g are 33, 44, 12, 25, 56. We see that e is 2-tame. Besides, H (e) = 0, as 13∧ 36 is a 2-cycle in degree 0 2 0 that is not a 2-boundary, as follows from the observation that e has an obvious nontrivial central extension in degree zero, given by at the level of groups by 1→ Z/Z→ E/Z→ E→ 1, 208 YVES CORNULIER, ROMAIN TESSERA where Z is the one-dimensional subgroup at position 26 (that is, the subgroup of Z con- ˜˜ sisting of matrices with u= 0), which is normalized by Ebut not by U. Since H (g ) ={0},byTheorem 11.C.4 there exists a hypercentral epimorphism 2 0 h→ g with non-polynomial distortion. By contrast, every central epimorphism h→ g has polynomial distortion. This follows from Proposition 11.E.1 and the following propo- sition. Proposition 11.E.4. —Let g be the above 13-dimensional triangulable Lie algebra. Then in H (g) ,the imageof (Ker(d )∩ (g∧ g )) is zero. 2 0 2 0 Proof. — To streamline the notation, we denote by ij the elementary matrix usually denoted by E ,with1at position (i, j) and zero everywhere else (including the diagonal). ij By considering each pair of nonzero opposite weights in (11.E.3), we can describe the map d on a basis of the 4-dimensional space (u∧ u ) . 2 0 13∧ 35 −→−15 13∧ 36 −→ 0, 14∧ 45 −→−15, 14∧ 46 −→ 0; accordingly a basis of (Ker(d )∩ (g∧ g )) is given by 2 0 13∧ 35− 14∧ 45, 13∧ 36, 14∧ 46; we have to check that these are all boundaries; let us snatch them one by one: 12∧ 25∧ 56 −→ 56∧ 15. 13∧ 34∧ 45 −→ 13∧ 45− 14∧ 45 13∧ 35∧ 56 −→ 56∧ 15+ 13∧ 36 14∧ 45∧ 56 −→ 56∧ 15+ 14∧ 46. Combining with Proposition 11.E.1,weget: Corollary 11.E.5. — We have H (g ) ={0}, but there is no central extension of Lie groups 2 0 1 −→ R −→ G −→ G −→ 1 with j(R) exponentially distorted in G. 12. G not 2-tame This section is totally independent from the previous ones. Its main result is Theo- rem 12.C.1, entailing Theorem E.2. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 209 Here we prove that any group satisfying the SOL obstruction has an at least expo- nential Dehn function. The method also provides the result that any group satisfying the non-Archimedean SOL obstruction is not compactly presented. Let us mention that, for instance, Theorem 12.C.1 reproves as particular case (us- ing an embedding as a cocompact lattice) the following theorem of Groves and Hermiller [GH01]: any strictly ascending HNN-extension of a ﬁnitely generated nilpotent group has an exponential Dehn function (the exponential upper bound being given by Corol- lary 3.B.6). 12.A. Combinatorial Stokes formula. Deﬁnition 12.A.1. —Let X be a set and let R be any commutative ring. We call a closed path a sequence c= (c ,..., c ) of points in X with c= c (so we can view it as indexed by Z/nZ). If 0 n 0 n α, β are functions X→ R,wedeﬁne β dα= β(c ) α(c )− α(c ) i i+1 i−1 i∈Z/nZ = β(c )α(c )− β(c )α(c ). i i+1 i+1 i i∈Z/nZ Clearly, this is invariant if we shift indices. The following properties are immediate consequences of the deﬁnition. • (Antisymmetry) We have , , β dα =− αdβ. c c • (Concatenation) If c= c= c and we write c= (c ,..., c ) and c= (c ,..., c ), 0 i n 0 i i n , , , β dα= β dα+ β dα. c c c • (Filiform vanishing) If n= 2 then the integral vanishes. More generally, the inte- gral vanishes when c is ﬁliform, i.e., n is even and c= c for all i. i n−i - - - Indeed, the difference β dα− β dα− β dα is equal to c c c β(c ) α(c )− α(c )+ α(c )− α(c ) 0 i+1 i−1 1 n−1 − α(c )− α(c )− α(c )− α(c )= 0. 1 i−1 i+1 n−1 The ﬁliform vanishing is immediate for n= 2 and follows in general by an induction based on the concatenation formula. 210 YVES CORNULIER, ROMAIN TESSERA Now let us deal with a Cayley graph of a group G with a generating set S, and we consider paths in the graph, that is sequences of vertices linked by edges. Thus any closed path based at 1 can be encoded by a unique element of the free group F , which is a relation (i.e. an element of the kernel of F→ G), and conversely, if r is a relation, we denote by[r] the corresponding closed path based at 1. Note that G acts by left translations on the set of closed paths. The above properties imply the following • (Product of relations) If r, r are relations, we have , , , β dα= β dα+ β dα. [rr][r][r] • (Conjugate of relations) If r is a relation and γ∈ F , , , β dα= β dα. −1 [γ rγ] γ·[r] • (Combinatorial Stokes formula) Suppose that a relation r is written as a product −1 r= γ r γ of conjugates of relations. Then i i i=1 i , , β dα= β dα. [r] γ·[r] i i i=1 The formula for products follows from concatenation if there is no simpliﬁcation in the product rr , and follows by also using the ﬁliform vanishing otherwise. The formula for conjugates also follows using the ﬁliform vanishing. The Stokes formula follows from the two previous by an immediate induction. Remark 12.A.2. — The above combinatorial Stokes formula is indeed analogous to the classical Stokes formula on a disc: here the left-hand term is thought of as an integral along the boundary, while the right-hand term is a discretized integral over the surface. 12.B. Loops in groups of SOL type. — Let K and K be two nondiscrete locally 1 2 compact normed ﬁelds. Consider the group G= (K× K ) −1 Z, 1 2 ( , ) where|| ≥|| > 1, with group law written so that the product depends afﬁnely on 2 K 1 K 2 1 the right term: n −n (x, y, n) x , y , n= x+ x , y+ y , n+ n . 1 2 Write|| =|| with 0 <μ≤ 1. 1 K 2 1 K Now we can also view x and y as the projections to the coordinates in the above description. In the next lemmas, we consider a normed ring K (whose norm is submul- tiplicative, not necessarily multiplicative), and functions A: K→ K,and B: K→ K, 1 2 yielding functions α, β: G→ K deﬁned by α= A◦ x and β= B◦ y. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 211 Lemma 12.B.1. — Suppose that A is 1-Lipschitz and that B satisﬁes the Hölder-like condition |B(s)− B s |≤|s− s| ,∀s, s∈ K . Then β dα is bounded on triangles of bounded diameter, that is, when c ranges over triples (c ) i i∈Z/3Z −1 with c∈ G such all c c belong to some given compact subset of G. i j Proof. — Let us consider a triangle T of bounded diameter (viewed as a closed path of length three), i.e. three points (g , g h, g h ),where h and h are bounded (but not g !). 0 0 0 0 Note that as a consequence of the antisymmetry relation, β dα does not change if we add constants to both α and β . We can therefore assume that α(g )= β(g )= 0. Hence 0 0 β dα= β(g h)α g h− β g h α(g h). 0 0 0 0 In coordinates, suppose that g= (x , y , n ), h= (x, y, n) and h= (x , y , n ).Then g h= 0 0 0 0 0 n−n n−n 0 0 0 0 (x+ x, y+ y, n+ n),and g h= (x+ x , y+ y , n+ n ). Since A(x )= 0 0 0 0 0 0 0 0 1 2 1 2 B(x )= 0, we have −n n−n n 0 0 0 0 β dα= B y+ y A x+ x− B y+ y A x+ x . 0 0 0 0 2 1 2 1 By our assumptions on A and B, we have . . . . −n μ n −n μ n 0 0 0 0 . . β dα ≤| y|| x |+| y|| x| 2 1 2 1 . . μ μ =|y||x |+|y||x|, which is duly bounded when h, h are bounded. Fix n≥ 1. We consider the relation n−n n−1−n−1 γ= t xt yt x t y; 1,n this deﬁnes a closed path of length 4n+ 4, where x, y and t denote (by abuse of notation) the elements (1, 0, 0) , (0, 1, 0) and (0, 0, 1) of G= K× K Z. 1 2 Lemma 12.B.2. —Consider A, B,α,β as introduced before Lemma 12.B.1. Suppose that A(0)= B(1)= 0. Then we have β dα= 2B(0)A . 1,n 212 YVES CORNULIER, ROMAIN TESSERA Proof of Lemma 12.B.2. — To simplify the notation, let us denote by c the closed path of length 4n+ 4deﬁned by γ . In the integral β dα, only those points c for 1,n i which “β dα” is nonzero, i.e. both α(c ) = α(c ) and β(c ) = 0, do contribute. In this i+1 i−1 i example as well as the forthcoming ones, this will make most terms be equal to zero. The closed path c can be decomposed as c= (0, 0, 0), (0, 0, 1),...,(0, 0, n− 1), (0, 0, n)= c , 0 n n n n n c= , 0, n , , 0, n− 1 ,..., , 0, 1 , , 0, 0= c , n+1 2n+1 1 1 1 1 n n n n c= , 1, 0 , , 1, 1 ,..., , 1, n− 1 , , 1, n= c , 2n+2 3n+2 1 1 1 1 c= (0, 1, n), (0, 1, n− 1),...,(0, 1, 1), (0, 1, 0)= c . 3n+3 4n+3 We see that x(c ) = x(c ) only for i= n, n+ 1, 3n+ 2, 3n+ 3. Moreover, for i+1 i−1 i= 3n+ 2, 3n+ 3, y(c )= 1, so B(y(c ))= 0. Thus i i n+1 β dα= B y(c ) A x(c )− A x(c )= 2B(0) A − A(0); i i+1 i−1 i=n whence the result. To illustrate the interest of the notions developed here, note that this is enough to obtain the following result. Proposition 12.B.3. — Under the assumptions above, if 2 = 0 in K , the group G= (K× K ) −1 Z 1 2 ( , ) has at least exponential Dehn function, and if both K and K are ultrametric then G is not compactly 1 2 presented. Remark 12.B.4. — Actually the lower bound in Proposition 12.B.3 is optimal: if either K or K is Archimedean, then G has an exponential Dehn function. Indeed, the 1 2 upper bound follows from Corollary 3.B.6. We use the following convenient language: in a locally compact group G with a compact system of generators S, we say that a sequence of null-homotopic words (w ) in F has asymptotically inﬁnite area if for every R, there exists N(R) such that no w for S n n≥ N(R) is contained in the normal subgroup of F generated by null-homotopic words of length≤ R. By deﬁnition, the non-existence of such a sequence is equivalent to G being compactly presented. Proof of Proposition 12.B.3.—Set I= β dα. 1,n Deﬁne K= K× K . We need to use suitable functions A: K→ K and B: K→ 1 2 1 2 K satisfying the hypotheses of Lemmas 12.B.1 and 12.B.2.For x∈ K ,deﬁne o(x)∈ K 2 1 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 213 to be equal to 1 if|x| < 1and to 0 if|x|≥ 1; also deﬁne (x)= max(1 −|x|, 0)∈ R; also keep in mind that any Archimedean locally compact ﬁeld naturally contains R as a closed subﬁeld. We deﬁne A and B according to whether K and K are Archimedean: 1 2 • K non-Archimedean (K arbitrary): A(x )= (x , 0) and B(x )= (o(x ), 0); 2 1 1 1 2 2 • K and K both Archimedean: A(x )= (x , 0) and B(x )= ((x ), 0); 1 2 1 1 2 2 • K Archimedean, K non-Archimedean: A(x )= (0,|x|) and B(x )= 2 1 1 1 2 (0,(x )). (Note that the cases K Archimedean and not K , and vice versa, cannot be treated 1 2 simultaneously because of the dissymmetry resulting from the condition μ≤ 1.) Since A(0)= B(1)= (0, 0), Lemma 12.B.2 implies that I= 2B(0)A( ),and thus n n I= (2 , 0) in the ﬁrst two cases, and I= (0, 2|| ) in the last case. It is clear that in each case, A is 1-Lipschitz. We have to check the Hölder-like condition for B, which is clear when K is non-Archimedean. If K is Archimedean, 2 2 we need to check the Hölder-like condition for f (x )= max(0, 1 −|x|)∈ R.Indeed,if 2 2 s, s∈ K ,ifboth|s|,|s |≥ 1, then f (s)= f (s );if|s|≤ 1 ≤|s|,then|f (s)− f (s )|= 1 −|s|; then since 1 ≤|s |≤|s− s|+|s|, we deduce 1 −|s|≤|s− s|.If|s− s |≤ 1, we deduce that |f (s)− f (s )|≤|s− s |≤|s− s| If|s− s |≥ 1, we directly see|f (s)− f (s )|= 1 −|s|≤ 1 ≤|s− s| . Finally if both|s|,|s |≤ 1, we have|f (s)− f (s )|=|s|−|s |≤ 1. Thus the inequality is again clear if|s− s |≥ 1, and otherwise|f (s)− f (s )|=|s|−|s |≤|s− s |≤ |s− s| . Therefore by Lemma 12.B.1, for each R there is a bound C(R) on the norm of β dα over any triangle of diameter≤ R. We now again discuss: and K both non-Archimedean, I= (2 , 0): by ultrametricity of K ,and • K 1 2 1 the combinatorial Stokes theorem, C(R) is a bound for the norm of β dα over an arbitrary loop that can be decomposed into triangles of diameter≤ R. Since (|2||| ) goes to inﬁnity, this shows that for every R there exists n such that K 1 0 1 K for every n≥ n , the loop γ cannot be decomposed into triangles of diameter 0 1,n ≤ R. Thus the sequence (γ ) has asymptotically inﬁnite area. This shows that 1,n G is not compactly presented. • K Archimedean, K arbitrary, I= (2 , 0). Fix R so that every combinatorial 1 2 loop in G can be decomposed into triangles of diameter≤ R. Suppose that γ can be decomposed into j triangles of diameter≤ R. By the combinatorial 1,n n Stokes formula (see Section 12.A) and Lemma 12.B.1,|I|≤ C(R)j . It follows that j≥ 2|| /C(R).Hence theareaof γ grows at least exponentially, so n 1 1,n the Dehn function of G grows at least exponentially. • K Archimedean, K non-Archimedean: I= (0, 2|| ). The argument is ex- 2 1 1 actly as in the previous case. Remark 12.B.5. — The assumption that K does not have characteristic two can be removed, but in that case we need to redeﬁne β dα as β(c )(α(c )− α(c )). i i+1 i The drawback of this deﬁnition is that the integral is not invariant under conjugation. 214 YVES CORNULIER, ROMAIN TESSERA However, with the help of Lemma 2.D.2, it is possible to conclude. Since we are not concerned with characteristic two here, we leave the details to the reader. However Proposition 12.B.3 is not enough for our purposes, because we do not only wish to bound below the Dehn function of the group G, but also of various groups H mapping onto G. In general, the loop γ does not lift to a loop in those groups, so we 1,n consider more complicated loops γ in G, which eventually lift to the groups we have in k,n mind. However, to estimate the area, we will go on working in G, because we know how to compute therein, and because obviously the area of a loop in H is bounded below by the area of its image in G. Deﬁne by induction −1−1 γ= γ g γ g . k,n k−1,n k k−1,n k Here, g denotes the element (0, y , 0) in the group G, where the sequence (y ) in K k k i 2 satisﬁes the following property: y= 1 and for any non-empty ﬁnite subset I of integers, . . . . . . y≥ 1. . . i∈I For instance, if K is ultrametric, this is satisﬁed by y= ;if K= R, the constant se- 2 i 2 quence y= 1 works. The sequence (y ) will be ﬁxed once and for all. i i Fix n. We wish to compute, more generally, β dα. Write the path γ as (c ). k,n i k,n Note that for given n, c does not depend on k (because γ is an initial segment of γ ). i k,n k+1,n Write the combinatorial length of γ as λ (λ= 4n+ 4, λ= 2λ+ 2). k,n k,n 1,n k+1,n k,n Lemma 12.B.6. — The number n being ﬁxed, we have (1) There exists a sequence ﬁnite subsets F of the set of positive integers, such that for all i, we have y(c )= y , and satisfying in addition: for all i <λ and all k≥ 1, we have i j k,n j∈F F ⊂{1,..., k}.Moreover y(c ) = 0 (and thus F =∅), unless either i i i – i≤ 2n+ 2,or – i= λ for some j . j,n (2) Assume that 1≤ i≤ n− 1,or n+ 2≤ i≤ 2n+ 2,or i= λ for some j . Then j,n x(c )= x(c ). i−1 i+1 Proof. (1) The sequence (F ) is constructed for i <λ , by induction on k.For k= 1, we i k,n set F =∅ if i≤ 2n− 1and F ={1} if 2n+ 2≤ i≤ 4n+ 3= λ− 1, and it i i 1,n satisﬁes the equality for y(c ) (see the proof of Lemma 12.B.2,where c is made i i explicit for all i≤ λ= 4n+ 4). 1,n GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 215 Now assume that k≥ 2and that F is constructed for i <λ= λ with the i k−1,n required properties. We set F =∅; since c= (0, 0, 0), the condition holds for λ λ i= λ. It remains to deal with i when λ< i <λ ; in this case c= g c ,so k,n i k 2λ−i y(c )= y+ y(c ). Thus if we set F ={k }∪ F , remembering by induction that i k i i 2λ−i F ⊂{1,..., k− 1}, we deduce that y(c )= y ; clearly F ⊂{1,..., k}. 2λ−i i j i j∈F (2) This was already checked for i≤ 2n+ 2 (see the proof of Lemma 12.B.2). In the case i= λ ,wehave c= g and c= g ,so x(c )= x(c ). j,n i−1 j i+1 j+1 i−1 i+1 Lemma 12.B.7. — Under the assumptions of Lemma 12.B.2, we have β dα =|2| . k,n Proof. — By Lemma 12.B.6,ifboth|y(c )| < 1and x(c ) = x(c ),then i= n or i i−1 i+1 i= n+ 1. It follows that the desired integral on γ is the same as the integral on γ k,n 1,n computed in Lemma 12.B.2. 12.C. Groups with the SOL obstruction. Theorem 12.C.1. —Let G be a locally compact, compactly generated group, and suppose there is a continuous surjective homomorphism G→ G= (K× K ) −1 Z, 1 1 2 ( , ) where 0 = 2 in K . Suppose that G has a nilpotent normal subgroup H whose image in G contains 1 1 K× K . Then 1 2 • the Dehn function of G is at least exponential. • if both K and K are ultrametric, then G is not compactly presented. 1 2 Proof.—Let k be the nilpotency length of H and ﬁx k≥ k . Using Lemma 12.B.7 0 0 and arguing as in the proof of Proposition 12.B.3 (using Lemma 12.B.7 instead if Lemma 12.B.2), we obtain that the loops γ , which have linear length with respect to k,n n, have at least exponential area, and asymptotically inﬁnite area in case K and K are 1 2 both ultrametric. n−n ˜˜˜ Lift x, y,and g to elements x˜, y˜, g˜ in H and t to an element t in G; set X= t x˜t ; k k n since H is normal, X∈ H. This lifts γ to a path γ+ based at 1; let v be its value at λ , n k,n k,n k,n k,n −1−1 −1−1 ˜˜ so v= X y˜X y˜ and v= v g˜ v g˜ . We see by an immediate induction that 1,n n k+1,n k,n k k n k,n v belongs to the (k+ 1)th term in the lower central series of H. Since k≥ k , we see that k,n 0 γ+ is a loop of G, of linear length with respect to n, mapping to γ . In particular, its area k,n k,n is at least the area of γ . So we deduce that γ+ has at least exponential area with respect k,n k,n to n, and has asymptotically inﬁnite area in case K and K are both ultrametric. 1 2 We will also need the following variant, in the real case. 216 YVES CORNULIER, ROMAIN TESSERA Theorem 12.C.2. —Let G be a locally compact, compactly generated group, and suppose there is a continuous homomorphism with dense image G→ G= (R× R) R, −t so that the element t∈ R acts by the diagonal matrix ( , ) (≥ > 0). Suppose that G has a 2 1 1 1 2 nilpotent normal subgroup H whose image in G contains R× R. Then the Dehn function of G is at least exponential. Proof. — Since the homomorphism has dense image containing R× R,there ex- ists some element t mapping to an element t of the form (0, 0,τ) with τ> 0. Changing the parameterization of G if necessary (replacing by for i= 1, 2), we can suppose that τ= 1. Then lift x and y and pursue the proof exactly as in the proof of Theo- rem 12.C.1. Remark 12.C.3. — To summarize the proof, the lower exponential bound is ob- tained by ﬁnding two functions α, β on G such that the integral β dα is bounded on triangles of bounded diameter, and a sequence (γ ) of combinatorial loops of linear di- ameter such that β dα grows exponentially. This approach actually also provides a lower bound on the homological Dehn function [Ger92, BaMS93, Ger99] as well. Let us recall the deﬁnition. Let G be a locally compact group with a generating set S and a subset R of the kernel of F→ G consist- ing of relations of bounded length, yielding a polygonal complex structure with oriented edges and 2-faces. Let A denote either Z or R.For i= 0, 1, 2, let C (G, A) be the A- module freely spanned by the set of vertices, resp. oriented edges, resp. oriented 2-faces. Endow each C (G, A) with the norm. There are usual boundary operators ∂ ∂ 2 1 C (G, A)→ C (G, A)→ C (G, A). 2 1 0 satisfying ∂◦ ∂= 0. If Z (G, A) is the kernel of ∂ , then it is easy to extend, by linearity, 1 2 1 1 the deﬁnition of β dα (from Section 12.A)to c∈ Z (G, A). Following [Ger99], deﬁne, for c∈ Z (G, A) HFill (c)= inf$P$: P∈ C (G, A), ∂ (P)= c . 1 2 2 G,S,R and A A Hδ (n)= sup HFill(z): z∈ Z (G, Z),$z$≤ n . 1 1 G,S,R G,S,R Clearly, if c is a basis element (so that its area makes sense) R Z G,S,R HFill(c)≤ HFill(c)≤ area (c) ≤∞; G,S,R G,S,R it follows that R Z Hδ (n)≤ Hδ (n)≤ δ (n), G,S,R G,S,R G,S,R GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 217 where δ is the smallest superadditive function greater or equal to δ . (Examples of G,S,R G,S,R non-superadditive Dehn functions of ﬁnite presentations of groups are given in [GS99]; however the question, raised in [GS99], whether any Dehn function of a ﬁnitely pre- sented group is asymptotically equivalent to a superadditive function, is still open.) The function HFill(c) is called the A-homological Dehn function of (G, R, S) G,S,R (the function HFill(c) is called abelianized isoperimetric function in [BaMS93]). If G,S,R ﬁnite, it can be shown by routine arguments that its≈-asymptotic behavior only depends on G. Some Bestvina–Brady groups [BeBr97] provide examples of ﬁnitely generated groups with ﬁnite homological Dehn function but inﬁnite Dehn function. Until recently, no example of a compactly presented group was known for which the integral (or even real) homological Dehn function is not equivalent to the integral homological Dehn func- tion; the issue was raised, for ﬁnitely presented groups, both in [BaMS93, p. 536] and [Ger99, p. 1]; the ﬁrst examples have ﬁnally been obtained by Abrams, Brady, Dani and Young in [ABDY13]. 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Publisher: Springer Journals
Copyright: Copyright © 2016 by IHES and Springer-Verlag Berlin Heidelberg
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-016-0087-3
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Abstract

GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS by YVES CORNULIER and ROMAIN TESSERA ABSTRACT We study the Dehn function of connected Lie groups. We show that this function is always exponential or poly- nomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local ﬁelds, uses and extends Abels’ theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these groups. CONTENTS 1. Introduction ...................................................... 80 1.1. Dehn function of Lie groups ......................................... 80 1.2. Riemannian versus combinatorial Dehn function of Lie groups ...................... 82 1.3. Main results . ................................................. 83 1.4. The obstructions as “computable” invariants of the Lie algebra ...................... 88 1.5. Examples .................................................... 89 1.6. Description of the strategy ........................................... 93 1.7. Sketch of the proof of Theorem F ....................................... 94 1.8. Introduction to the Lie algebra chapters ................................... 97 1.9. Guidelines ................................................... 100 2. Dehn function: preliminaries and ﬁrst reductions .................................. 101 2.A. Asymptotic comparison ............................................ 101 2.B. Deﬁnition of Dehn function .......................................... 101 2.C. Dehn vs Riemannian Dehn .......................................... 103 2.D. Two combinatorial lemmas on the Dehn function .............................. 105 2.E. Gromov’s trick ................................................. 106 3. Geometry of Lie groups: metric reductions and exponential upper bounds . . . ................. 108 3.A. Reduction to triangulable groups ....................................... 108 3.B. At most exponential Dehn function ...................................... 108 3.C. Reduction to split triangulable groups .................................... 112 4. Gradings and tameness conditions .......................................... 114 4.A. Grading in a representation .......................................... 114 4.B. Grading in a standard solvable group .................................... 116 4.C. Tameness conditions in standard solvable groups .............................. 118 4.D. Cartan grading and weights .......................................... 121 5. Algebraic preliminaries about nilpotent groups and Lie algebras ......................... 124 5.A. Divisibility and Malcev’s theorem ....................................... 124 5.B. Powers and commutators in nilpotent groups ................................ 125 5.C. Lazard’s formulas ............................................... 129 6. Dehn function of tame and stably 2-tame groups .................................. 130 6.A. Tame groups .................................................. 130 6.B. Length in nilpotent groups .......................................... 134 6.C. Application of Gromov’s trick to standard solvable groups ......................... 138 6.D. Dehn function of stably 2-tame groups .................................... 139 6.E. Generalized tame groups ........................................... 140 7. Estimates of areas using algebraic presentations ................................... 143 7.A. Afﬁne norm on varieties over normed ﬁelds ................................. 143 7.B. Area of words of bounded combinatorial length . .............................. 146 8. Central extensions of graded Lie algebras ...................................... 153 8.A. Basic conventions ............................................... 153 The authors are supported by ANR Project GAMME (ANR-14-CE25-0004). DOI 10.1007/s10240-016-0087-3 80 YVES CORNULIER, ROMAIN TESSERA 8.B. The blow-up .................................................. 156 8.C. Homology and restriction of scalars ..................................... 159 8.D. Auxiliary descriptions of H (g) and Kill(g) ................................ 165 2 0 0 9. Abels’ multiamalgam ................................................. 173 9.A. 2-tameness . .................................................. 173 9.B. Lemmas related to 2-tameness ........................................ 173 9.C. Multiamalgams of Lie algebras ........................................ 175 9.D. Multiamalgams of groups ........................................... 179 10. Presentations of standard solvable groups and their Dehn function ........................ 182 10.A. Outline of the section . ............................................ 182 10.B. Abels’ multiamalgam ............................................. 184 10.C. Algebraic and geometric presentations of the multiamalgam ........................ 185 10.D. Presentation of the group in the 2-tame case . ................................ 187 10.E. Quadratic estimates and concluding step for standard solvable groups with zero Killing module .... 190 10.F. Area of welding relations ........................................... 191 10.G. Concluding step for standard solvable groups ................................ 196 10.H. Dehn function of generalized standard solvable groups ........................... 198 11. Central and hypercentral extensions and exponential Dehn function ....................... 198 11.A. Introduction of the section .......................................... 199 11.B. FC-Central extensions ............................................. 199 11.C. Hypercentral extensions ............................................ 201 11.D. Proof of Theorem 11.C.4 ........................................... 203 11.E. An example without central extensions .................................... 206 12. G not 2-tame ...................................................... 208 12.A. Combinatorial Stokes formula ........................................ 209 12.B. Loops in groups of SOL type ......................................... 210 12.C. Groups with the SOL obstruction . ..................................... 215 Acknowledgements ..................................................... 217 References ......................................................... 217 1. Introduction 1.1. Dehn function of Lie groups. — The object of study in this paper is the Dehn function of connected Lie groups. For a simply connected Lie group G endowed with a left-invariant Riemannian metric, this can be deﬁned as follows: the area of a loop γ is the inﬁmum of areas of ﬁlling discs, and the Dehn function δ (r) is the supremum of areas of loops of length at most r. The asymptotic behavior of δ (when r →+∞) actually does not depend on the choice of a left-invariant Riemannian metric. If G is an arbitrary connected Lie group and K a compact subgroup such that G/K is simply connected (e.g. K is a maximal compact subgroup, in which case G/K is diffeomorphic to a Euclidean space), we can endow G/K with a G-invariant Riemannian metric and thus deﬁne the Dehn function δ (r) in the same way; its asymptotic behavior depends G/K only on G, neither on K nor on the choice of the invariant Riemannian metric, and is called the Dehn function of G. Let us provide some classical illustrating examples. If for some maximal compact subgroup K, the space G/K has a negatively curved G-invariant Riemannian metric, then the Dehn function of G has exactly linear growth. Otherwise, G is not Gromov- hyperbolic, and by a very general argument due to Bowditch [Bo95](notspeciﬁc to Lie groups), the Dehn function is known to be at least quadratic. On the other hand, the Dehn function is at most quadratic whenever G/K can be endowed with a non-positively curved GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 81 invariant Riemannian metric, notably when G is reductive. It is worth emphasizing that many simply connected Lie groups G fail to have a non-positively curved homogeneous space G/K and nevertheless have a quadratic Dehn function. Characterizing Lie groups with a quadratic Dehn function is a very challenging problem, even in the setting of nilpotent Lie groups. Indeed, although connected nilpotent Lie groups have an at most polynomial Dehn function, there are examples with Dehn function of polynomial growth with arbitrary integer degree. Besides, the Dehn function of a connected Lie group is at most exponential, the prototypical example of a Lie group with an exponential Dehn function being the three-dimensional SOL group. A simple main consequence of the results we describe below is the following theo- rem. Theorem A. —Let G be a connected Lie group. Then the Dehn function of G is either exponential or polynomially bounded. Using the well-known fact that polycyclic groups are virtually cocompact lattices in connected Lie groups, we deduce Corollary B. — The Dehn function of a polycyclic group is either exponential or polynomially bounded. Let us mention that in both results, “polynomially bounded” cannot be improved to “of polynomial growth”, since S. Wenger [We11] has exhibited some simply connected 2 2 nilpotent Lie groups (with lattices) with a Dehn function satisfying n≺ δ(n) n log n. Our results are more precise than Theorem A: we characterize algebraically which ones have a polynomially bounded or exponential Dehn function. In order to state the re- sult, we describe below two “obstructions” implying exponential Dehn function; the ﬁrst being related to SOL, and the second to homology in degree 2; these two obstructions appeared in a related context in Abels’ seminal work on p-adic algebraic groups [Ab87]. We prove that each of these obstructions indeed implies that the Dehn function has an exponential growth, and conversely that if none of these obstructions is fulﬁlled, then the group has an at most polynomial Dehn function, proving in a large number of cases that the Dehn function is at most quadratic or cubic. These results can appear as unexpected. Indeed, it was suggested by Gromov [Gro93,5.A ] that the only obstruction should be related to SOL. This has been proved in several important cases [Gro93, Dru04, LP04] but turns out to be false in general. Our approach relies on the dynamical structure arising from the action of G on itself by conjugation. A crucial role is played by some naturally deﬁned subgroups that are contracted by suitable elements. The presence of these subgroups is a non-discrete feature, which is invisible in any cocompact lattice (when such lattices exist). In particular, it sounds unlikely that Corollary B can be proved directly with no reference to ambient 82 YVES CORNULIER, ROMAIN TESSERA Lie groups, and the obstructions themselves are not convenient to state directly in terms of the structure of those discrete polycyclic groups. The remainder of this introduction is organized as follows: in Section 1.2,wede- ﬁne a combinatorial Dehn function for compactly presented locally compact groups, and use it to state a version of Theorem A for algebraic p-adic groups. Then Section 1.3 is dedicated to our main results. The most difﬁcult part of the main theorem is the fact that in the absence of the two obstructions, the Dehn function is polynomially bounded. In Section 1.4, we describe a useful characterization of these obstructions in terms of Lie alge- bra gradings. We apply these results to various concrete examples in Section 1.5. We outline the general strategy in Section 1.6. We introduce an additional condition under which we prove that the Dehn function is quadratic (Theorem F). In Section 1.7, we sketch the proof of this theorem; this allows us emphasize the key ideas of our approach. A substan- tial part of our work is purely algebraic and possibly of interest for other purposes, we introduce it independently in Section 1.8. 1.2. Riemannian versus combinatorial Dehn function of Lie groups. — The previous ap- proaches consisted of either working with groups admitting a cocompact lattice and using combinatorial methods, or using the Riemannian deﬁnition. Our approach, initiated in [CT10], consists of extending the combinatorial language and methods to general locally compact compactly generated groups. In particular, Lie groups are treated as combina- torial objects, i.e. groups endowed with a compact generating set and the corresponding Cayley graph. The object of study is the combinatorial Dehn function, usually deﬁned for discrete groups, which turns out to be asymptotically equivalent to its Riemannian coun- terpart. This unifying approach allows to treat p-adic algebraic groups and connected Lie groups on the same footing. We now give the combinatorial deﬁnition of Dehn function (rechristening the above deﬁnition of Dehn function as Riemannian Dehn function); see Section 2.B for more details. Let G be a locally compact group, generated by a compact subset S. Let F be the free group over the (abstract) set S and F→ G the natural epimorphism, and K its kernel (its elements are called relations). We say that G is compactly presented if for some ,K is generated, as a normal subgroup of F , by the set K of elements with length at most −1 with respect to S, or equivalently if K is generated, as a group, by the union gK g g∈F of conjugates of K in F ; this does not depend on the choice of S; the subset K is called a set of relators. Assuming this, if x∈ K, the area of x is by deﬁnition the number area(x) de- −1 ﬁned as its length with respect to gK g . Finally, the Dehn function of G is deﬁned g∈F as δ(n)= sup area(x): x∈ K,|x|≤ n . In the discrete setting (S ﬁnite), this function takes ﬁnite values, and this remains true in the locally compact setting. If G is not compactly presented, a good convention is to set δ(n) =+∞ for all n. The Dehn function of a compactly presented group G depends on GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 83 the choices of S and , but its asymptotic behavior does not. In addition, for a simply connected Lie group, the combinatorial and Riemannian Dehn functions have the same asymptotic behavior, see Proposition 2.C.1. With this deﬁnition at hand, we can now state a version of Theorem A in a non- Archimedean setting. Theorem C. —Let G be an algebraic group over some p-adic ﬁeld. Then the Dehn function of G is at most cubic, or G is not compactly presented. Before providing more detailed statements, let us compare Theorems A and C.Itis helpful to have in mind a certain analogy between Archimedean and non-Archimedean groups, where exponential Dehn function corresponds to not compactly presented. On the other hand, a striking difference between these two theorems is the absence for p- adic groups of polynomial Dehn functions of arbitrary degree. The explanation of this fact can be summarized as follows. In the connected Lie group setting, Dehn functions of “high polynomial degree” witness to the presence of simply connected non-abelian nilpotent quotients, see Theorem 10.H.1 for a precise statement. By way of contrast, any totally disconnected, compactly generated locally compact nilpotent group is compact- by-discrete; and if a compactly generated locally compact group G is isomorphic to the -points of some p-adic algebraic group, then any nilpotent quotient of G is group of Q compact-by-abelian. 1.3. Main results. — We now turn to more comprehensive statements. Let us ﬁrst introduce the two main classes of groups we will be considering in the sequel. Deﬁnition 1.1. —A real triangulable group is a Lie group isomorphic to a closed connected group of real triangular matrices. Equivalently, it is a simply connected solvable group in which for every g, the adjoint operator Ad(g) has only real eigenvalues. It can be shown that every connected Lie group G is quasi-isometric to a real triangulable Lie group. Namely, there exist compactly generated Lie groups G , G , G 1 2 3 and maps G← G→ G← G , 1 2 3 where G is real triangulable, and each arrow is a proper continuous homomorphism with cocompact image and thus is a quasi-isometry, see Lemma 3.A.1. Let A be an abelian group and consider a representation of A on a K-vector space V, where K is a ﬁnite product of complete normed ﬁelds. Let V be the largest A-equivariant quotient of V on which A acts with only eigenvalues of modulus one. Deﬁnition 1.2. — A locally compact group is a standard solvable group if it is topologically isomorphic to a semidirect product U A so that 84 YVES CORNULIER, ROMAIN TESSERA (1) A is a compactly generated locally compact abelian group (2) U decomposes as a ﬁnite direct product U ,where each U is normalized by A and can i i be written as U= U (K ),where U is a unipotent group over some nondiscrete locally i i i i compact ﬁeld of characteristic zero K ; (3) (U/[U, U]) ={0}. For a group G satisfying (1) and (2), condition (3) implies that G is compactly gener- ated, and conversely if U is totally disconnected, the failure of condition (3) implies that G is not compactly generated. If G is a compactly generated p-adic group as in Theorem C, then it has a Zariski closed cocompact subgroup which is a standard solvable group (with a single i and K= Q ). Many real Lie groups have a closed cocompact standard solvable i p group; however, for instance, a simply connected nilpotent Lie group is not standard solv- able unless it is abelian. We now introduce a very special but important class of standard solvable groups. Deﬁnition 1.3. —A group of SOL type is group U A,where U= K× K ,where K , 1 2 1 ∗∗ K are nondiscrete locally compact ﬁelds of characteristic zero, and A⊂ K× K is a closed subgroup 1 2 ∗∗ of K× K containing, as a cocompact subgroup, the cyclic group generated by some element (t , t ) with 1 2 1 2 |t| > 1 >|t|. Note that this is a standard solvable group. We call it a non-Archimedean group 1 2 of SOL type if both K and K are non-Archimedean. 1 2 −1 Example 1.4. —If K= K= K and A is the set of pairs (t, t ), then G is the usual 1 2 k− group SOL(K). More generally, A is the set of pairs (t , t ) where (k,) is aﬁxedpairof positive integers, then this provides another group, which is unimodular if and only k= . Another example is (R× Q ) Z,where Z acts as the cyclic subgroup generated by (p, p) (note that|p| > 1 >|p| ); the latter contains the Baumslag-Solitar group Z[1/p] Z as R Q a cocompact lattice. Also, deﬁne, for λ> 0, the group SOL as the semidirect product R R where λ∗ 2 R is identiﬁed with the subgroup{(t, t ): t > 0} of (R ) .Notethat SOL has index 2 in SOL(R); there are obvious isomorphisms SOL SOL−1,and the SOL ,for λ≥ 1, λ λ λ are pairwise non-isomorphic. These are the only real triangulable groups of SOL type. Deﬁnition 1.5. — (SOL obstruction) A locally compact group has the SOL obstruction (resp. non-Archimedean SOL obstruction) if it admits a homomorphism with dense image to a group of SOL type (resp. non-Archimedean SOL type). That the SOL obstruction implies that the Dehn function grows at least exponen- tially is the contents of the forthcoming Theorems E.1 and E.2. In the positive direction, we start with following result, which generalizes [Gro93, 5.A ], [Dru04, Theorem 1.1 (2)], [LP04]and [CT10]. Theorem D. —Let G= U A be a standard solvable group. Suppose that every closed subgroup of G containing A (thus of the form V A with V a closed A-invariant subgroup of U)is GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 85 a standard solvable group and does not satisfy the SOL obstruction. Then G has an at most quadratic Dehn function. The main result of [CT10] is essentially the case when the normal subgroup U is abelian (but on the other hand works in arbitrary characteristic), which itself generalizes results of Gromov [Gro93], Dru¸ tu [Dru04], Leuzinger-Pittet [LP04]. Theorem D turns out to be a particular instance of the much more general Theo- rem F below, but is considerably easier: the material is the length estimates of the begin- ning of Section 6 and Gromov’s trick described in Section 2.E. A direct proof of Theo- rem D is given in Section 6.D. In [Gro93,5.B ], quoth Gromov, “We conclude our discussion on lower and upper bounds for ﬁlling area by a somewhat pessimistic note. The present methods lead to satisfactory results only in a few special cases even in the friendly geometric surroundings of solvable and nilpotent groups.” Indeed, for standard solvable groups without the SOL obstruction, it seems that Theorem D is the best result that can be gotten without bringing forward new ideas. The ﬁrst example of a standard solvable group without the SOL obstruction but not covered by Theorem D is Abels’ group A (K) (see the previous subsection). In this particular ex- ample, the authors obtain a quadratic upper bound for the Dehn function in [CT13]. In this case, the group is tractable enough to work with explicit matrices, but such a pedes- trian approach becomes hopelessly intricate in an arbitrary group as in Theorem E.4. In a more general context, we have to deal with groups without the SOL ob- struction but not satisfying the assumptions of Theorem D. In this context, we have to introduce the 2-homological obstruction. For this, we need to recall a fundamental no- tion introduced and studied by Guivarc’h [Gui80]and laterbyOsin[Os02]. Let G be a real triangulable group. Its exponential radical G is deﬁned as the intersection of its lower central series and actually consists of the exponentially distorted elements in G. Let g be its Lie algebra. In the case of a standard solvable group, the role of exponential radical is played by U itself (it can be checked to be equal to the derived subgroup of G, so is a characteristic subgroup). Deﬁnition 1.6 (2-homological obstruction). • The real triangulable group G is said to satisfy the 2-homological obstruction if ∞∞ H (g ) ={0}, or equivalently if the A-action on H (g ) has some nonzero invariant 2 0 2 vector. • The standard solvable group G= U A is said to satisfy the 2-homological obstruction if H (u) ={0}, that is to say, H (u ) ={0} for some j . If moreover j can be chosen so that 2 0 2 j 0 K is non-Archimedean, we call it the non-Archimedean 2-homological obstruction. In most cases, including standard solvable groups, the 2-homological obstructions can be characterized by the existence of suitable central extensions. For instance, if a real 86 YVES CORNULIER, ROMAIN TESSERA triangulable group G has a central extension G, also real triangulable, with nontrivial kernel Z such that Z⊂ (G) then it satisﬁes the 2-homological obstruction. The converse is true when G admits a semidirect decomposition G N, but nevertheless does not in general, see Section 11.E. We are now able to state our main theorem, which immediately entails Theo- rems A and C. Theorem E. —Let G be a real triangulable group, or a standard solvable group. • if G satisﬁes one of the two non-Archimedean (SOL or 2-homological) obstructions, then G is not compactly presented; • otherwise G is compactly presented and has an at most exponential Dehn function. Moreover, in this case – if G satisﬁes one of the two (SOL or 2-homological) obstructions, then G has an exponential Dehn function; – if G satisﬁes none of the obstructions, then it has a polynomially bounded Dehn function; in the case of a standard solvable group, the Dehn function is at most cubic. This result can be seen as both a generalization and a strengthening of the follow- ing seminal result of Abels [Ab87]. Theorem (Abels). — Let G be a standard solvable group over a p-adic ﬁeld. Then G is compactly presented if and only if it satisﬁes none of the non-Archimedean obstructions. Let us split Theorem E into several independent statements. The ﬁrst two provide lower bounds and the last two provide upper bounds on the Dehn function. Theorem E.1. —Let G be a standard solvable or real triangulable group. If G satisﬁes the SOL (resp. non-Archimedean SOL) obstruction, then G has an at least exponential Dehn function (resp. is not compactly presented). We provide a uniﬁed proof of both assertions of Theorem E.1 in Section 12. Note that the non-Archimedean case is essentially contained in the “only if ” (easier) part of Abels’ theo- rem above, itself inspired by previous work of Bieri-Strebel, notably [BiS78,Theorem A]. Part of the proof consists in estimating the size of loops in the groups of SOL type, where our proof is inspired by the original case of the real 3-dimensional SOL group, due to Thurston [ECHLPT92], which uses integration of a well-chosen differential form. Our method in Section 12 is based on a discretization of this argument, leading to both a simpliﬁcation and a generalization of the argument. Theorem E.2. —Let G be a standard solvable or real triangulable group. If G satisﬁes the 2-homological (resp. non-Archimedean 2-homological) obstruction, then G has an at least exponential Dehn function (resp. is not compactly presented). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 87 The case of standard solvable groups reduces, after a minor reduction, to a simple and classical central extension argument, see Section 11.B. The case of real triangulable groups is considerably more difﬁcult; in the absence of splitting of the exponential rad- ical, we construct an “exponentially distorted hypercentral extension”. This is done in Section 11. Theorem E.3. —Let G be a real triangulable group. Then G has an at most exponential Dehn function. ◦◦ Let G= U A be a standard solvable group and U the identity component in U.If G/U is compactly presented, then G is compactly presented with an at most exponential Dehn function. Since compact presentability is stable under taking extensions [Ab72], for an arbi- trary locally compact group G with a closed connected normal subgroup C, it is true that G is compactly presented if and only G/C is compactly presented. We do not know if this can be generalized to the statement that if G/C has Dehn function f (n), then G has Dehn function max(f (n), exp(n)).Theorem E.3, which follows from Theorem 3.B.1 and Corollary 3.B.6, contains two particular instances where the latter assertion holds, which are enough for our purposes. The ﬁrst instance, namely that every connected Lie group has an at most exponential Dehn function, was asserted by Gromov, with a sketch of proof [Gro93, Corollary 3.F ]. Example 1.7. —Fix n∈ Z with|n|≥ 2. Consider the group G= (R× Q ) Z, n n n where Q is the product of Q where p ranges over distinct primes divisors of n.Here n p ◦◦ R and G/U Q Z, which, as a hyperbolic group (it is an HNN ascending n n extension of the compact group Z ), has a linear Dehn function. So, by Theorem E.3,G n n has an at most exponential Dehn function. Since|n|≥ 2, it admits a prime factor p,so G admits the group of SOL type (R× Q ) Z as a quotient, and therefore G satisﬁes the p n n SOL obstruction and thus has an at least exponential Dehn function by Theorem E.1. We conclude that G has an exponential Dehn function. This provides a new proof that its lattice, the Baumslag-Solitar group −1 n BS(1, n)= t, x| txt= x , has an exponential Dehn function. This is actually true for arbitrary Baumslag-Solitar groups BS(m, n),|m| =|n|, for which the exponential upper bound was ﬁrst established in [ECHLPT92, Theorem 7.3.4 and Example 7.4.1] and independently in [BGSS92], and the exponential lower bound, attributed to Thurston, was obtained in [ECHLPT92, Example 7.4.1]. Let us provide a useful corollary of Theorems E.1 and E.3. Corollary E.3.a. —Let G= U A be a standard solvable group in which A has rank 1 (i.e., has a closed inﬁnite cyclic cocompact subgroup). Then exactly one of the following occurs 88 YVES CORNULIER, ROMAIN TESSERA • G satisﬁes the non-Archimedean SOL obstruction and thus is not compactly presented; • G satisﬁes the SOL obstruction but not the non-Archimedean one; it is compactly presented with an exponential Dehn function; • G does not satisfy the SOL obstruction; it has a linear Dehn function and is Gromov-hyperbolic. It is indeed an observation that if A has rank 1 and G does not satisfy the SOL obstruction, then some element of A acts on G as a compaction (see Deﬁnition 4.B.4) and it follows from [CCMT15] that G is Gromov-hyperbolic, or equivalently has a linear Dehn function. In this special case where A has rank 1, the 2-homological obstruction, which may hold or not hold, implies the SOL obstruction and is accordingly unnecessary to consider; see also Theorem D. Turning back to Theorem E, the fourth and most involved of all the steps is the following. Theorem E.4. —Let G be a standard solvable (resp. real triangulable) group not satisfying nei- ther the SOL nor the 2-homological obstructions. Then G has an at most cubic (resp. at most polynomial) Dehn function. The proof of Theorem E.4 for standard solvable groups is done in Section 6, relying on algebraic preliminaries, occupying Sections 8 and 9. We actually obtain, with some additional work, a similar statement for “generalized standard solvable groups”, where A is replaced by some nilpotent compactly generated group N (see Theorem 10.H.1). The case of real triangulable groups requires an additional step, namely a reduction to the case where the exponential radical is split, in which case the group is general- ized standard solvable. This reduction is performed in Section 3.C, and relies on results from [C11]. 1.4. The obstructions as “computable” invariants of the Lie algebra. — The obstructions were introduced above in a convenient way for expository reasons, but the natural frame- work to deal with them uses the language of graded Lie algebras, which we now de- scribe. Let G be either a standard solvable group U A or a real triangulable group with exponential radical also denoted, for convenience, by U. Let u be the Lie algebra of U; it is a Lie algebra (over a ﬁnite product of nondiscrete locally compact ﬁelds of characteristic zero). It is, in a natural way, a graded Lie algebra. In both cases, the grading takes values into a ﬁnite-dimensional real vector space, namely Hom(G/U, R). It is introduced in Section 4.B, relying on Theorem 4.A.2 (see Section 1.5 for a representative particular case.) In this setting, there are useful restatements (see Propositions 4.C.3 and 4.D.9)ofthe obstructions. We say that α∈ Hom(G/U, R) is a weight of G (or of U, when u is endowed with the grading) if u = 0 and is a principal weight of G if α is a weight of U/[U, U].The deﬁnitions imply that 0 is not a principal weight (although it can be a weight). We say GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 89 that two nonzero weights α, β are quasi-opposite if 0 ∈[α, β], i.e., β =−tα for some t > 0. We write U= U× U as the product of its Archimedean and non-Archimedean parts. a na Then we have the following restatements: • G satisﬁes the SOL obstruction⇔ U admits two quasi-opposite principal weights (Propositions 4.C.3 and 4.D.9); • G satisﬁes the non-Archimedean SOL obstruction⇔ U admits two quasi- na opposite principal weights (Proposition 4.C.3 applied to G/G ); • G satisﬁes the 2-homological obstruction⇔ H (u) ={0}; 2 0 • G satisﬁes the non-Archimedean 2-homological obstruction⇔ H (u ) ={0}; 2 na 0 • G fails to satisfy the assumption of Theorem D⇔ 0 belongs to the interval joining some pair of weights. Here, H (u) denotes the homology of the Lie algebra u; the grading on u in the real vector space Hom(G/U, R) canonically induces a grading of H (u) in the same space (see Section 8.A), and H (u) is its component in degree zero. 2 0 Another important module associated to u is Kill(u), the quotient of the second symmetric power u u by the submodule generated by elements of the form[x, y] z− x [y, z]; thus, in case of a single ﬁeld (that is, U= U in Deﬁnition 1.2), the invariant quadratic forms on u are elements in the dual of Kill(u). Theorem F. —Let G= U A be a standard solvable group not satisfying any of the SOL or 2-homological obstructions. Suppose in addition that Kill(u) ={0}. Then G has an at most quadratic Dehn function (thus exactly quadratic if A has rank at least two). Theorem F is proved in Section 10.E.Partofitisprovedalong with Theorem E.4 and involves the same difﬁculties, except that the welding relations do not appear. The condition Kill(u) ={0} corresponds to the existence of certain central exten- sions of G as a discrete group. Using asymptotic cones, it will be shown in a subsequent paper that it implies, in many cases (without the SOL and 2-homological obstructions), that the Dehn function grows strictly faster than a quadratic function. 1.5. Examples. — Let us give a few examples. All are standard solvable con- nected Lie groups G= U A so that the action of A on the Lie algebra u of U is R-diagonalizable. In this context, we call Hom(A, R) the weight space. The grading of the Lie algebra u in Hom(A, R) is given by −1 α(v) u= u∈ u |∀v∈ A,v uv= e u . When we write the set of weights, we use boldface for the set of principal weights. We underline the zero weight (or just denote to mark zero if zero is not a weight). 90 YVES CORNULIER, ROMAIN TESSERA 1.5.1. Groups of SOL type. — For a group of SOL type (Deﬁnition 1.3), the weight space is a line and the weights lie on opposite sides of zero: 1 2 By deﬁnition it satisﬁes the SOL obstruction. On the other hand, it satisﬁes the 2-homological obstruction only in a few special cases. For instance, (R× Q ) Z does p p not satisfy the 2-homological obstruction, and the real group SOL (see Example 1.4) satisﬁes the 2-homological obstruction only for λ= 1. 3 2 2 3 1.5.2. Gromov’s higher SOL groups. — For the group R R where R acts on R as the group of diagonal matrices with positive diagonal entries and determinant one (often called higher-dimensional SOL group, but not of SOL type nor even satisfying the SOL obstruction according to our conventions), the weight space is a plane in which the weights form a triangle whose center of gravity is zero: Since there are no opposite weights, we have (u⊗ u)= 0 and therefore H (u) 0 2 0 and Kill(u) (which are subquotients of (u⊗ u) ) are also zero. It was stated with a sketch 0 0 of proof by Gromov that this group has a quadratic Dehn function [Gro93,5.A ]. Dru¸ tu 3+ε obtained in [Dru98, Corollary 4.18] that it has a Dehn function n ,and then ob- tained a quadratic upper bound in [Dru04, Theorem 1.1], a result also obtained by Leuzinger and Pittet in [LP04]. The quadratic upper bound can also be viewed as an illustration of Theorem D. (The assumption that 0 is the center of gravity is unessential: the important fact is that 0 belongs to the convex hull of the three weights but does not lie in the segment joining any two weights.) 1.5.3. Abels’ﬁrstgroup.— The group G= A (K) consists of matrices of the form ⎛ ⎞ 1 u u u 12 13 14 ⎜ ⎟ 0 s u u 2 23 24× ⎜ ⎟ ; s∈ K , u∈ K. i ij 00 s u 3 34 0 001 Its weight conﬁguration is given by 13 24 12 34 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 91 This example is interesting because it does not satisfy the SOL obstruction but admits opposite weights. A computation shows that H (u)= 0and Kill(u)= 0 (see 2 0 0 Abels [Ab87, Example 5.7.1]). ¯¯ If G= U A is the quotient of G by its one-dimensional center, then G does not satisfy the SOL obstruction but satisﬁes H (u¯) = 0, i.e. satisﬁes the 2-homological 2 0 obstruction. This example was studied speciﬁcally by the authors in the paper [CT13], where it is proved that it has a quadratic Dehn function, which also follows from Theo- rem F. 1.5.4. Abels’ second group. — This group was introduced in [Ab87, Example 5.7.4]. Consider the group U A, where A R and U is the group corresponding to the quotient of the free 3-nilpotent Lie algebra generated by the 3-dimensional K-vector space of basis (X , X , X ) by the ideal generated by[X ,[X , Y]] for all i, j ∈{1, 2, 3}. 1 2 3 i i j Its weight structure is as follows 12 23 ∗∗ (The sign∗∗ indicates that the degree 0 subspace u is 2-dimensional.) A compu- tation (see Remark 8.D.3)shows that H (u)= 0and Kill(u) is 1-dimensional. 2 0 0 This example was introduced by Abels as typically difﬁcult because although H (u) vanishes, U is not the multiamalgam of its tame subgroups (see Section 1.7); as 2 0 we show in this paper, this is reﬂected in the fact that Kill(u) is nonzero. This results in a signiﬁcant additional difﬁculty in order to estimate the Dehn function, which is at most cubic by Theorem E.4. 1.5.5. Semidirect products with SL .— We consider the groups V(K) SL (K), 3 3 where V is one of the following three irreducible modules: V= V ,the standard 3- dimensional module; V= Sym (V ), the 6-dimensional second power of V and V , 20 10 10 11 the 8-dimensional adjoint representation. (The notation is borrowed from [FuH,Lec- ture 13], writing V instead of .) These groups are not solvable but have a cocompact ij ij K-triangulable subgroup, namely V(K) T (K),where T (K) is the group of lower tri- 3 3 angular matrices. It is a simple veriﬁcation that for an arbitrary nontrivial irreducible representation, the group V(K) T (K) admits exactly three principal weights, namely the two principal negative roots r , r of SL itself, and the highest weight of the rep- 21 32 3 resentation V , which can be written as aL− bL ,where±L , L , L are the minimal ab 1 3 1 2 3 92 YVES CORNULIER, ROMAIN TESSERA vectors of the weight lattice, arranged as in the following picture. −L L L 2 1 −L−L 1 2 r r 31 32 The three principal weights always form a triangle with zero contained in its interior. In particular, V(K) T (K) does not satisfy the SOL obstruction. Let us write the weight diagram for each of the three examples (we mark some other points in the weight lattice as· for the sake of readability). More speciﬁcally, for V , the weights conﬁguration looks like L L 2 1 .. . L r r 31 32 We see that there are no quasi-opposite weights at all. This is accordingly a case for which Theorem D applies directly. Thus K SL (K) has a quadratic Dehn function (it can be checked to also hold for K SL (K) for d≥ 3). For V , writing L= L+ L , the weights are as follows 20 ij i j 2L L 2L 2 12 1 .. r . L L 23 13 .. . r r 31 32 . 2L . Thus there are quasi-opposite weights but no opposite weights. Theorem 10.E.1 implies that V (K) SL (K) has a quadratic Dehn function. 20 3 For V , writing L= L− L , the weights are as follows 11 ij i j L L 23 13 .. . .. r , L∗∗ L 21 21 12 .. .. . r , L r , L 31 31 32 32 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 93 In this case, there are opposite weights, there is an invariant quadratic form in degree zero (akin to the Killing form), deﬁned by φ(r , L )= 1for all i < j and all other products ji ij being zero, so Kill(u) = 0. However, a simple computation shows that H (u)= 0. So 0 2 0 Theorem E.4 implies that sl (K) SL (K) has an at most cubic Dehn function. 3 3 1.6. Description of the strategy. — Let us now describe the main ideas that underlie the proof of Theorems E.4 and F. We then proceed to sketch, with more details, the proof of Theorem F. General picture. — A central idea, already essential in Gromov’s approach is to use non- positively curved subgroups (called tame subgroups in the sequel). It was previously used in Abels’ work on compact presentability of p-adic groups [Ab87]. Abels considers a certain abstract group, obtained by amalgamating the tame subgroups over their intersections (more details are given in 1.7). Our combinatorial approach to the Dehn function allows us to take advantage of the consideration of this “multiamalgam” G. One similarly deﬁnes a multiamalgam of the tame Lie subalgebras, denoted by gˆ . Strategy. — To simplify the discussion, let us assume that G is standard solvable over a single nondiscrete locally compact ﬁeld of characteristic zero K, satisfying none of the SOL and 2-homological assumptions. Roughly speaking, the strategy is as follows. Ideally, one would like to obtain G= G; this turns out to be true under the additional assump- tion that Kill(u) vanishes. In general we obtain that G is a central extension of G, and describe generators of the central kernel. Second, we need to be able to decompose any combinatorial loop into boundedly many loops corresponding to relations in the tame subgroups. It turns out that both steps are quite challenging. While Abels’ work provides substantial material to tackle the ﬁrst step, we had to introduce completely new ideas to solve the second one. The ﬁrst step: giving a compact presentation for G.— Under the assumption that the group G does not satisfy the SOL obstruction, we show, improving a theorem of Abels, that ˆˆˆ the multiamalgam is a central extension of G. Thus we have G= U A, where Uis a central extension of U, whose central kernel is centralized by A. At ﬁrst sight, it seems that the condition H (u)= 0 should be enough to ensure that G= G. However, it turns 2 0 out that in general, U is a “wild” central extension, in the sense that it does not carry any locally compact topology such that the projection onto U is continuous. This strange phenomenon is easier to describe at the level of the Lie algebras. There, we have that gˆ= uˆ a,where uˆ is a central extension in degree 0 of u, seen as Lie algebras over Q.Now, if in the last statement, we could replace Lie algebras over Q by Lie algebras over K, then clearly H (u)= 0 would imply that uˆ= u. This happens if and only if the natural 2 0 morphism H (u)→ H (u) is an isomorphism; we show in Section 8 that this happens 0 2 0 if and only if the module Kill(u) (see Section 1.4) vanishes. In fact, there are relatively 0 94 YVES CORNULIER, ROMAIN TESSERA simple examples, already pointed out by Abels where Kill(u) does not vanish (see the previous subsection). As a consequence, even when none of the obstructions hold, we need to complete the presentation of G with a family of so-called welding relations, which encode generators of the kernel of U→ U. At the Lie algebra level, these relations encode K-bilinearity of the Lie bracket. The second step: reduction to special relations. — The second main step is to reduce the esti- mation of area of arbitrary relations to that of relations of a special form (e.g., relations inside a tame subgroup). This idea amounts to Gromov and is instrumental in Young’s approach for nilpotent groups [Y13a]and for SL (Z) [Y13b]; it is also used in [CT10, d≥5 CT13]. The main difference in this paper is that we have to perform such an approach without going into explicit calculations (which would be extremely complicated for an arbitrary standard solvable group, since the unipotent group U is essentially arbitrary). Our trick to avoid calculations is to use a presentation of U that is stable under “ex- tensions of scalars”. Let us be more explicit, and write U= U(K),sothat U(A) makes sense for any commutative K-algebra A. We actually provide a presentation, based on Abels’ multiamalgam and welding relations, of U(A) for any K-algebra A. When apply- ing it to a suitable algebra A of functions of at most polynomial growth, we obtain area estimates (in the sketch of Section 1.7, we do this with the algebra A of sequences of el- ements of at most exponential growth). This is the core of our argument; it is performed in Section 10.E (in the setting of Theorem F) and in Section 10.G in general, relying on the general approach developed in Section 7.B. The presentation itself is established in Sections 8 and 9. Last step: computation of the area of special relations. — For a standard solvable group not satisfying the SOL and 2-homological obstructions, these relations are of two types: those that are contained (as loops) in a tame subgroup and thus have an at most quadratic area; and the more mysterious welding relations. We show that welding relations have an at most cubic area. When the welding relations are superﬂuous, namely when Kill(u) vanishes, Theorem 10.E.1 asserts that G then has an at most quadratic Dehn function. A study based on the asymptotic cone, in a paper in preparation by the authors, will show that conversely, in some cases where Kill(u) does not vanish, the Dehn function of G grows strictly faster than quadratic. We mention this to enhance the important role played by the welding relations in the geometry of these groups. We suspect that they might be relevant as well in the study of the Dehn function of nilpotent Lie groups. 1.7. Sketch of the proof of Theorem F.— In the following, we emphasize the main ideas, omitting some technical issues occurring in the detailed, rigorous proof completed in Section 10.E. To begin with, let us explain in detail the notion of multiamalgam. Let G be a group and (H ) a family of subgroups. The multiamalgam Gof the H is the quotient of i i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 95 the free product of the H by the obvious amalgamation relations (identifying H∩ H to i i j its image in both H and H for all i, j ). We say that G is the multiamalgam of the H if i j i the canonical homomorphism G→ G is an isomorphism. Basic example: suppose that G= G . Then G is the multiamalgam of the family of subgroups (G⊕ G ) . This is because G is the quotient of the free product by the i j i =j commutation relations, and each of these commutation relations lives within one pair of summands. Consider now a standard solvable group G= U A satisfying the hypotheses of Theorem F. We call tame subgroup of G a subgroup of the form V A, where V is a closed A-invariant subgroup such that some element of A acts on V as a contraction. Let (U A) be the family of maximal tame subgroups. There are ﬁnitely i 1≤i≤ν many. We now deﬁne a presentation of the multiamalgam G. Let S be a compact sym- metric neighborhood of 1 in U , T a compact symmetric generating subset of A with nonempty interior, and S= S T. Let R be the set of bounded length relators of the following four types: • bounded length relators of (A, T), • relators of length 3 inside S, −1−1 • relators of the form ts t s for s, s∈ S, t∈ T, −1 • (amalgamation relators) relators of the form s s when s∈ S , s∈ S have the i j same image in U∩ U . i j Fix k≥ 1and let ℘ ∈{1...,ν} be a k-tuple of indices. Let W (n) be the set of −1 group words in the free group F of the form t s t where t∈ F with|t |≤ n and S j j j T j j=1 j s∈ S .Let δ (n) be the supremum of areas (for the presentationS| R) of those null- j ℘ ℘ homotopic words in W (n). Gromov’s trick (see Section 2.E) says that there exists k and ℘ such that δ (n) 2 2 n implies that the Dehn function of G is n . Showing that Gromov’s trick can be performed requires some length estimates which take some work but we do not insist on this here. So we ﬁx k and are reduced to showing δ (n) n . For each n,weﬁxsome w(n)∈ W (n); we decompose it as −1 w(n)= t (n)s (n)t (n) . j j j i=1 Let us proceed to show that the area of w(n) is ﬁnite and quadratically bounded in terms −1 of n.For each n,deﬁne σ (n) as the element of U represented by t (n)s (n)t (n) .Then j ℘ j j j σ (n),viewedasanelement of thefreeproduct U ∗ ··· ∗ U , has a trivial image j 1 ν i=1 in U. exp Let K be the K-algebra of sequences in K with at most exponential growth. exp Viewing σ as a function of n, simple estimates show that σ∈ U (K ). j j ℘ j 96 YVES CORNULIER, ROMAIN TESSERA Using the universal property of the multiamalgam, there is a canonical well-deﬁned exp exp exp homomorphism from the multiamalgam U(K ) of the U (K ) to U(K ). It is at this point that we use in an essential way the assumptions of Theorem F, namely that G does not satisfy the SOL and 2-homological obstructions, and Kill(u) ={0}. These assump- tions, thanks to the preliminary algebraic work of Sections 8 and 9, imply that this homo- exp exp morphism U(K )→ U(K ) is an isomorphism (Corollary 9.D.4). For q, r ∈{1,...,ν}, exp denote by η the inclusion of U∩ U into U . So, the multiamalgam of the U (K ) qr q r q i is the quotient of the free product by the normal subgroup generated by the relators −1 exp exp η (x)η (x) ,when (q, r) ranges over pairs and x ranges over U (K )∩ U (K ). qr rq q r exp Therefore, there exists a ﬁnite family (μ (n)) of elements of U(K ),and m 1≤m≤p elements q , r ∈{1,...,ν},suchthat μ (n)∈ U∩ U ,elements (g (n)) in the free m m m q r m m m exp product of the U (K ) such that −1−1 σ= g η (μ )η (μ ) g . j m q r m r q m m m m m m j=1 m=1 Thus for every n we have, in U ∗ ··· ∗ U 1 ν −1−1 σ (n)= g (n)η (μ )(n)η (μ )(n) g (n) . j m q r m r q m m m m m m j=1 m=1 Denote by g (n) a choice of word of length≤ Cn representing g (n) in U A, m m q in the generators S∪ T; this canbedonebecause g (n) has exponential size, so can q m −cn cn be written as t st with s∈ S and t∈ T, and c independent of n.Wealsodeﬁne −cn cn−cn cn simultaneously η (μ )(n) and η (μ )(n) to be of the form t st and t s t ,with q r m r q m m m m m s∈ S , s∈ S having the same image in U∩ U . q r q r m m m m So the word −1 −1 −1 w (n)= w(n) g (n) η (μ )(n) η (μ )(n) g (n) . m q r m r q m m m m m m m=1 is null-homotopic in (U ∗· · · ∗ U ) A and has an at most linear length. 1 ν −1 Let us obtain a uniform bound on the area of η (μ )(n)η (μ )(n) for each q r m r q m m m m m m.Wehave −cn −1 cn −1 η (μ )(n)η (μ )(n)= t ss t; q r m r q m m m m m −1 since ss is a relator, its area is equal to 1. Thus the area of w(n)w (n) is≤ p (indepen- dently of n). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 97 This shows that we are reduced to proving that w (n) has an at most quadratic area. For convenience, we rewrite w (n)= h (n), m=1 where h is a word in T∪ S for some ν and ν -tuple ℘ , independent of n. Let us show m ℘ that the area of a word w= h that is null-homotopic in (U ∗ ··· ∗ U ) Ais at m 1 ν m=1 most quadratic with respect to the total length|h| (with a constant depending only on ℘ ). This is shown by induction on ν . Since w is null-homotopic, there exists either m such that h is null-homotopic, or there exists m such that ℘= ℘ . In the second case, m m+1 this reduces to the case of ν− 1. In the ﬁrst case, it also reduces to the case of ν− 1, at the cost of replacing h with 1, which has at most quadratic cost since U A has an m ℘ at most quadratic Dehn function. So w (n) has an at most quadratic area. It follows that w(n) has an at most quadratic area. Since this holds for every choice of w, this shows that δ (n) n . Thus G has an at most quadratic Dehn function. This completes the sketch of proof of Theorem F. The ﬁrst two conditions (absence of SOL and 2-homological obstructions) were necessary as otherwise, the Dehn function is at least exponential. On the other hand, in the proof of Theorem E.4 we also have to proceed when the Killing module is not assumed zero in degree 0. In this case, U can be described as a quotient of the multiamalgam by some welding relators,inawaythatis exp inherited to U(K ) (see 10.D). So, in the above proof, some further relations appear (in −1 addition to the η (μ )(n)η (μ )(n) ), and we establish (Section 10.F) that these fur- q r m r q m m m m m ther relations have at most cubic area with respect to some compact presentation (where the additional relators are some welding relations of bounded length). 1.8. Introduction to the Lie algebra chapters. — Although Sections 8 and 9 can be viewed as technical sections when primarily interested in the results about Dehn func- tions, they should also be considered as self-contained contributions to the theory of graded Lie algebras. Recall that graded Lie algebras form a very rich theory of its own interest, see for instance [Fu, Kac]. Let us therefore introduce these chapters indepen- dently. In this context, the Lie algebras are over a given commutative ring A,withno ﬁniteness assumption. This generality is essential in our applications, since we have to consider (ﬁnite-dimensional) Lie algebras over an inﬁnite product of ﬁelds. We actually consider Lie algebras graded in a given abelian group W , written additively. 1.8.1. Universal central extensions. — Recall that a Lie algebra is perfect if g =[g, g]. It is classical that every perfect Lie algebra admits a universal central extension. We pro- vide a graded version of this fact. Say that a graded Lie algebra is relatively perfect in degree zero if g ⊂[g, g] (in other words, H (g) ={0}). In Section 8.B,toany graded 0 1 0 Lie algebra g, we canonically associate another graded Lie algebra g˜ along with a graded 98 YVES CORNULIER, ROMAIN TESSERA Lie algebra homomorphism τ: g˜→ g, which has a central kernel, naturally isomorphic to the 0-component of the 2-homology module H (g) . 2 0 Theorem G (Theorem 8.B.4). — Let g be a graded Lie algebra. If g is relatively perfect in degree zero, then the morphism g˜→ g is a graded central extension with kernel in degree zero, and is universal among such central extensions. An important feature of this result is that it applies to graded Lie algebras that are far from perfect: indeed, in our case, the Lie algebras are even nilpotent. 1.8.2. Restriction of scalars. — In Section 8.C, we study the behavior of H (g) un- 2 0 der restriction of scalars. We therefore consider a homomorphism A→ B of commuta- tive rings. If g is a Lie algebra over B, then it is also a Lie algebra over A and therefore to A B avoid ambiguity we denote its 2-homology by H (g) and H (g) according to the choice 2 2 of the ground ring. There is a canonical surjective A-module homomorphism H (g)→ A,B H (g);wecallits kernelthe welding module and denote it by W (g).Itisagraded A- 2 2 A,B A B module, and W (g) is also the kernel of the induced map H (g)→ H (g) . 0 0 0 2 2 2 These considerations led us to introduce the Killing module.Let g be a Lie algebra ⊗3 over the commutative ring B. Consider the homomorphism T from g to the symmetric square g g,deﬁnedby T (u⊗ v⊗ w)= u [v, w]−[u,w] v (all tensor products are over B here). The Killing module is by deﬁnition the cokernel of T . The terminology is motivated by the observation that for every B-module m, Hom (Kill (g), m) is in natural bijection with the module of invariant B-bilinear forms g× g→ m.If g is graded, then Kill(g) is canonically graded as well. Theorem H. —Let Q⊂ K be ﬁelds of characteristic zero, such that K has inﬁnite transcendence degree over Q.Let g be a ﬁnite-dimensional graded Lie algebra over K, relatively perfect in degree zero (i.e., H (g) ={0}). Then the following are equivalent: 1 0 Q,K (i) W (g) ={0}; Q,R (ii) W (g⊗ R) ={0} for every commutative K-algebra R; K 0 (iii) Kill (g) ={0}. In particular, assuming moreover that H (g) ={0}, these are also equivalent to: 2 0 (iv) H (g) ={0}; (v) H (g⊗ R) ={0} for every commutative K-algebra R. K 0 The interest of such a result is that (iii) appears as a checkable criterion for the vanishing of a complicated and typically inﬁnite-dimensional object (the welding mod- ule). Let us point out that this result follows, in case g is deﬁned over Q, from the results GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 99 of Neeb and Wagemann [NW08]. In our application to Dehn functions (speciﬁcally, in the proof of Theorems 10.E.1 and 10.G.1), we make an essential use of the implication (iii)⇒(v), where Q= Q, K is a nondiscrete locally compact ﬁeld, and R is a certain al- gebra of functions on K. That (iii) implies the other properties actually does not rely on the speciﬁc hypotheses (restriction to ﬁelds, ﬁnite dimension), see Corollary 8.C.8.The converse, namely that the negation of (iii) implies the negation of the other properties, follows from Theorem 8.C.13. 1.8.3. Abels’ multiamalgam. — Section 9 is devoted to the study of Abels’ multia- malgam gˆ and to its connection with the universal central extension g˜→ g.Here, we again consider arbitrary Lie algebras over a commutative ring R, but we now assume that the abelian group W is a real vector space. Given a graded Lie algebra, a Lie subalge- bra is called tame if 0 does not belong to the convex hull of its weights. Abels’ multiamal- gam gˆ is the (graded) Lie algebra obtained by amalgamating all tame subalgebras of g along their intersections (see 9.C for details); it comes with a natural graded Lie algebra homomorphism gˆ→ g. Abels deﬁnes a 2-tame nilpotent graded Lie algebra to be such that 0 does not belong to the convex hull of any pair of principal weights (this condition is related to the condition that the SOL-obstruction is not satisﬁed). A more general notion of 2-tameness, for arbitrary W -graded Lie algebras, is introduced in 9.A. Although Abels works in a speciﬁc framework (p-adic ﬁelds, ﬁnite-dimensional nilpotent Lie algebras), his methods imply with minor changes the following result. Theorem (essentially due to Abels, see Theorem 9.C.2). — If g is 2-tame, then gˆ→ g is the universal central extension in degree 0. In other words, gˆ is canonically isomorphic to g˜ . This means that in this case, gˆ is an excellent approximation of g, the discrepancy being encoded by the central kernel H (g) . 2 0 We actually need the translation of this result in the group-theoretic setting, which involves signiﬁcant difﬁculties. Assume now that the ground ring R is a commutative al- gebra over the ﬁeld Q of rationals. Recall that the Baker-Campbell-Hausdorff formula deﬁnes an equivalence of categories between nilpotent Lie algebras over Q and uniquely divisible nilpotent groups. Then g is the Lie algebra of a certain uniquely divisible nilpo- tent group G, and we can deﬁne the multiamalgam G of its tame subgroups, i.e. those subgroups corresponding to tame subalgebras of g. Note that it does not follow from ab- ˆˆ stract nonsense that G is controlled in any way by g, because G is deﬁned in the category of groups and not of uniquely divisible nilpotent groups. In a technical tour de force, Abels [Ab87, §4.4] managed to prove that G is nilpotent and asked whether the exten- sion G→ G is central. The following theorem, which we need for our estimates of Dehn function, answers the latter question positively. Theorem I (Theorem 9.D.2). — Let g be a 2-tame nilpotent graded Lie algebra over a commuta- ˆˆ tive Q-algebra R.If g is 2-tame, then G is nilpotent and uniquely divisible, and G→ G corresponds to 100 YVES CORNULIER, ROMAIN TESSERA gˆ→ g under the equivalence of categories. In particular, the kernel of G→ G is central and canonically isomorphic to H (g) . 1.8.4. On the “computational” aspect. — Let g be a Lie algebra (over some com- mutative ring) graded in some abelian group. Theorems E and F involve the condi- tions H (g) ={0} and Kill(g) ={0}. Typically, in the cases of interest, g is a ﬁnite- 2 0 0 dimensional nilpotent Lie algebra over a ﬁeld, and these conditions mean the surjectivity of some explicit matrices. There are simple families of examples for which g has dimen- 3 2 sion n , while g= g has dimension n , so that most of the contribution to α =0 dim(g) is in degree zero. Hence, when computing H (g) and Kill(g) , the main con- 2 0 0 tribution comes from those terms involving g . It turns out that under mild assumptions fulﬁlled in both theorems (implied by the failure of the SOL obstruction), we can get rid of these terms. Namely, recalling that the homology module H (g) is the cokernel of the map d: 2 3 3 2 3 2 g→ g, we observe (see Section 8.D.1 for details) that d maps ( g )→ g ; 3 0 let H (g) be the cokernel of the latter module homomorphism; it is endowed with a canonical map H (g)→ H (g) . 0 2 0 ⊗3 2 Similarly, Kill(u) is the cokernel of the map T: g→ S g (the second symmetric ⊗3 2 power) mapping x⊗ y⊗ z to[x, y] z− x [y, z], which maps ((g ) ) into (S g ) ; 0 0 let Kill (g) be the cokernel of the latter; it admits a canonical module homomorphism into Kill(g) . The following theorem is contained in Theorem 8.D.2 for H and in Theo- rem 8.D.9 for the Killing module. Theorem J. —Let g be a nilpotent Lie algebra (over any commutative ring) graded in some real vector space. Suppose that for every α we have g=[g , g]. Then the natural module 0 β−β β/∈{0,α,−α} homomorphisms H (g)→ H (g) and Kill (g)→ Kill(g) are isomorphisms. 0 2 0 0 0 The assumption is satisﬁed if g is nilpotent and 2-tame, that is, g/[g, g] has no pair {α, β} of weights such that 0 belongs to the segment[α, β] [Ab87, Lemma 4.3.1(c)]. 1.9. Guidelines. • The reader interested in the proof of Theorem E.1 candirectlygotoSection 12, which is essentially independent. • The reader interested in the proof of Theorem E.2 can directly go to Sec- tion 11, and refer when necessary to the preliminaries of Sections 4, 5,and, more scarcely, Section 8. • The reader interested in the proof of the polynomial upper bounds will ﬁnd the ﬁrst important steps in Section 3 and the weight deﬁnitions in Section 4,and then can proceed to Sections 6, 7,and 10, and refer when necessary to all the preceding sections. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 101 • The reader interested speciﬁcally in the algebraic results, as described in Sec- tion 1.8, can directly go to Sections 8 and 9, and refer when necessary to the preliminary Section 5. Although being of preliminary nature here, Sections 2 (introducing the Dehn func- tion) and 4 (weights and tameness conditions, Cartan grading) can be read for their own interest. 2. Dehn function: preliminaries and ﬁrst reductions This section contains general preliminaries on the Dehn function. In particular, we state Gromov’s trick in Section 2.E, essentially following arguments from [CT10]. 2.A. Asymptotic comparison. Deﬁnition 2.A.1. —If f , g are real functions deﬁned on any set where∞ makes sense (e.g., in a locally compact group, it usually refers to convergence to the point at inﬁnity in the 1-point compactiﬁca- tion), we say that f is asymptotically bounded by g and write f g if there exists a constant C≥ 1 such that for all x close enough to∞ we have f (x)≤ Cg(x)+ C; if f g f we write f g and say that f and g have the same asymptotic behavior,orthe same -asymptotic behavior. Deﬁnition 2.A.2. —If f , g are non-decreasing non-negative real functions deﬁned on R or N, we say that f is bi-asymptotically bounded by g and write f g if there exists a constant C≥ 1 such that for all x f (x)≤ Cg(Cx)+ C; if f g f we write f≈ g, and say that f and g have the same bi-asymptotic behavior, or the same ≈-asymptotic behavior. Besides the setting, the essential difference between -asymptotic and≈-asymp- n n totic behavior is the constant at the level of the source set. For instance, 2≈ 3 but n n 3 . 2.B. Deﬁnition of Dehn function. — If G is a group and S any subset, we denote by |g| the (possibly inﬁnite) word length of g∈ G with respect to S; the length |·| takes S S ﬁnite values on the subgroup generated by S. If H is a group and R⊂ H, we deﬁne the area of an element w∈ H as the (pos- −1 sibly inﬁnite) word length of w with respect to the union hRh ;wedenoteitby h∈H area (w). It takes ﬁnite values on the normal subgroup generated by R. R 102 YVES CORNULIER, ROMAIN TESSERA Now let S be an abstract set and F thefreegroup over S, and π: F→ Ga S S surjective homomorphism. Its kernel is denoted by K; elements in K are called relations. If a subset R of K is given, the elements in the normal subgroup generated by R are called null-homotopic. Thus any null-homotopic word is a relation, and conversely, if R generates K as a normal subgroup (then R is called a system of relators), then relations are null-homotopic. By a common abuse of terminology, elements in a system of relators are called relators. We deﬁne the Dehn function δ (n)= maxn/2, sup area (w): w∈ K,|w|≤ n . S,π,R R S (The term n/2 is not serious, and only avoids some pathologies. Intuitively, it corresponds n−n to the idea that the area of a word s s , which is zero by deﬁnition, should account for at least n.) We like to think of δ as the Dehn function of G, but in this general setting, S,π,R this function as well as its asymptotic behavior can depend on the choice of S, π and R. It can also take inﬁnite values. Note that often S is a given subset of G and since π is obviously chosen as the unique homomorphism extending the identity of S, we write the Dehn function as δ . S,R Let now G be a compactly generated LC-group (LC-group means locally compact group), and S a compact generating symmetric subset whose interior contains 1. View S as an abstract set and consider the surjective homomorphism F→ G which is the identity on S, and K its kernel. Let K(d) be the intersection of K with the d -ball in (F , |·| ). We say that G is compactly presented if for some d≥ 0, the subset K(d) generates S S K as a normal subgroup. This does not depend on the choice of S (but the minimal value of d can depend). If so, the function δ takes ﬁnite values (see Lemma 2.B.2 below), S,K(d) and its≈-asymptotic behavior does not depend on the choice of S and d . Thelatteristhencalledthe Dehn function of G. If G is not compactly presented, we say by convention that the Dehn function is inﬁnite. For instance, when we say that a compactly generated LC-group G has a Dehn function f (n), we allow the possibility that G is not compactly presented, i.e. its Dehn function is (eventually) inﬁnite. Proposition 2.B.1. —The≈-behavior of the Dehn function is a quasi-isometry invariant of the compactly generated locally compact group. On the proof. — This follows from a more general statement for arbitrary connected graphs. In this case, the Dehn function can be deﬁned as soon that the graph is large-scale simply connected, i.e., can be made simply connected by adding 2-cells with a bounded number of edges, thus deﬁning a notion of area for arbitrary loops. The quasi-isometry invariance of the asymptotic behavior of the Dehn function for arbitrary connected graph is similar to the usual one showing that the Dehn function is a quasi-isometry invariant among ﬁnitely generated groups, see [Al91]or[BaMS93, Theorem 26]. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 103 Lemma 2.B.2. —Asabove,let G be an LC-group with a compact symmetric generating subset S,and K the kernel of the canonical homomorphism F→ G. Assume that, for some d≥ 4,the intersection K(d) of the d -ball in F with K generates K as a normal subgroup. Then δ (n) is S S,K(d) ﬁnite for all n. Proof.—Fix n.Denoteby S the product of n copies of S, with the product topology n n (rather than the n-ball in F ). Consider the product map p: S→ F ,and let (S ) be the S S 0 n n n inverse image of K(n) in S ;thus (S ) is a closed subset of S , hence is compact. By assumption, the area function area◦ p takes ﬁnitevalueson (S ) .Wehavetoshow K(d) 0 that it is bounded on (S ) ; thus it is enough to show that this function is locally bounded. n n Indeed, given s= (s ,..., s )∈ (S ) , there exists a neighborhood V of s in (S ) such that 1 n 0 0 −1 for all s= (s ,..., s )∈ Vand all1≤ k≤ n, we have the element (s ... s ) (s ... s ) 1 k 1 n 1 k represents an element t∈ S. Note that t= 1. k n Fix s∈ V, and deﬁne t ,..., t∈ Sas above. Set t= 1, and deﬁne, for 0≤ k≤ n 1 n 0 n+1 s= s ,..., s , t , s ,..., s∈ S (k) 1 k k k+1 n Then p(s )= p(s ), p(s )= p(s),and forall 0≤ k≤ n− 1 (0) (n) −1 −1 −1 −1 area p s s= area s ... s s t s t s ... s K(d) (k) K(d) k+1 k+1 (k−1) k+2 n k+1 k k+2 n −1 −1 = area s t s t≤ 1; K(d) k+1 k+1 k+1 k −1 it follows that area (p(s ) p(s))≤ n. Therefore K(d) area p s≤ area p(s)+ n,∀s∈ V, K(d) K(d) proving the local boundedness. 2.C. Dehn vs Riemannian Dehn. — Let X is a Riemannian manifold. If γ is a Lips- chitz loop in X, deﬁne the area of γ as the inﬁmum of areas of Lipschitz disc ﬁllings of γ . Deﬁne the ﬁlling function of X as the function mapping r to the supremum F(r) of areas of all Lipschitz disc ﬁllings of Lipschitz loops of length≤ r. Proposition 2.C.1. —Let G be a locally compact group with a proper cocompact isometric action on a simply connected Riemannian manifold X. Then G is compactly presented and the Dehn function of G satisﬁes δ(n)≈ max F(n), n . (The max(·, n) is essentially technical: unless X has dimension≤ 1or is compact, it can be shown that F(r) grows at least linearly.) The proof is given, for G discrete, by Bridson [Bri02, Section 5]. Here we only pro- vide a complete proof of the easier inequality F(n) δ(n), because the proof in [Bri02] 104 YVES CORNULIER, ROMAIN TESSERA makes a serious use of the assumption that G is ﬁnitely presented. For the converse in- equality δ(n) max(F(n), n), the (highly technical) proof given in [Bri02] uses general arguments of ﬁlling in Riemannian manifolds and a general cellulation lemma, and the remainder of the proof carries over our broader context, so we shall only provide a brief sketch of the converse. Lemma 2.C.2. —Let X be a simply connected Riemannian manifold with Isom(X) acting cocompactly on X. Then the Riemannian area of loops of bounded length is bounded. Proof.—Write G= Isom(X). By cocompactness, there exists r such that for every x∈ X, the exponential is (1/2, 2)-bilipschitz from the r -ball in T X to X. In particular, 0 x given a loop of length≤ r , it passes through some point x; its inverse image by the exponential at x has length≤ 2r and can be ﬁlled by a disc of area≤ π r in T X, and its 0 x image by the exponential is a ﬁlling of area≤ 4π r in X. −1 Now ﬁx a positive integer m≥ 6/r . LetDbe an m -metric lattice in X, that 0 0 −1 is, a maximal subset in X in which any two distinct points have distance≥ m .Note −1 that every point in X is at distance≤ m to some element in D. Also note that, by properness of X, the intersection of D with any bounded subset is ﬁnite. For any x, y∈ D −1 with d(x, y)≤ 3m , ﬁx a geodesic path S(x, y) between x and y.Consideracompact subset such that G= X. Consider a 1-Lipschitz loop f :[0, k]→ X and let us bound the Riemannian area of f by some constant depending only of k. By homogeneity, we can suppose that −1−1 f (0)∈ .For all 0≤ n≤ km ,let x be a point in D that is m -close to f (nm ).Fix 0 n 0 0 −1 geodesic paths joining x and f (nm ). So there is a homotopy from f to the concate- −1 nation of the S(x , x ) through km “squares” of perimeter at most 6m≤ r .Bythe n n+1 0 0 above, each of these km squares can be ﬁlled with area≤ π 4r . The remaining loop is a concatenation of k segments of the form S(x , x ) with x∈ D (with n taken modulo n n+1 n km ). For given k, there are only ﬁnitely many such loops, since all the points x are at 0 n −1 distance≤ k/2+ m to . Since X is simply connected, each of these loops has ﬁnite area. So the remaining loop has area≤ a for some a <∞. So we found a Lipschitz k k ﬁlling of the original loop, of area≤ a+ km π r . k 0 Partial proof of Proposition 2.C.1. — Fix a compact symmetric generating set S in G and a set R of relators. Fix x∈ X. Set r= sup d(x , sx ).If s∈ S, ﬁx a r-Lipschitz map 0 0 0 s∈S j :[0, 1]→ X mapping 0 to x and 1 to sx .If w= s ... s ,deﬁne j :[0, k]→ Xas s 0 0 1 k w follows: if 0≤ ≤ k− 1and 0≤ t≤ 1, j (+ t)= s ... s j (t). It is doubly deﬁned for w 1 s an integer, but both deﬁnitions coincide. So j is r-Lipschitz. If w represents 1 in G, then j (0)= j (k). w w If w is a word in the letters in S and represents the identity, let A(w) be the area of the loop j . w GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 105 By Lemma 2.C.2,A(w) is bounded when w is bounded. Also, it is clear that −1 A(gwg )= A(w) forall groupwords g. This shows that there exists a constant C > 0, namely C= sup A(r),suchthat A(w)≤ Carea(w) for some constant C. r∈R For some r , every point in X is at distance≤ r of a point in Gx .Considera 0 0 0 loop of length k in X, given by a 1-Lipschitz function u :[0, k]→ X. For every n (modulo k), let g x be a point in Gx with d(g x , u(n))≤ r .Wehave d(g x , g x )≤ 2r+ 1. n 0 0 n 0 0 n 0 n+1 0 0 −1 N By properness, there exists N (depending only on r )suchthat g g∈ S .If σ is a 0 n+1 n −1 word of length N representing g g ,and σ= σ ...σ ,thenwepassfrom u to j n+1 0 k−1 σ by a homotopy consisting of k squares of perimeter≤ 4r+ 2. By Lemma 2.C.2,there is a bound M on the Riemannian area of such squares. So the Riemannian area of u is bounded by kM+ A(j )≤ kM+ Carea(σ )≤ kM+ δ (Nk). This shows that 0 σ 0 0 S,R F(k)≤ kM+ δ (Nk). 0 S,R For the (more involved) converse inequality, we only give the following sketch: let ρ> 0 be such that each point in X is at distance <ρ/8toGx , and assume in addition that ρ> ρ= ,where κ is an upper bound on the sectional curvature of X. Let S be 2 κ the set of elements in G such that d(gx , x )≤ ρ and R the set of words in F ,oflength 0 0 S at most 12 and representing 1 in G. Then the proof in [Bri02, §5.2] shows thatS| R is a presentation of G with Dehn function δ(n) bounded above by 4λ (F(ρn)+ ρn+ 1), where λ= 1/ min(4 κ/π, α(r,κ)) and α(r,κ) is the area of a disc of radius r in the standard plane or sphere of constant curvature κ . 2.D. Two combinatorial lemmas on the Dehn function. — This subsection contains sev- eral general lemmas about the Dehn function, which will be used at some precise parts of the paper. 2.D.1. Free products. — Recall that a function R→ R is superadditive if it satisﬁes ≥0 f (x+ y)≥ f (x)+ f (y) for all x, y. For instance, the function r → r is superadditive for every α≥ 1. Lemma 2.D.1. — Let f be a superadditive function. Let (G ) be a ﬁnite family of (abstract) groups, each with a presentationS| R, with Dehn function≤ f . Then the free product H of the G i i i has Dehn function δ≤ f with respect to the presentation S| R. i i Proof.—Let w= s ... s be a null-homotopic word with n≥ 1. Because H is a free 1 n product, there exists i and 1≤ j≤ j+ k− 1≤ n such that every letter s for j≤ ≤ j+ k− 1 is in S and s ... s represents the identity in G . So, with the corresponding cost, i j j+k−1 i which is≤ f (k), we can simplify w to the null-homotopic word s ... s s ... s .Thus 1 j−1 k n δ(n)≤ f (k)+ δ(n− k) (with 1≤ k≤ n). Using the property that f is superadditive, we can thus prove by induction on n that δ(n)≤ f (n) for all n. (This argument is used in [GS99] for ﬁnitely generated groups.) 106 YVES CORNULIER, ROMAIN TESSERA 2.D.2. Conjugating elements. Lemma 2.D.2. —LetS| R be a group presentation, and r a bound on the length of the words in R. Then for every null-homotopic w∈ F , with length n and area α, we can write, in F , S S −1±1 w= g r g with r∈ R ∪{1} and g∈ F , with the additional condition|g|≤ n+ rα. i i i i S i S i=1 i Proof. — We start with the following claim: consider a connected polygonal planar complex, with n vertices on the boundary (including multiplicities); suppose that the num- ber of polygons is α and each has at most r edges. Fix a base-vertex. Then the distance in the one-skeleton of the base-vertex to any other vertex is≤ n+ rα. Indeed, pick an injective path: it meets at most n boundary vertices. Other vertices belong to some face, but each face can be met at most r times. So the claim is proved. Now a van Kampen diagram for a null-homotopic word of size n and area α with relators of length≤ r satisﬁes these assumptions, the distance from the identity to some vertex corresponds to the length to the conjugating element that comes into play. Thus the g can be chosen with|g|≤ n+ rα. i i S 2.E. Gromov’s trick. Deﬁnition 2.E.1. —Let F be the free group over an abstract set S,let G be an arbitrary group, and let π: F→ G be a surjective homomorphism. We call (linear) combing of (G,π) (or, informally, of G if π is implicit), a subset F⊂ F such that 1∈ F and some constant C≥ 1,we n Cn have the property that π(S)⊂ π(S∩ F) for all n≥ 0. Example 2.E.2. — Let T be any generating subset of a ﬁnitely generated abelian group A. Suppose that{t ,..., t }⊂ T is also a generating subset. Then the set of words { t} where (m ) ranges over Z , is a combing of F→ A. If every element of T is j T j=1 j equal or inverse to some t , the constant C can be taken equal to 1. The following is, up to minor changes explained below, established in [CT10, Proposition 4.3]: Theorem 2.E.3 (Gromov’s trick). — Let f: R→ R be a function such that for some >0 >0 α> 1, the function r → f (r)/r is non-decreasing. Let S be an abstract set, let G be an arbitrary group, and let π: F→ G be a surjective homomorphism. Let R⊂ F be a subset contained in Ker(π). Consider a combing F⊂ F . S S Assume that for all n, the area with respect to R of any w∈ Ker(π) of the form w= w w w 1 2 3 with max|w |≤ n with w∈ F is≤ f (n).ThenS| R is a presentation of G (i.e., R generates i i i Ker(π) as a normal subgroup) and for some constants C , C > 0, the Dehn function of G with respect to R is≤ C f (C n). Remark 2.E.4. — The statement in [CT10, Proposition 4.3] is awkward because it purportedly considers a locally compact group without specifying a presentation and GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 107 gives a conclusion on its Dehn function as a function (and not an asymptotic type of func- tion). Actually the local compactness assumption is irrelevant and the correct statement is the one given here, the proof given in [CT10] applying without modiﬁcation. The setting is just that of a group presentation; it is even not necessary to assume that the relators have bounded length. Second, the assumption was for words w of length≤ n, while here we assume that all w have length≤ n; this is more natural and handy, and requires no change in the proof. Third, [CT10, Proposition 4.3] is stated with the function n → n for α> 1, but it relies on [CT10, Lemma 4.1], which allows much more general functions. At the time [CT10] it was not obvious to the author how to state a general result without a compli- cated technical assumption, but it turns out that the non-decreasing assumption of f is very general: we are not even aware of any non-hyperbolic compactly presented locally compact group whose Dehn function is not≈-equivalent to a function f such that, say, 3/2 f (r)/r is non-decreasing. Proof of Theorem 2.E.3. — We ﬁrst claim that for any k≥ 3, any n and any w= w ...w∈ Ker(π) with w∈ F and sup|w |≤ n,wehave area (w)≤ f (n)= 1 k i i R k kf (Ckn/2). Indeed, for every i the length of π(w ...w ) is≤ n/2, so we can ﬁnd u∈ F 1 i i with π(u )= π(w ...w ) and|u |≤ Cn/2, with u= u= 1. Then i 1 i i 0 k k k −1 w= w= u w u; i i−1 i i=1 i=1 −1−1 since π(u w u )= 1, by assumption we have area (u w u )≤ f (Ckn/2). i−1 i R i−1 i i i Then, denoting δ= δ ,by[CT10, Lemma 4.2], we have, for all n≥ k, S,R δ(n)≤ kδ 2(C+ 1)n/k+ f C! n/k" ,∀n≥ k (where we have f (C! n/k") rather than f (Ck! n/k") because we consider here sup|w| k k i rather than|w|). So, using that! n/k"≤ 2n/k,wehave δ(n)≤ kδ 2(C+ 1)n/k+ kf C n ,∀n≥ k. If we deﬁne u (r)= kf (C r), then we claim that zu (y)/u (yz) tends to zero uni- k k k formly in y≥ 1, k≥ 1, when z →∞. Indeed, writing f (r)= r v(r) with v non- decreasing, we have 2 α 2 2 zu (y) z(C y) v(C y) v(C y) 1 ==≤ . 2 α 2 α−1 2 α−1 u (yz) (C yz) v(C yz) z v(C yz) z This uniform convergence to zero allows to use [CT10, Lemma 4.1], which yields that for some r ,some k, and some constant c > 0, we have f (r)≤ cu (r) for all r≥ r .Thus 0 k 0 δ(r)≤ ckf (C r) for all r≥ r . 0 108 YVES CORNULIER, ROMAIN TESSERA (Here the value of the constant C= ck is hidden behind the estimates from the proof [CT10, Lemma 4.1] but the constant C= C is explicit, C being given in Deﬁni- tion 2.E.1.) 3. Geometry of Lie groups: metric reductions and exponential upper bounds In the study of upper bounds for the Dehn function of Lie groups, there are two im- portant reduction steps, described in this section. The ﬁrst (Section 3.A) allows to reduce the problem to triangulable groups, with no cost on the Dehn function. The second (Sec- tion 3.C) passes to a generalized standard solvable group, with a polynomial cost on the Dehn function. In between (Section 3.B), we describe a general retraction argument in Lie groups, proving in particular that connected Lie groups have an at most exponential Dehn function. 3.A. Reduction to triangulable groups. — The following lemma is essentially borrowed from [C08]. Lemma 3.A.1. — For every connected Lie group G, there exists a series of proper homomor- phisms with cocompact images G← G→ G← G , with G triangulable. In particular, G is 1 2 3 3 3 quasi-isometric to G and the Dehn functions of G and G are≈-equivalent. Proof. — Let N be the nilradical of G. By [C08, Lemma 6.7], there exists a closed cocompact solvable subgroup G of G containing N, and a cocompact embedding G⊂ 1 1 H with H a connected solvable Lie group, such that H is generated by G and its center 2 2 2 1 Z(H ). In particular, every normal subgroup of G is normal in H . Let W be the largest 2 1 2 compact normal subgroup of H and deﬁne L= H /W, so L is a connected solvable 2 2 2 2 Lie group whose derived subgroup has a simply connected closure. By [C08, Lemma 2.4], there are cocompact embeddings L⊂ G⊃ G ,with G and G connected Lie 2 2 3 2 3 groups, and G triangulable. The latter statement follows from the fact that continuous proper homomorphisms with cocompact images are quasi-isometries, along with Proposition 2.B.1, which says that the≈-behavior of the Dehn function is a quasi-isometry invariant. 3.B. At most exponential Dehn function. Theorem 3.B.1 (Gromov). — Every connected Lie group has an at most exponential Dehn function. As usual, Gromov gives a rough sketch of proof [Gro93, Corollary 3.F ], but we are not aware of a complete written proof. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 109 The basic idea is that the law can be described using polynomials and exponentials, but this is not enough because iterated exponentials would be an issue. A workable lemma is the following. n n Recall that an exponential polynomial R→ R is a real-valued function each of whose n coordinates is a polynomial in the coordinates x ,..., x and in exponentials 1 n λ x j i e for suitable complex scalar numbers λ . Lemma 3.B.2. — Every simply connected solvable Lie group is isomorphic to a Lie group described as (R× R ,∗) with the law of the form (u ,v )∗ (u ,v )= P(v ,v , u , u ), v+ v; 1 1 2 2 1 2 1 2 1 2 −1 (u,v)= T(v, u),−v , where P is an exponential polynomial depending polynomially on the variable (u , u ) (that is, not 1 2 involving exponentials in the coordinates of u and u ), and where T is an exponential polynomial 1 2 depending polynomially on the variable u. For instance, the law of SOL can be described as v−λv 2 2 (x , y ,v )∗ (x , y ,v )= e x+ x , e y+ y ,v+ v 1 1 1 2 2 2 1 2 1 2 1 2 (here (, m)= (2, 1)). Proof. — Let N be the derived subgroup of G and H a Cartan subgroup; both are a simply connected nilpotent Lie group. By [Bou, Chap. 7, §1, 2, 3] (see also Proposi- tion 4.D.4), we have NH= G. Consider a supplement subspace of the Lie algebra of M= N∩ H in the Lie algebra of H, and deﬁne V as its exponential (it is usually not a subgroup). Set m= dim(V) and = dim(N). We identify V and N to R and R through the exponential and a choice of bases of log(V) and log(N). The product map M× V→ H is an algebraic isomorphism of varieties, which yields a decomposition H= M× V, for which the law is given by polynomials; for the moment we just need to use the fact that the product (0,v)(0,v ) can be described as (Q(v, v ), v+ v ) for some polynomial Q on 2k variables. The adjoint action of h on n is given by some Lie algebra homomorphism φ: h→ End(n). If h∈ Hand n∈ N, we have −1 hnh= exp Ad(h) log(n)= exp exp φ log(h) log(n) . Beware that in this expression, the ﬁrst exp is the exponential of the Lie algebra N, with inverse log (both being polynomials), while the second is the operator exponentiation on n. The operator exp(φ(log(h))) is a polynomial in the entries and exponentials of 110 YVES CORNULIER, ROMAIN TESSERA the coefﬁcients of log(h) (the exponentials appear because φ(log(h)) is not necessary nilpotent), possibly with complex coefﬁcients. Hence after identiﬁcation of H and N to real vector spaces using the Lie algebras, −1 we can write hnh= R(h, n), where R is a polynomial in the coefﬁcients of h and n,and in some (possibly complex) exponentials of the coefﬁcients of h. Also, describe the product of 3 elements in N by a polynomial: nn n= S(n, n , n ). The product map yields a decomposition G= VN, and for v, v∈ V, n, n∈ N, we −1 have (nv)(n v )= nvn v v(vv ), so in coordinates, we have −1 (n,v) n ,v= S n,vn v , Q v, v ,v+ v = S n, R v, n , Q v, v ,v+ v . −1−1−1−1 That the inverse map has the given form is similar, since (nv)= (v n v)v . Lemma 3.B.3. —If G is a simply connected solvable Lie group with a left-invariant Rieman- nian metric, there is an exponentially Lipschitz strong deformation retraction of G to the trivial subgroup, i.e. a map F: G ×[0, 1]→ G such that for all g, F(g, 0)= gand F(g, 1)= 1, and such that for some constant C,if B(n) is the n-ball in G then F is exp(Cn)-Lipschitz in restriction to B(n) ×[0, 1]. Proof. — We identify G with R× R and write its law as −1 (u ,v )∗ (u ,v )= u+ u , P(u , u ,v ,v ); (u,v)= (−u,−v), 1 1 2 2 1 2 1 2 1 2 with P as in Lemma 3.B.2. Deﬁne, for ((u,v), t,τ)∈ G ×[0, 1] , s (u,v), t,τ= (tu,τv). We need an upper bound on the differential of s at a given point (u ,v , t ,τ ). 0 0 0 0 Let L denote the left translation by g,given by h → g∗ h. Then −1 L◦ s L (u,v), t,τ (u ,v ) 0 0 (t u ,τ v ) 0 0 0 0 = T(τ v , t u ),−τ v∗ tP(v ,v, u , u), τ v+ τv 0 0 0 0 0 0 0 0 0 = P−τ v ,τv+ τv, T(τ v , t u ), tP(v ,v, u , u) , 0 0 0 0 0 0 0 0 0 −τ v+ τv+ τv 0 0 0 We can write it as Q(τ v ,τv ,τv,v ,v; t , t, u , u),where Q= (Q , Q ),Q is an expo- 0 0 0 0 0 0 1 2 1 nential polynomial depending polynomially on the last four variables and Q is a poly- nomial. It follows that each partial derivative of Q with respect to the variables (u,v, t,τ) is an exponential polynomial R in the same variables, polynomial on the last four GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 111 variables (with coefﬁcients depending only the group law). Each such partial deriva- tive, when (u,v, t,τ)= (0, 0, t ,τ ), is equal to R(τ v ,τ v , 0,v , 0; t , t , u , 0), which 0 0 0 0 0 0 0 0 0 0 canberewrittenas R (τ v ,v; t , u ),with R an exponential polynomial that is poly- 0 0 0 0 0 nomial in the last two variables. Hence, keeping in mind that τ , t belong to[0, 1], 0 0 the partial derivatives of Q at (0, 0, t ,τ ) at any (0, 0, t ,τ ) are bounded above by 1 0 0 0 0 c$v$ c 2 0 3 c e (1 +$u$) ,where c , c are positive constants depending only on the group law 1 0 1 2 and the choice of left-invariant Riemannian metric μ. Therefore, the differential of s at any (u ,v , t ,τ ), for the left-invariant Rieman- 0 0 0 0 nian metric μ, has the same bound. If (u ,v )∈ B(2n) (the 2n-ball around 1 in G), then$v $≤ n (up to rescaling μ) 0 0 0 c n 2 and$u $≤ e (for some ﬁxed constant c ). So, for every (u ,v , t ,τ )∈ B(2n) ×[0, 1] , 0 4 0 0 0 0 Cn the differential of s at (u ,v , t ,τ ) is bounded by e , for some positive constant C only 0 0 0 0 depending on (G,μ). In particular, since any two points in B(n) can be joined by a 2 Cn geodesic within B(2n), we deduce that the restriction of s to B(n) ×[0, 1] is e -Lipschitz. The function (g, t) → s(g, t, t) is the desired retraction. (We used an extra variable τ by anticipation, in order to reuse the argument in the proof of Proposition 3.B.5.) Proof of Theorem 3.B.1. — By Lemma 3.A.1, we can restrict to the case of a simply connected solvable (actually triangulable) group G. Given a loop of size n in G based at the unit element, Lemma 3.B.3 provides a Lipschitz homotopy with exponential area to the trivial loop. Remark 3.B.4. — Using Guivarc’h’s estimates on the word length in simply con- nected solvable Lie groups [Gui73, Gui80], we see that there exists a constant C such that if B(n) is the n-ball in G, then F(B(n),[0, 1]) is contained in the ball B(C n) (here F is the function constructed in the proof of Lemma 3.B.3). Thus in particular, F provides a ﬁlling of every loop of linear size, with exponential area and inside a ball of linear size. In particular, any virtually connected Lie group (and cocompact lattice therein) has a linear isodiametric function. d d Let G= (U× U ) Z be a standard solvable group. The group G= U Z a na 1 a can be embedded as a closed cocompact subgroup into a virtually connected Lie group G with maximal compact subgroup K. Consider a left-invariant Riemannian metric on the connected manifold G /K. The composite map G→ G /K is a G-equivariant 2 1 2 quasi-isometric injective embedding; endow G with the induced metric. Endow U Z 1 na with a word metric with respect to a compact generating subset, and endow G with a metric induced by the natural quasi-isometric embedding G→ (U Z )× G . na 2 Proposition 3.B.5. —Let G be a standard solvable group of the form (U× U ) Z , with a na the above metric. Then there is an exponentially Lipschitz homotopy between the identity map of G and its natural projection π to U Z . Namely, for some constant C,there is amap na d d σ: (U× U ) Z ×[0, 1]→ (U× U ) Z , a na a na 112 YVES CORNULIER, ROMAIN TESSERA such that for all g∈ G, σ(g, 0)= gand σ(g, 1)= π(g),and σ(g, t)= gif g∈ U Z ,and σ na Cn is e -Lipschitz in restriction to B(n) ×[0, 1]. Proof. — As in the deﬁnition of standard solvable group, write G= U Z and U= U× U . Since U is a simply connected nilpotent Lie group, we can identify it to a na a its Lie algebra thorough the exponential map. Deﬁne, for (v, w, u)∈ U× U Z and a na t ∈[0, 1], σ(v,w, u, t)= (tv, w, u). By the computation in the proof of Lemma 3.B.3, Cn σ is e -Lipschitz in restriction to B(n) ×[0, 1]; the presence of w does not affect this computation. Corollary 3.B.6. — Under the assumptions of the proposition, if G/U U A is com- na pactly presented with at most exponential Dehn function, then G has an at most exponential Dehn function. Proof. — Given a loop γ of size n in G, the retraction of Proposition 3.B.5 inter- polates between γ and its projection γ to G/G . The interpolation has an at most ex- ponential area because the retraction is exponentially Lipschitz; γ has linear length and hence has at most exponential area by the assumption. So γ has an at most exponential area. 3.C. Reduction to split triangulable groups. Proposition 3.C.1 ([C11]). — Let G be a triangulable real group and E= G its exponential ˘˘ radical. There exists a triangulable group G= E V and a homeomorphism φ: G→ G,sothat, denoting by d and d left-invariant word distances on G and G G˘ • φ restricts to the identity E→ E, • E is the exponential radical of G • V is isomorphic to the simply connected nilpotent Lie group G/E, • the map φ quasi-preserves the length: for some constant C > 0, −1 C|g|≤|φ(g)|≤ C|g|;∀g∈ G and is logarithmically bilipschitz −1 D(|g |+|h|) d (g, h)≤ d (g, h)≤ D(|g |+|h|)d (g, h);∀g, h∈ G, G G where C > 0 is a constant and where D is an increasing function satisfying D(n)≤ C log(n) for large n. Corollary 3.C.2. —Set{H, L}={G, G}. Suppose that the Dehn function δ of L satisﬁes δ (n) n . Then for any ε> 0, the Dehn function δ of H satisﬁes α+ε 2α+ε α δ (n) log(n) δ n log(n) log(n) n . H L GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 113 Proof. — Suppose, more precisely, that every loop of length n in L can be ﬁlled with area δ(n) in a ball of radius s(n);note that s can be chosen to be asymptotically equal to δ (by Lemma 2.D.2). −1 Start with a combinatorial loop γ of length n in H. It maps (by φ or φ )toa “loop” in L, in the Cn-ball, in which every pair of consecutive vertices are at distance ≤ Clog(n). Join those pairs by geodesic segments and ﬁll the resulting loop γ of length ≤ Cn log(n) by a disc consisting of δ(Cn log(n)) triangles of bounded radius (say,≤ C), inside the s(Cn log(n))-ball. Map this ﬁlling back to H. We obtain a “loop” γ consisting of Cn log(n) points, each two consecutive being at distance≤ Clog(s(Cn log(n))),with a ﬁlling by δ(Cn log(n)) triangles of diameter at most C log(s(Cn log(n))). Interpolate γ by geodesic segments, so as to obtain a genuine loop γ .So γ is ﬁlled by γ and Cn log(n) “small” loops of size ≤ Clog s Cn log(n)+ 1≤ Clog C Cn log(n)+ 1≤ C log(n). The loop γ itself is ﬁlled by δ(Cn log(n)) triangles of diameter at most Clog(s(Cn log(n))) and thus of size≤ 3C log(n). cn We know that H has its Dehn function bounded above by C e . So each of these small loops has area≤ C exp(3c(C log(n)))= C n .Wededucethat γ can be ﬁlled by 2 1 c 1+c+max(1,α) Cn log(n)+ δ Cn log(n) C n n triangles of bounded diameter. We deduce that H has a Dehn function of polynomial growth, albeit with an out- rageous degree, the constants c being out of control. Anyway, this provides a proof that Hhas a Dehn function≤ C n for some q, and we now repeat the above argument with this additional information. The small loops of size C log(n) therefore have area≤ C (C log(n)) and the 1 3 1 δ(Cn log(n)) triangles ﬁlling γ can now be ﬁlled by≤ C (3C log(s(Cn log(n)))) triangles of bounded diameter. We deduce this time that γ can be ﬁlled by at most q q Cn log(n)C C log(n)+ 3CC log s Cn log(n) δ Cn log(n) 3 1 3 ≈ log s n log(n) δ n log(n) triangles of bounded diameter. We have 2 2α log s(n log n)≤ log s n log n log(n), so we deduce that the Dehn function of γ is log(n) δ n log(n) . Since δ(n) n , the previous reasoning can be held with q of the form α+ ε for any ε> 0. This proves the desired result. 114 YVES CORNULIER, ROMAIN TESSERA Remark 3.C.3. — A variant of the proof of Corollary 3.C.2 shows that if the Dehn function of G is exponential, then the Dehn function of G is exp(n/ log(n) ), but is not strong enough to show that the Dehn function of G is exponential, nor even exp(n/ log(n) ) for small α≥ 0. 4. Gradings and tameness conditions This preliminary section describes basics on the weight decomposition of standard solvable groups and real triangulable groups. In Section 4.A, we state a general weight decomposition theorem for ﬁnite- dimensional representations of nilpotent groups over complete normed ﬁelds. We use it to introduce weights in standard solvable groups in Section 4.B, where we provide some use- ful characterizations. We introduce the tameness conditions in Section 4.C, which allow to reinterpret the SOL obstructions in a more conceptual way. In Section 4.D,wedeal with the Cartan grading in real triangulable groups, which is also needed in Section 11. 4.A. Grading in a representation. Lemma 4.A.1. —Let H⊂ G be an inclusion of ﬁnite index between nilpotent groups. Then any homomorphism f: H→ R has a unique extension f: G→ R. k k Proof.—If g∈ Gand g∈ H, the element f (g )/k does not depend on k,wedeﬁne ˜˜ it as f (g); note that this is the only possible choice for f and already proves uniqueness. To prove the existence, we need to check that f is a homomorphism. Since checking ˜˜˜ f (xy)= f (x)+ f (y) only involves two elements, we can suppose that G and H are ﬁnitely generated. We can also suppose that they are torsion-free, as the problem is not modiﬁed if we mod out by the ﬁnite torsion subgroups. So G and H have the same rational Mal- cev closure, and every homomorphism H→ R extends to the rational Malcev closure. ˜˜ Necessarily, the extension is equal to f in restriction to G, so f is a homomorphism. (Note that R could be replaced in the lemma by any torsion-free divisible nilpotent group.) Recall that for every complete normed ﬁeld K, the norm on K extends to every ﬁnite extension ﬁeld in a unique way [DwGS, Theorem 5.1, p. 17]. Theorem 4.A.2. —Let K be a non-discrete complete normed ﬁeld. Let N be a nilpotent topolog- ical group and V a ﬁnite-dimensional vector space with a continuous linear N-action ρ: N→ GL(V). Then there is a canonical decomposition V= V , α∈Hom(N,R) GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 115 where, for α∈ Hom(N, R), the subspace V is the sum of characteristic subspaces associated to α(ω) irreducible polynomials whose roots have modulus e for all ω∈ N;moreoverwehave, for α∈ Hom(N, R) n 1/n α(ω) V ={0}∪ v∈ V {0}:∀ω∈ N, lim$ρ(ω)· v$= e n→+∞ n 1/n α(ω) = v∈ V :∀ω∈ N, lim$ρ(ω)· v$≤ e n→+∞ (Notethatwedonot assume that ρ(N) has a Zariski-connected closure.) Proof. — Let us begin with the case when K is algebraically closed. Let N be the Zariski closure of ρ(N); decompose its identity component N= D× U into diagonaliz- able and unipotent parts. Consider the corresponding projections d and u into D and U. 0−1 0 Deﬁne N= ρ (N ); it is an open subgroup of ﬁnite index in N. Let D be the (ordinary) closure of the projection d(ρ(N )). We can decompose, with respect to D, the space V into weight subspaces: V= V ,where V ={v∈ V :∀d∈ D, d· v= γ γ γ∈Hom(D,K ) γ(d)v}.Write (log|γ|)(v)= log(|γ(v)|),so log|γ |∈ Hom(D, R).For δ∈ Hom(D, R), deﬁne V= V ,sothat V= V .If δ∈ Hom(D, R),then δ γ δ {γ: log|γ|=δ} δ∈Hom(D,R) δ◦ d◦ ρ∈ Hom(N , R). So by Lemma 4.A.1, it uniquely extends to a homomorphism δ: N→ R, which is continuous because its restriction δ◦ d◦ ρ to N is continuous. Note that δ → δ is obviously injective. If v∈ V {0} and ω∈ N , write it as a sum v= v where 0 = v∈ V , δ γ γ γ γ∈I where I is a non-empty ﬁnite subset of Hom(D, K ), actually consisting of elements γ for which log|γ |= δ. Then, changing the norm if necessary so that the norm is the supremum norm with respect to the norms on the V (which does not affect the limits because of the exponent 1/n), we have n n n 1/n 1/n $ρ(ω) v$= sup$u ρ(ω) d ρ(ω) v$ γ∈I 1/n =|γ d ρ(ω)| sup$u ρ(ω) v$ γ∈I 1/n = exp δ(ω) sup$u ρ(ω) v$; γ∈I the spectral radii of both u(ρ(ω)) and its inverse being equal to 1 and v = 0, we deduce n 1/n n 1/n 0 that lim$u(ρ(ω)) v$= 1. It follows that lim$ρ(ω) v$= exp(δ(ω)) for all ω∈ N k 0 n 1/n and v∈ V {0}.If ω∈ N, there exists k such that ω∈ N . Writing f (n) =$ρ(ω) v$ , δ ω we therefore have 1/k 1/k k ˆˆ lim f (kn)= lim f k(n)= exp δ ω= exp δ(ω); ω ω n→∞ n→∞ on the other hand a simple veriﬁcation shows that lim f (n+ 1)/f (n)= 1, and it n→∞ follows that lim f (n)= exp(δ(ω)). n→∞ ω 116 YVES CORNULIER, ROMAIN TESSERA ±1 It follows in particular that the spectral radius of ρ(ω) on V is exp(±δ(ω)) and since the δ are distinct, it follows that V is the sum of common characteristic subspaces associated to eigenvalues of modulus exp(δ(ω)) for all ω. Conversely, suppose that v∈ V and that there exists α∈ Hom(N, R) such that for n 1/n α(ω) all ω∈ Nwe have lim$ρ(ω)· v$≤ e , and let us check that v∈ V . Observe n→+∞ n−1 n 1/n that lim ($ρ(ω)· v $$ρ(ω )· v$)≤ 1. Write v= v with v∈ V and suppose δ δ δ δ∈J n→+∞ ˆˆ by contradiction that J contains two distinct elements δ ,δ .So δ = δ and there exist 1 2 1 2 ˆˆ ω∈ Nsuch that δ (ω) > δ (ω).Then 0 1 2 n 1/n n−1 1≥ lim$ρ(ω )· v $$ρ ω· v$ n→+∞ n 1/n−1 1/n ≥ lim$ρ(ω )· v$$ρ ω· v$ 0 δ δ 1 0 2 n→+∞ ˆˆ = exp δ (ω )− δ (ω ) > 1, 1 0 2 0 a contradiction; thus v∈ V . This concludes the proof in the algebraically closed case. Now let K be arbitrary. The above decomposition can be done in an algebraic clo- sure of K, and is deﬁned on a ﬁnite extension L of K,so W= V⊗ L= W , K α α∈Hom(N,R) where W satisﬁes all the characterizations. Consider a K-linear projection π of L onto K;itextends to a K-linear projection π= id⊗ π of W= V⊗ L onto V, V K K which commutes with the action of ρ(N). Clearly V= π (W ). Since K is a com- plete normed ﬁeld, π is Lipschitz. It then follows from the deﬁnition that π maps W into itself (where we naturally consider the inclusion V⊂ W): this follows from the characterization of those elements of W as those v∈ Wsuch that for all ω∈ N n 1/n α(ω) we have lim$ρ(ω)· v$≤ e . It follows that V= V ,where V= π (W )= α α α n→+∞ (W∩ V). So the proof is complete. Note that the proof, as a byproduct, characterizes the elements v of V α∈Hom(N,R) n−n 1/n as those in V for which, for all ω∈ N, we have lim ($ρ(ω)· v $$ρ(ω)· v$)≤ 1 n→+∞ (which is actually a limit, and equal to 1, if v = 0). 4.B. Grading in a standard solvable group. — Let G= U A be a standard solvable group in the sense of Deﬁnition 1.2. It will be convenient to consider the product ring K= K ; it is endowed with the supremum norm, and view U as U(K).Wethuscall j=1 G a standard solvable group over K. In a ﬁrst reading, the reader can assume there is a single ﬁeld K= K . Since K is a ﬁnite product of ﬁelds, a ﬁnite length K-module is the same as a direct sum V= V ,where each V is a ﬁnite-dimensional K -vector space. The length of V j j j as a K-module, is equal to dim V . K j Let u be the Lie algebra of U .So u= u is a Lie algebra over K and the j j j exponential map, which is truncated by nilpotency, is a homeomorphism u→ U. This GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 117 conjugates the action of D on U to a linear action on u, preserving the Lie algebra structure; for convenience we denote it as an action by conjugation. We endow u with the action of A, and with the grading in Hom(A, R), as intro- duced in Theorem 4.A.2.Thus u itself is graded by u= u . The ﬁnite-dimensional α j,α vector space W= Hom(A, R) is called the weight space. This is a Lie algebra grading: [u , u ]⊂ u ,∀α, β∈ Hom(A, R). α β α+β Note that since A is a compactly generated locally compact abelian group, it is d d 1 2 isomorphic to R× Z× K for some integers d , d , and K a compact abelian group. In 1 2 particular, if d= d+ d , then the weight space W= Hom(A, R) is a d -dimensional real 1 2 vector space. If we split u as the direct sum u= u⊕ u of its Archimedean and non- a na Archimedean parts, the weights of u and u , respectively, are called Archimedean weights a na and non-Archimedean weights. Example 4.B.1. — Assume that the action of A on u is diagonalizable. If K= R and the diagonal entries are positive, we have −1 α(v) (u )= x∈ u :∀v∈ A,v xv= e x j α j and if K= Q and the diagonal entries are powers of p,wehave j p −1−α(v)/ log(p) (u )= x∈ u :∀v∈ A,v xv= p x . j α j Example 4.B.2 (Weights in groups of SOL type). —Let G= (K× K ) Abe a group 1 2 of SOL type as in Deﬁnition 1.3, where A contains as a cocompact subgroup the cyclic subgroup generated by some element (t , t ) with|t| > 1 >|t|. Then the weight space 1 2 1 2 is a one-dimensional real vector space, and with a suitable normalization, the weights are α= log(|t|)> 0and α= log(|t|)< 0, and U= K . It is useful to think of the 1 1 2 2 α i weight α with multiplicity q , namely the dimension of K over the closure of Q in K i i i i (which is isomorphic to R or Q for some p). In particular, G is unimodular if and only if q α+ q α= 0. 1 1 2 2 For instance, if G= (R× Q ) Z,then u= R× Q , u= R ×{0}, u= p p p log(p)− log(p) {0}× Q . For an arbitrary standard solvable group, we deﬁne the set of weights W= α: u ={0}⊂ Hom(A, R). u α It is ﬁnite. Weights of the abelianization u/[u, u] are called principal weights of u. Lemma 4.B.3. — Every weight of u is a sum of≥ 1 principal weights; 0 is not a principal weight. 118 YVES CORNULIER, ROMAIN TESSERA Proof. — The ﬁrst statement is a generality about nilpotent graded Lie algebras. If P is the set of principal weights and v= u ,then u= v +[u, u], i.e., v gener- α∈P ates u modulo the derived subalgebra. A general fact about nilpotent Lie algebras (see Lemma 6.B.3 for a reﬁnement of this) then implies that v generates u. The ﬁrst assertion follows. The condition that 0 is not a principal weight is a restatement of Deﬁni- tion 1.2(3). Deﬁnition 4.B.4. —If U is a locally compact group and v a topological automorphism, v is called a compaction if there exists a compact subset ⊂ U that is a vacuum subset for v, in the sense that for every compact subset K⊂ U there exists n≥ 0 such that v (K)⊂ .Ifevery neighborhood of 1 is a vacuum subset, we say that v is a contraction. Proposition 4.B.5. —Let U A be a standard solvable group. Equivalences: (i) some element of A acts as a compaction of U; (ii) some element of A acts as a contraction of U; (iii) 0 is not in the convex hull in W of the set of weights; (iv) 0 is not in the convex hull in W of the set of principal weights. Proof. — Automorphisms of U are conjugate, through the exponential, to linear au- tomorphisms; in particular, contractions and compactions coincide, being characterized by the condition that all eigenvalues have modulus < 1. Thus (i)⇔(ii). By Lemma 4.B.3, weights are sums of principal weights, and thus (iii)⇔(iv). If v∈ A acts as a contraction of U, then α(v) < 0 for every weight α.Thus α → α(v) is a linear form on W , which is positive on all weights. Thus (ii)⇒(iii). Conversely, let L be the set of linear forms of W= Hom(A, R) such that (α) > 0 for every weight α∈ W . Suppose that 0 is not in the convex hull of W , or equivalently u u ∗∗ that L =∅. Since the image of A in the bidual W linearly spans W , it has a cocompact closure in bidual W ; since L is a nonempty open convex cone, it follows that it has a non-empty intersection with the image of A in the bidual W . Thus there exists v∈ A such that α(v) > 0 for every weight α. So (iii)⇒(ii) holds. 4.C. Tameness conditions in standard solvable groups. — Motivated by Proposition 4.B.5, we introduce the following terminology: Deﬁnition 4.C.1. — We say that the standard solvable group G= U A (or the graded Lie algebra u)is • tame if 0 is not in the convex hull of the set of weights; • 2-tame if 0 is not in the segment joining any pair of principal weights; • stably 2-tame if 0 is not in the segment joining any pair of weights. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 119 To motivate the adjective “stably”, observe that u is stably 2-tame if and only if every graded subalgebra of u is 2-tame. The importance of 2-tameness, ﬁrst brought forward by Abels, will appear gradu- ally in the paper, starting with Proposition 4.C.3. Remark 4.C.2. — Clearly tame⇒ stably 2-tame⇒ 2-tame; the converse implications do not hold in general; however they hold when W is 1- dimensional, i.e. when A has a discrete cocompact inﬁnite cyclic subgroup. Also, when u is abelian, then 2-tame and stably 2-tame are obviously equivalent. The following characterization of 2-tameness will be needed to show in Sec- tion 4.C.3 that if G is not 2-tame then it has an at least exponential Dehn function. Proposition 4.C.3. —Let G= U A be a standard solvable group. Then G is not 2-tame if and only if there exists a group V E of type SOL (see Deﬁnition 1.3) and a homomorphism f into V E whose image contains V and is dense. Proof. — In general, let V E be a standard solvable group, and consider a contin- uous homomorphism f: U A→ V E with dense image (assuming that the closure of the image is cocompact would be enough). Since U and V are the derived subgroups of thetwo groups,wehave f (U)⊂ V. Since f (U)⊂ V, passing to the quotients, f induces a continuous homomorphism A→ E with dense image, which induces an injective linear map f: Hom(E, R)→ Hom(A, R). Denote by f the Lie algebra map it induces u→ v. If we consider the weight decomposition u= u , we see that either f (u ) is zero, or α has the form α α α∈Hom(A,R) f (β) and f (u )⊂ v . In particular, f (u) is contained in the sum of v ,where β ranges α β β over elements in Hom(E, R) such that f (β) is a weight of U A. In particular, if we assume that f (U)= V, we deduce that f maps weights to weights. Assume now that V E is a group of type SOL; since E is abelian, f factors through a homomorphism U/[U, U] A→ V E. Recall from Example 4.B.2 that for a group of SOL type, the set of weights consists of a quasi-opposite pair, i.e. a pair of nonzero elements such that the segment joining them contains zero. Since f is injective R-linear and maps weights to weights, we deduce that U/[U, U] A admits a quasi-opposite pair of weights. Conversely, suppose that U A, where U/[U, U] has two nonzero weights α, β with β= tα for some t < 0. To construct the map f , ﬁrst mod out by[U, U], so we are reduced to the case when U is abelian. Modding out if necessary by all other nonzero weights, we can suppose that u= u⊕ u .Write u= u ,and modout all u except one, so that u is a Lie α β α j,α j,α α j 120 YVES CORNULIER, ROMAIN TESSERA algebra over a single ﬁeld K ; again modding out by a maximal invariant subspace and possibly replacing K with a ﬁnite extension, we can assume that u is 1-dimensional over j α K , with a scalar action. Similarly we can assume that u is 1-dimensional over K ,with j β j ∗∗ a scalar action. We can mod out by the kernel of the homomorphism A→ K× K ; j j let B be the closure of its image. Since α and β are proportional, E contains a cyclic cocompact subgroup. The resulting group V E, where V= K× K ,isoftypeSOL j j and the resulting homomorphism U A→ V E has a dense image containing V. Remark 4.C.4. — The homomorphism to a group of type SOL cannot always be 2 2 chosen to have a closed image. For instance, let Z act on R by −m−n m n (m, n)· (x, y, z)= 2 3 x, 2 3 y . 2 2 Let G= R Z be the corresponding standard solvable group. Then every nontrivial normal subgroup of G contains either R ×{0} or{0}× R; thus every proper quotient of G is tame and it follows that no quotient of G is of SOL type. Let G= U A is a standard solvable group in the sense of Deﬁnition 1.2.The Lie algebra u of U admits a grading in the weight space W= Hom(A, R), introduced in Section 4.B. Deﬁne the set of weights of u as the ﬁnite subset W ={α∈ W: u ={0}}. u α We say that a subset of W is conic if it is of the form W∩ C, with C an open u u convex cone not containing 0. Let C be the set of conic subsets of W .If C∈ C,deﬁne u= u; C α α∈C this is a graded Lie subalgebra of u; clearly it is nilpotent. Let U be the closed subgroup of U corresponding to u under the exponential map and G= U A. In particular, if C C C v∈ A, deﬁne H(v) ={α∈ W: α(v) > 0}. Deﬁnition 4.C.5. —The G are called the essential tame subgroups of G.The G H(v) C are called the standard tame subgroups of G. Note that each essential tame subgroup is standard tame, and there ﬁnitely many standard tame subgroups; moreover every standard tame subgroup is a ﬁnite intersection of essential tame subgroups. Remark 4.C.6. — These notions can be developed in a broader context, avoiding references to gradings and Lie algebras. Let G be a locally compact group with a ﬁxed semidirect product decomposition G= U A. A tame subgroup of G (relative to the given semidirect decomposition) is deﬁned as a subgroup of the form V A, which is tame, i.e., in which some element of A acts on V as a compaction (in the sense of Deﬁnition 4.B.4). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 121 For v∈ A, if we deﬁne U as the contraction subgroup −n n (4.C.7) x∈ U: lim v xv= 1 , n→+∞ then v acts as a compaction on U ([CCMT15, Prop. 6.17]) and therefore U Ais v v tame. We call this a essential tame subgroup.Then standard tame subgroups aredeﬁnedasﬁ- nite intersections of essential tame subgroups; this can be shown to match the previous deﬁnition in the setting of standard solvable groups. 4.D. Cartan grading and weights. — Here we introduce the notion of Cartan grading, especially in the context of real triangulable Lie algebras; this will especially be needed in Section 11. Triangulable Lie groups are not standard solvable in general and although the treatment is analogous to that in Section 4.B, we have to face some speciﬁc difﬁculties. All Lie algebras in this Section 4.D are ﬁnite-dimensional over a ﬁxed ﬁeld K of characteristic zero. ∞ k Deﬁnition 4.D.1. —If g is a Lie algebra, let g= g be the intersection of its lower k≥1 central series, so that g/g is the largest nilpotent quotient of g. Deﬁnition 4.D.2. —If G is a triangulable Lie group with Lie algebra g, deﬁne its exponen- ∞∞ tial radical G as the intersection of its lower central series (so that its Lie algebra is equal to g ). We need to recall the notion of Cartan grading of a Lie algebra, which is used in Section 11.D and Section 11.E.Let n is a nilpotent Lie algebra; denote by n the space of homomorphisms from n to K (that is, the linear dual of n/[n, n]). Let v be an n-module (ﬁnite-dimensional) with structural map ρ: n→ gl(v).If α∈ n , deﬁne the characteristic subspace v= v∈ v :∀g∈ n, ρ(g)− α(g) v= 0 . k≥1 The subspaces v generate their direct sum; we say that v is K-triangulable if v= v ; α α α∈n this is automatic if K is algebraically closed. If v is a K-triangulable n-module, the above decomposition is called the natural grading (in n )of v as an n-module. If v= v ,wecall v a nilpotent n-module. Deﬁnition 4.D.3 ([Bou]). — A Cartan subalgebra of g is a nilpotent subalgebra that is equal to its normalizer. We use the following proposition, proved in [Bou, Chap. 7, §1, 2, 3] (see Deﬁni- tion 4.D.1 for the meaning of g ). 122 YVES CORNULIER, ROMAIN TESSERA Proposition 4.D.4. — Every Lie algebra admits a Cartan subalgebra. If n is a Cartan subal- gebra, then g= n+ g and n contains the hypercenter of g (the union of the ascending central series). If g is solvable, any two Cartan subalgebras of g are conjugate by some elementary automorphism exp(ad(x)) with x∈ g (here ad(x) is a nilpotent endomorphism so its exponential makes sense). Let us provide a simple way to recognize a Cartan subalgebra. Proposition 4.D.5. —Let g be a Lie algebra graded in some abelian group W (see Section 8.A if necessary). Assume that g is nilpotent and that for every α = 0, we have[g , g ]= g . Then g is 0 0 α α 0 a Cartan subalgebra in g. Proof. — We have to show that the normalizer h of g in g is g itself. This is a 0 0 graded subalgebra; hence we have to show that h= 0for all α = 0. Since[g , h] is α 0 α contained in both g and g , we deduce that[g , h ]= 0. 0 α 0 α To show that h= g , it is enough to do it in the case K is algebraically closed. For α = 0, we can decompose the module g with respect to the action of the nilpotent Lie algebra g as a sum of characteristic subspaces V , for homomorphisms λ: g→ K. 0 λ 0 Then V= 0 since the contrary would contradict[g , g ]= g . On the other hand, since 0 0 α α h is contained in the centralizer of g , it is contained in V . We deduce that it is zero. α 0 0 Example 4.D.6. —For λ∈ K, let g(λ) be the 4-dimensional Lie algebra with basis (u, x, y, z) and nonzero brackets[x, y]= z,[u, x]= x,[u, y]= λy,[u, z]= (1+ λ)z.Then a Cartan subalgebra is given as follows: • if λ/ ∈{−1, 0}, it can be taken to be the 1-dimensional subalgebra generated by u; • if λ =−1, as the abelian subalgebra generated by u, z; • if λ= 0, as the abelian subalgebra generated by u, y. Indeed, we can consider the grading in Z for which u has weight 0, x has weight 1, y has weight λ,and z has weight 1+ λ. Then it satisﬁes the condition of Proposition 4.D.5. Other illustrating examples of Cartan subalgebras in triangulable Lie algebras can be found in [C08, Ex. 4.1 and 4.2]. We now turn to a partial converse for Proposition 4.D.5. Let g be a Lie algebra and n a Cartan subalgebra. For the adjoint representation, g is an n-module. Assume that g is K-triangulable as an n-module (e.g., this holds if K is algebraically closed). The corresponding natural grading is called the Cartan grading of g (relative to the Cartan subalgebra n); moreover the Cartan grading determines n,namely n= g .Wecall weights the set of α such that g = 0. The Cartan grading is a Lie algebra 0 α grading, i.e. satisﬁes[g , g ]⊂ g for all α, β∈ n . In addition it satisﬁes[g , g ]= g α β α+β 0 α α for all α = 0; hence it fulﬁlls the conditions of Proposition 4.D.5. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 123 ∞∞ We have n+ g= g,and g ⊂[g, g], so the projection n→ g/[g, g] is surjective, ∨∨ inducing an injection (g/[g, g])⊂ n . Now assume that g is K-triangulable. Then all the weights of the Cartan grading lie inside the subspace (g/[g, g]) of n; the advantage is that this space does not depend on n, allowing to refer to a weight α without reference to a the choice of a Cartan subalgebra (although the weight space g still depends on this choice). In view of Proposition 4.D.4, any two Cartan gradings of g are conjugate. In particular, the set of weights, viewed as a ﬁnite subset of (g/[g, g]) , does not depend on the Cartan grading. Actually, the subspace linearly spanned by weights is exactly (g/r) ,where r ⊃[g, g] is the nilpotent radical of g.The principal weights of g are by deﬁnition the weights of the Lie algebra ∞∞∞∞∞ g /[g , g] (that is, the nonzero weights of the Lie algebra g/[g , g]); note that every nonzero weight is a sum of principal weights, as a consequence of the following lemma, which relies on the (basic) results of Section 8.A.2. Lemma 4.D.7. —Deﬁne g= g .Then g is the subalgebra generated by g . α =0 Proof. — It is clear from the deﬁnition that[n, g ]= g for all α = 0 and therefore α α ∞∞ g⊂ g . Conversely, since g= n is nilpotent, Lemma 8.A.5 implies that g is contained in the subalgebra generated by g . If m is a g-module, deﬁne m ={x∈ m :∀g∈ g, gx= 0}. Lemma 4.D.8. —Let g be a K-triangulable Lie algebra with a Cartan grading, and i≥ 1. ∞ g∞ Then H (g ) ={0} if and only if H (g ) ={0}. i i 0 ∞ g∞ Proof. — By deﬁnition of Cartan grading, we have H (g )⊂ H (g ) ; this pro- i i 0 vides one implication. ∞∞∞ Note, that H (g ) is a trivial g -module, hence can be viewed as a (g/g )- module, and for any α,H (g ) is a nonzero g -submodule; since it is stable under both i α 0 ∞∞∞∞ g and g ; since g= g+ g we deduce that each H (g ) is a g-submodule of H (g ) 0 0 i α i ∞ g∞ g and H (g )= H (g ) . i i ⊗i On the other hand, it follows from the deﬁnition of Cartan grading that (g ) is a nilpotent g -module; hence its subquotient H (g ) is also a nilpotent g -module 0 i 0 0 and hence is a nilpotent g-module by the previous remark. Hence if H (g ) = 0, then i 0 H (g ) is nonzero as well. Now consider a real triangulable Lie algebra g, or the corresponding real triangu- lable Lie group. We call weights of g the weights of graded Lie subalgebra g endowed with the Cartan grading from g (note that since g= g+ g , this is precisely the set of weights of the graded Lie algebra g, except possibly the 0 weight). We deﬁne a real triangulable Lie algebra g, or the corresponding real triangulable Lie group G, to be tame, 2-tame,or stably 2-tame exactly in the same fashion as in Deﬁni- tion 4.C.1. In the case G is also a standard solvable group U D, we necessarily have 124 YVES CORNULIER, ROMAIN TESSERA u= g and its grading is also the Cartan grading; in particular whether G is tame (resp. 2-tame, stably 2-tame) does not depend on whether G is viewed as a real triangulable Lie group or a standard solvable Lie group. For α> 0, deﬁne SOL as the semidirect product R R, where the action is t−αt given by t· (x, y)= (e x, e y). Note that apart from the obvious isomorphisms SOL SOL , they are pairwise non-isomorphic. 1/α In a way analogous to Proposition 4.C.3,wehave Proposition 4.D.9. —Let G be a real triangulable group. Then G is not 2-tame if and only if it admits SOL as a quotient for some α> 0. Proof. — The “if ” part can be proved in the same lines as the corresponding state- ment of Proposition 4.C.3 and is left to the reader. ∞∞ Conversely, assume that G is not 2-tame. Clearly, G/[G , G] is not 2-tame, so we ∞∞ can assume that G is abelian. Endow g with a Cartan grading; then since g is abelian, ∞∞∞ 0isnot aweightof g and hence g= g g .Let G= G G be the corresponding 0 0 decomposition of G. In the same way as in the proof of Proposition 4.C.3, we can pass to a quotient that is still not 2-tame, for which the new G is 2-dimensional. Since the action of G on G is given by two proportional weights, its kernel has codimension 1 in N; in particular this kernel is normal in G; we see that the quotient is necessarily isomorphic to SOL for some α> 0. 5. Algebraic preliminaries about nilpotent groups and Lie algebras 5.A. Divisibility and Malcev’s theorem. — Recall that a group G is divisible (resp. uniquely divisible)iffor every n≥ 1, the power map G→ G mapping x to x ,issurjec- tive (resp. bijective). The following lemma is very standard. Lemma 5.A.1. — Every torsion-free divisible nilpotent group is uniquely divisible. k k Proof. — We have to check that x= y⇒ x= y holds in any torsion-free nilpotent k k group. Assume that x= y and embed the ﬁnitely generated torsion-free nilpotent group =x, y into the group of upper unipotent matrices over the reals. Since the latter is uniquely divisible (the k-th extraction of root being deﬁned by some explicit polynomial), we get the result. Let N be the category of nilpotent Lie algebras over Q (of possibly inﬁnite di- mension) with Lie algebras homomorphisms, and N the category of nilpotent groups with group homomorphisms, and N its subcategory consisting of uniquely divisible (i.e. divisible and torsion-free, by Lemma 5.A.1) groups. If g∈ N , consider the law on g Q g deﬁned by the Campbell-Baker-Hausdorff formula. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 125 Theorem 5.A.2 (Malcev [Mal49a, St70]). — For any nilpotent Lie algebra g over Q, (g, ) is agroup andif f: g→ h is a Lie algebra homomorphism, then f is also a group homomorphism (g, )→ (h, ). In other words, g → (g, ),f → f is a functor from N to N .Moreover, g k g Q this functor induces an equivalence of categories N→ N . Q Q The contents of the last statement is that • any group homomorphism (g, )→ (h, ) is a Lie algebra homomorphism; g h • any uniquely divisible nilpotent group (G,•) has a unique Q-Lie algebra struc- ture g= (G,+,[·,·]) such that =•. Recall that an element in a group is divisible if it admits n-roots for all n≥ 1. Lemma 5.A.3 (see Lemma 3 in [Ho77]). — In a nilpotent group, divisible elements form a subgroup. Lemma 5.A.4 (see Theorem 14.5 in [Ba60]). — Let G be nilpotent and uniquely divisible, n n with lower central series (G ). Then G/G is torsion-free (hence uniquely divisible) for all n. The following lemma is needed in the proof of Theorem 9.D.2. Lemma 5.A.5. — In the category of s-nilpotent groups, any free product of uniquely divisible groups is uniquely divisible. Proof. — First, the free groups in this category are torsion-free: to see this, it is enough to consider the case of a free group of ﬁnite rank in this category; such a group is of the form F/F with F a free group; it is indeed torsion free for all i: this is a result about the lower central series of a non-abelian free group and is due to Magnus [Mag35] (see also [Se, IV.6.2]). Let G , G be torsion-free uniquely divisible s-nilpotent groups and G∗ G their 1 2 1 s 2 s-nilpotent free product, which is divisible by Lemma 5.A.3.Denoteby (N ) the lower s+1 central series of G∗ G .Let g be a non-trivial element in G∗ G= (G∗ G )/N 1 2 1 s 2 1 2 and let us show that g is not torsion in this group. By [Mal49b], a free product of torsion- free nilpotent groups is residually torsion-free nilpotent, and therefore there exists t≥ s+ 1 such that the image of g is not torsion in (G∗ G )/N . Applying Lemma 5.A.4 to 1 2 t s+1 (G∗ G )/N , we see that (G∗ G )/N is torsion-free, so since g is non-trivial, it is not 1 2 1 2 torsion. So G∗ G is uniquely divisible by Lemma 5.A.1. 1 s 2 5.B. Powers and commutators in nilpotent groups. — This subsection includes several lemmas, which will be used at various places of the paper. Denote by ((·,·)) group commutators, namely −1−1 ((x, y))= x y xy, 126 YVES CORNULIER, ROMAIN TESSERA and n-fold group commutators (5.B.1) ((x ,..., x ))= ((x ,((x ,..., x )))). 1 n 1 2 n Deﬁne similarly n-fold Lie algebra brackets. When n≥ 2 is not speciﬁed, we just call them iterated group commutators or Lie algebra brackets. 1 i+1 i If G is a group, its lower central series is deﬁned by G= Gand G= ((G, G )) (the group generated by commutators ((x, y)) when (x, y) ranges over G× G ). The group s+1 Gis s-nilpotent if G ={1}. In particular, 0-nilpotent means trivial, 1-nilpotent means abelian, and more generally, s-nilpotent means that (s+ 1)-fold group commutators van- ish in G. Similarly, if g is a Lie algebra, its lower central series is deﬁned in the same way (and does not depend on the ground commutative ring), and s-nilpotency has the same meaning. The following lemma will be used in the proof of Lemma 6.E.4. Lemma 5.B.2. —Let N be an s-nilpotent group, and let i be an integer. Then there exists an integer m= m(i, s) such that for all x, y∈ N, we can write ((x, y ))= w ...w , 1 m ±1±1 where each w (1≤ j≤ m) is an iterated commutator (or its inverse) whose letters are x or y , that ±1±1 is, w= ((t ,..., t )) for some k≥ 2 and t ∈{x , y}. j j,1 j,k j j,i Example 5.B.3. — In a 3-nilpotent group N, we have, for all x, y∈ Nand i∈ Z i i−i(i−1)/2 ((x, y ))= ((x, y)) ((y, x, y)) . Proof of Lemma 5.B.2. — We shall prove the lemma by induction on s. The statement is obvious for s= 0 (i.e. when N is the trivial group), so let us suppose s≥ 1. Applying the induction hypothesis modulo the s-th term of the descending series of N, one can write ((x, y ))= wz,where w has the form w ...w where m= m(i, s− 1),and where z lies in 1 m the s-th term of the lower central series of the subgroup generated by x and y, which will ±1±1 i be denoted by H. Since the word length according to S ={x , y} of both ((x, y )) and w is bounded by a function of i and s, this is also the case for z. Now H is generated by the set of iterated commutators T ={((x , x ..., x ))| x ,..., x∈ S}. Therefore, z can be 1 2 s 1 s ±1 written as a word in T , whose length only depends on s and i. So the lemma follows. The following lemma will be used in the proof of Lemma 6.B.3. Lemma 5.B.4. — In any uniquely divisible s-nilpotent group, if x= exp v i i ((x ,..., x ))= exp[v ,v ,...,v]. 1 s 1 2 s GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 127 Proof. — (Refer to (5.B.1) for the conventions deﬁning iterated Lie brackets or group commutators.) Use the convenient convention to identify the group and the Lie algebra through the exponential and with this convention, the lemma simply states that ((x ,..., x )) =[x , x ,..., x ]∀x ,..., x∈ G. 1 s 1 2 s 1 s We prove the result by induction on s;itistrivial for s= 1 (abelian groups); assume it holds for s− 1. By induction, ((x ,..., x )) =[x ,..., x ]+ O(s),where O(s) means some 2 s 2 s combination of s-fold Lie algebra brackets. It follows from the Baker-Campbell-Hausdorff formula that if[x, y] is central then ((x, y)) =[x, y]. We can apply this to x= x and y= ((x ,..., x )). This yields 1 2 s ((x ,((x ,..., x ))))= x ,[x ,..., x ]+ O(s) =[x ,..., x]. 1 2 s 1 2 s 1 s The following proposition is the key new feature in the proof of Theorem 9.D.2. Proposition 5.B.5. —Let x, y be elements of a uniquely divisible nilpotent group G.Let N r−r be the normal subgroup generated by the elements of the form x y , where r ranges over Q. Then N is divisible. Equivalently, G/N is torsion-free. We need the following lemma. Lemma 5.B.6. —Let x, y be elements of a uniquely divisible nilpotent group G,and n an −1 1/n 1/k−1/k integer. Then (xy ) is contained in the normal subgroup generated by{x y: k∈ Z}. Proof.—Let G be s-nilpotent. By [BrG11, Lemma 5.1] (see however Re- mark 5.B.7), there exists a sequence of rational numbers a , b ,..., a , b such that in 1 1 k k any uniquely divisible s-nilpotent group H and any u,v∈ H, we have 1/n −1 1/n−1/n a b i i uv= u v u v . i=1 In particular, picking (H, u,v)= (R, 1, 2), we see that a= b= 0. Therefore, if i i a b 1/n−1/n i i d is a common denominator to a ,..., b and n, u v as well as u v belong to 1 k i=1 1/d−1/d−1 1/n the normal subgroup generated by u v ,hence (uv ) as well. In particular, this applies to (H, u,v)= (G, x, y). Remark 5.B.7. — In the above proof, we used [BrG11, Lemma 5.1] to be concise. However, this is not very natural, because the latter is proved using the Hall-Petrescu formula; the problem is that in this formula, exponents are put outside the commutators. The proof of the Hall-Petrescu formula can easily be modiﬁed to prove by induction on the degree of nilpotency a similar formula with exponents inside the commutators. In 128 YVES CORNULIER, ROMAIN TESSERA [BrG11], in order to shorten the argument (as we also do), instead of processing this in- duction, they work with a much simpler induction based on the Hall-Petrescu formula; this is very unnatural, because if we do not allow ourselves to use the Hall-Petrescu for- mula, to go through the latter is very roundabout; moreover the exponents a , b obtained k k in [BrG11]dependon s, while in a direct induction, we pass from the s-nilpotent case to the (s+ 1)-nilpotent case by multiplying on the right by some suitable iterated commu- tator of powers. r−r ρ Proof of Proposition 5.B.5. — By Lemma 5.B.6, N contains elements (x y ) for any r−r ρ ρ∈ Q. So N is generated as a normal subgroup by the divisible subgroups N ={(x y ): ρ∈ Q}, and therefore N is divisible by Lemma 5.A.3. The following lemma is used in the proof of Lemma 9.B.1. Lemma 5.B.8. —Let g be a Lie algebra over the commutative ring R.Let m be a generating [1][i][1][i−1] R-submodule of g.Deﬁne g= m, and by induction the submodule g =[g , g] for i≥ 2 (namely, the submodule generated by the brackets of the given form). Let (g ) be the lower central series i[m] of g. Then for all i we have g= g . m≥i Proof.—Letuscheck that [i][j][i+j] (5.B.9) g , g⊂ g∀i, j≥ 1. Note that (5.B.9) holds when either i= 1or j= 1. We prove (5.B.9) in general by induction on k= i+ j≥ 2, the case k= 2 being already settled. So suppose that k≥ 3 and that the result is proved for all lesser k. We argue again by induction, on min(i, j), the case min(i, j)= 1 being settled. Let us suppose that i, j≥ 2and i+ j= k and let us check that (5.B.9) holds. We can suppose that j≤ i. By the Jacobi identity and then the induction hypothesis [i][j][i][1][j−1] [g , g ]= g , g , g [1][i][j−1][j−1][1][i] ⊂ g , g , g+ g , g , g [1][i+j−1][j−1][1+i] ⊂ g , g+ g , g [i+j][j−1][1+i] ⊂ g+ g , g Since j≤ i,wehave min(j− 1, i+ 1)< min(i, j), the induction hypothesis yields [j−1][1+i][i+j] [g , g ]⊂ g and (5.B.9)isproved. It follows from (5.B.9) that, deﬁning for all i≥ 1 {i}[j] g= g , j≥i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 129 {i} the graded submodule g is actually a Lie subalgebra of g. Let us check by induction on i{i}[1]{1} 1 i≥ 1that g= g . Since g generates g, it follows that g= g= g . Now, suppose i≥ 2 and the equality holds for i− 1. Then g is, in our notation, the Lie subalgebra generated by i−1{i−1}[j] g, g= g, g= g, g j≥i−1 [j][j+1]{i} = g, g= g= g; j≥i−1 j≥i−1 {i} i{i} since g is a Lie subalgebra we deduce that g= g . 5.C. Lazard’s formulas. — Lazard’s formulas (which will be used in Section 10.D and Section 10.F) are, in a sense, inversion formulas for the Baker-Campbell-Hausdorff formula: roughly speaking, they express the Lie algebras laws in terms of the group law; actually it involves taking roots and we ﬁnd it convenient to state them by “canceling denominators” as below. Here, F denotes the free group on 2 generators). Theorem 5.C.1 (Lazard [La54]). — For every s≥ 1, there exist group words A , B∈ F s s 2 and positive integers q , q , such that for every simply connected s-nilpotent Lie group G with Lie algebra g,and all x, y∈ G, writing X= log(x)∈ g, Y= log(y)∈ g, we have log A (x, y)= q (X+ Y) and log B (x, y)= q[X, Y] s s s Here A stands for Add, and B for Bracket. Remark 5.C.2. — We see that we have the formal equality of group words A (x, 1)= x . Indeed, A (x, 1) has the form x for some ∈ Z, and by evaluation in s s R we obtain = q . Similarly, we have the formal equality B (x, 1)= 1. s s The usual Lie ring axioms are reﬂected in the following proposition: Proposition 5.C.3. — In every s-nilpotent group G, abbreviating A= A , B= B ,q= q , s s s we have identities∀x, y, z∈ G, −1 A(x, y)= A(y, x); B(x, y)= B(y, x); B A(x, y), z= A B(x, z), B(y, z); q q A A(x, y), z= A x , A(y, z); k k k B x , y= B x, y= B(x, y)∀k∈ Z. 130 YVES CORNULIER, ROMAIN TESSERA Proof. — If G is a simply connected nilpotent Lie group, this follows from the cor- responding identities in the Lie algebra: commutativity of addition, anti-commutativity of the bracket, distributivity, associativity of addition; in the last equality if follows from the fact that log(x )= k log(x). Therefore this holds in every subgroup of such a group G, and in particular in any ﬁnitely generated free s-nilpotent group, and therefore in any s-nilpotent group, by substitution. 6. Dehn function of tame and stably 2-tame groups In Section 6.A, we introduce the class of tame groups, showing in particular that their Dehn function is at most quadratic. In Section 6.C, we formulate and prove a ver- sion of Gromov’s trick for standard solvable groups involving its tame subgroups. This result relies on some lengths estimates on nilpotent algebraic groups over local ﬁelds, which are obtained in Section 6.B.ThenSection 6.D is dedicated to the proof of Theo- rem D from the introduction (which is reformulated in terms of 2-tameness). Finally we extend this result to generalized 2-tame groups in Section 6.E. 6.A. Tame groups. — We introduce here a class of locally compact groups, called tame groups, including tame standard solvable groups from Deﬁnition 4.C.1. Tame groups are, for our purposes, the best-behaved. Using a “large-scale Lip- schitz homotopy”, these groups are shown to be compactly presented with an at most quadratic Dehn function and a have simple presentation (Theorem 6.A.4 and Corol- lary 6.A.7). While standard solvable groups are “usually” not tame, their geometrical study will be based on the family of their tame subgroups. Deﬁnition 6.A.1. —Wecalla tame group any locally compact group with a semidirect product decomposition G= U A,where A is a compactly generated abelian group, such that some element of A acts as a compaction of U (in the sense of Deﬁnition 4.B.4), in which case we call U A a tame decomposition. Recall that this means that there exist v∈ A and a compact subset (called vac- n−n uum subset) of U such that v v= U uniformly on compact subsets, in the n≥0 n−n sense that for every compact subset K of U there exists n such that K⊂ v v .If −n n we apply this to itself, so that v v⊂ ,wesee,byreplacing with the larger −n n−n n−1 v v= v v , that we can suppose v v⊂ , in which case we call 0≤k≤n−1 k≥0 a stable vacuum subset. Remark 6.A.2. —If G= U A is a tame decomposition, and if W is the largest compact subgroup in A, then W admits a direct factor A in A (isomorphic to R× Z for some k,≥ 0), so that G= UW A ;if x∈ A actsasacompaction on U thenit GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 131 also acts as a compaction on UW, and so does the projection of x on A (because the set of compactions of a given locally compact group is stable by multiplication by inner automorphisms [CCMT15, Lemma 6.16]). This shows that assuming that A= R× Z is not really a restriction. We are going to prove that tame groups are compactly presented with an at most quadratic Dehn function; in order to make quantitative statements, we use the following language. Let v∈ A be an element acting (by conjugation on the right) as a compaction of U. Let m be an integer. Let S be a compact symmetric subset of U with unit. Let T be a compact symmetric generating subset of A with unit. Deﬁnition 6.A.3. — We say that (m, S , T) is adapted to U A and v if v∈ T,the −1 subset S is a stable vacuum subset for the right conjugation u → v uv by v, and there exist non- −k 2 k −1 negative integers k,≥ 1 with k+ ≤ msuch that v S v⊂ S and (v w) S (v w)⊂ S U U U for all w∈ T. There always exists such an adapted triple; more precisely, for every symmetric generating subset T of A containing{1,v} and every symmetric stable vacuum subset S⊂ Uof v, (m, S , T) is adapted for all m large enough. Also, if S is a stable vacuum U U U ±k} k subset for some v∈ A and T is arbitrary, then (2, S , T ∪{v ) is adapted to v for large enough k. Theorem 6.A.4. —Let G be a tame group with a tame decomposition U A, with an element v acting as a compaction of U. Denote by π the canonical projection G→ A. Let (m, S , T) be adapted to U A and v in the above sense. Then S= S∪ T is a compact U U generating subset of G,and G being endowed with the corresponding word metric, the function γ: G× N→ G; (g, n) → gv is 1-Lipschitz in each variable and satisﬁes: γ(g, 0)= g, and d(γ (g, n), v π(g))≤ 1 whenever n≥ m|g|. The function γ should be understood as “locally” a large-scale homotopy between the identity of G and its projection onto A. This is only “local” since the time needed to reach A depends on the size of the element. See however Remark 6.A.10. −1−1 Proof of Theorem 6.A.4. — Since v S v⊂ S ,wehave v Sv⊂ S. So, for n≥ 0, U U −n n n the automorphism g → v gv is 1-Lipschitz, and since the left multiplication by v is an isometry, we deduce that γ(·, n) is 1-Lipschitz. Also, assuming that v∈ T, it is immediate that γ(g,·) is 1-Lipschitz. 132 YVES CORNULIER, ROMAIN TESSERA It remains to prove the last statement. Consider an element in U, of size at most n. We can write it as g= s u i i i=1 with s∈ Tand u∈ S . Deﬁning t= s ... s , we deduce i i U i i n −1 g= t t u t . n i i i=1 So t= π(g) and −n n−n−1 n v gv= π(g) v t u t v . i i i=1 −1 −n n By deﬁnition of , the element v= v t u t v belongs to S .So i i i U −n n v gv= π(g) v . i=1 −k 2 k−kj n kj Since v (S )v⊂ S , setting j =! log (n)",wehave v (S )v⊂ S .Thuswe U U U 2 U obtain −n−k! log (n)" n+k! log (n)" 2 2 v gv∈ π(g)S , mn mn since for all n we have! log (n)"≤ n and k+ ≤ n,weobtain d(gv ,v π(g))≤ 1. A ﬁrst consequence is an efﬁcient way of writing elements in a tame group: Corollary 6.A.5. — Under the assumptions of Theorem 6.A.4,for every x∈ G of with|x|= mn−mn n, we can write x= π(x)v sv and s∈ S . mn−1 Proof.—Deﬁne s= (v π(x)) γ(x, mn).Bythe theorem, s∈ S , while x= mn−mn π(x)v sv . The corollary allows us to introduce the following deﬁnition, which will be used in the sequel. Let G= U A betameand v∈ A act as a compaction on U. Let (m, S , T) be adapted to U Aand v in the sense of Deﬁnition 6.A.3,and S= S∪ T. By Corol- −mn mn lary 6.A.5,if x∈ Uand n =|x| , the element s= v xv belongs to S . S U Deﬁnition 6.A.6. —For x∈ U with|x|= nas above, we deﬁne x∈ F as the word S S mn−mn v sv , which has length 2mn+ 1 and represents x. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 133 We now write the consequence of Theorem 6.A.4, that tame groups have a “nice” presentation and a good control on the Dehn function. Corollary 6.A.7. — Under the assumptions of Theorem 6.A.4, G admits a presentation with generating set S in which the relators are the following (1) All relators of the form s s s for s , s , s∈ S ,whenever s s s= 1 in U; 1 2 3 1 2 3 U 1 2 3 −1−1 (2) all relators of the form ws w s whenever w∈ T,s , s∈ A and ws w s= 1 in G; 1 2 1 2 1 2 (3) a ﬁnite set R of deﬁning relators of A (with respect to the generating subset T) including all commutation relators. The Dehn function of this presentation is bounded above by mn+ δ (n),where δ is the Dehn T,R T,R 3 3 function of the presentationT| R of the abelian group A. Corollary 6.A.8. —If G is a tame locally compact group, then it has an at most quadratic Dehn function. This follows since A has an at most quadratic Dehn function. Remark 6.A.9. — Since G has a 1-Lipschitz retraction onto A (namely the canon- ical projection), we deduce that the Dehn function of G is exactly quadratic when A has rank at least 2. On the other hand, if A has rank one (i.e., contains Z as a cocompact lattice), then G is hyperbolic and thus its Dehn function is linear [CCMT15]. Proof of Corollary 6.A.7. — Start from a combinatorial loop γ= (u ) of length n. 0 i i This is a combinatorial loop in the Cayley graph of (G, S), i.e., d(u , u )≤ 1. Since γ: i i+1 n j (g, n) → gv is 1-Lipschitz by Theorem 6.A.4,each γ= (u v ) is also a loop of length n. j i i Deﬁne μ as the loop (v π(u )) in A. We know that for j≥ mn,wehave d(μ (i), γ (i))≤ 1 j i i j j for all i. So the “homotopy” consists in γ γ ··· γ μ 0 1 mn mn and is ﬁnished by a homotopy from μ to a trivial loop inside A. So we have to describe mn each step of this homotopy. To go from γ to γ ,weuse n squares, each with vertices of the form j j+1 j j j+1 j+1 u v , u v , u v , u v i i+1 i+1 i −1 for i= 1,... n (modulo n). To describe this square, we discuss whether u u belongs to i+1 S or T: in the ﬁrst case this is a relator of the form (2) and in the second case it is a relator of the form (3). To go from γ to μ ,weuse n squares with vertices mn mn mn mn mn mn u v , u v ,π(u )v ,π(u )v . i i+1 i+1 i 134 YVES CORNULIER, ROMAIN TESSERA −1−1 We discuss again: if u u∈ T, this is a relator of the form (3). If u u∈ S , actually i+1 i+1 U i i π(u )= π(u ) and this square actually degenerates to a triangle of the form (1). i i+1 Finally, the loop μ ,oflength n can be homotoped within A to a constant loop mn with area≤ δ (n). T,R Remark 6.A.10 (Asymptotic cones of tame groups are contractible). —Let X beamet- ric space and ω a nonprincipal ultraﬁlter on the set of positive integers. If (x ), (y ) n n are sequences in X, deﬁne d ((x ), (y ))= lim d(x , y )/n ∈[0,∞].The asymptotic cone ω n n ω n n (Cone (X), d ) is deﬁned as the metric space consisting of those sequences (x ) with ω ω n d ((x ), (x )) <∞, modulo identiﬁcation of (x ) and (y ) whenever d ((x ), (y ))= 0. ω n 0 n n ω n n A straightforward corollary of Theorem 6.A.4 is that if G is a tame locally compact group, then all its asymptotic cones are contractible. Indeed, the large-scale Lipschitz mapping γ induces a Lipschitz map γ˜: Cone (G)× R→ Cone (G) ω≥0 ω (x ), t → γ x ,tn n n such that γ(˜ x, 0)= x and γ(˜ x, t)∈ Cone (A) if t≥ C|x|. Deﬁning h(x, t) =˜γ(x, t|x|), then h is continuous, h(x, 0)= x and h(x, C)∈ Cone (A) for all x∈ Cone (G).Inother ω ω words, h is a homotopy between the identity of Cone (G) and a map whose image is contained in Cone (A). Since Cone (A) is (bilipschitz) homeomorphic to a Euclidean ω ω space, this shows that Cone (G) is contractible. 6.B. Length in nilpotent groups. — As in Deﬁnition 1.2,let G= U Abe a standard solvable group. We ﬁx a decomposition U= U ,where U= U (K ),where K is (j) (j) (j) j j j=1 some nondiscrete locally compact ﬁeld. We write K= K . We say that a subgroup j=1 VofUis K-closed if it is a direct product V ,where V= V∩ U is Zariski-closed (j) (j) (j) j=1 in U . (j) The condition that V is K-closed is equivalent to the requirement that log(V) is a Lie K-subalgebra of the Lie K-algebra u of U. When K is R or Q for some prime p,itis just equivalent to the condition that V is closed (for the ordinary topology) and divisible. We ﬁx on G an admissible ﬁnite family U ,..., U of tame subgroups of U, where 1 ν by admissible we mean it fulﬁlls all the following conditions: (1) each U is K-closed in U (2) U is generated by U . i=1 (3) each U is normalized by A. For instance, we can choose them to be the standard tame subgroups (there are ﬁnitely many, see Deﬁnition 4.C.5), but in the proof of Theorem 6.D.1 we will make another choice. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 135 We can ﬁnd a compact symmetric subset T⊂ A containing 1 and compact sym- metric subsets S of U containing 1, such that for every i,S is a stable vacuum subset for i i i some element of T. Deﬁne S= T∪ S . We assume that S= S∩ U for all i (we can i i i always replace S with S∩ U so that it holds). i i Notation 6.B.1. —For r ∈[−∞,+∞] and n∈ N,let T⊂ F be the n-ball of F and S T (r) −1 max(0,r) deﬁne S as the set elements of F of the form tst , where s∈ S and t∈ T .Let π be the S i canonical projection F→ G. Theorem 6.B.2. — Fix a standard solvable group G, an admissible family of tame subgroups and generating subset as above. There exist constants λ> 0, μ ,μ > 1, a positive integer κ and a 1 2 κ -tuple (q ,..., q ) of elements in{1,...,ν} satisfying 1 κ • for every small enough norm $·$ on u, denoting by U[r] the exponential of the r-ball in (u, $·$), we have, for all n, the inclusions n n n n n n n 2n S⊂ U μ T , U μ⊂ S , U μ T⊂ S . 2 1 1 n (λn) (λn) n • the n-ball S of G is contained in the projection π(S ... S T ) (which is itself contained q q 1 κ in the (κλ+ 1)n-ball). We ﬁrst need a general lemma. Lemma 6.B.3. — Fix an integer s≥ 1.Let K= K be a ﬁnite product of nondiscrete locally compact ﬁelds of characteristic zero (or of characteristic p > s). Let u be a s-nilpotent ﬁnite length Lie algebra over K (ﬁnite length just means that under the canonical decomposition u= u , each u is ﬁnite-dimensional over K ) and ﬁx a submultiplicative norm $·$ on u.Let U be the corresponding nilpotent topological group; if x∈ U, write$x$=$ log(x)$. Let (u ) be K-subalgebras of u; suppose that the u generate u as a K-subalgebra modulo i 1≤i≤c i [u, u].Let U= exp(u )⊂ U be the corresponding subgroups. i i Then there exists an integer κ and a constant K such that every element x∈ U can be written x ... x with x∈ u ,and sup$x $≤ Kmax($x$, 1). 1 κ k i k i k Proof. — We argue by induction on s.If s= 1, the assumption is that u is abelian and generated by the u . So there exist subspaces h⊂ u such that u= h ,soif x∈ u i i i i and p is the projection to h ,then x= p (x), and if K= sup$p$ (operator norm) i i i 0 i i=1 then$p (x)$≤ K$x$. i 0 Suppose now s > 1 and that the result is proved for s− 1. Denoting by (u ) the lower central series, we choose the norm on u/u to be the quotient norm. We use the in- duction hypothesis modulo U , so that there exist d and K≥ 1such that every x∈ Ucan be written as y ... y z,with$y $≤ K max($x$, 1) and z∈ U . By the Baker-Campbell- 1 d i Hausdorff formula, there exists a constant C > 0 (depending on s and d )suchthatfor all 136 YVES CORNULIER, ROMAIN TESSERA x ,... x in u,wehave 1 d+1 $x ··· x $≤ Csup max$x$ ,$x$ . 1 d+1 i i −1−1 Since z= y ... y x, we deduce d 1 ! " s s $z$≤ Cmax sup max$y$ ,$y$ , max$x$ ,$x$ i i s s s s s s ≤ Cmax max K , K$x$ , max$x$ ,$x$= C K max$x$ , 1 . s s Consider the K-multilinear map (u/[u, u])→ u given by the s-fold bracket. Since its image generates u as a K-submodule and since the u generate u modulo[u, u],wecan ﬁnd a ﬁnite sequence (i ) and a ﬁxed ﬁnite family (κ ) with κ∈ u such j 1≤j≤j jk 1≤j≤j ,1≤k≤s jk i 0 0 j that, setting ζ =[κ ,...,κ] we have u= Kζ (we can normalize so that$ζ $= 1). j j1 js j j Denote C= sup$κ$.Let K be the supremum of the norm of projections onto the jk j,k submodules Kζ . Then we can write z= λ ζ with λ∈ K, satisfying sup|λ |≤ K$z$. j j j j If L is a normed ﬁeld, denote β(L)= inf{|x|:|x|∈ L,|x| > 1}.Deﬁne α= sup β(K )≥ 1 (it only depends on K). 1/s Hence for any element λ∈ K, there exist μ ,...,μ∈ K with sup|μ |≤|λ| α , 1 s k 0 s s 1/s such that λ= μ . We apply this to write λ= μ ,with|μ |≤|λ| α . k j jk jk j 0 k=1 k=1 Now identify the Lie algebra and the group through the exponential map. By Lemma 5.B.4,wehave λ ζ= ((μ κ ,...,μ κ )). j j j1 j1 js js We have 1/s 1/s 1/s 1/s 1/s |μ κ |≤ α C K$z$≤ α C CK max K$x$, K$x$ jk jk 0 0 and x= y ... y ((μ κ ,...,μ κ )), 1 d j1 j1 js js which is a bounded number of terms in U ,eachwithnorm 1/s ≤ max A$x$, B$x$ , 1/s 1/s where A= K max(α C (CK ) , 1) and B= α C (CK K ) . 0 0 Proof of Theorem 6.B.2. — We can identify U with its Lie algebra u through the exponential map. Use an embedding of U into matrices over K to deﬁne a matrix norm $·$ on U, which is submutiplicative. Since both the embedding of U into its image and GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 137 its inverse are polynomial maps, there is a polynomial control of the two norms $·$ and $·$ with respect to each other, say$u$≤ max(A$u$ , B) and$u$≤ max(A$u$ , B ), with A, A≥ 1, B, B≥ 0. n n n Let us now prove the existence of μ > 1 such that the inclusion S⊂ U[μ]T n n holds for all n. Any element x in S can be rewritten as a product ua,where a∈ T and −1 n u is a product of at most n elements, each of which has the form tat ,where t∈ T and a∈ U∩ S. If C≥ 1 is an upper bound for$a$ when a∈ U∩ Sand λ≥ 2 is an upper 1 1 bound for the Lipschitz constant of elements of T viewed as operators on u,weobtain −1 n−1 k nk $tat $≤ C λ .Hence$tat$≤ max(A C λ , B ). Since $·$ is submultiplicative, it 1 1 1 k nk k k n follows that$u$≤ max(nA C λ , n). Deﬁning λ= 2A C λ ,weobtain$u$≤ λ for 1 1 1 1 2 k kk nkk k all n. In turn, we deduce$x$≤ max(An A C c , An , B). Thus there exists μ and n 2 0 n n such that for all n≥ n and all x= ua∈ S we have$u$≤ μ ; enlarging μ if necessary 0 2 n n n we can ensure n= 0. Hence S⊂ U[μ]T holds for all n. The U generate U modulo[U, U] (because u ⊂[u, u] and u is contained in one i 0 α of the u for every nonzero weight α). Hence we can apply Lemma 6.B.3: there exists c≥ 1and a κ -tuple (q ,..., q ) ∈{1,...,ν} such that every x∈ U can be written as 3 1 κ x with x∈ U with sup$x $≤ e max(1,$x$). Then there exists c such that for q 4 =1 every i and every y∈ U , there exists t∈ Tand m∈ N such that m≤ c max(1, log$y$) i 4 (c max(1,log$y$)) m−m and t yt∈ S .Inother words, forevery i and y∈ U ,wehave y∈ S . i i In particular, setting c= c c 5 3 4 (c max(1,log$x$)) (c[c+max(0,log$x$)]) (c max(1,log$x$)) 4 4 3 5 x∈ S⊂ S⊂ S; q q q thus if we set c= c c we have 5 3 4 (c max(1,log$x$)) x∈ S Hence, for every r > 0we have r (c max(1,r)) κ c max(1,r) 5 5 U e⊂ S⊂ S; =1 −1 in particular, if C= κ c and μ= e , n max(n,C) U μ⊂ S n n n−C n−C n Thus U[μ ]⊂ S for all n≥ C. In particular, U[μ ]⊂ S⊂ S for all n≥ 0. Multi- −C n−C n plying the norm by μ redeﬁnes U[μ] to be U[μ] and hence with this new norm n n we have U[μ ]⊂ S for all n. For the second inclusion, let us go back to the inclusion of U[e], which yields (for all n≥ 0), with λ= c log μ > 0 5 2 n (max(c ,λn)) U μ⊂ S; 2 q =1 as in the previous case, multiplying the norm allows to assume c= 0. 5 138 YVES CORNULIER, ROMAIN TESSERA 6.C. Application of Gromov’s trick to standard solvable groups. — As above, let U ,..., U 1 ν be an admissible family of tame subgroups in U. If c is a positive integer and ℘ is a c-tuple of elements in{1,...,ν}, we can consider the product U= U . ℘ ℘ =1 Then we can deﬁne, for r ≤∞, a set of null-homotopic words (r) U (r)= w= w ...w| π(w)= 1,∀, w∈ S⊂ F . ℘ 1 c S If R is a subset of the kernel of F→ G consisting of words of bounded length, deﬁne δ (n)= sup area (w) ∈[0,∞]. ℘,S,R S,R w∈U (n) (The area can a priori take inﬁnite values, since we do not assume that R normally generates the kernel.) The following theorem is an essential reduction in the study of the Dehn function of standard solvable groups. Theorem 6.C.1. —Let G= U A be a standard solvable group. Fix an admissible family U ,..., U of tame subgroups and generating subsets as in Section 6.B;alsoﬁxasubset R of the kernel 1 ν of π: F→ G. Let f be a function such that r → f (r)/r is non-decreasing for some α> 1.Suppose that for every c and every c-tuple ℘ ∈{1,...,ν} we have δ (n) f (n). Then G is compactly ℘,S,R presented byS| R and has Dehn function f (n). Remark 6.C.2. — The proof actually even shows that the following holds, G, S and R being ﬁxed as well α> 1: There exists c and a c-tuple ℘ satisfying: if for some f we have δ (n) f (n),then G is compactly presented byS| R and has Dehn function f (n). ℘,S,R However, we will only use the result in the slightly weaker form given in Theorem 6.C.1. (λn) (λn) n Proof.—Denote W= S ... S T ,with λ, k as given by Theorem 6.B.2 (we use q q 1 κ the notation of 6.B.1). Hence π(W ) contains the n-ball in G; note that W is contained n n in the[(2κλ+ 1)n+ κ]-ball of F , which itself is contained in the κ n-ball, where κ= 2κλ+ κ+ 1. A simple observation is that (λn) (λn) (λn+n) (λn+n) (λn+2n) (λn+2n) 3n W W W⊂ S ... S S ... S S ... S T; n n n q q q q q q κ κ κ 1 1 1 in particular, setting λ= λ+ 2, and deﬁning ℘= q for 1≤ ≤ κ , p ∈{0, 1, 2},we +pκ have (λ n) (λ n) W W W∩ Ker(π)⊂ S ... S∩ Ker(π)= U λ n . n n n ℘ ℘ ℘ 1 3κ Thus the assumption implies that words in W W W∩ Ker(π) have an area n n n f (n).ByTheorem 2.E.3, it follows thatS| R is a presentation of G and has a Dehn function f (n). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 139 6.D. Dehn function of stably 2-tame groups. — The following is a reformulation of Theorem D from the introduction. Theorem 6.D.1. —Let G= U A be a stably 2-tame standard solvable group (see Def- inition 4.C.1). Then G has a linear or quadratic Dehn function (linear precisely when A has rank one). Remark 6.D.2. —Theorem 6.D.1 is actually a corollary of Theorem 10.E.1,be- causefor astably2-tamegroup,wehave H (u)= 0and Kill(u)= 0 (because both are 2 0 0 subquotients of (u⊗ u) which is itself trivial if u is stably 2-tame). The point is that The- orem 10.E.1 is considerably more difﬁcult, since it relies on the work of Section 7.B and the algebraic work of Section 8 and Section 9. Proof of Theorem 6.D.1. — If A has rank one, then G is tame and thus hyperbolic, see Remark 6.A.9, and thus has a linear Dehn function. Otherwise, A being a retract of G, the Dehn function has a quadratic lower bound. Now let us prove the quadratic upper bound. We prove the quadratic upper bound by arguing by induction on the length of u as a K-module. If ≤ 2, then G is tame, hence has an at most quadratic Dehn function. Assume now that ≥ 2 and the result is proved for lesser . In order to apply Theorem 6.C.1, we ﬁrst need to choose a suitable admissible family of tame subgroups. Let P be the (ﬁnite) set of principal weights, that is, the set of weights of u/[u, u]. Given a principal weight α;deﬁne u= u ; this is a graded [α] tα t>0 Lie algebra (beware that u need not be a Lie subalgebra). Let α ,...,α be representa- α 1 μ tives of the principal weights modulo positive collinearity; write U= exp(u ).Then [α][α] i i we choose U ,..., U as admissible family of tame subgroups, and we let S ,... S [α][α] 1 ν 1 μ be ﬁnite generating subsets of the U ’s satisfying the conditions of Section 6.B.Welet [α] (n) (n) (β n) β≥ 1 be an integer such that for all n≥ 1, S S⊂ S (the existence of β follows 1 1 1 from the second part of Theorem 6.B.2 applied to U A, for the admissible fam- [α] ily{U}). [α] Let v be the sum of[u, u] and all u when β ranges over all weights not positively collinear to α ;then v is a graded ideal. Note that u= u+ v.Deﬁne V= exp(v). 1[α] Let V ,..., V be the standard tame subgroups of V and denote by acompact 1 ν i subset of V (with all the previous requirements we made for S ). i i Fix k≥ 1, ℘ ∈{1,...,μ} . For positive integers n ,..., n ,consideranelement 1 k (n ) (n ) 1 k w in F belonging to S ... S , we do the following: if there are any two consecutive ℘ ℘ 1 k (n) (m) (β max(m,n)) occurrences of S , we glue them using the inclusion S S⊂ S in G. The cost 1 1 1 of this is at most quadratic since it lies in the tame subgroup U A. If not, we con- [α] sider i maximal such that ℘= 1. If i= 1or i does not exist, say we are done. Otherwise (n ) (n ) (n ) j (n ) (n ) j j (n ) i i i j= ℘ = 1. We ﬁrst observe that S S is contained in S S ((S , S )) (inclusion i−1 j 1 j 1 j 1 140 YVES CORNULIER, ROMAIN TESSERA in F ). Then α and α belong to a single standard tame subgroup, and hence the com- S 1 j mutator belongs to a single standard tame subgroup of V A. So the commutator can be rewritten in V A, thanks to the second part of Theorem 6.B.2,asanelement of (c(n+n )) (c(n+n )) 1 2 1 2 ... ,where c and κ only depend on G. We note that doing this, we q q 1 κ make some generators in appear, but they stay at the right of any occurrence of S . i 1 Therefore we can reiterate some bounded number of times (say≤ k ) until we obtain a (c n) word w w with|w |≤ c n (n= n ), w∈ S ,and w awordin S∪ ∪ T. In 1 2 2 i 1 2 i i 1 i≥2 i particular, assuming that w represents the trivial element in G, w represents an element of V. There exists a standard tame subgroup of U containing U ; then its intersection [α] with V is a standard tame subgroup, say V .So w can be replaced with some element w 1 1 −1 c n of with quadratic cost since the word w w lies in a tame subgroup of G. So from 1 1 1 w we passed with quadratic cost to a word in V of length≤ C|w|. Since by induction V has an at most quadratic Dehn function, we deduce that w has an at most quadratic area with respect to n. Since this works for every ℘ (with constants possibly depending on ℘ ), we can apply Theorem 6.C.1 to conclude that G has an at most quadratic Dehn function. 6.E. Generalized tame groups. Deﬁnition 6.E.1. — A locally compact group is generalized tame if it has a semidirect product decomposition U N where some element c of N acts on U as a compaction, and N is nilpotent and compactly generated. Thus, the assumption that N is abelian in tame groups is relaxed to nilpotent. If G= U N is a tame generalized standard solvable group it is tempting to believe that, in a way analogous to Section 6.A, there is a large-scale Lipschitz deformation retraction of G onto N. However, the proof only carries over when the element c of N acting as a compaction of N can be chosen to be central in N. Unfortunately, this can not always be assumed and, in order to get an upper bound on the Dehn function, we need a more complicated approach. Theorem 6.E.2 (The generalized tame case). — Consider a generalized standard solvable group G= U N such that there exists c∈ N acting on U as a compaction. Let δ be the Dehn function of N, and let f be a function such that δ fand r → f (r)/r is non-decreasing for some α> 1. Then the Dehn function δ of G satisﬁes δ δ f . In particular, if f can be chosen to be G N G ≈-equivalent to the Dehn function of N, then δ≈ δ . G N Note that in all examples we are aware of, we can indeed choose f to be≈- equivalent to δ ,suchthat r → f (r)/r is non-decreasing (unless N admits Z as a lattice, in which case G is tame and has a linear Dehn function). In general, we can always ﬁx α−α α> 1 and consider the function r → f (r)= r sup s δ (s). 1≤s≤r GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 141 Remark 6.E.3. —Theorem 6.E.2 has some similarity with a theorem of Varopoulos [Var00, Main theorem, p. 57] concerning connected Lie groups. Namely, for a simply connected Lie group of the form U NwithU, N simply connected nilpotent Lie groups such that N contains an element acting as a contraction on N, he proves that there exists a “polynomially Lipschitz” homotopy from the identity of G to its projection on N. To prove the theorem, we need the following lemma, whose proof is much more complicated than we could expect at ﬁrst sight (see Remark 6.E.6). Lemma 6.E.4. — Relations of the form s t, where s and t belong to S and where w is a word of length nin S have area δ (n). N N Proof. — For the sake of readability, we ﬁrst consider the (easier) case when N is 2-nilpotent. Denote j= in,where i is an integer to be determined latter in the proof, but that will only depend on G and S (hence is to be considered as bounded). Up to conjugating by a power of c, it is enough to evaluate the area of the relation j j wc c s t . Since conjugation by c is a contraction (see Deﬁnition 4.B.4), it turns out that j j c−1 c t= u∈ S . It is straightforward to check that the relation t u has area j.Hence we wc are left to consider the relation s u. n−1−n n Denoting y= ((w, c ))= w c wc ,wehave j j j j i (6.E.5) wc= c w((w, c ))= c y . j−i Moreover the area of the relation ((w, c ))y is controlled by the Dehn function of N, so we are reduced to compute the area of wy s u. Denote N the Zariski closure of the range of N in Aut(U). The algebraic group N a a decomposes as a direct product AV where A (resp. V) is semisimple (resp. unipotent). Let us write c= c c and w= w w according to this decomposition. Endow the Lie algebra d v d v n n of U with some norm. The crucial observation is that y= ((w, c ))= ((w , c )). It follows that the matrix norm of y (acting on the Lie algebra of U) is at most Cn for some C, D depending only on G and S. Let K > 1 be a constant such that the matrix norm of every subword of wy is at n i k most K .Let z be a preﬁx of wy :itisofthe form ry ,where r is a subword of wy,and D n k≤ i. The matrix norm of z is therefore at most Cin K . Since c acts as a contraction, one can choose i be such that the matrix norm of c −2 j D−n is less than K .Hence thematrixnormof c z is less than Cin K which is bounded by j i some function of i, C, D and K. It follows that for any preﬁx r of c wy has bounded matrix j i norm. Let a for q= 1, 2,... be the sequence of letters of the word c wy ,and let z= q q 142 YVES CORNULIER, ROMAIN TESSERA z−1 a ... a . It follows that the elements s are bounded in U. Now let t be words in S of 1 q U bounded length representing the elements s . We conclude by reducing successively the relations (t ) t whose area are bounded. This solves the case where N is 2-nilpotent. q−1 q i i If N is not assumed to be 2-nilpotent, then the relation ((a, b ))= ((a, b)) does not j n i hold anymore. Therefore we cannot simply replace ((w, c )) by ((w, c )) ,aswedid above. We shall use instead a more complicated formula, namely the one given by Lemma 5.B.2. According to that lemma, one can write ((w, c )) as a product of m iterated commutators ±1±n (or their inverses) in the letters w and c . The rest of the proof is then identical to the i n i 2-nilpotent case, replacing in the previous proof, the power of commutators y= ((w, c )) by this product of (iterated) commutators. Remark 6.E.6. — Lemma 6.E.4 is not as trite as it may look at ﬁrst sight, and fails when N is not nilpotent. For instance, consider the free group F =s, t, and the semidi- rect product U F ,where U R is written as N ={(u )} is written multiplicatively, 2 x x∈R −1 −1 2−n n−1 sxs= x for all x∈ Nand[t, N]= 1. Write m= s ts .Then m v m v can be shown n n 1 n 1 to have an exponential area. Lemma 6.E.7. — Consider a semidirect product of groups G= U N;let c beanelement −1−n n of N and a symmetric subset of U such that ⊂ c cand c c= U.Let T be a n≥0 −1−1 symmetric generating subset of N containing{1, c}, such that tt⊂ c c. Then S= ∪ T t∈T n−2n 2n n 5n+1 generates G and S⊂ (c c )T⊂ S . Proof. — It is immediate that S generates G. By an immediate induction, we have 2−k k n−! log n"! log n" 2 2 ⊂ c c . In particular, ⊂ c c . −1 Consider a word w of length n in S. Then it can be rewritten as ( t ω t )t, i i i=1 i where each t and t belong to T and each ω belong to . Thus, in G, we have i i n−n n n−n n n n S⊂ c c T= c c T −n−! log n" n+! log n" n−2n 2n n 5n+1 2 2 ⊂ c c T⊂ c c T⊂ S Proof of Theorem 6.E.2. — Since N is a Lipschitz retract of G, we have δ δ ;let N G us prove that δ f . We pick the 1-element family (U ),where U , as an admissible family of sub- 1 1 groups; we let S be a stable vacuum subset for some element c∈ T acting as a com- paction on U. We ﬁrst bound (assuming c∈ T) the area of relations of length≤ n,ofthe form −n n i i w= ( c ω c )w with ω∈ and w a word in T (necessarily n≤ n). Since w is i i i i=1 a relation in N, with cost≤ f (n), we can replace w with the trivial word. Then after n n−n n−n n−n n−n i i i i conjugation by c , we are reduced to the word c ω c . The element c ω c i i i=1 n−n n−n i i represents an element s∈ , and by Lemma 6.E.4, the cost to pass from c ω c to s i i i is f (n).Hence theareaof w is f (n). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 143 We can apply Lemma 6.E.7 (if necessary, replace c with a large power c of it and ±k replace T with T ∪{c} for its hypotheses to be fulﬁlled. Thanks to the length estimates of Lemma 6.E.7, we can use Gromov’s trick (Theorem 2.E.3) to conclude that the Dehn function of G is f . We also need a generalization of Theorem 6.B.2. We deﬁne an admissible family of subgroups as in 6.B (A being replaced with N), and let T be a compact symmetric gen- erating subset of N. Since the proof of the following theorem is an immediate adaptation of that of Theorem 6.B.2, we omit the proof. Theorem 6.E.8. — Fix a generalized standard solvable group G, an admissible family of tame subgroups and generating subset as above. There exists a constant λ> 0, a positive integer κ and a κ -tuple (q ,..., q ) of elements in{1,...,ν} such that for every n, the n-ball of G is contained in the 1 κ (λn) (λn) n projection π(S ... S T ). q q 1 κ 7. Estimates of areas using algebraic presentations In this section, we provide a new method allowing to obtain upper bounds on the Dehn function of groups with “enough algebraic structure”. The basic idea is to consider a group G given by an “algebraic presentation”, that is, it is isomorphic to the quotient of a free product U (K) of algebraic groups U ∗ i i i=1 over some locally compact ﬁeld K (or, more generally, a ﬁnite product of such ﬁelds), by the subgroup generated by ﬁnitely many “algebraic families of relators”. An algebraic family of relators is an algebraic subvariety of some U × ··· × U ,where (i ,..., i ) is a i i 1 ﬁxed sequence of integers from 1, .., ν , which is interpreted as a set of words of length of the free product. In order to get estimates, we need to suppose that G itself is of the form G(K), and assume that the above presentation holds in a very strong sense. Namely, we need to assume that for every commutative K-algebra A,the group G(A) is the quotient of U (A) by the relators (evaluated in A). ∗ i i=1 Our main result consists in controlling the size and word length of a set of relations that belong to a same algebraic family. More precisely, consider some algebraic family M of relations: namely a subvariety of some U ×· · · × U such that for every K-algebra A, j j 1 k the A-points of this variety are mapped to the neutral element of G . Then there exists an integer N such that every element x= (x ,..., x ) of M can be written as a word of length 1 k at most N in U (A) consisting of a product of conjugates of relators. Moreover the ∗ i i=1 norms of the letters of this word are polynomially controlled by the norms of the x ’s. 7.A. Afﬁne norm on varieties over normed ﬁelds. 7.A.1. Norms. — In this paragraph, let (K, |·|) be a normed ﬁeld. The afﬁne d d space A (K)= K is endowed with the sup norm $·$. 144 YVES CORNULIER, ROMAIN TESSERA Let X be an afﬁne K-variety with basepoint x (a ﬁxed element of X(K)). For every d and every pointed closed K-embedding φ: X→ A (pointed means mapping x to 0), we can deﬁne a “length function” on X(K) by (x) =$φ(x)$. The following proposition asserts that up to polynomial distortion, this length is unique. Proposition 7.A.1. —Let φ, ψ be two choices of pointed embeddings. Then for some positive constants C, c > 0, (x)≤ Cmax (x) , 1 ,∀x∈ X(K) ψ φ d d Proof. — If the embeddings are into A and A ,withsay d≤ d , using that the d d trivial embedding A (K)⊂ A is isometric, we can enlarge d to assume that d= d .Now −1 consider the isomorphism η= ψ◦ φ: φ(X)→ ψ(X).Then η extends to a regular map d d d η: A→ A : indeed, ﬁrst view η as a regular map φ(X)→ A , which is the same as the data of d regular maps φ(X)→ A and remind that regular maps on φ(X) are by deﬁnition restrictions of regular maps of the afﬁne space. Then, viewing A embedded 2d d into A as the left factor A ×{0},wecan extend η to a K-automorphism 2d 2d η˜: A→ A (x, y) → η (x)+ y, x Now if c is the total degree of η˜ , there exist a positive constant C such that for all u∈ 2d A (K) we have $˜η(u)$≤ Cmax$u$ , 1; therefore for all x∈ X(K) we have (x) =$ψ(x)$=$η◦ φ(x)$≤ Cmax$φ(x)$, 1 = Cmax (x) , 1 . Note that the analogue for distances is not true: for instance if we take the disjoint union of two lines K ×{0, 1}, setting φ(x, t)= (x, t) (two parallel lines) and ψ(x, t)= (x, tx ) (a line and a parabola), then considering the sequence of pairs of points (n, 0) and (n, 1) we see that$ψ(x)− ψ(y)$ cannot be bounded in terms of$φ(x)− φ(y)$. Example 7.A.2. — Assume that K has characteristic zero. Let U be a unipotent group over K. Given an embedding ψ of U as a Zariski-closed subgroup of SL ,and ﬁxing a norm on the algebra of matrices, we obtain two norms on U: the one given by this embedding, and the one given by the embedding ψ of the Lie algebra u into sl , pulled back to U through the exponential. Each of these two norms on U is polynomially bounded by the other, by Proposition 7.A.1. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 145 Remark 7.A.3. — We can deﬁne the logarithmic length on X(K) associated to φ as L (x)= log 1+ (x) . φ φ As a consequence of Proposition 7.A.1, for any two choices ψ and φ,wehave L L , ψ φ i.e. there exist positive constants c≥ 1and C≥ 0such that −1 c L (x)− C≤ L (x)≤ cL (x)+ C,∀x∈ X(K). φ ψ φ 7.A.2. Ring of polynomial growth functions. — Let K be a normed ﬁeld. Endow K with the supremum norm and consider the algebra P= P (K) of functions from Y× R Y Y≥0 to K that have most polynomial growth with respect to the real variable, uniformly in Y, that is, satisfying ∃c,α > 0, c∈ R,∀t≥ 0,∀y∈ Y,$f (y, t)$≤ ct+ c . Lemma 7.A.4. —Let Y be a set. Then for any K-afﬁne variety X, there are canonical bijections ∼∼ Y Y X K→ X(K) , X P (K)→ P X(K) Y Y Proof. — The ﬁrst equality is formal: if A= K[X], then, Hom being in the category of K-algebras, we have Y Y Y X K= Hom K[X], K= Hom K[X], K= X(K) . Given a K-closed embedding of X into the d -dimensional afﬁne space, it is just given by d Y d Y the function A (K )→ A (K) (f ,..., f ) → y → f (y),..., f (y) . 1 d 1 d Y Y d d Then (X(K ))= X(K) and (A (P (K)))= P (A (K)), and hence Y Y Y d P X(K)= X(K)∩ P A (K) Y Y Y d = X K∩ A P (K)= X P (K) . Y Y 7.A.3. Remark. — All the results from this Section 7.A immediately to the case of X(K) when K is a ﬁnite product of normed ﬁelds (endowed with the sup norm). The only difference lies in the language: K-afﬁne variety should be replaced with: afﬁne scheme of ﬁnite type over K;if K= K , this is just the data of one K -afﬁne variety over K for j j j j=1 each j . 146 YVES CORNULIER, ROMAIN TESSERA 7.B. Area of words of bounded combinatorial length. 7.B.1. The setting: generators (*). — By K we mean a ﬁnite product of normed ﬁelds (in a ﬁrst reading, we can assume it is a single normed ﬁeld). We recall the reader familiar with varieties but not schemes that it is not much here: over a single ﬁeld, “scheme of ﬁnite type” can be thought of as “variety” and “afﬁne group scheme of ﬁnite type” as “linear algebraic group”. Over a ﬁnite product K of ﬁelds, the datum of a scheme is just the datum of one scheme over each of the ﬁelds K . Nevertheless, we stick to the word “scheme” because it is the only rigorous setting in which the assertions we state are precise. Let U ,..., U be afﬁne K-group schemes of ﬁnite type. Write U= U (K). 1 ν i i We need to introduce some “norm function” on each U . For this, as explained in Section 7.A, we ﬁx a closed K-embedding of U into SL , so that the norm, written$u$, i q of any element u of U makes sense (using the operator norm on q× q matrices, where K is endowed with the sup norm). 7.B.2. The setting: relators (**). — We are going to introduce some functors X from the category of commutative (associative unital) K-algebras to the category of sets, de- noted by A → X[A], we use brackets rather than parentheses to emphasize that these objects are possibly not representable by a scheme. We need to consider words of some given length whose letters belong to the disjoint union of the U . To do so, we ﬁx integers 1≤ ω ,...,ω≤ ν . Hence we can consider i 1|ω| |ω| the product U= U .Thus ω ω =1 |ω| U (A)= U (A) ω ω =1 is an obvious way to parameterize the set of words u ... u with u∈ U (A) for all . ω ω ω ω 1|ω| We are going to consider closed subschemes of U , which can then be used to parameterize sets of words. Formally, this is done as follows. Let H[A] be the free product U (A). There is an obvious product map ∗ i i=1 π: U (A)→ H[A],(u ,..., u ) → u ... u . ω 1|ω| 1|ω| Now ﬁx ﬁnitely many closed subschemes R ,..., R⊂ U (they will play the role 1 ξ ω of algebraically parameterized relators). For convenience, write R[A]= R (A) ∪ ··· ∪ R (A) for any K-algebra A (if K were a ﬁeld, it would be representable by a closed subscheme and we could then assume ξ= 1; anyway this is not an issue); clearly the π together deﬁne a map π: R[A]→ H[A]. Deﬁne Q[A] as the quotient of H[A] by the normal subgroup generated by π (R[A]). Informally, Q is a group generated by algebraic generators and algebraic sets of relators. A priori, Q is not representable by a group scheme over K (for instance, if R is empty, Q[A] is the free product H[A]). GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 147 7.B.3. A family of relations (***). — Now given an integer c≥ 0and a c-tuple ℘= (℘ ,...,℘ ) of integers in{1,...,ν}, so we have the product 1 c U (A)= U (A) ℘ ℘ i=1 and we deﬁne L[A]= (f ,..., f )∈ U (A): f ... f≡ 1in Q[A] . ℘ 1 c ℘ 1 c (It should rather be denoted L[A]) but since R is ﬁxed we omit it in the notation.) In R,℘ other words, denoting by N[A] the normal subgroup of H[A] generated by R[A],we −1 have L[A]= (π ) (N[A]). We cannot a priori represent L as a K-closed subscheme of U . We obtain results ℘ ℘ in case it turns out to be a closed subscheme, or more generally when it contains a K- closed subscheme. Namely, let M⊂ U be a K-closed subscheme and assume that it is contained in L , in the sense that for every (reduced) commutative K-algebra A we have M(A)⊂ L[A] (equality as subsets of U (A)). Also assume that the unit element of ℘ ℘ U (K) belongs to M(K) (this is no restriction since it belongs to L[K]). ℘ ℘ Deﬁne V[A] as the union of all U (A) in H[A]. The following lemma roughly says that any element x= (x ,..., x )∈ M(K) can 1 c be written as a product of boundedly many conjugates of relators whose coefﬁcients are controlled by polynomials in the size of x. Lemma 7.B.1. —Fix U ,..., U as in (*), ω and R as in (**). Given any ℘ and any 1 ν family M of relations as in (***), there exist positive integers m,μ,α satisfying the following: for all x= (x ,..., x )∈ M(K), there exist 1 c • elements ρ (x)∈ U (K), 1≤ k≤ m, 1≤ ≤|ω|, such that setting k ω |ω| ρ˜ (x)= ρ (x),...,ρ (x)∈ U (K), k k1 k|ω| ω =1 we have ρ˜ (x)∈ R[K] for every k; • elements h (x) in V[K], 1≤ k≤ m, 1≤ ≤ μ, satisfying the inequalities $h (x)$,$ρ (x)$≤ (2 +$x$)− 2,∀k,; k k and such that, setting |ω| μ ρ (x)= π ρ˜ (x)= ρ (x), h (x)= h (x), k K k k k k =1 =1 148 YVES CORNULIER, ROMAIN TESSERA we have the equality in H[K]: −1 (7.B.2) x ... x= h (x)ρ (x)h (x) . 1 c k k k k=1 Remark 7.B.3. —Thatfor every x we can write an equality as in (7.B.2) follows directly from the fact that M(K)⊂ L[K], but this gives such an equality with m,μ de- pending on x, and with an efﬁcient upper bound on the norm of relators and conjugating elements. Lemma 7.B.1 is a uniform version of this fact, and relies on the stronger in- clusion M(A)⊂ L[A] for some well-chosen K-algebra A. While a suitable choice of exp A would be the algebra K (as in the sketch of proof of Section 1.7), we ﬁnd more convenient to work with the algebra P (K) introduced in Section 7.A.2. The difference is unessential, and especially allows us to avoid to argue by contradiction as we did in Section 1.7. Proof of Lemma 7.B.1. — Let us ﬁx an abstract set Y; we will assume that its cardinal is at least equal to that of K. The ring P (K) (Section 7.A.2) can be described as the set of functions Y× R→ K satisfying ≥0 ∃α> 0,∀t≥ 0,∀y∈ Y,$f (y, t)$≤ (2+ t)− 2. For any afﬁne K-scheme X of ﬁnite type (with a given embedding in an afﬁne space), X(K) inherits of a norm and the set P (X(K)) is well-deﬁned. There is an obvious inclusion L[P (K)]⊂ P[L (K)]. It is not necessarily an equality (see Example 7.B.8); ℘ Y Y ℘ however the equality M(P (K))= P (M(K)) holds (see Lemma 7.A.4). Y Y For x∈ U (K),recallthat$x$= max$x$. Now assume that Y is inﬁnite of ℘ 1≤i≤c i cardinal at least equal to that of K. We claim that there exists a function f= (f ,..., f ): (Y× R )→ M(K) 1 c≥0 satisfying (a)$f (y, t)$≤ t for all (y, t)∈ Y× R ; ≥0 (b) for every x= (x ,..., x )∈ M(K), there exists y∈ Ysuch that f (y,$x$)= x. 1 c To construct such an f , consider an arbitrary injective map x → y(x) from U (K) to Y (it exists because of the cardinality assumption on Y); for each x∈ M(K),deﬁne f (y(x),$x$)= x and deﬁne f (y, t)= 1for every (y, t) not of the form (y(x),$x$),where 1 here denotes the unit element (1,..., 1) of U (K), which belongs to M(K) by assump- tion. By construction, f takes values in M(K).By(a), f∈ P (M(K)). Hence f∈ M(P (K))= M(P), where write P= P (K) as a shorthand. So, by the Y Y inclusion M(P)⊂ L (P),wehave f∈ L (P).Hence π (f )= f ... f∈ H[P] belongs to ℘ ℘ 1 c the normal subgroup generated by π (R[P]). This means that there exist P GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 149 • integers m,μ; • ρ∈ U (P),1≤ k≤ m,1≤ ≤|ω|, such that setting ρ˜= (ρ ,...,ρ )∈ k i k k1 k|ω| U (P),wehave ρ˜∈ R[P] for every k; ω k • elements h in V[P],1≤ k≤ m,1≤ ≤ μ, such that, setting |ω| μ ρ= π (ρ˜ )= ρ , h= h , k P k k k k =1 =1 we have, in H[P], the equality −1 (7.B.4) f ... f= h ρ h . 1 c k k k=1 Hence for all y∈ Yand t∈ R , we have the equality in H[K] ≥0 −1 f (y, t)... f (y, t)= h (y, t)ρ (y, t)h (y, t) , 1 c k k k k=1 μ μ where ρ (y, t)= ρ (y, t) and h (y, t)= h (y, t). Since each of the elements k k k k =1 =1 ρ , h belongs to some U (P), there exists α> 0such that for all k, and all y, t,wehave k k i α α $ρ (y, t)$≤ (2+ t)− 2,$h (y, t)$≤ (2+ t)− 2. k k Hence for all x= (x ,..., x )∈ M(K), choosing one y= y(x) satisfying (b) and 1 c t =$x$, and using the shorthands ρ (x)= ρ y(x),$x$ , h (x)= h y(x),$x$ , k k k k μ μ ρ (x)= ρ y(x),$x$= ρ (x), h (x)= h y(x),$x$= h (x), k k k k k k =1 =1 we precisely obtain that the norms of both ρ (x) and h (x) is bounded above by (2+ k k $x$)− 2 and that the equality (7.B.2) holds in H[K]. 7.B.4. Generating subsets and presentations (****). — Keep the previous setting (with U , R given). Assume that we have, in addition, a group H, subgroups H ,..., H gener- i 1 ν ating H, and homomorphisms ψ: U→ H (in the examples we have in mind, all ψ are i i i i given as inclusions). We assume that the resulting homomorphism H[K]→ His trivial on R[K], or equivalently factors through Q[K]. LetS| be a presentation of H . We suppose that for each i and for every i i i x∈ U , the word length of ψ (x) with respect to the generating subset S of H is i i i i log(1 +$x$).For x∈ U , ﬁx a representing word x in S of ψ (x),ofsize log(1 +$x$). i i i 150 YVES CORNULIER, ROMAIN TESSERA Assume in addition the following: • all presentationsS| have Dehn function bounded above by some superad- i i ditive function δ ; • there is a subset R⊂ R[K] and a function δ such that for every r= (r ,..., r )∈ R[K],the area of r ... r with respect to S| ∪ R 1|ω| 1|ω| i i i i |ω| is ﬁnite and≤ δ (max|r| ). 2 S =1 i 7.B.5. The fundamental theorem. Theorem 7.B.5. — In the above setting (*), (**), (****), and given a family of relations as in (***), there exist constants s, s , m > 0 such that for every x= (x ,..., x )∈ M(K), denoting 1 c n= max|x| , the area of x ... x with respect to S| ∪ R is≤ δ (sn)+ mδ (s n). i i S 1 c i i 1 2 i i i Proof.—Fix x as above. As in Lemma 7.B.1, write the equality in H[K] −1 (7.B.6) x ... x= h (x)ρ (x)h (x) 1 c k k k k=1 (with all the additional features of the statement of the lemma). Deﬁning[x]= log(2 +$x$) and similarly[u]= log(2 +$u$) for any i and u∈ U (K), the upper bounds on$h (x)$ and$ρ (x)$ provided by the lemma can be i k k rewritten as h (x) , ρ (x)≤ α[x]. k k |ω| μ Deﬁne S= S .Write ρˆ (x)= ρ (x) and h (x)= h (x).Deﬁne the i k k k k =1 =1 word in the free group over S −1−1 ˆˆ w= (x ... x ) h (x)ρˆ (x)h (x) 1 c k k k k=1 Hence, by (7.B.6), the image of w in H[K] is trivial. Denoting by |·| the word length in H[K] with respect to S. By assumption, there −1 exist γ,ζ≥ 1such that for every i and u∈ U (K),wehave ζ[u]≤|u|≤ γ[u];in i S particular|u|≤ γ[u]. Also deﬁne n= max|x| .Hence i S i=1 −1−1 n≥ ζ max[x ]≥ ζ[x] and c m|ω| μ |w|≤|x |+|ρ (x)|+ 2|h (x)| i k k i=1 k=1 =1 =1 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 151 c m|ω| μ ≤ γ[x ]+ ρ (x)+ 2 h (x) i k k i=1 k=1 =1 =1 ≤ γ c+ mα(|ω|+ 2μ)[x]≤ sn, where s= ζγ (c+ mα(|ω|+ 2μ)). By Lemma 2.D.1 (Dehn function of free products of presentations), the area of w with respect to S| is≤ δ (sn). i i 1 We have, by deﬁnition of w: −1−1 ˆˆ x ... x= h (x)ρˆ (x)h (x) w; 1 c k k k k=1 the area of ρˆ (x) with respect to the presentation P = S| ∪ R is k i i ! " ! " |ω||ω| ≤ δ max|ρ (x)|≤ δ max γ ρ (x)≤ δ γα[x]≤ δ (γ αζ n) 2 k S 2 k 2 2 =1 =1 −1 ˆˆ so the area of h(x)ρˆ (x)h (x) with respect to the presentation P is k k k=1 ≤ mδ (γ αζ n). Therefore the area of x ... x with respect to P is 1 c ≤ δ (sn)+ mδ (γ αζ n). 1 2 7.B.6. Restatement of the theorem involving a group word. — Let F be the free group on the generators t ,..., t . We say that w∈ F is essential if all t appear in the reduced form 1 c c i of w (possibly with a negative exponent). We call the word t ... t , which is essential, the 1 c tautological word; denote it by τ . Let w be an essential word in F ,oflength c (necessarily c≥ c if w is essential). Let ℘ be a c -tuple of elements of{1,...,ν}. −1 Write w as a reduced word, namely w= w ...w with w ∈{t , t , 1≤ j≤ c};if 1 c i j ±1 w= t , we write W(i)= j and w[i]=±1. Let us deﬁne a c-tuple ℘= w• ℘ of elements in{1,...,ν}:for 1≤ i≤ c, write −1 ℘= j if w ∈{t , t}. For instance τ• ℘= ℘ ; another less trivial example is i i j −2−1 w(t , t , t )= t t t t ,℘= (5, 6, 3)⇒ w•℘= (6, 5, 5, 3, 6). 1 2 3 2 3 1 2 Deﬁne (w• U )(A) as w[k] w[] −1 (u ,..., u )∈ U (A) :∀j∈ 1,..., c ,∀k,∈ W{j} , u= u . 1 c ℘ Thus w• U is a closed subscheme of U . In the above example, we have ℘ ℘ −1 (w• U )(A)= u , u , u , u , u| (u , u , u )∈ (U× U× U )(A) . ℘ 6 5 5 3 3 5 6 3 5 6 6 152 YVES CORNULIER, ROMAIN TESSERA When w is essential, we have an isomorphism φ of schemes U → w• U ,de- w ℘ ℘ ﬁned on U (A) by w[1] w[c] φ (u ,..., u )= u ,..., u w 1 c W(1) W(c) Note that φ is the identity. If M is a closed subscheme of U ,wedeﬁne w• M= φ (M); τ ℘ w this is a closed subscheme of w• U . The following theorem seems to be a generalization of Theorem 7.B.5 (namely when w= t ... t ), but is actually a simple consequence. It will be useful to refer to it. 1 c Let ℘ be a c-tuple of elements in{1,...,ν}.If w∈ F ,denoteby L[A] the set of (x ,..., x )∈ U (A) such that w(x ,..., x ) equals 1 in Q[A]. 1 c ℘ 1 c Theorem 7.B.7. — In the above setting (*), (**), (****), let c be a positive integer and ℘ a c -tuple of elements of{1,...,ν}.Let M be a closed subscheme of U and let w be an essential word in F , such that M (A)⊂ L[A] for all A. Then there exist constants s, s , m > 0 such that for every x= (x ,..., x )∈ M (K),de- 1 c noting n= max|x| , the area of w(x ,..., x ) with respect to S| ∪ R is≤ i i S 1 c i i δ (sn)+ mδ (s n). 1 2 Proof.—Deﬁne ℘= w• ℘ and M= w• M as above. Thus π (M(A)) is precisely the set elements of the form w(x ,..., x )∈ H[A] with 1 c (x ,..., x )∈ M (A). In particular, M(A)⊂ L[A].Hence M satisﬁes the assumption of 1 c (***) and Theorem 7.B.5 can be applied. Y Y Example 7.B.8. — Here is an example in which the inclusion L[K ]⊂ L[K] is ℘ ℘ strict. We choose K= R, ν= 1, and U (A)= A for every A (so U is the 1-dimensional 1 1 unipotent group). We choose R= R to be the singleton{1} (formally, R(A) ={1}⊂ A). Hence Q(A) is equal to A/Z1 ,and for ℘ ={1},wehave L[A]= Z1 . In particular, A ℘ A Y Y the inclusion L[K ]⊂ L[K] is proper as soon as Y contains two distinct elements. ℘ ℘ Anyway this example is not really serious because assuming that R is a subscheme con- Y Y taining 0 (which could easily be arranged), we would have L[K ]= L[K] for every ℘ ℘ ﬁnite Y, and the serious issue is when Y is inﬁnite. Indeed, now deﬁning R= R as equal to{0, 1} (namely R= Spec(R[t]/(t− t))), we have R(A) equal to the set of idempotents in A, and hence L[A] is the additive subgroup of A generated by idempotents. In par- Y Y ticular, L[R] is equal to the set of bounded functions Y→ Z, while L[R] is equal to ℘ ℘ the set of all functions Y→ Z, which differs from the former if Y is inﬁnite. The inclu- sion L[P (R)]⊂ P (L (R)) is also proper, since then L[P (R)] is equal to the set of ℘ Y Y ℘ ℘ Y bounded functions Y→ Z, while P (L (R)) is equal to the set of all functions Y→ Z of Y ℘ at most polynomial growth, uniformly in Y. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 153 8. Central extensions of graded Lie algebras This section contains results on central extensions of graded Lie algebras, which will be needed in Section 9.Let Q⊂ K be ﬁelds of characteristic zero (for instance, Q= Q and K is a nondiscrete locally compact ﬁeld). To any graded Lie algebra, we associate a central extension in degree zero, which we call its “blow-up”, whose study will be needed in Section 9. We are then led to following problem: given a Lie algebra g over K, we need to compare the homologies H (g) and H (g) of g viewed as a Lie algebra over K and as a Lie algebra over Q by restriction of scalars. When g is deﬁned over Q, i.e. g= K⊗ l, this problem has been tackled in several papers [KL82, NW08]. Here most of the work is carried out over an arbitrary commutative ring; this generality will be needed as we need to apply the results over suitable rings of functions. 8.A. Basic conventions. — The following conventions will be used throughout this chapter. The letter R denotes an arbitrary commutative ring (associative with unit). Un- less explicitly stated, modules, Lie algebras are over the ring R and are not assumed to be ﬁnitely generated. The reader is advised not to read this part linearly but rather refer to it when necessary. Gradings. — We ﬁx an abelian group W , called the weight space. By graded mod- ule, we mean an R-module V endowed with a grading,namelyan R-module decomposi- tion as a direct sum V= V . α∈W Elements of V are called homogeneous elements of weight α.An R-module homomor- phism f: V→ W between graded R-modules is graded if f (V )⊂ W for all α.If V is α α a graded module and V is a subspace, it is a graded submodule if it is generated by homo- geneous elements, in which case it is naturally graded and so is the quotient V/V .By the weights of V we generally mean the subset W⊂ W consisting of α∈ W such that V ={0}.Weuse thenotation V= V . α =0 By graded Lie algebra we mean a Lie algebra g endowed with an R-module grading g= g such that[g , g ]⊂ g for all α, β∈ W . α α β α+β Tensor products. — If V, W are modules, the tensor product V⊗ W= V⊗ Wis deﬁned in the usual way. The symmetric product V V is obtained by modding out by the R-linear span of all v⊗ w− w⊗ v and the exterior product V∧ V is obtained by 154 YVES CORNULIER, ROMAIN TESSERA modding out by the R-linear span of all v⊗ w+ w⊗ v (or equivalently all v⊗ v if 2 is invertible in R). More generally the nth exterior product V ∧· · · ∧ V is obtained by modding out the nth tensor product V⊗ ...⊗ Vby all tensors v ⊗· · · ⊗ v+ w ⊗ ··· ⊗ 1 n 1 w , whenever for some 1≤ i = j≤ n,wehave w= v , w= v and w= v for all k = i, j . n i j j i k k If W , W are submodules of V, we will sometimes denote by W∧ W (resp. W 1 2 1 2 1 W ) the image of W⊗ W in V∧ V (resp. V V). In case W= W= W, the latter map 2 1 2 1 2 factors through a module homomorphism W∧ W→ V∧ V (resp. W W→ V V), and this convention is consistent when this homomorphism is injective, for instance when W is a direct factor of V. If V, W are graded then V⊗ W is also graded by (V⊗ W)= V⊗ W . α β γ {(β,γ ):β+γ=α} When V= W, we see that V∧Vand V V are quotients of V⊗ V by graded submodules and are therefore naturally graded; for instance if W has no 2-torsion (V∧ V)= (V∧ V )⊕ V⊗ V . 0 0 0 α−α α∈(W−{0})/± Homology of Lie algebras. — Let g be a Lie algebra (always over the commutative ring R). We consider the complex of R-modules d d d 4 3 2 1 ··· g∧ g∧ g∧ g −→ g∧ g∧ g −→ g∧ g −→ g −→ 0 given by d (x , x ) =−[x , x] 2 1 2 1 2 d (x , x , x )= x ∧[x , x ]+ x ∧[x , x ]+ x ∧[x , x] 3 1 2 3 1 2 3 2 3 1 3 1 2 and more generally the boundary map i+j d (x ,..., x )= (−1)[x , x ]∧ x ∧ ··· ∧# x ∧· · · ∧# x ∧ ··· ∧ x; n 1 n i j 1 i j n 1≤i≤j≤n and deﬁne the second homology group H (g)= Z (g)/B (g),where Z (g)= Ker(d ) is 2 2 2 2 2 the set of 2-cycles and B (g)= Im(d ) is the set of 2-boundaries. (We will focus on d 2 3 i for i≤ 3 although the map d will play a minor computational role in the sequel. This is of course part of the more general deﬁnition of the nth homology module H (g)= Ker(d )/Im(d ), which we will not consider.) If A→ R is a homomorphism of n n n+1 commutative rings, then g is a Lie A-algebra by restriction of scalars, and its 2-homology as a Lie A-algebra is denoted by H (g).If g is a graded Lie algebra, then the maps d are graded as well, so H (g) is naturally a graded R-module. 2 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 155 Iterated brackets. — In a Lie algebra, we deﬁne the n-fold bracket as the usual bracket for n= 2 and by induction for n≥ 3as [x ,..., x ]= x ,[x ,..., x] . 1 n 1 2 n Central series and nilpotency. — Deﬁne the lower central series of the Lie algebra g by 1 i+1 i s+1 g= g and g =[g, g] for i≥ 1. We say that g is s-nilpotent if g ={0}. 8.A.1. The Hopf bracket. — Consider a central extension of Lie algebras 0→ z→ g→ h→ 0. Since z is central, the bracket g∧ g→ g factors through an R-module homomorphism B: h∧ h→ g, called the Hopf bracket. It is unique for the property that B(p(x)∧ p(y))= [x, y] for all x, y∈ g (uniqueness immediately follows from surjectivity of g∧ g→ h∧ h). Lemma 8.A.1. —For all x, y, z, t∈ h we have B([x, y]∧[z, t]) =[B(x∧ y), B(z∧ t)]. Proof. — Observe that if x¯, y¯ are lifts of x and y then B(x∧ y) =[x ¯, y¯],and that [¯ x, y¯] is a lift of[x, y]. In view of this, observe that both terms are equal to[[x¯, y¯], [¯ z, t]]. 8.A.2. 1-tameness. Deﬁnition 8.A.2. — We say that a Lie algebra g is 1-tame if it is generated by g= g . α =0 Lemma 8.A.3. —Let g be a graded Lie algebra. Then g is 1-tame if and only if we have g=[g , g],where β ranges over nonzero weights. 0 β−β Proof. — One direction is trivial. Conversely, if g is 1-tame, then g is generated as an abelian group by elements of the form x =[x ,..., x] with k≥ 2and x homogeneous 1 k i of nonzero weight. So x =[x , y] with y =[x ,..., x ]∈ g and x∈ g . 1 2 k 1 Lemma 8.A.4. —Let g be a graded Lie algebra. Then the ideal generated by g coincides with the Lie subalgebra generated by g and in particular is 1-tame. Proof.—Let h be the subalgebra generated by g ; it is enough to check that h is an ideal, and it is thus enough to check that[g , h]⊂ h.Set h= g and h =[g , h], 0 1 d d−1 so that h= h . It is therefore enough to check that[g , h ]⊂ h for all d≥ 1. This d 0 d d d≥1 is done by induction. The case d= 1is clear. If d≥ 2, x∈ g , y∈ g , z∈ h , then using 0 d−1 the induction hypothesis x,[y, z]= y,[x, z]−[y, x], z ∈[g , h ]⊂ h d−1 d and we are done. 156 YVES CORNULIER, ROMAIN TESSERA We use this to obtain the following result, which will be used in Section 11. Lemma 8.A.5. —Let g be a graded Lie algebra with lower central series (g ), and assume s+1 that g is s-nilpotent. Then g is contained in the subalgebra generated by g . In particular, if g is 0 0 ∞ i∞ nilpotent and g= g , then g is contained in the subalgebra generated by g . s+1 Proof. — By Lemma 8.A.4, it is enough to check that g is contained in the ideal s+1 s+1 j generated by g . It is sufﬁcient to show that g∩ g⊂ j.Eachelement x of g∩ g 0 0 can be written as a sum of nonzero (s+ 1)-fold brackets of homogeneous elements. Each of those brackets involves at least one element of nonzero degree, since otherwise all its entries would be contained in g and it would vanish. So x belongs to the ideal generated by g . 8.B. The blow-up. Deﬁnition 8.B.1. —Let g be an arbitrary graded Lie R-algebra (the grading being in the abelian group W ). Deﬁne the blow-up graded algebra g˜ as follows. As a graded R-module, g˜= g α α for all α = 0,and g˜= (g∧ g) /d (g∧ g∧ g) . 0 0 3 0 Deﬁne a graded R-module homomorphism τ: g˜→ g by τ(x)= xif x∈ g˜ and τ(x∧ y)= [x, y] if x∧ y∈ g˜ (which of course factors through 2-boundaries). Let us deﬁne the Lie algebra structure[·,·] on g˜ . Suppose that x∈ g˜ , y∈ g˜ . α β • if α+ β = 0,deﬁne[x, y] =[τ(x), τ (y)]; • if α+ β= 0,deﬁne[x, y]= τ(x)∧ τ(y). Lemma 8.B.2. — With the above bracket, g˜ is a Lie algebra and τ is a Lie algebra homomor- phism, whose kernel is central and naturally isomorphic to H (g) . Its image is the ideal g +[g, g] 2 0 of g. Proof.—Letusﬁrstcheck that τ is a homomorphism (of non-associative algebras). Let x, y∈ g˜ have weight α and β . In each case, we apply the deﬁnition of[·,·] and then of τ .If α+ β = 0 τ[x, y]= τ τ(x), τ (y)= τ(x), τ (y); if α+ β= 0then τ[x, y]= τ τ(x)∧ τ(y)= τ(x), τ (y) . By linearity, we deduce that τ is a homomorphism. Since τ is an isomorphism for α = 0 and τ =−d , the kernel of τ is equal by deﬁnition to H (g) . Moreover, by deﬁnition 0 2 2 0 the bracket[x, y] only depends on τ(x)⊗ τ(y), and it immediately follows that Ker(τ ) is central in g˜ . GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 157 Let us check that the bracket is a Lie algebra bracket; the antisymmetry being clear, we have to check the Jacobi identity. Take x∈ g˜ , y∈ g˜ , z∈ g˜ . From the deﬁnition α β γ above, we obtain • if α+ β+ γ = 0,[x,[y, z]] =[τ(x),[τ(y), τ (z)]]; • if α+ β+ γ= 0,[x,[y, z]]= τ(x) ∧[τ(y), τ (z)] (the careful veriﬁcation involves discussing on whether or not β+ γ is zero). Therefore, the Jacobi identity for (x, y, z) immediately follows from that of g in the ﬁrst case, and from the fact we killed 2-boundaries in the second case. We have τ(g˜ )= g and τ(g˜ ) =[g, g] . Therefore the image of τ is equal to 0 0 g +[g, g]. The latter is an ideal, since it contains[g, g]. Deﬁnition 8.B.3. — We say that a graded Lie algebra g is relatively perfect in degree zero if it satisﬁes one of the following (obviously) equivalent deﬁnitions (a) 0isnot aweightof g/[g, g]; (b) g ⊂[g, g]; (c) g= g +[g, g]; (d) g˜→ g is surjective (in view of Lemma 8.B.2). (e) g is generated by g +[g , g]; 0 0 Note that if g is 1-tame, then it is relatively perfect in degree zero, but the converse is not true, as shows the example of a nontrivial perfect Lie algebra with grading concen- trated in degree zero. The interest of this notion is that it is satisﬁed by a wealth of graded Lie algebras that are very far from perfect (e.g., nilpotent). Theorem 8.B.4. —Let g be a graded Lie algebra. If g is relatively perfect in degree zero (e.g., g is 1-tame), then the blow-up g˜→ g is a graded central extension with kernel in degree zero, and is universal among such central extensions. That is, for every surjective graded Lie algebra homomorphism h→ g with central kernel z= z , there exists a unique graded Lie algebra homomorphism φ: g˜→ h so that the composite map g˜→ h→ g coincides with the natural projection. Proof. — By Lemma 8.B.2, g˜→ g is a central extension with kernel in degree zero. Denote by B: g∧ g→ h the Hopf bracket associated to h→ g (see Section 8.A). Let us show uniqueness in the universal property. Clearly, φ is determined on g . ˜˜˜˜ So we have to check that φ is also determined on[g, g]. Observe that the map g∧ g→ h, x∧ y → φ([x, y]) =[φ(x), φ(y)] factors through a map w: g∧g→ h,so w(τ(x)∧τ(y))= [φ(x), φ(y)] for all x, y∈ g˜ . Since p◦ φ= τ and since h is generated by the image of φ and by its central ideal Ker(p), we deduce that for all x, y∈ h,wehave w(p(x)∧ p(y)) =[x, y]. By the uniqueness property of the Hopf bracket (see Section 8.A), we deduce that w= B . So for all x, y∈ g˜ , φ([x, y]) is uniquely determined as B (τ (x)∧ τ(y)). h 158 YVES CORNULIER, ROMAIN TESSERA −1 Now to prove the existence, deﬁne φ: g˜→ h to be p on g˜ ,and φ(x∧ y)= B (x∧ y) if x∧ y∈ g˜ .Itisclear that φ is a graded module homomorphism and that h 0 p◦ φ= τ . Let us show that φ is a Lie algebra homomorphism, i.e. that the graded module homomorphism σ: g˜∧ g˜→ h, (x∧ y) → φ([x, y] ) −[φ(x), φ(y)] vanishes. Since p◦ φ is a homomorphism and p is bijective, we have (p◦ σ)= 0, so it is enough to check that σ vanishes in degree 0. If x and y have nonzero opposite weights, noting that p◦ φ is the identity on g , −1−1 φ[x, y]= φ(x∧ y)= B (x∧ y)= p (x), p (y)= φ(x), φ(y) . If x∧ y and z∧ t belong to g˜= (g∧ g) , then, using Lemma 8.A.1,wehave 0 0 φ[x∧ y, z∧ t]= φ[x, y]∧[z, t]= B[x, y]∧[z, t] = B (x∧ y), B (z∧ t)= φ(x∧ y), φ(z∧ t) . h h By linearity, we deduce that σ= 0 and therefore φ is a Lie algebra homomorphism. ˜˜ Corollary 8.B.5. —If g is relatively perfect in degree zero then g= g. Proof. — Observe that g˜→ g has kernel z concentrated in degree zero, so it immediately follows that[z, g ]= 0. We need to show that z is central in g˜ . Since ˜˜ g =[g, g]+ g , it is enough to show that[z,[g˜, g˜]]={0}. By the Jacobi identity, ˜˜˜˜˜ [z,[g˜, g˜]]⊂[g˜,[z, g˜]]. Now since g˜→ g has a central kernel,[z, g˜] is contained in the ˜˜˜˜˜ kernel of g˜→ g˜ , which is central in g˜.So[g˜,[z, g˜]]={0}.Thus z is central in g˜.The universal property of g˜ then implies that g˜→ g˜ is an isomorphism. Remark 8.B.6. — There is an immediate generalization of this blow-up construc- tion, where we replace{0} by an arbitrary subset X⊂ W , deﬁning g˜ to be equal to g if α α 2 3 α/∈ X and to ( g/d ( g)) if α∈ X . Then immediate adaptations of Lemma 8.B.2, 3 α Theorem 8.B.4 and its corollary hold, with essentially no change in the proofs. (Deﬁni- tion 8.B.3 is valid replacing g with g= g and g with g ,exceptthatwe 0 X α WX α∈X have to remove (the analogue of) Condition (e) of Deﬁnition 8.B.3. Nevertheless, under the assumption that no weight in X is the sum of a weight X and of a weight in W X , Condition (e) is still equivalent; in particular this works if X is a subgroup of W and in particular for X ={0}. Otherwise, the reader can ﬁnd examples with gradings in{0, 1} and X ={1}, where the ﬁrst four conditions hold but not (e).) Lemma 8.B.7. —Let g be a graded Lie algebra and g its blow-up. If g is 1-tame, then so is g. Proof. — Suppose that g is 1-tame. Then by linearity, it is enough to check that for every x∈ g and u,v of nonzero opposite weights, the element x ∧[u,v] belongs to 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 159 [g˜ , g˜]. This is the case since modulo 2-boundaries, this element is equal to u ∧[x,v]+ v ∧[u, x]∈ (g∧ g ) . Lemma 8.B.8. —Let g be ﬁnitely many graded Lie algebras (all graded in the same abelian group) and g˜ their blow-up. If g= g satisﬁes the assumption that g/[g, g] has no i i opposite weights, then the natural homomorphism g→ g˜ is an isomorphism. Equivalently, i i H ( g )→ H (g ) is an isomorphism. 2 i 0 2 i 0 Proof.—Foreach i, there are homomorphisms g→ g→ g , whose composi- i j i tion is the identity, and hence H (g )→ H ( g )→ H (g ) , whose composition is the 2 i 0 2 j 0 2 i 0 identity again. So we obtain homomorphisms ! " H (g )→ H g→ H (g ) , 2 i 0 2 i 2 i 0 whose composition is the identity. To ﬁnish the proof, we have to check that H (g )→ 2 i 0 H ( g ) is surjective, or equivalently that in Z ( g ) , every element x is the sum of 2 i 0 2 i 0 an element in Z (g ) and a boundary. Now observe that 2 i 0 ! " Z g= Z (g )⊕ (g∧ g ) . 2 i 2 i 0 i j 0 i<j So we have to prove that for i = j , g∧ g consists of boundaries. Given an element x∧ y i j (x∈ g , y∈ g homogeneous), the assumption on g/[g, g] implies that, for instance, y is a i j sum[z ,w] of commutators. Projecting if necessary into g , we can suppose that all k k j z and w belong to g .So k k j x∧ y= x∧[z ,w ]= d (x∧ z∧ w ). k k 3 k k k k 8.C. Homology and restriction of scalars. — We now deal with two commutative rings A, B coming with a ring homomorphism A→ B (we avoid using R as the previous re- sults will be used both with R= A and R= B). If g is a graded Lie algebra over B,itcan then be viewed as a graded Lie algebra over A by restriction of scalars. This affects the deﬁnition of the blow-up. There is an obvious surjective graded Lie algebra homomor- A B phism g˜→ g˜ . The purpose of this part is to describe the kernel of this homomorphism A B (or equivalently of the homomorphism H (g)→ H (g) ), and to characterize, under 0 0 2 2 suitable assumptions, when it is an isomorphism. Our main object of study is the following kernel. A,B Deﬁnition 8.C.1. —If g is a Lie algebra over B,wedeﬁne the welding module W (g) A B as the kernel of the natural homomorphism H (g)→ H (g), or equivalently of the homomorphism 2 2 A,B A B (g∧ g)/B (g)→ (g∧ g)/B (g).If g is graded, it is a graded module as well, and W (g) A B 0 2 2 2 A B then also coincides with the kernel of g˜→ g˜ . 160 YVES CORNULIER, ROMAIN TESSERA The following module will also play an important role. Deﬁnition 8.C.2. —If g is a Lie algebra over B, deﬁne its Killing module Kill(g) (or Kill (g) if the base ring need be speciﬁed) as the cokernel of the homomorphism T: g⊗ g⊗ g→ g g u⊗ v⊗ w → u [v, w]+ v [u,w]. If g is graded, it is graded as well. Note that we can also write T (u⊗ w⊗ v) =[u,v] w− u [v, w],and thus we see that the set of B-linear homomorphisms from Kill(g) to any B-module M is naturally identiﬁed to the set of the so-called invariant bilinear maps g× g→ M. Lemma 8.C.3. —Let v be a B-module. Then the kernel of the natural surjective A-module homomorphism v⊗ v→ v⊗ v is generated, as an abelian group, by elements A B (8.C.4) λx⊗ y− x⊗ λy A A with λ∈ B,x, y∈ v. The same holds with⊗ replaced by or∧. Proof.—Endow v⊗ v with a structure of a B-module, using the structure of B-module of the left-hand v,namely λ(x⊗ y)= (λx⊗ y) if λ∈ B, x, y∈ v. Let W be the subgroup generated by elements of the form (8.C.4); it is clearly an A-submodule, and is actually a B-submodule as well. The natural A-module surjective homomorphism φ: (v⊗ v)/W→ v⊗ v is B-linear. To show it is a bijection, we A B observe that by the universal property of v⊗ v,wehavea B-module homomorphism ψ: v⊗ v→ (v⊗ v)/W mapping x⊗ y to x⊗ y modulo W. Clearly, ψ and φ are B A B A inverse to each other. Let us deal with∧,the case of being similar. The group v∧ v is deﬁned as the quotient of v⊗ v by symmetric tensors (i.e. by the subgroup generated by elements of the form x⊗ y+ y⊗ x), and the group v∧ v is deﬁned the quotient of v⊗ v by B B symmetric tensors. By the case of⊗, this means that v∧ v is the quotient of v⊗ v B A by the subgroup generated by symmetric tensors and elements (8.C.4). This implies that v∧ v is the quotient of v∧ v by the subgroup generated by elements (8.C.4) (with⊗ B A replaced by∧) Proposition 8.C.5. — For any Lie algebra g over B,the A-module homomorphism A,B : B⊗ g⊗ g→ W (g) A A λ⊗ x⊗ y → λx∧ y− x∧ λy A,B is surjective. If g is graded and is 1-tame, then : B⊗ (g⊗ g)→ W (g) is surjective in 0 A A 0 0 restriction to B⊗ (g⊗ g ) . 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 161 Proof.—Thegroup (g∧ g)/B (g) is deﬁned as the quotient of g∧ g by 2- A A boundaries, and (g∧ g)/B (g) is the quotient of g∧ g by 2-boundaries, or equivalently, B B by Lemma 8.C.3, is the quotient of g∧ g by 2-boundaries and elements of the form (8.C.6) λx∧ y− x∧ λy. A A A B It follows that the kernel of (g∧ g)/B (g)→ (g∧ g)/B (g) is generated by elements A B 2 2 of the form (8.C.6), proving the surjectivity of . For the additional statement, deﬁne W= (B⊗ (g⊗ g )) .Wehavetoshow that any element λx∧ y− x∧ λy as in (8.C.6)with x, y of zero weight belongs to W .By A A linearity and Lemma 8.A.3, we can suppose that y =[z,w] with z,w of nonzero opposite weight. So, modulo boundaries, λx ∧[w, z]− x∧ λ[w, z]=−w ∧[z,λx]− z ∧[λx,w]− x∧ λ[w, z] =−w∧ λ[z, x]+ λw ∧[z, x], which belongs to W . Proposition 8.C.7. —Let g be a Lie algebra over B. Then the map A,B : B⊗ g⊗ g→ W (g) A A of Proposition 8.C.5 factors through the natural projection B⊗ g⊗ g→ B⊗ Kill (g); A A A moreover in restriction to B⊗ g⊗[g, g], it factors through B⊗ Kill (g). In particular, if g is A A A graded and relatively perfect in degree 0 (e.g., 1-tame), then factors through B⊗ Kill (g). 0 A Proof. — All⊗,∧, are meant over A. λ λ λ Write (x⊗ y)= (λ⊗ x⊗ y). It is immediate that (y⊗ x)= (x⊗ y) (even before modding out by 2-boundaries), and thus factors through g g.Let us now check that factors through Kill (g). Modulo 2-boundaries: x [y, z]− y [z, x]= λx ∧[y, z]− x∧ λ[y, z] − λy ∧[z, x]+ y∧ λ[z, x] =−z ∧[λx, y]+ z ∧[x,λy]= 0. λ λ For the last statement, by Lemma 8.C.3, we have to show that (μx [y, z])= (x μ[y, z]) for all x, y, z∈ g and μ∈ B. Indeed, using the latter vanishing we get λ λ λ μx [y, z]= y [z,μx]= y [μz, x] = x [y,μz] . 162 YVES CORNULIER, ROMAIN TESSERA In turn, we obtain, as an immediate consequence: Corollary 8.C.8. —Let g be a graded Lie algebra over B; assume that g is relatively perfect in A,B degree 0. If Kill (g) ={0} then W (g) ={0}. 0 0 Proof. — By Propositions 8.C.5 and 8.C.7, induces a surjection B⊗ 0 A A,B Kill(g)→ W (g) . Under the additional assumptions that A= Q is a ﬁeld of characteristic = 2and g is deﬁned over Q, i.e. has the form l⊗ B for some Lie algebra l over Q, Corol- lary 8.C.8 easy follows from [NW08, Theorem 3.4]. We are essentially concerned with ﬁnite-dimensional Lie algebras g over a ﬁeld K of characteristic zero (K playing the role of B)and A= Q, but nevertheless in general we cannot assume that g be deﬁned over Q. We will also use the more speciﬁc application. Corollary 8.C.9. — Assume that A= Q and B= K= K is a ﬁnite product of locally j=1 compact ﬁelds K , each isomorphic to R or some Q .Let B be the closed unit ball in K . Assume that j p K j g is ﬁnite-dimensional over K, that is, g= g with g ﬁnite-dimensional over K . Suppose that g j j j is 1-tame and g/[g, g] has no opposite weights. For every j and weight α,let V be a neighborhood j,α Q,K K of 0 in g= (g ) . Then the welding module W (g)= Ker(H (g)→ H (g) ) is generated j,α j α 2 2 2 0 0 by elements of the form (λ⊗ x⊗ y) with λ∈ B and x∈ V ,y∈ V ,where α ranges over K j,α j,−α nonzero weights and j= 1,...,τ . A,B Proof. — By Proposition 8.C.5,W (g) is generated by elements of the form (λ⊗ x⊗ y ),with λ∈ K and x , y∈ g are homogeneous of nonzero opposite weight. By linearity and Lemma 8.B.8, these elements for which λ , x , y∈ K× g× g j j,α j,−α j α =0 are enough. Given such an element (λ , x , y )∈ K× g× g , using that K= Q+ j j,α j,−α j B , write λ= ε+ λ with ε∈ Q and λ∈ B . Also, since g= QV , write x= μx K K j,±α j,±α and y= γ y with μ, γ∈ Q, x∈ V , y∈ V . Clearly, by the deﬁnition of ,wehave j,α j,−α (ε⊗ x⊗ y )= 0, so ⊗ x⊗ y= (λ⊗ μx⊗ γ y)= μγ (λ⊗ x⊗ y), and (λ⊗ x⊗ y) is of the required form. 8.C.1. Construction of 2-cycles. — We now turn to a partial converse to Corol- lary 8.C.8. Deﬁne HC (B) (or HC (B) if A is implicit) as the quotient of B∧ B by the A- 1 A submodule generated by elements of the form uv∧ w+ vw∧ u+ wu∧ v. This is the ﬁrst GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 163 cyclic homology group of B (which is usually deﬁned in another manner; we refer to [NW08] for the canonical isomorphism between the two). Lemma 8.C.10. — Assume that K is a ﬁeld of characteristic zero and Q a subﬁeld. Suppose that K contains an element t that is transcendental over Q. Assume that either K has characteristic zero, Q Q −1 or K⊂ Q((t)). Then HC (K) ={0}.Moreprecisely,the imageof t∧ t in HC (K) is nonzero. 1 1 Proof. — We denote W by the Q-linear subspace of K∧ K generated by elements of the form uv∧ w+ vw∧ u+ wu∧ v. Let us begin with the case of Q((t)). Deﬁne a Q-bilinear map F: Q((t))→ Qby ! " i j (8.C.11)F x t , y t= kx y . i j k−k k∈Z −1 Observe that the latter sum is ﬁnitely supported. (In Q[t, t], the above map appears in the deﬁnition of the deﬁning 2-cocycle of afﬁne Lie algebras, see [Fu].) We see that F is −1 alternating by a straightforward computation, and that F(t, t )= 1. Setting f (x∧ y)= F(x, y),if x= x t , etc., we have f (xy∧ z)= k(xy) z k−k k∈Z = k x y z i j−k k∈Z i+j=k =− kx y z; i j k i+j+k=0 thus f (xy∧ z+ yz∧ x+ zx∧ y) =− (k+ i+ j)x y z= 0. i j k i+j+k=0 −1 Hence f factors through a Q-linear map from HC (Q((t)))→ Q mapping t∧ t to 1. The proof is thus complete if K⊂ Q((t)). −1 Now assume that K has characteristic zero; let us show that t∧ t has a nontrivial image in HC (K). Since K has characteristic zero, the above deﬁnition (8.C.11) imme- −∞ 1/n−1 diately extends to the ﬁeld Q((t ))= Q((t )) of Puiseux series and hence t∧ t n>0 −∞ has a nontrivial image in HC (Q((t ))) for every ﬁeld Q of characteristic zero. To prove the general result, let (u ) be a transcendence basis of K over Q(t). j j∈J Replacing Q by Q(t: j∈ J) if necessary, we can suppose that K is algebraic over Q(t). −∞ ˆˆ Let Q be an algebraic closure of Q. By the Newton-Puiseux Theorem, Q((t )) is al- gebraically closed. By the Steinitz Theorem, there exists a Q(t)-embedding of K into Q Q −∞ L= Q((t )). This induces a Q-linear homomorphism HC (K)→ HC (L) mapping 1 1 164 YVES CORNULIER, ROMAIN TESSERA Q Q −1−1 the class of t∧ t in HC (K) to the class of t∧ t in HC (L); the latter is mapped in 1 1 Q Q −1−1 turn to the class t∧ t in HC (L), which is nonzero. So the class of t∧ t in HC (K) 1 1 is nonzero. Theorem 8.C.12. —Let g be a Lie algebra over B, which is deﬁned over A, i.e. g B⊗ g A A for some Lie algebra g over A. Consider the homomorphism A A A ϕ: (g∧ g)/B (g)→ M= HC (B)⊗ Kill (g ) A A 2 1 (λ⊗ x)∧ (μ⊗ y) → (λ∧ μ)⊗ (x y). A,B Then ϕ is well-deﬁned and surjective, and ϕ(W (g))= 2M. In particular, if 2 is invertible in A, A,B g is agradedLie algebraand M= HC (B)⊗ Kill(g ) ={0}, then W (g) ={0}. A 0 1 A 0 0 The above map ϕ was considered in [NW08], for similar motivations. Assum- ing that A is a ﬁeld of characteristic zero, the methods in [NW08]can providea more precise description (as the cokernel of an explicit homomorphism) of the kernel A,B A B W (g)= Ker(H (g)→ H (g)). Since we do not need this description and in order 2 2 not to introduce further notation, we do not include it. Proof of Theorem 8.C.12.—Letusﬁrstview ϕ as deﬁned on g∧ g. The surjectivity is A,B trivial. By Proposition 8.C.5 (with grading concentrated in degree 0), we see that W (g) is generated by elements of the form λx∧ μy− μx∧ λy with x, y∈ g (we omit the⊗ signs, which can here be thought of as scalar multiplication); the image by ϕ of such an element is 2(λ∧ μ)(x y), which belongs to 2M. Conversely, since 2M is generated by A,B elements of the form 2(λ∧ μ)(x y), we deduce that ϕ(W (g))= 2M. Let us check that ϕ vanishes on 2-boundaries (so that it is well-deﬁned on g∧ g modulo 2-boundaries): ϕ tx ∧[uy,vz]= (t∧ uv)⊗ x [y, z]; ϕ uy ∧[vz, tx]= (u∧ vt)⊗ y [z, x]= (u∧ vt)⊗ x [y, z]; ϕ vz ∧[tx, uy]= (v∧ tu)⊗ z [x, y]= (v∧ tu)⊗ x [y, z] and since t∧ uv+ u∧ vt+ v∧ tu= 0inHC (A), the sum of these three terms is zero. The last statement clearly follows. 8.C.2. The characterization. — Using all results established in the preceding para- graphs, we obtain Theorem 8.C.13. —Let g be a ﬁnite-dimensional graded Lie algebra over a ﬁeld K of char- acteristic zero, assume that g/[g, g] has no opposite weights. Let Q be a subﬁeld of K, so that K has inﬁnite transcendence degree over Q. We have equivalences GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 165 Q,K Q • W (g) ={0} (i.e., H (g)→ H (g) is an isomorphism); 0 0 0 2 2 2 • Kill (g) ={0}. Corollary 8.C.14. — Under the same assumptions, we have equivalences • H (g) ={0} (i.e. the blow-up g˜→ g is an isomorphism); • H (g)= Kill (g) ={0}. 0 0 The interest is that in both the theorem and the corollary, the ﬁrst condition is a problem of linear algebra in inﬁnite dimension, while the second is linear algebra in ﬁnite dimension (not involving Q) and is therefore directly computable in terms of the structure constants of g. Proof of Theorem 8.C.13. — Suppose that Kill (g)= 0. By Corollary 8.C.8,the induced homomorphism H (g)→ H (g) is bijective. 0 2 0 Conversely, suppose that Kill (g) = 0. Since g is ﬁnite-dimensional over K, there exists a subﬁeld L⊂ K, ﬁnitely generated over Q, such that g is deﬁned over L, i.e. we can K L L write g= g⊗ K. Obviously, Kill (g)= Kill (g )⊗ K, so Kill (g ) = 0. Let (x y) L L L L L L be the representative of a nonzero element in Kill (g ).Let λ be an element of K, tran- −1 scendental over L. By Lemma 8.C.10, the element λ∧ λ has a nontrivial image in HC (K).ByTheorem 8.C.12 (applied with (A, B)= (L, K)), we deduce that −1−1 c= λx∧ λ y− λ x∧ λy L L L is not a 2-boundary, i.e. is nonzero in H (g) . In particular the element c (written as c 0 Q L K L with∧ instead of∧ )isnonzero in H (g) since its image in H (g) is c , while its image Q L 0 L 2 2 c in g∧ g and hence in H (g) is obviously zero. K K 0 8.D. Auxiliary descriptions of H (g) and Kill(g) .— We now prove the result in- 2 0 0 cluding Theorem J as a particular case. In this subsection, all Lie algebras are over a ﬁxed commutative ring R. Deﬁnition 8.D.1. —Let g be a graded Lie algebra. We say that g is doubly 1-tame if for every α we have g=[g , g]. 0 β−β β/∈{0,α,−α} In view of Lemma 8.A.3, doubly 1-tame implies 1-tame, and the reader can easily ﬁnd counterexamples to the converse. This deﬁnition will be motivated in Section 9, because it is a consequence of 2-tameness (Lemma 9.B.1(1)), which will be introduced therein. The purpose of this subsection is to provide descriptions of H (g) and Kill(g) . 2 0 0 166 YVES CORNULIER, ROMAIN TESSERA 8.D.1. The tame 2-homology module. — Let us begin with the trivial observation that if α+ β+ γ= 0and α, β, γ = 0, then α+ β, β+ γ,γ+ α = 0. It follows that d maps (g∧ g∧ g ) into g∧ g .Deﬁne the tame 2-homology module H (g)= Ker(d )∩ (g∧ g ) /d (g∧ g∧ g ) . 0 2 0 3 0 We are going to prove the following result. Theorem 8.D.2. —Let g be a graded Lie algebra. The natural homomorphism H (g)→ H (g) induced by the inclusion (g∧ g )→ (g∧ g) is surjective if g is 1-tame, and is an 2 0 0 0 isomorphism if g is doubly 1-tame. Remark 8.D.3. — In Abels’ second group (Section 1.5.4), (g∧ g) and (g∧ g∧ g) 0 0 have dimension 4 and 5, while (g∧ g ) and (g∧ g∧ g ) have dimension 3 and 2. 0 0 Thus, we see that the computation of H (g) is in practice easier than the computation of H (g) . 2 0 Lemma 8.D.4. —Let g be any graded Lie algebra. If g is 1-tame, then (1) g∧ g⊂ Im(d )+ (g∧ g ); 0 0 3 0 (2) g∧ g∧ g⊂ Im(d )+ g∧ (g∧ g ); 0 0 0 4 0 0 Proof. — Observe that (1) is a restatement of Lemma 8.B.7. The second assertion is similar: if u,v have nonzero opposite weights and x, y∈ g , then, modulo Im(d ), the element x∧ y ∧[u,v] is equal to y∧ u ∧[v, x]−[x, y]∧ u∧ v+ x∧ v ∧[u, y]− y∧ v ∧[u, x]− x∧ u ∧[v, y], which belongs to g∧ (g∧ g ) . 0 0 Proposition 8.D.5. — Consider the following R-module homomorphism : g⊗ g⊗ g⊗ g→ (g∧ g )/d (g∧ g∧ g ) u⊗ v⊗ x⊗ y → x∧ y,[u,v]− y∧ x,[u,v] . If g is doubly 1-tame, then there exists an R-module homomorphism : g⊗ g→ (g∧ g ) /d (g∧ g∧ g ) 0 0 0 3 0 such that whenever α, β are non-collinear weights, we have [x, y]⊗[u,v]= (u⊗v⊗ x⊗ y),∀x∈ g , y∈ g , u∈ g ,v∈ g . α−α β−β Moreover, is antisymmetric, i.e. factors through g∧ g . 0 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 167 Proof. — Suppose that α, β are non-collinear weights and that x∈ g , y∈ g , α−α u∈ g , v∈ g .Wehave d◦ d (x∧ y∧ u∧ v)= 0. If we write this down in (g∧ β−β 3 4 g ) /d (g∧ g∧ g ) , four out of six terms vanish and we get 0 3 0 d[x, y]∧ u∧ v+ d[u,v]∧ x∧ y= 0, 3 3 which expands as (8.D.6) (u⊗ v⊗ x⊗ y)+ (x⊗ y⊗ u⊗ v)= 0. Deﬁne, for w∈ g , (w)= x ∧[y,w]− y ∧[x,w]. The mapping 0 x,y (x, y) → ∈ Hom g ,(g∧ g ) /d (g∧ g∧ g ) x,y 0 0 3 0 is bilinear and in particular extends to a homomorphism σ: (g⊗ g )→ Hom g ,(g∧ g ) /d (g∧ g∧ g ) . 0 0 0 3 0 Since g is doubly 1-tame, any w∈ g can be written as[u ,v] with u∈ g , 0 i i i β v∈ g , β ∈{ / 0,±α}, so, using (8.D.6) i−β i (w)= (u⊗ v⊗ x⊗ y) x,y i i =− (x⊗ y⊗ u⊗ v )= [x, y] . i i u ,v i i i i This shows that σ(x⊗ y) only depends on[x, y], i.e. we can write σ(x⊗ y)= σ ([x, y]). Deﬁne, for z,w∈ g (z⊗ w)= σ (z)(w). By construction, whenever z =[x, y] and w =[u,v],with x∈ g , y∈ g , u∈ g , v∈ g α−α β−β and α, β are not collinear, we have [x, y]⊗[u,v]= (u⊗ v⊗ x⊗ y); from (8.D.6) we see in particular that is antisymmetric. Proof of Theorem 8.D.2.—If g is 1-tame, the surjectivity immediately follows from Lemma 8.D.4(1). Now to show the injectivity of the map of the theorem, suppose that c∈ (g∧ g ) is a 2-boundary and let us show that c belongs to d (g∧ g∧ g ) .Inviewof 0 3 0 Lemma 8.D.4(2), we already know that c belongs to d (g∧ g∧ g ) ,and letuswork 3 0 168 YVES CORNULIER, ROMAIN TESSERA again modulo d (g∧ g∧ g ) , so that we can suppose that c belongs to d (g∧ (g∧ 3 0 3 0 g ) ), and we wish to check that c= 0. Since g is doubly 1-tame, we can write c= d[u ,v ]∧ x∧ y 3 i i i i with x∈ g , y∈ g , u∈ g , v∈ g , α ,β nonzero and α =±β .Write w =[u ,v]. i α i−α i β i−β i i i i i i i i i i i Then c= w ∧[x , y]+ x ∧[y ,w ]+ y ∧[w , x] , i i i i i i i i i i i the ﬁrst term belongs to g∧ g and the second to (g∧ g ) /d (g∧ g∧ g ) ; since c 0 0 0 3 0 is assumed to lie in (g∧ g ) we deduce that (8.D.7) w ∧[x , y ]= 0 i i i in g∧ g . Therefore 0 0 c= x ∧[y ,w ]+ y ∧[w , x ]= (u⊗ v⊗ x⊗ y ). i i i i i i i i i i i i Now using Proposition 8.D.5 we get c= w ∧[x , y]= w ∧[x , y]= 0 i i i i i i i i again by (8.D.7). 8.D.2. The tame Killing module. Deﬁnition 8.D.8. —Let g be a graded Lie algebra over R. Consider the R-module homomor- phism T: (g g)⊗ g→ g g R R R u v⊗ w → u [v, w]+ v [u,w]. By deﬁnition, Kill(g) is the cokernel of T .Wedeﬁne Kill (g) as the cokernel of the restriction of T to (g g )⊗ g→ (g g ) . Note that T satisﬁes the identities, for all x, y, z T (x y⊗ z)+ T (y z⊗ x)+ T (z x⊗ y)= 0. There is an canonical homomorphism Kill (g)→ Kill(g) . 0 0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 169 Theorem 8.D.9. —Let g be a graded Lie algebra. If g is 1-tame then the homomorphism Kill (g)→ Kill(g) is surjective; if g is doubly 1-tame then it is an isomorphism. 0 0 Lemma 8.D.10. —Let g be an arbitrary Lie algebra. Then we have the identity T w, x,[y, z]= T[x, z],w, y− T[x, y],w, z − T[w, y], x, z+ T[w, z], x, y; Proof. — Use the four equalities T[x, z],w, y= w [x, z], y +[x, z] [w, y], T[y, x],w, z= w [y, x], z +[y, x] [w, z], T[w, z], x, y= x [w, z], y +[w, z] [x, y], T[y,w], x, z= x [y,w], z +[y,w] [x, z]; the sum of the four right-hand terms is, by the Jacobi identity and cancellation of ([·,·] [·,·])-terms, equal to −w [z, y], x− x [z, y],w= T w, x,[z, y] . Lemma 8.D.11. —Let g be an arbitrary graded Lie algebra. • Let α, α ,β,β be nonzero weights, with α+ β, α+ β ,α+ β, α+ β = 0,and (x, x , y, y )∈ g× g × g× g .Then, modulo T (g g⊗ g ), we have α α β β T x, x , y, y= 0. (1) and T x, x , y, y= T y, y , x, x . (2) • Let α ,β,γ,γ be weights with β, γ, γ = 0, α ∈{ / −γ,−γ}.For all w∈ g , x∈ 0 0 0 α g , y∈ g , y∈ g we have β γ γ T w , x, y, y= T x, y ,w , y− T[x, y],w , y . (3) 0 0 0 • Let α, α ,β,β be nonzero weights with α+ α ,β+ β = 0.For all x∈ g , x∈ g , y∈ α α g , y∈ g we have β β T x, y, x , y= T x, y , y, x− T y, x , x, y . (4) Proof. — These are immediate from the formula given by Lemma 8.D.10 applied to (x, x , y, y ), resp. (x, y, x , y ), resp. (w , x, y, y ), resp. (x, y, x , y ). 0 170 YVES CORNULIER, ROMAIN TESSERA Lemma 8.D.12. —Let g be a doubly 1-tame graded Lie algebra. • Let α, β, γ be weights with α, β = 0 and α+ β+ γ= 0,and (x, y, z)∈ g× g× g . α β γ Then, modulo T (g g⊗ g ), we have T (x, y, z)= 0. (1) • Let α, α ,β,β be weights, with α, β = 0 and α+ α+ β+ β= 0,and (x, x , y, y )∈ g× g × g× g . Then, modulo T (g g⊗ g ), we have α α β β T x, y , y, x= T y, x , x, y . (2) Proof. — Let us check the ﬁrst assertion. If γ = 0 this is trivial, so assume γ= 0 (so α =−β ). Since g is doubly 1-tame, we can write z=[u ,v] with u ,v of opposite i i i i weights, not equal to±α. Then Lemma 8.D.11(1) applies. Let us check the second assertion. We begin with two particular cases • α+ β = 0. Then α+ β = 0and by (1), modulo T (g g⊗ g ),both T ([x, y], y, x ) and T ([y, x], x, y ) are zero. • α+ β= 0, α+ α = 0. Then α ,β are nonzero, so we obtain the assertion by applying Lemma 8.D.11(4)and then (1) of the current lemma. Finally, the only remaining case is when (α,β,α ,β ) = (α, α,−α,−α), as we assume now. To tackle this last case, let us use this to prove ﬁrst the following: if γ is a nonzero weight, w∈ g , z∈ g ,z∈ g ,then T (w , z, z )+ T (w , z , z)= 0. Indeed, since 0 0 γ−γ 0 0 g is doubly 1-tame, this reduces by linearity to w =[u, u] with u∈ g , u∈ g and 0 δ−δ δ/ ∈{0,±γ}. So, using twice (2) in one of the cases already proved, we obtain T w , z, z= T u, u , z, z = T z, z , u, u =−T z , z , u, u =−T u, u , z , z =−T w , z , z Now suppose that (α,β,α ,β )= (α, α,−α,−α). Then, using the antisymmetry property above and again using one last time one already known case of (2), we obtain T x, y , y, x =−T x, y , x , y =−T x , y , x, y = T y, x , x, y . GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 171 Lemma 8.D.13. —Let g be a 1-tame graded Lie algebra. Then (1) (g g)= T (g g⊗ g)+ (g g ) ; 0 0 0 (2) T (g g⊗ g )= T (g g⊗ g) . 0 0 Proof. — Suppose that x, y∈ g . To show that x y belongs to the right-hand term in (1), it sufﬁces by linearity to deal with the case when y =[u,v] with u,v homogeneous of nonzero opposite weight. Then x [u,v]= T (x, u,v)− u [x,v], which is the sum of an element in ((g g)⊗ g) and an element in (g g ) .So(1) is 0 0 proved. Let us prove (2). By linearity, it is enough to prove that any element T (x, y,[u,v]), where x, y have weight zero and u,v have nonzero opposite weight, belongs to T (g⊗ g⊗ g ) : the formula in Lemma 8.D.10 expresses T (x, y,[u,v]) as a sum of four terms in T (g⊗ g⊗ g ) . Proposition 8.D.14. —Let g be a doubly 1-tame graded Lie algebra. Then T (g g⊗ g)∩ (g g )⊂ T (g g⊗ g ) . 0 0 Proof.—Fix u,v∈ n , n (β = 0) and consider the R-module homomorphism β−β : g→ M= (g g) /T (g g⊗ g ) u,v 0 0 0 w → u [v, w]. The mapping (u,v) → ∈ Hom (g , M) is bilinear. Therefore it extends to a map- u,v R 0 ping s → deﬁned for all s∈ (g⊗ g ) . s 0 If s= x⊗ y∈ g∧ g, we writes=[x , y].Now,for s, s∈ (g⊗ g ) ,deﬁne i i i i 0 ˆˆ (s⊗ s )= (s)∈ M. In other words, (u⊗ v)⊗ (x⊗ y)= u v,[x, y] . We have (u⊗ v)⊗ (x⊗ y)= u v,[x, y] =−T[x, y], u,v +[x, y] [u,v] and similarly (x⊗ y)⊗ (u⊗ v) =−T[u,v], x, y +[u,v] [x, y], Given a map deﬁned on a tensor product such as , we freely view it as a multilinear map when it is convenient. 172 YVES CORNULIER, ROMAIN TESSERA so ˆˆ (u⊗ v)⊗ (x⊗ y)− (x⊗ y)⊗ (u⊗ v) = T[u,v], x, y− T[x, y], u,v= 0 ˆˆˆ by Lemma 8.D.12(2). Thus, is symmetric and we can write (s⊗ s )= (s s ).Note ˆˆ that (trivially) (s s )= 0 whenever s is a 2-cycle (i.e.s = 0), so by the symmetry factors through a map : g g→ Msuch that 0 0 s s= s s for all s, s∈ (g∧ g ) . In other words, we can write u v,[x, y]= [u,v] [x, y] . Now consider some element in T (g g⊗ g)∩ (g g ) . By Lemma 8.D.13,it can be taken in T (g g⊗ g ). We write it as τ= T (x , y , z ), i i i with x , y , z homogeneous and y , z of nonzero weight. Since we work modulo T (g i i i i i g⊗ g ) , we can suppose x is of weight zero for all i, and we have to prove that τ= 0. 0 i So τ= x [y , z]+ y [x , z] , i i i i i i i i the ﬁrst term belongs to g∧ g and the second to the quotient Kill (g) of (g∧ g ) 0 0 0 0 by T (g g⊗ g ) ,so (8.D.15) x [y , z ]= 0. i i i Now, writing x=[u ,v] with u ,v of nonzero opposite weight i ij ij ij ij τ= y [x , z] i i i = y z ,[u ,v] i i ij ij i,j = [y , z] [u ,v] i i ij ij i,j = [y , z] x= [y , z] x=0by(8.D.15). i i i i i i i i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 173 Proof of Theorem 8.D.9. — The ﬁrst statement follows from Lemma 8.D.13(1) and the second from Proposition 8.D.14. 9. Abels’ multiamalgam 9.A. 2-tameness. — In this section, we deal with real-graded Lie algebras, that is, Lie algebras graded in a real vector space W.AsinSection 8, Lie algebras are, unless explicitly speciﬁed, over the ground commutative ring R. Let g be a real-graded Lie algebra. We say that P⊂ W is g-principal if g is gen- erated, as a Lie algebra, by g= g (note that this only depends on the structure P α α∈P of Lie ring, not on the ground ring R). We say that P (or (g, P))is k-tame if when- ever α ,...,α∈ P , there exists an R-linear form on W such that (α )> 0for all 1 k i i= 1,..., k.Notethat P is 1-tame if and only if 0∈ / P and is 2-tame if and only if for all α, β∈ P we have 0 ∈[ / α, β].Notethat k-tame trivially implies (k− 1)-tame. We say that the graded Lie algebra g is k-tame if there exists a g-principal k-tame subset. Note that for k= 1 this is compatible with the deﬁnition in Section 8.A.2. Example 9.A.1. —Asusual,wewrite W ={α: g ={0}}. g α • g= sl with its standard Cartan grading, W ={α: 1≤ i = j≤ 3}∪{0} (with 3 g ij α= e− e , (e ) denoting the canonical basis of R ); then{α ,α ,α} and ij i j i 12 23 31 {α ,α ,α} are g-principal and 2-tame. 21 13 32 • If P is the set of weights of the graded Lie algebra g/[g, g],thenany g-principal set contains P ; conversely if g is nilpotent then P itself is g-principal. Thus if g 1 1 is nilpotent, then g is k-tame if and only if 0 is not in the convex hull of k weights of g/[g, g]. 9.B. Lemmas related to 2-tameness. — This subsection gathers a few technical lemmas needed in the study of the multiamalgam in Section 9.C and Section 9.D.The reader can skip it in a ﬁrst reading. The following lemma was proved by Abels under more speciﬁc hypotheses (g nilpo- tent and ﬁnite-dimensional over a p-adic ﬁeld). As usual, by[g , g] we mean the module β γ generated by such brackets. In a real vector space, we writeα, β for the segment joining α and β (so as to avoid any confusion with Lie brackets). Lemma 9.B.1. —Let g be a real-graded Lie algebra and P⊂ W a g-principal subset. Suppose that (g, P) is 2-tame. Then (1) for any ω∈ W , we have g=[g , g], with β ranging over W− Rω; 0 β−β This paper will not deal with k-tameness for k≥ 3 but this notion is relevant to the study of higher-dimensional isoperimetry problems. 174 YVES CORNULIER, ROMAIN TESSERA (2) if R α∩ P =∅, then g=[g , g], with (β, γ ) ranging over pairs in W− Rα + α β γ such that β+ γ= α. Lemma 9.B.1 is a consequence of the more technical Lemma 9.B.2 below (with i= 1). Actually, the proof of Lemma 9.B.1 is based on an induction which makes use of the full statement of Lemma 9.B.2. Besides, while Lemma 9.B.1 is enough for our pur- poses in the study of the multiamalgam of Lie algebras in Section 9.C, the statements in Lemma 9.B.2 involving the lower central series are needed when studying multiamalgams of nilpotent groups in Section 9.D. Lemma 9.B.2. — Under the assumptions of Lemma 9.B.1,let (g ) be the lower central series i i i i i of g and g= g∩ g (note that g thus means (g ) and can be distinct from (g ) ). Then α 0 0 α 0 i k (1) for any ω∈ W , we have g=[g , g], with β ranging over W Rω; 0 j+k=i β β−β i k (2) if R α∩ P =∅, then g=[g , g], with (β, γ ) ranging over pairs in α j+k=i β γ W Rα such that β+ γ= α; i k (3) if α/∈ P , then g=[g , g], with (β, γ ) ranging over pairs in W such that α β γ j+k=i β+ γ= α and 0 does not belong to the segmentβ, γ. [1][i][1][i−1] Proof.—Deﬁne g= g , and by induction the submodule g =[g , g] α∈P [i][i] for i≥ 2; note that this depends on the choice of P.Deﬁne g= g∩ g .Notethateach [i] g is a graded submodule of g. Let us prove by induction on i≥ 1 the following statement: in the three cases, [j] [i][k] (9.B.3) g⊂ g , g α β γ j,k≥1, j+k=i β+γ=α... where in each case (β, γ ) satisﬁes the additional requirements of (1), (2), or (3) (we encode this in the notation ). β+γ=α... Since in all cases, α/∈ P,the case i= 1 is an empty (tautological) statement. Sup- [i] pose that i≥ 2 and the inclusions (9.B.3) are proved up to i− 1. Consider x∈ g .By [1] [i−1] deﬁnition, x is a sum of elements of the form[y, z] with y∈ g , z∈ g with β+ γ= α. β γ If β and γ are linearly independent over R, the additional conditions are satisﬁed and we are done. Otherwise, since β∈ P,wehave β = 0 and we can write γ= rβ for some r∈ R. There are three cases to consider: • r > 0. Then we are in Case (3), and 0∈ /β, γ, so the additional condition in (3) holds. • r= 0. Then β= α = 0 is a principal weight, which is consistent with none of Cases (1), (2) or (3). • r < 0. Then since β∈ P,wehave R γ∩ P =∅, so we can apply the induction [j] hypothesis of Case (2) to z; by linearity, this reduces to z =[u,v] with u∈ g , δ GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 175 [k] v∈ g , j+ k= i− 1, δ+ ε= rβ,and δ, ε not collinear to β.Notethat α= β+ δ+ ε. By the Jacobi identity, [j][k+1][j+1] [k] (9.B.4) x= y,[u,v]∈ g , g+ g , g . α−ε δ α−δ ε Note that none of δ, α− δ, ε, α− ε belongs to Rα.Hence we aredonein both Cases (2) or (3). In case (1), if ω∈ Rβ , then since δ and ε are not collinear to ω, we see that (9.B.4) satisﬁes the additional conditions of (1). However, the case ω/∈ Rβ is trivial since then the writing x =[y, z] itself already satisﬁes the additional condition of (1). At this point, (9.B.3) is proved. By Lemma 5.B.8,for all i≥ 1, we have g= [] i[] g . In particular g= g and we have ≥i α ≥i α j j [] k k g⊂ g , g⊂ g , g , α β γ β γ j+k= β+γ=α... j+k=i β+γ=α... so i k g⊂ g , g , α β γ j+k=i β+γ=α... k i and since the inclusion[g , g ]⊂ g is clear, we get the desired equality β γ α i k g= g , g . α γ j+k=i β+γ=α... 9.C. Multiamalgams of Lie algebras. 9.C.1. The deﬁnition. — By convex cone in a real vector space, we mean any subset stable under addition and positive scalar multiplication (such a subset is necessarily con- vex). Let C be the set of convex cones of W not containing 0. Let g be a real-graded Lie algebra over the ring R.If C∈ C,deﬁne g= g; C α α∈C this is a graded Lie subalgebra of g (if W is ﬁnite, g is nilpotent). Denote by x →¯ x the g C inclusion of g into g. Deﬁnition 9.C.1. —Deﬁne gˆ= gˆ as the multiamalgam (or colimit) of all g ,where C ranges over C . 176 YVES CORNULIER, ROMAIN TESSERA This is by deﬁnition an initial object in the category of Lie algebras h endowed with compatible homomorphisms g→ h. It can be realized as the quotient of the Lie R-algebra free product of all g ,bythe ideal generated by elements x− y,where x, y range over elements in g , g such that x¯ =¯ y C D and C, D range over C . Note that among those relators, we can restrict to homogeneous x, y as the other ones immediately follow. Since the free product as well as the ideal are ˆˆ graded, g is a graded Lie algebra; in particular, g is also the multiamalgam of the g in the category of Lie algebras graded in W . The inclusions g→ g induce a natural homomorphism gˆ→ g. 9.C.2. Link with the blow-up. — Let g be the blow-up introduced in Section 8.B.For every C∈ C , the structural homomorphism g˜→ g is an isomorphism, and therefore C C we obtain compatible homomorphisms g→ g˜⊂ g, inducing, by the universal property, C C a natural graded homomorphism gˆ→ g˜ . R R Theorem 9.C.2. —If g is 1-tame, then the natural Lie algebra homomorphism κ: gˆ→ g˜ is surjective, and if g is 2-tame, κ is an isomorphism. In particular, if g is 2-tame then the kernel of R R ˆˆ g→ g is central in g and is canonically isomorphic (as an R-module) to H (g) . To prove the second statement, we need the following lemma. Lemma 9.C.3 (Abels). — If g is 2-tame, then gˆ→ g has central kernel, concentrated in degree zero. Proof of Theorem 9.C.2 from Lemma 9.C.3.—If g is 1-tame, then so is g˜ by Lemma 8.B.7, and the surjectivity of κ follows, proving the ﬁrst assertion. If g is 2-tame, then by Lemma 9.C.3, gˆ→ g has central kernel concentrated in degree zero, so by the universal property of the blow-up (Theorem 8.B.4), there is a section s of the natural map κ . By uniqueness in the universal properties, s◦ κ and κ◦ s are both identity and we are done. The last statement is then an immediate consequence of Lemma 8.B.2. Proof of Lemma 9.C.3.—If α is a nonzero weight of g, then there exists C∈ C such that α∈ C. By the amalgamation relations, the composite graded homomorphism g→ g→ gˆ does not depend on the choice of C. We thus call it i and call its image m . α C α α Let us ﬁrst check that whenever α, β are non-zero non-opposite weights, then we have the following inclusion in gˆ (9.C.4)[m , m ]⊂ m . α β α+β We begin with the observation that if γ,δ are nonzero weights with δ/∈ R γ ,then (9.C.5) i[g , g] =[m , m]. γ+δ γ δ γ δ GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 177 Indeed, in the above deﬁnition of i, we can choose C to contain both γ and δ.Inpar- ticular if α/∈ R β,then(9.C.4) is clear. Let us prove (9.C.4) assuming that α∈ R β.By −− 2-tameness, we can suppose that R β∩ P =∅, so by Lemma 9.B.1(2), g⊂[g , g], + β γ δ where (γ, δ) ranges over the set Q(β) of pairs of weights such that γ+ δ= β and γ,δ are not in Rβ(= Rα). So, applying (9.C.5), we obtain m⊂[m , m]. Therefore β γ δ (γ,δ)∈Q(β) [m , m ]⊂ m ,[m , m] . α β α γ δ (γ,δ)∈Q(β) Note that γ and α+ δ are not collinear, since otherwise we would infer that α =−β . If we ﬁx such (γ, δ)∈ Q(β), we get, by the Jacobi identity and using that α, γ, δ are pairwise non-collinear [m ,[m , m]]⊂ m ,[m , m]+ m ,[m , m] α γ δ γ δ α δ α γ ⊂[m , m ]+[m , m] γ δ+α δ α+γ ⊂ m= m , γ+δ+α α+β and ﬁnally[m , m ]⊂ m .So(9.C.4)isproved. α β α+β Now deﬁne v as the submodule of gˆ (9.C.6) v=[m , m]. 0 γ−γ γ =0 We are going to check that for every α = 0 (9.C.7)[m , v ]⊂ m α 0 α and (9.C.8)[v , v ]⊂ v . 0 0 0 Before proving (9.C.7), let us ﬁrst check that for every nonzero α,if X (α) is the set of nonzero weights not collinear to α then (9.C.9)[m , m ]⊂[m , m]. α−α β−β β∈X (α) Indeed, by 2-tameness, we can suppose that R (−α)∩P =∅, and apply Lemma 9.B.1(2), so g⊂[g , g]; −α γ δ (γ,δ)∈Q(−α) 178 YVES CORNULIER, ROMAIN TESSERA by (9.C.5) we deduce m⊂[m , m]; −α γ δ (γ,δ)∈Q(−α) so [m , m ]⊂ m ,[m , m] . α−α α γ δ (γ,δ)∈Q(−α) If (γ, δ)∈ Q(−α), we have, by the Jacobi identity and then (9.C.4) [m ,[m , m]]⊂ m ,[m , m]+ m ,[m , m] α γ δ γ δ α δ α γ ⊂[m , m ]+[m , m] γ δ+α δ α+γ =[m , m ]+[m , m]; γ−γ δ−δ we thus deduce (9.C.9). We can now prove (9.C.7), namely if α, γ = 0, then[m ,[m , m]]⊂ m .By α γ−γ α (9.C.9) we can assume that α and γ are not collinear, in which case (9.C.7) follows from (9.C.4) by an immediate application of Jacobi’s identity. To prove (9.C.8), we need to prove that v is a subalgebra, or equivalently that for every α = 0, we have v ,[m , m]⊂ v . 0 α−α 0 Indeed, using the Jacobi identity and then (9.C.7) [v ,[m , m]]⊂ m ,[m , v]+ m ,[v , m] 0 α−α α−α 0−α 0 α ⊂[m , m ]+[m , m ]⊂ v . α−α−α α 0 so (9.C.8)isproved. We can now conclude the proof of the lemma. By the previous claims (9.C.4), (9.C.7), and (9.C.8), if v is deﬁned as in (9.C.6) the submodule ( m )⊕ v is a Lie 0 α 0 α =0 subalgebra of gˆ . Since the m for α = 0 generate gˆ by deﬁnition, this proves that this Lie subalgebra is all of gˆ . Therefore, if φ: gˆ→ g is the natural map, we see that φ is the natural isomor- phism m→ g .Thus Ker(φ) is contained in gˆ .If z∈ Ker(φ) and x∈ m for some α α 0 α α = 0, then φ[z, x]= φ(z), φ(x)= 0,φ(x)= 0, −1 so[z, x]= φ (φ([z, x]))= 0. Thus z centralizes m for all α; since these generate gˆ,we deduce that z is central. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 179 9.C.3. On the non-2-tame case. — There is a partial converse to Theorem 9.C.2:if R is a ﬁeld and g/[g, g] is not 2-tame, then gˆ has a surjective homomorphism onto a free Lie algebra on two generators. In particular, if g is nilpotent (as in all our applica- tions), g˜ is nilpotent as well but gˆ is not. The argument is straightforward: the assumption implies that there is a graded surjective homomorphism g→ h,where h is the abelian 2-dimensional algebras with weights α, β with β∈ R α, inducing a surjective homo- morphism gˆ→ h. Now it follows from the deﬁnition that h is thefreeproductofthe 1-dimensional Lie algebras h and h and hence is not nilpotent. α β 9.D. Multiamalgams of groups. — In this part, g is a nilpotent real-graded Lie algebra over a commutative Q-algebra R of characteristic zero (although the results would work in characteristic p > s+ 1, where s is the nilpotency length of the Lie algebra involved). For C∈ C ,let G and G be the groups associated to g and g by Malcev’s equiv- g C C alence of categories between nilpotent Lie algebras over Q and uniquely divisible nilpo- tent groups, described in Theorem 5.A.2. Let the embedding G→ G corresponding to g→ g be written as g →¯ g.Let G be the corresponding amalgam, namely the group −1 generated by the free product of G for C∈ C , modded out by the relators xy when- C g ever x¯ =¯ y. The following result was proved by Abels [Ab87, Cor. 4.4.14] assuming that R= Q and that g is ﬁnite-dimensional. This is one of the most delicate points in [Ab87]. We provide a sketch of the (highly technical) proof, in order to indicate how his proof works over our general hypotheses. Theorem 9.D.1 (Abels). — Suppose that g is 2-tame and s-nilpotent. Then G is (s+ 1)- nilpotent. For the purpose of Section 6, we need a stronger result. Abels asked [Ab87, 4.7.3] whether G→ G is always a central extension. This is answered in the positive by the following theorem. Theorem 9.D.2. — Under the same hypotheses, the nilpotent group G is uniquely divisible, and its Lie algebra is gˆ in the natural way. In particular, the homomorphism G→ G has a central kernel, naturally isomorphic to H (g) as a Q-linear space. Proof. — Our proof is based on Theorem 9.D.1 and some generalities about nilpo- tent groups, which are gathered in Section 5. Since by Theorem 9.D.1, G is nilpotent, and since it is generated by divisible sub- groups, it is divisible (see Lemma 5.A.3). Therefore by the (standard) Lemma 5.A.1,to ˆˆ check that G is uniquely divisible, it is enough to check that it is torsion-free. Since G is known to be (s+ 1)-nilpotent, we see that G is the multiamalgam (=colimit) of the G within the category K of (s+ 1)-nilpotent groups. Therefore, G is the quotient of the free C 180 YVES CORNULIER, ROMAIN TESSERA product W in K of the G by amalgamations relations. By Lemma 5.A.5, W is a uniquely −1 divisible torsion-free nilpotent group. Since, for x, y∈ W, whenever xy is an amalgama- r−r tion relation, x y is an amalgamation relation as well for all r∈ Q,Proposition 5.B.5 applies to show that the normal subgroup N generated by amalgamation relations is di- visible. Therefore G/N is torsion-free, hence uniquely divisible. It follows that G is also the multiamalgam of the G in the category K of uniquely C 0 divisible (s+ 1)-nilpotent groups. By Malcev’s Theorem 5.A.2, this category is equivalent to the category of (s+ 1)-nilpotent Lie algebras over Q. Therefore, if H is the group associated to gˆ and H→ G are the homomorphisms associated to g→ gˆ,then H and C C the family of homomorphisms G→ H satisfy the universal property of multiamalgam in the category K , and this gives rise to a canonical isomorphism G→ H. In particular, ˆˆ by Theorem 9.C.2, the kernel W of G→ Gis central in G, and isomorphic to H (g) . Corollary 9.D.3. — Under the same hypotheses, if moreover H (g) ={0} then the central kernel of G→ G is generated, as an abelian group, by elements of the form −1 exp[λx, y] exp[x,λy] ,λ∈ R, x, y∈ g . If R= R is a ﬁnite product of Q-algebras (so that g= g canonically), then those elements j j j=1 j −1 exp([λx, y]) exp([x,λy]) with λ∈ R and x, y∈ g are enough. j j Proof. — The additional assumption and Theorem 9.D.2 imply that the kernel of Q,R G→ G is naturally isomorphic, as a Q-linear space, to the kernel W (g) of the natural map H (g)→ H (g) . 0 0 2 2 Let N be the normal subgroup of G generated by elements of the form −1 exp[λx, y] exp[x,λy] , x, y∈ g ,λ∈ R; clearly N is contained in the kernel W of G→ G. By Proposition 5.B.5, N is divisible, i.e., N is a Q-linear subspace of W. Therefore G/N is uniquely divisible and its Lie algebra can be identiﬁed with gˆ/N. By deﬁnition of N, in G/Nwe haveexp([λx, y])= exp([x,λy]) for all x, y∈ g . So in the Lie algebra of G/N, which is equal to gˆ/N we have[λx, y]=[x,λy]. Since by Proposition 8.C.5, the subgroup W is generated by elements of the form exp([λx, y]−[x,λy]) when x, y range over G and λ ranges over R, it follows that N= W. It remains to prove the last statement. By Lemma 8.B.8,H (g) can be identiﬁed 2 0 with the product H (g ) .Inparticular, by Proposition 8.C.5, it is generated by ele- 2 j 0 ments of the form[λx, y]−[x,λy] when x, y∈ g , λ∈ R ,and j= 1,...,τ . Then, we j,C j can conclude by a straightforward adaptation of the above proof. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 181 R R Corollary 9.D.4. — Under the same hypotheses, if moreover H (g)= Kill (g) ={0}, 0 0 then G→ G is an isomorphism. Proof. — As observed in the proof of Corollary 9.D.3, since H (g)= 0, the kernel Q,R of G→ G is isomorphic to W (g) . Since Kill (g)= 0, Corollary 8.C.8 implies that 0 0 Q,R W (g)= 0. Proof of Theorem 9.D.1 (sketched). — We only sketch the proof; the proof in [Ab87] (with more restricted hypotheses) is 8 pages long. Here the hypotheses are more general since Abels works with p-adic groups. The theorem is stated here assuming that R is a Q- algebra, but more generally, it works under the assumption that R is a Z[1/s!]-module, and in particular when it is a vector space over a ﬁeld of characteristic p > s. Fix a 2-tame g-principal subset P.Let S denote the set of all half-lines R w >0 i i i (w = 0) in W . Consider the lower central series (G ).Let M be the image of G∩ G 1 i i ˆˆ in Gand M= M .Let A be the normal subgroup of G generated by all M ,where C C C ranges over S . Abels also introduces more complicated subgroups. Let L denote the set of lines of W (i.e. its projective space). Each line L∈ L contains exactly two half lines: L= S ∪{0}∪ S . 1 2 Deﬁne the following subgroups of G M= M =M∪ M [L] S S 1 2 [L] and, by induction % & i i i k M= M∪ M∪ M , M , [L] [L] S S[L] 1 2 j+k=i where ((·,·)) denote the subgroup generated by group commutators. Abels proves the following lemma [Ab87, 4.4.11]: if L∈ L and C is a open cone in W such that L+ C⊂ C, then j j+k (9.D.5) M , M⊂ M . C[L] C j+k The interest is that M is a complicated object, while the right-hand term M is a [L] reasonable one. Abels obtains [Ab87, Prop. 4.4.13] the following result, which can appear as a group version of Lemma 9.B.2(1): for any L∈ L and any i,the ith term of the lower i i i central series G is generated, as a subgroup and modulo A ,bythe M where L ranges [L] over L −{L}, or, in symbols, % & i i i (9.D.6) G= M A . [L] L =L 0 182 YVES CORNULIER, ROMAIN TESSERA It is important to mention here that all three items of Lemma 9.B.2 are needed in the proof of (9.D.6) (encapsulated in the proof of [Ab87, Prop. 4.4.7]). This is also the step where the Baker-Campbell-Hausdorff formula is used to convey properties from the Lie algebra g to G. To conclude the proof, suppose that G is s-nilpotent. We wish to prove that s+1 ˆˆˆ ((G, G )) ={1}. Since G is easily checked to be generated by M for S ranging over S , it is enough to check, for each S∈ S s+1 (9.D.7) M , G ={1}. s+1 s+1 Now since G is s-nilpotent, A ={1}, so we can forget “modulo A ”in(9.D.6) (with i= s+ 1), so (9.D.7) follows if we can prove s+1 M , M ={1} [L] for all L∈ L with maybe one exception L ; namely, we choose L to be the line generated 0 0 by S. Then S+ Lis an open cone, (9.D.5) applies and we have s+1 s+1 s+2 M , M⊂ M , M⊂ M ={1}. S S+L [L][L] S+L Remark 9.D.8. — As we observed, Theorem 9.D.1 works when R is only assumed to be a Z[1/s!]-module. It follows that Theorem 9.D.2 works when R is assumed to be a Z[1/(s+ 1)!]-module, and in particular when it is a vector space over a ﬁeld of characteristic p > s+ 1. 10. Presentations of standard solvable groups and their Dehn function 10.A. Outline of the section. — This section, which is the most important of the pa- per, contains the conclusions of the proofs of Theorems E.4 and F and relies on essentially all the preceding sections. Let G= U A be a standard solvable group. In Section 10.B we consider the multiamalgam G of (standard) tame subgroups (which are ﬁnitely many) along their in- tersections (regardless of any topology), as originally introduced by Abels. This group admits a canonical homomorphism onto G. Under the assumption that G is “2-tame” (which means that G does not satisfy the SOL obstruction, which is one of the hypotheses of Theorem E), we use the algebraic results of Section 9 to check in Section 10.D that the kernel of the canonical homomorphism G→ G is central in G. Moreover, assuming that G does not satisfy the 2-homological obstruction, the work of Section 10.B provides an explicit generating family of the central kernel, called welding relations. This provides us with a ﬁrst “algebraic” presentation of G, deﬁned as follows. Let U A,..., U A be the standard tame subgroups. We consider the presentation P 1 ν GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 183 whose set of generators is the disjoint union A U , i=1 and whose set of relators are the following • all tame relations, namely relations of length 3 deﬁning the group law in each of the U A; • all amalgamation relations of length 2, namely if some u∈ U∩ U , then in this i j disjoint union it corresponds to two generators u∈ U and u∈ U ,and the i i j j −1 relator is then u u ; • all “algebraic” welding relations, introduced in Section 10.B (where “algebraic” means that these are words with letters in the alphabet U ). Next we obtain a compact presentation P of G, informally by replacing elements of the various U with efﬁcient representative words, and restricting to generators and relators in a large enough compact subset. It is given by a set of generators S= T S , i=1 where T is in bijection with some compact generating subset of A and S is in bijection with some compact subset of U so that S∪ T generates U A. From Section 6.C,we i i i have a way to associate to each i and each u∈ U ,aword u in the generators S T. i i Replacing each u with u is a way to pass from any word in the previous generators to a word in these new generators, which we call its geometric realization. The reader should keep in mind that even when we consider algebraic words of bounded length, their geometric realization can be unbounded since the length of u with respect to S is unbounded when u varies. The relators of P are: • deﬁning relators of each U A of bounded length; −1 • amalgamation relations of length 2, of the form s s for s∈ S and s∈ S rep- j i i j j resenting the same element in U∩ U ; i j • geometric realizations of algebraic welding relators whose length is bounded by some suitable number. To obtain an upper bound of the Dehn function of P , we proceed in 3 steps (1) we ﬁrst bound the area of a special class of relations, namely those geometric realizations of those algebraic relators; namely for the geometric realization of tame and amalgamation relations we obtain a quadratic upper bound (with respect to the word length with respect to S), and for the geometric realization of welding relations we obtain in Section 10.F a cubic upper bound. 184 YVES CORNULIER, ROMAIN TESSERA (2) the general method developed in Section 7.B allows to use these estimates so as to bound a more general class of relations, the so-called “relations of bounded combinatorial length”, that is, when n is ﬁxed, we consider all geometric real- izations of algebraic relations of length≤ n , and for such relations we show an asymptotic cubic upper bound (the constant depending on n ). (3) We conclude thanks to Gromov’s trick, which ensures that it is enough to con- sider such relations of bounded combinatorial length, using that the efﬁcient words u indeed have a bounded combinatorial length. The conclusions of these steps are written down in Section 10.E in the absence of welding relators, thus proving Theorem F, and in Section 10.G otherwise, ending the proof of Theorem E.4. Let us pinpoint that Section 10.E also provides information in the case where there are welding relators, in order to get quadratic estimates on the area of relations that follow only from the other types of relators. This is crucial in Section 10.F, where the cubic estimate is obtained by using n times a homotopy of area n . 10.B. Abels’ multiamalgam. — Let G be any group and consider a family (G ) of subgroups. We deﬁne the multiamalgam (or colimit) to be an initial object in the cate- gory of groups H endowed with homomorphisms H→ Gand G→ H, such that all composite homomorphisms G→ H→ G are equal to the inclusion. Such an object is deﬁned up to unique isomorphism commuting with all homomorphisms. It can be explic- itly constructed as follows: consider the free product H of all G ,denoteby κ: G→ H i i i −1 the inclusion and mod out by the normal subgroup generated by the κ (x)κ (x) when- i j ever x∈ G∩ G .Ifthe family (G ) is stable under ﬁnite intersections, it is enough to mod i j i −1 out by the elements of the form κ (x)κ (x) whenever x∈ G and G⊂ G . Clearly, the i j i i j multiamalgam does not change if we replace the family (G ) by a larger family (G ) such that each G is contained in some G ; in particular it is no restriction to assume that the family is closed under intersections. Also, the image of G→ G is obviously equal to the subgroup generated by the G . Remark 10.B.1. — Our (and, originally, Abels’) motivation in introducing the mul- tiamalgam is to obtain a presentation of G. Therefore, in this point of view, the ideal case is when G→ G is an isomorphism. In many interesting cases, which will be studied in the sequel, G→ G is a central extension. On the other hand, if the G have pairwise trivial intersection, the multiamalgam of the G is merely the free product of the G , and this generally means that the kernel of i i G→ G is “large”. Example 10.B.2. — Consider a group presentation G =S| R in which every relator involves at most two generators. If we consider the family of subgroups gener- ated by 2 elements of S, the homomorphism G→ G is an isomorphism. Instances of GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 185 such presentations are presentations of free abelian groups, Coxeter presentations, Artin presentations. Lemma 10.B.3. — In general, let G be any group, (G ) any family of subgroups. Suppose that each G has a presentationS| R and that S∩ S generates G∩ G for all i, j. Then the i i i i j i j multiamalgam of all G has a presentation with generators S , relators R and, for all (i, j) the i i i relators of size two identifying an element of S and of S whenever they are actually equal (if (G ) is i j i closed under ﬁnite intersections, those (i, j) such that G⊂ G are enough). i j Proof. — This is formal. Following a fundamental idea of Abels, we introduce the following deﬁnition. Deﬁnition 10.B.4. —Let G be a locally compact group G with a semidirect product decompo- sition G= U A (as in Section 4.C). Consider the family (G= U A) of its tame subgroups. i i 1≤i≤ν Let U be the multiamalgam of the U along their intersections; it inherits a natural action of A,and we ˆˆ deﬁne G= U A. Remark 10.B.5. —Byaneasyveriﬁcation, G is the multiamalgam of the G 10.C. Algebraic and geometric presentations of the multiamalgam. — Let G= U Abe a standard solvable group and (G ) the family of its standard tame subgroups, G= i 1≤i≤ν i U A. Consider the free product H= U . There is a canonical surjective homomor- ∗i phism p: H→ U. Besides, there are canonical homomorphisms j: U→ H, so that p◦ j i i i is the inclusion U⊂ U. Recall that u is deﬁned as equal to u for some subset C(i) of the weight space, i C(i) stable under addition and not containing 0. Lemma 10.C.1 (Algebraic presentation of the multiamalgam). — The multiamalgam U is, −1 through the canonical map p, the quotient of H by the normal subgroup generated by pairs j (x)j (x) i i where (i, i ) ranges over pairs such that C(i)⊂ C(i ) and x ranges over U . Our goal is to translate this presentation into a compact presentation of G. Let ¯¯¯ S be disjoint copies of the S . Deﬁne the disjoint union S= S Tand F the free i i i S ¯¯ group over S; note that there is a canonical surjection S→ S= S T. Denote by ζ i i ¯¯ the canonical bijection S→ S . There is a canonical surjection ζ: S→ Sgiven as the i i identity on T and as ζ on S . i i There is a canonical surjective homomorphism π¯: F→ H A, S 186 YVES CORNULIER, ROMAIN TESSERA and by composition, p ◦¯ π is a canonical surjection F→ U. If ι is the inclusion of S into i i F ,then p ◦¯ π◦ ι is the inclusion of S into U. i i Let us introduce some important sets of elements in the kernel of the map F¯→ U. We ﬁx a presentation of A over T, including all commutation relators between generators. (See Section 2.B for basic conventions about the meaning of relations and relators.) ¯¯¯ • R= R ,where R consists of all elements in Ker(π)∩ F .We tame tame,i tame,i TS call these tame relations. 1 1 1 ¯¯¯¯ • R= R ,where R consists of all elements in R that are relators tame,i tame i tame,i tame,i in the presentation of G given in Corollary 6.A.7,aswellasthe relators in R i tame,i of length two; we call these tame relators. −1 • R consists of the elements of the form σ (w) σ (w),where w ranges over amalg i i F ,and where (i, i ) ranges over pairs such that C(i)⊂ C(i ),and σ: F→ ST i S∪T i i F is the unique homomorphism mapping every s∈ S to ζ (s) and every t∈ T i i to itself. We call these amalgamation relations; • R consists of those amalgamation relations for which w∈ S . These are amalg words of length at most two. We call these amalgamation relators. Note that the quotient of F by R is just the free group F .Wedeﬁne R and S S tame amalg 1 1 ¯¯ R as the images of R and R in F . tame tame tame S Proposition 10.C.2 (Geometric presentation of the multiamalgam). — The multiamalgam U is, 1 1 through the canonical map p ◦¯ π , the quotient of F¯ by the normal subgroup generated by R∪ R , tame amalg and is also the quotient of F by the normal subgroup generated by R . tame Proof. — The quotient of F by the (normal subgroup generated by) tame relations is obviously isomorphic to ( U ) A. Since by Corollary 6.A.7,for each i, R is ∗i i tame,i contained in the normal subgroup generated by R , it follows that ( U ) Ais also ∗i i tame,i the quotient of F by the tame relators. Denote by J the inclusion of U into U . By Lemma 10.B.3, the quotient of i i∗ i −1 U by the relations J (u) J (u) for (i, i ) ranging over pairs such that C(i)⊂ C(i ),is ∗ i i i ˆˆ the multiamalgam U. It follows that the quotient of ( U ) Aby the R is U, and ∗ i amalg since obviously R is normally generated by R , we deduce that the quotient of amalg amalg ( U ) Aby the R is also U. ∗i amalg 1 1 ˆ¯ Hence U is naturally the quotient of F¯ by R∪ R . The second assertion tame amalg follows. Proposition 10.C.3 (Quadratic ﬁlling of tame and amalgamation relations). — For the presen- tation 1 1 ¯¯ S| R∪ R tame amalg GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 187 of G, the tame relations have an at most quadratic area with respect to their length, and the amalgamation −1 relations σ (w) σ (w) have an area bounded above by the length of w. In the presentation i i S| R tame of G, the tame relations have an at most quadratic area with respect to their length. Proof. — The tame relations have an at most quadratic area by Corollary 6.A.7. −1 Let us consider an amalgamation relation. It has the form σ (w) σ (w) with w i i of length n; then with cost≤ n we can replace all letters ζ (s) for s∈ S with ζ (s);the i i i −1 resulting word is then equal to σ (w) σ (w)= 1. i i The second assertion immediately follows. Proposition 10.C.2 is a ﬁrst step towards a compact presentation of G. 10.D. Presentation of the group in the 2-tame case. — The following two theorems, which use the notion of 2-tameness introduced in Deﬁnition 4.C.1, are established in Section 9 (Theorem 9.D.2 and Corollary 9.D.3), relying on Section 8. In the follow- ing statements, we identify U with its image in U. Also, recall that the decomposition K= K induces a decomposition u= u ,and u= u . j (j) i i(j) j=1 j j Theorem 10.D.1 (Presentation of a 2-tame group, weak form). — Assume that G= U A is a 2-tame standard solvable group. Then the homomorphism G→ G has a central kernel. If moreover the degree zero component of the second homology group H (u) vanishes, the kernel of G→ G is 2 0 generated by elements of the form −1 exp[λx, y] exp[x,λy] , 1≤ j≤ d, x, y∈ u ,λ∈ K ,. i(j) j We pinpoint the striking fact that H (u) ={0} does not imply that G→ Gis an 2 0 isomorphism. This was pointed out by Abels [Ab87, 5.7.4], and relies on the fact that the space H (u) of 2-homology of u,where u is viewed as a (huge) Lie algebra over the rationals, can be larger than H (u) . On the other hand, the fact that G→ Ghas 2 0 a central kernel is a new result, even in Abels’ framework (u ﬁnite-dimensional nilpotent Lie algebra over Q ); Abels however proved that Uis (s+ 1)-nilpotent if U is s-nilpotent and this is a major step in the proof. Actually, in order to use the results of Section 7.B, we need a (stronger) stable form of Theorem 10.D.1, namely holding over an arbitrary commutative K-algebra. We can view U as the group of K-points of U,where U is an afﬁne algebraic group over the ring K= K . Thus, for every commutative K-algebra A, U(A) is the group associated to the nilpotent Lie Q-algebra u⊗ A. Similarly, U is deﬁned so that U (A)⊂ U(A) is K i i the exponential of u⊗ A,and we deﬁne U[A] as the corresponding multiamalgam of i K the U (A). i 188 YVES CORNULIER, ROMAIN TESSERA Theorem 10.D.2 (Presentation of a 2-tame group, strong (stable) form). — Assume that G= U A is a 2-tame standard solvable group. Then for every K-algebra A, the homomorphism U[A]→ U(A) has a central kernel, which remains central in U[A] A. If moreover the zero degree component of the second homology group H (u) vanishes, the kernel of U[A]→ U(A) is generated by elements 2 0 of the form −1 exp[λx, y] exp[x,λy] , x, y∈ U (A), λ∈ A , j= 1..., d. i(j) j Note that we could state the theorem without decomposing along the decomposi- tion K= K , but we really need this statement when we estimate the area of welding relations in Section 10.F. Note that Theorem 10.D.1 is equivalent to the case A= K of Theorem 10.D.2, ˆˆ given the trivial observation that the kernels of G→ Gand U→ U coincide. Theorem 10.D.1 involves the Lie algebra bracket; in order to translate it into a purely group-theoretic setting, we need to use Lazard’s formulas from Section 5.C. We can now restate the second statement of Theorem 10.D.1 with no reference to the Lie algebra in the conclusion (except taking real powers): Theorem 10.D.3. —Let G= U A be a 2-tame standard solvable group such that H (u) ={0}. Choose s so that U is s-nilpotent. Then the (central) kernel of G→ G is generated by 2 0 elements of the form −1 λ λ B x , y B x, y , x, y∈ U ,λ∈ K , j= 1..., d, s+1 s+1 i(j) j where x denotes exp(λ log(x)). Again we need a stable form of the latter result. Theorem 10.D.4. —Let G= U A be a 2-tame standard solvable group such that H (u) ={0}. Choose s so that U is s-nilpotent. Then for every commutative K-algebra A, denoting 2 0 A= A⊗ K (so that A= A canonically), the (central) kernel of U[A]→ U(A) is generated j K j j by elements of the form −1 λ λ B x , y B x, y , x, y∈ U (A), λ∈ A , j= 1..., d, s+1 s+1 i(j) j where x denotes exp(λ log(x)). We can view the elements −1 λ λ B x , y B x, y s+1 s+1 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 189 as elements of the free product H= U ;now x, y range over the disjoint union x, y∈ ∗i i U . We call these welding relations in the free product H. i(j) By substitution, Theorem 10.D.3 gives rises to the set of relations in F ( ) −1 λ λ (10.D.5)R= B x , y B x, y: x, y∈ U ,λ∈ K , j= 1..., d . weld s+1 s+1 i(j) j in the free group F . We call these welding relations. We deﬁne the set R of welding relators weld as those welding relations for which$x$ ,$y$ ,|λ|≤ 1, where $·$ is the prescribed Lie algebra norm. It follows from Corollary 8.C.9 that G has a presentation with relators those of G (given in Lemma 10.B.3) along with welding relators. At this point, this already reproves Abels’ result. Corollary 10.D.6 (Compact presentation of G). — Let G be a standard solvable group. Assume that G is 2-tame and H (u ) ={0}.Then: 2 na 0 (a) (Abels) G is compactly presented; (b) if H (u) ={0}, a compact presentation of G is given by 2 0 1 1 1 ¯¯ (10.D.7) S| R∪ R∪ R . tame amalg weld (c) if H (u) ={0} and Kill(u) ={0}, a compact presentation of G is given by 2 0 0 1 1 ¯¯ S| R∪ R . tame amalg Proof. — First assume that H (u) ={0}. The above remarks show that if π is the 2 0 ˆˆ natural projection F→ G, then π (R ) generates normally the kernel of G→ G. weld Therefore, by Proposition 10.C.2, G admits the compact presentation (10.D.7). This proves (b). To obtain (a), It remains to prove that G is compactly presented only assuming H (u ) ={0},but then G/G is compactly presented by the previous case, and it follows 2 na 0 that G is compactly presented. 1 1 Finally (c) is by combining Proposition 10.C.2, which says thatS| R∪ R tame amalg is a presentation of G (for an arbitrary standard solvable group), and Corollary 9.D.4, ˆˆ which says that the natural projection U→ U is an isomorphism (and hence G→ Gas well). Remark 10.D.8. — Let us pinpoint that this does not coincide, at this point, with Abels’ proof. Abels did not prove that the kernel of G→ G is generated by welding relators. Instead (assuming that G is totally disconnected), he considered the multiamal- ˙˙˙ gamated product Uof all U and a compact open subgroup of U, and G= U A, i 190 YVES CORNULIER, ROMAIN TESSERA ˆ˙ˆ˙ assuming that is generated by the intersections ∩ U ,sothat U→ Uand G→ G are surjective. Deﬁning S= U∩ and using S= S∪ T as a generating subset of G (recall i i i that T is a compact generating subset of A), we see that is boundedly generated by S ˙ˆ (for instance, by the Baire category theorem). It follows that G is the quotient of Gby a normal subgroup normally generated by generators of bounded length. Hence Gis boundedly presented by S. There is a natural projection G→ G. Since welding relators are killed by the amal- gamation with , we know that the natural projection G→ G is an isomorphism. Not having the presentation by welding relators, Abels used instead topological arguments [Ab87, 5.4, 5.6.1] to reach the conclusion that G→ G is indeed an isomorphism; thus G is compactly presented. This approach, with the use of a compact open subgroup is, however, “unstable”, in the sense that it does not yield a presentation of U(A) Awhen A is an arbitrary commutative K-algebra, and the presentation with welding relators will be needed in the sequel in a crucial way when we obtain an upper bound on the Dehn function. 10.E. Quadratic estimates and concluding step for standard solvable groups with zero Killing module. — Let us call the rank of A the unique d such that A admits a copy of Z as a cocompact lattice. Theorem 10.E.1. —Let G= U A be a standard solvable group. If G is 2-tame and H (u) and Kill(u) both vanish, then the Dehn function of G is quadratic (or linear in case A has 2 0 0 rank≤ 1). If A has rank 0 then G is compact; if A has rank 1, then 2-tame implies tame, and in that case, the Dehn function is linear (see Remark 6.A.9). Otherwise, if A has rank at least 2, the Dehn function is at least quadratic. So the issue is to obtain a quadratic upper bound. Fix c and a c-tuple (℘ ,...,℘ ) of elements in{1,...,ν}.Recallthat U (r)⊂ F¯ 1 c ℘ (r) denotes the set of null-homotopic words of the form w ...w ,with w∈ S .Theo- 1 c rem 6.C.1 reduces the proof of Theorem 10.E.1 to proving that, for each given ℘,words in U (r) have an at most quadratic area with respect to r. We now proceed to prove this. We also prove a statement holding for more speciﬁc relations but without the vanishing assumptions, which will be used in the sequel. Theorem 10.E.2. —Let G= U A be a 2-tame standard solvable group. Let U ,..., U 1 ν be its standard tame subgroups. Then (1) Assume that H (u) and Kill(u) both vanish. Fix an integer c and any c-tuple 2 0 0 (℘ ,...,℘ ) of elements in{1,...,ν}. Then for r≥ 0, the area of words in U (r)⊂ F 1 c ℘ S is quadratically bounded in terms of r. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 191 (s+1) (2) Let s be such that U is s-nilpotent (that is, U ={1}). Then for every group word (s+2) w(x ,..., x ) that belongs to the (s+ 2)-th term F of the lower central series of the 1 c free group F , and for any u ,..., u in U , the relation w(u ,..., u ) in G has an at c 1 c i 1 c most quadratic area with respect to its total length (the constant not depending on c). Proof.—Theorem 10.E.1 makes use of the results of Section 7.B. With the notation of Section 7.B,for any K-algebra A we have H[A]= U (A). ∗ i i=1 Let us encode the amalgamation relations. For any 1≤ i = j≤ ν , write U (A)= ij U (A)∩ U (A),and let s be the inclusion of U (A) into U (A). Deﬁne a closed subscheme i j ij ij i R of U by deﬁning ij i i=1 R (A)= (u ,..., u ): u= 1∀k ∈{ / i, j}, u , u∈ U (A), and u u= 1 . ij 1 ν k i j ij i j Deﬁne R[A]= R (A). Thus, taking the quotient of H[A] by π (R[A]) means amal- ij A i<j gamating the U along their intersections; we thus denote this quotient as U[A] (in Sec- 1 1 tion 7.B it was denoted as Q[A]). Deﬁne R= R∪ R . tame amalg Using this, we now prove (1) and (2) separately. We begin with (1). Since u is 2-tame and H (u) and Kill(u) vanish, for every 2 0 0 A K commutative K-algebra A,wehave H (u⊗ A)= H (u)⊗ A ={0},and similarly K 0 0 K 2 2 Kill (u⊗ A) ={0}. Thus by Corollary 9.D.4, U[A]→ U(A) is an isomorphism for K 0 every A. It follows that L (deﬁned in Section 7.B.3) is a closed subscheme of U ,sowe ℘ ℘ can apply Theorem 7.B.5 with M= L . It gives an upper bound of the areas of elements in M(K) in terms of upper bounds on the Dehn functions of the U and of the areas of the relations of the form u ... u with (u ,..., u )∈ R[K]. In this case, the U have at most 1 ν 1 ν i quadratic Dehn function by Corollary 6.A.8, and the amalgamation relations have an at most linear area by Proposition 10.C.3. Thus by Theorem 7.B.5,elementsof M(K) have an at most quadratic area with respect to their length and R. To prove of (2), we argue as follows. By Theorem 9.D.1, U[A] is (s+ 1)-nilpotent for every A; thus, deﬁning M= U ,wehave M (A)⊂ L (A) (with the notation of The- orem 7.B.7. By Proposition 10.C.3, the tame and amalgamation relations have an at most 1 1 ¯¯ quadratic area with respect to the presentationS| R∪ R. We are thus in position tame amalg to apply Theorem 7.B.7, which yields the desired result. 10.F. Area of welding relations. — We now bound the area of welding relations, mak- ing use of the quadratic bound on the area of general nilpotency relations from Sec- tion 10.E. Theorem 10.F.1. — If the standard solvable group G= U A is 2-tame, then welding relations in G have an at most cubic area. More precisely, there exists a constant K such that for all j , all x, y∈ U , and all λ∈ K , i(j) j the welding relation (10.D.5) has area at most Kn , with n= log(1 +$x$+$y$+|λ|). 192 YVES CORNULIER, ROMAIN TESSERA Let us assume that the subgroup U, in Theorem 10.F.1,is s-nilpotent. In all this subsection, we write A= A ,B= B , q= q .Theorem 10.E.2(2) applies to the s+1 s+1 s+1 group words corresponding to equalities of Proposition 5.C.3, providing Theorem 10.F.2. —If G is 2-tame and U is s-nilpotent, for all x∈ U ,y∈ U and C C 1 2 z∈ U , the relations −1 A(x, y)= A(y, x); B(x, y)= B(y, x) B A(x, y), z= A B(x, z), B(y, z); q q A A(x, y), z= A x , A(y, z) k k k B x , y= B x, y= B(x, y) 1 1 ˆ¯¯ have an at most quadratic area in G for the presentationS| R∪ R. tame amalg −1 (By “the relation w= w has an at most quadratic area, we mean the relation w w has an at most quadratic area.) Note that in the last case, the quadratic upper bound is of the form c n ,where c may depend on k. k k Although we only use it as a lemma in the course of proving Theorem 10.F.1,we state Theorem 10.F.2 as a theorem, because it provides a geometric information which is not encoded in the Dehn function, namely a quadratic area for certain types of loops. We now proceed to prove Theorem 10.F.1. Using (with quadratic cost) the “bilin- earity” of B (or restricting scalars from the beginning), we can suppose that K is equal to R or Q (although the forthcoming argument can be adapted with unessential modiﬁ- cations to their ﬁnite extensions). If K= Q ,deﬁne π= p.If K= R,deﬁne π= 2. We j p j j j consider the following ﬁnite subset of Q: ( ) n−1 1 i (n)= λ= ε π: ε ∈{−π+ 1,...,π− 1}⊂ Q⊂ K . i i j j j j j i=0 Lemma 10.F.3. —For every κ> 0, there exists K > 0 such that for all n∈ N, all j , all x, y∈ U with log(1 +$x$+$y$)≤ n and all λ∈ (κ n), the welding relation i(j) −1 λ λ B x , y B x, y has area≤ Kn . Proof. — We can work for a given j , so we write π= π .Write κ n−1 λ= ε π ε ∈{−π+ 1,...,π− 1} . i i i=0 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 193 If we set κ n−1 k−i λ= ε π , i k k=i we have λ= λ, λ= 0, and, for all i 0 κ n λ= πλ+ ε i i+1 i −1 i Set z= q y and σ= ε π (so σ= 0); consider the word i i−1 j=1 = A B λ x, π z , B(x, σ z) i i i−1 Here, for readability, we write λx (etc.) instead of x . This is natural since x can be identiﬁed to its Lie algebra logarithm. For i= 0, σ= 0 so, since 1= 1 and, formally, i−1 B(x, 1)= 1and A(x, 1)= x (see Remark 5.C.2), we have = B(λx, z) . For i= κ n, λ= 0and σ= λ so this is (using that formally B(1, y)= 1and n i−1 A(1, y)= y ) = B(x, λz) . κ n Let us show that we can pass from to with quadratic cost. In the following i i+1 computation, each means one operation with quadratic cost, i.e., with cost≤ K n for some constant K only depending on the group presentation. The tag on the right explains why this quadratic operation is valid, namely: • (1) means both the homotopy between loops lies in one tame subgroup. • (2) means the operation follows from Theorem 10.F.2;tobespeciﬁc: – (2) for distr left B A(x, y), z= A B(x, z), B(y, z) and similarly (2) on the right distr right – (2) for an equality of the type k k k B x, y= B x , y= B(x, y) , with k an integer satisfying|k|≤ max(q,π). q q – (2) for the identity A(A(x, y), z )= A(x , A(y, z)). assoc • Or a tag referring to a previous computation, written in brackets. 194 YVES CORNULIER, ROMAIN TESSERA −1−1 (10.F.4) λ x= q πλ q x+ ε q x i i+1 i −1−1 πλ q x+ ε q x (1) i+1 i −1−1 πλ q x+ ε q x (1) i+1 i −1−1 A πλ q x, ε q x (1) i+1 i So by substitution we obtain i−1−1 i (10.F.5) B λ x, π z B A πλ q x, ε q x , π z[10.F.4] i i+1 i −1 i−1 i A B πλ q x, π z , B ε q x, π z (2) i+1 i distr left −1 i−1 i A B πλ q x, π z , B q x, ε π z (2) i+1 i Q Independently we have −1 (10.F.6) B(x, σ z) B q x , σ z (1) i−1 i−1 −1 B q x, σ z (2) i−1 Q Again by substitution, this yields (10.F.7) = A B λ x, π z , B(x, σ z) i i i−1 −1 i−1 i−1 A A B πλ q x, π z , B q x, ε π z , B q x, σ z i+1 i i−1 [10.F.5,10.F.6] −1 i−1 i−1 A B πλ q x, π z , A B q x, ε π z , B q x, σ z i+1 i i−1 (2) assoc −1 i−1 i (10.F.8) A B πλ q x, π z , B q x, A ε π z, σ z i+1 i i−1 (2) distr right By similar arguments (2) q Q −1 i i+1 B πλ q x, π z B λ x, π z , i+1 i+1 and −1 i−1 i B q x, A ε π z, σ z B q x, q ε π z+ σ z (1) i i−1 i i−1 B x, ε π z+ σ z (2) i i−1 Q = B(x, σ z) i GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 195 so substituting from (10.F.8)weget i+1 A B λ x, π z , B(x, σ z)= i i+1 i i+1 in quadratic cost, say≤ K n (noting that each has length≤ K n forsomeﬁxedcon- 1 i 2 stant K ). Note that the constant K only depends on the group presentation, because 2 1 the above estimates use a quadratic ﬁlling only ﬁnitely many times, each among ﬁnitely many types (note that we used (2) only for k in a bounded interval, only depending on K and s). It follows that we can pass from to with cost≤ K κ n .Onthe other 0 κ n 1 hand, by substitution of type (2) , we can pass with quadratic cost from B(λx, y) to −1 q−1 q B(λx, q y)= and from = B(x, λq y) to B(x, λy). So the proof of the lemma is 0 κ n complete. Conclusion of the proof of Theorem 10.F.1.—Letnow (n) be the set of quotients λ /λ with λ ,λ∈ (n), λ = 0. Lemma 10.F.3 immediately extends to the case when 2 2 λ∈ . It follows from the deﬁnition that (κ n) contains all elements of the form λ= j j κ n ε π ,with ε ∈{−π ,...,π}. i i j j i=−κ n j 2−2κ n 2κ n 2 κ n If K= R, π (κ n) contains all integers between−π and π .Thus (κ n) j j j j j −κ n κ n κ n 2 contains a set which is|π|-dense in the ball of radius|π|.If K= Q , π (κ n) j p 2κ n κ n contains a|π|-dense subset of Z .Thus (κ n) contains a|π|-dense subset of the −κ n−1 2 ball of radius|π|. In both cases, deﬁning = max(|π|,|π| ),weobtainthat (κ n) j j j −κ n κ n contains a -dense subset of the ball of radius in K . j j κ n 2n We now ﬁx j and write = , π= π .Wepick κ= 2/ log(),sothat = e . j j We assume that n≥ log(1/|q|),where|q| is the norm of q in K (if K= R this is an empty j j −κ n−1 n condition). It follows that |q| e≤ 1. Now ﬁx λ∈ K with|λ|≤ e . We need to prove that we can pass from B(λx, y) to q q B(x, λy) with cubic cost; clearly it is enough to pass from B(λx, y) to B(x, λy) with cubic cost. n κ n 2−κ n Since|λ|≤ e≤ , we can write λ= μ+ qε with μ∈ (κ n) and|qε|≤ . −κ n−1−n So|ε|≤ |q|≤ e . Also, assume that n≥ log(). So we can ﬁnd an integer k with n/ log()≤ k≤ ±k 2n/ log(). Thus, if we deﬁne η= π , with the choice of sign so that|η| > 1; we have n 2n n−1 e ≤|η|≤ e≤ e|ε| . Using a computation as in (10.F.4), we obtain, with quadratic cost B(λx, y)= B (μ+ qε)x, y −1 B A μq x, εx , y [10.F.4] −1 A B μq x, y , B(εx, y) (2) distr left 196 YVES CORNULIER, ROMAIN TESSERA and similarly, with quadratic cost. −1 B(x, λy) A B x, μq y , B(x, εy) By thepreviouscase, with cubiccostwehave −1−1 B μq x, y B x, μq y So it remains to check that with cubic cost we have (10.F.9)B(εx, y) B(x, εy). 2 n If η is the element introduced above, observe that η∈ (κ n) and e ≤|η|≤ n−1 e|ε| . We have, with cubic cost −1−1 (10.F.10)B(εx, y) B ηεx, η y; B η x, ηεy B(x, εy). −1−n−1−1 Since max(|ηε|,|η| )≤ e , it follows that all four elements ηεx, η y, η x, εy have norm at most one, and it follows that we can perform −1−1 (10.F.11)B ηεx, η y B η x, ηεy by application of a single welding relator. So (10.F.9) follows from (10.F.10)and (10.F.11). 10.G. Concluding step for standard solvable groups. Theorem 10.G.1. —Let G be a standard solvable group. If G is 2-tame and H (u)= 0 2 0 then its Dehn function is at most cubic. Exactly as in the proof of Theorem 10.E.1,Theorem 6.C.1, along with the cubic bound on the area of welding relators provided by Theorem 10.F.1, allows to reduce the proof of Theorem 10.G.1 to the following: Theorem 10.G.2. —Let G= U A be a 2-tame standard solvable group with H (u)= 0; 2 0 let U ,..., U be its standard tame subgroups; ﬁx a positive function n → g(n) such that n → g(n)/n 1 ν is eventually non-decreasing. Assume that the welding relations in G of length n have an area g(n). Fix c and a c-tuple (℘ ,...,℘ ) of elements in{1,...,ν}. Then words in U (n) have an 1 c ℘ 1 1 1 ¯¯ area, for the presentationS| R∪ R∪ R, asymptotically bounded above by g(n). tame amalg weld (We state the theorem with an arbitrary function g(n), because we do not know if the cubic bound is optimal; this might depend on G even assuming Kill(u) = 0.) Proof.—Theproof follows thesamelinesasTheorem 10.E.2(1); let us highlight the differences. We need to include welding relations in the deﬁnition of R: namely, deﬁning GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 197 R as in the proof of Theorem 10.E.2, we will deﬁne R[A]= R[A]∪ R (A),where ij weld ij R is deﬁned as follows. weld Let s be such that U is s-nilpotent. Deﬁne w∈ F as equal to w(x, x , y, y )= −1 B (x, y)B (x , y ) ,where B is a word as given by Theorem 5.C.1. s+1 s+1 s+1 We wish to deﬁne (recall that K= K ) j=1 R[A]= W (A), weld ii (j) 1≤i,i≤ν,1≤j≤d and we have to deﬁne W ⊂ U . ii (j) i =1 Deﬁne, for V, V unipotent K-groups, the following subscheme M= M of V× V,V V× V× V deﬁned by M (A)= v ,v ,v ,v∈ V× V× V× V (A): 1 2 1 2 ∃λ∈ A: v= v ,v= v . 2 2 1 Here v means exp(λ log(v)). That it is a subscheme can be seen on the Lie algebra, where it corresponds to the 4-tuples (v ,v ,v ,v ) such that there exists λ such that 1 2 1 2 v= λv and v= λv . Then writing in coordinates, this is equivalent to the condition 1 2 2 1 that • v v − v v = v v− v v= 0for all i, i , 2i 1i 1i 2i 2i 1i 1i 2i • v v= v v for all i; 2i 1i 2i 1i hence it is indeed a subscheme. Using the notation of Section 7.B.6,wehave −1 λ λ w• M (A)= B x , y B x, y| x∈ V(A), y∈ V (A), λ∈ K . s+1 s+1 V,V Hence if we deﬁne W = w• M ,wehave. ii (j) U ,U i(j) i (j) −1 λ λ W (A)= B x , y B x, y| x∈ U (A), y∈ U (A), λ∈ K . ii (j) s+1 s+1 i(j) i (j) j Now the deﬁnition of R is complete, and by Theorem 10.D.4, the quotient of H[A] by the normal subgroup generated by R[A] is canonically U(A), for every commutative K-algebra A. Hence we can, as in the proof of Theorem 10.E.2(1), deﬁne M as equal to L . Fix c and a c-tuple ℘ of elements in{1,...,ν}.ByTheorem 10.D.4, denoting 1 1 1 R= R∪ R∪ R ,wehave R⊂ R[K];byTheorem 10.D.6(b), the kernel of the tame amalg weld quotient map from F to G is generated as a normal subgroup by R. By assumption (and Proposition 10.C.3), the words deﬁned by the relations in R[K] have area g(r) with respect to their length r. Hence by Theorem 7.B.5,the area of words x ... x with 1 c x∈ U is g(r),where r= max|x|. i ℘ i i Since this holds for every given ℘,byTheorem 6.C.1 (which encapsulates Gro- mov’s trick), we deduce that the Dehn function of G is g(n). 198 YVES CORNULIER, ROMAIN TESSERA 10.H. Dehn function of generalized standard solvable groups. — We deﬁne a generalized standard solvable group as a locally compact group of the form U N, where the deﬁni- tion is exactly as for standard solvable groups (Deﬁnition 1.2), except that N is supposed to be nilpotent instead of abelian. Recall from Section 6.E that such a group is generalized tame if some element c of N acts on U as a compaction. Clearly split triangulable Lie groups, i.e. of the form G= G N, are special cases of generalized standard solvable groups. Theorem 10.H.1. —Let G be a generalized standard solvable group not satisfying any of the (SOL or 2-homological) obstructions. Suppose that the Dehn function of N is bounded above by some function f such that r → f (r)/r is non-decreasing for some α> 1. Then the Dehn function of G satisﬁes δ (n) nf (n).Ifmoreover Kill(u) ={0}, then G 0 δ (n) f (n). Note that in all examples we are aware of, the function f can be chosen to be equivalent to the Dehn function of N. The proof follows similar steps as in the case of standard solvable groups, so we only sketch it. The ﬁrst step is an upper bound for the Dehn function of generalized tame groups, which was obtained in Section 6.E. Sketch of proof of Theorem 10.H.1.—Theorem 6.E.2 implies that the generalized standard tame subgroups of G (i.e., the U N, where U are the standard tame subgroups i i of U) have their Dehn function f (n). The remainder of the proof follows the same steps as Theorems 10.E.1 and 10.G.1, allowing a reduction to giving estimates on the area of amalgamation and welding relations of some given combinatorial length. The amalgamation relations have a linearly bounded area with respect to their length, by the same trivial argument as in Proposition 10.C.3.Incase Kill(u) ={0}, this provides δ 0 N as an upper bound for the Dehn function. In the context of generalized standard groups, the analogue of Theorem 10.E.2 holds (with the same proof using Gromov’s trick and the results of Section 7.B), with an estimate f (n) instead of n . Accordingly, the same holds for its corollary, The- orem 10.F.2. The remainder of the proof of the generalized tame analogue of Theo- rem 10.F.1 works in the same way: we perform n times a homotopy of area f (n) instead of n , which yields an area nf (n) for welding relators. The concluding step is exactly as in Section 10.G and yields a Dehn function nf (n). 11. Central and hypercentral extensions and exponential Dehn function In this section, unless explicitly speciﬁed, all Lie algebras are ﬁnite-dimensional over a ﬁeld K of characteristic zero. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 199 11.A. Introduction of the section. — The purpose of this section is to prove the neg- ative statements of the introduction in presence of 2-homological obstructions (Theo- rem E.2), gathered in the following theorem. Theorem 11.A.1. (1) If G is a standard solvable group (see Deﬁnition 1.2) satisfying the non-Archimedean 2-homological obstruction, then G is not compactly presented; (2) if G is a standard solvable group satisfying the 2-homological obstruction, then it has an least exponential Dehn function (possibly inﬁnite); (3) if G is a real triangulable group with the 2-homological obstruction, then it has an at least exponential Dehn function. (1) and (2) are proved, by an elementary argument relying on central extensions, in Section 11.B. (3) is much more involved. The reason is that the exponential radical g is not necessarily split in g and the non-vanishing of H (g) does not necessarily yield a 2 0 central extension of g in degree zero (an explicit counterexample is given in Section 11.E). We then need some signiﬁcant amount of work to show that it provides, anyway, a hyper- central extension. 11.B. FC-Central extensions. — We use the following classical deﬁnition, which is a slight weakening of the notion of central extension. Deﬁnition 11.B.1. — Consider an extension (11.B.2)1→ Z→ G→ G→ 1. We say that it is an FC-central extension if i(Z) is FC-central in G, in the sense that every compact subset of i(Z) is contained in a compact subset of G that is invariant under conjugation. This widely used terminology is (lamely) borrowed from the discrete case, in which it stands for “Finite Conjugacy (class)”. In the following, the reader can assume, in a ﬁrst reading, that the FC-central extensions (deﬁned below) are central. The greater generality allows to consider, for in- stance, the case when Z is a nondiscrete locally compact ﬁeld on which the action by conjugation is given by multiplication by elements of modulus 1. Consider now an FC-central extension as in (11.B.2), and assume that G generated −1 by a compact subset S (symmetric with 1); Fix k, set W= Z∩ S ,and W= gW g , k k * k g∈G which is a compact subset of Z by assumption. The following easy lemma, is partly a restatement of [BaMS93, Lemma 5] (which deals with ﬁnitely generated groups and assumes Z is central). 200 YVES CORNULIER, ROMAIN TESSERA Lemma 11.B.3. — Keep the notation of Deﬁnition 11.B.1.Let * γ be any path in the Cayley graph of G with respect to S, joining 1 to an element z of Z.Let γ be the image of * γ in the Cayley graph of G (with respect to the image of S). If * γ can be ﬁlled by m (≤ k)-gons, then z∈ W . Proof.—If * γ can be ﬁlled by m (≤ k)-gons, then z can be written (in the free group −1 ˜ * * over S, hence in G) as z= h r h ,where r , h∈ G, r∈ Ker(G→ G)= Z having i i i i i i=1 −1 ˇˇ length≤ k with respect to S, i.e. r∈ W .Thus h r h∈ W ;hence z∈ W . i k i i k k In Deﬁnition 11.B.1, the group Z may or not be compactly generated; if it is the case, let U a compact generating set of Z, and deﬁne the distortion of Z in G as d (n)= max n, sup{|g|: g∈ Z,|g|≤ n}; G,Z U S if Z is not compactly generated set d (n) =+∞. Note that this function actually de- G,Z pends on S and U as well, but its∼-equivalence class only depends on (G, Z). Proposition 11.B.4. — Given a FC-central extension as in Deﬁnition 11.B.1,if G is compactly presented, then Z is compactly generated and its Dehn function satisﬁes δ (n)0 d (n). G G,Z Proof. — By Lemma 11.B.3, if G is presented by S and relators of length≤ k,then Z is generated by W , which is compact. If U is a compact generating set for Z, then W⊂ U for some .Write d(n)= d (n) (relative to S and U) and δ(n)= δ (n).Consider g∈ Zwith|g|≤ n and|g|= G,Z G S U d(n). Taking * γ to be a path of length≤ n in G joining 1 and g as in Lemma 11.B.3,we obtain that the loop γ in G has length≤ n and area m, and Lemma 11.B.3 implies that d(n) =|g|≤ m.So δ(n)≥ d(n)/. Proof of (1) and (2) in Theorem 11.A.1. — In the setting of (1), we assume that G= U A satisﬁes the non-Archimedean 2-homological obstruction, so that for some j , K is non-Archimedean and the condition Z= H (u ) = 0 means that the action of A on U 2 j 0 j can be lifted to an action on a certain FC-central extension 1→ Z→ U→ U→ 1, j j ˜˜ so that i(Z) is contained and FC-central in[U , U];here u is the blow-up of the graded j j Lie algebra u,asdeﬁnedinSection 8.B,and Z H (u ). Thus it yields an FC-central 2 j extension (11.B.5)1→ Z→ U A→ U A→ 1, ˜˜˜ where U= U× U ;notethat U A is compactly generated. By Proposition 11.B.4, j j j =j it follows that U A is not compactly presented. In the setting of (2), the proof is similar, with K being Archimedean; there is a dif- ference however: in (11.B.5), Z need not be FC-central in U A, because of the possible GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 201 real unipotent part of the A-action. Note that i(Z) is central in U. We then consider an A-irreducible quotient Z= Z/Z of Z and consider the FC-central extension 1→ Z→ U A→ U A→ 1. Since the real group U is globally exponentially distorted (in the sense that all elements of exponential size in U have linear size in U A), it follows that i(Z ) is exponentially dis- j j torted as well and by Proposition 11.B.4, it follows that U A has an at least exponential Dehn function. 11.C. Hypercentral extensions. — To prove Theorem 11.A.1(3), the natural approach seems to start with a real triangulable group G with H (g ) = 0 and ﬁnd a central 2 0 extension of G with exponentially distorted center. If the exponential radical of G is split, i.e. if G= G A for some nilpotent group A, the existence of such a central extension follows by a simple argument similar to that in the proof of Theorem 11.A.1(2). Unfortunately, in general such a central extension does not exist; although there are no simple counterexamples, we construct one in Section 11.E. Nevertheless, in order to prove Theorem 11.A.1(3), the geometric part of the ar- gument is the following variant of Proposition 11.B.4. Proposition 11.C.1. —Let G be a connected triangulable Lie group and G its exponential radical (see Deﬁnition 4.D.2). Suppose that there exists an extension of connected triangulable Lie groups 1 −→ N −→ H −→ G −→ 1 with N hypercentral in H (i.e. the ascending central series of H covers N) and with N∩ H ={1}. Then G has an (at least) exponential Dehn function. Proof. — We ﬁx a compact generating set S in H, and its image S in G. Let h∈ N be an element of linear size n in H and exponential size e in N. Pick a path of size n joining 1 to h in H, i.e. represent h by a word γ= x ... x in H with x∈ S. Push this n 1 n i path forward to G to get a loop of size n in G; let a be itsarea. So in thefreegroup over S, we have −1 γ= g r g n nj nj nj j=1 where r is a relation of G (i.e. represents the identity in G) of bounded size. By a standard argument using van Kampen diagrams (see Lemma 2.D.2), we can choose the size of g nj to be at most≤ C(a+ n), where C is a positive constant only depending on (G, S ).Push this forward to H to get −1 h= g r g , n nj nj nj j=1 202 YVES CORNULIER, ROMAIN TESSERA where r here is a bounded element of N, and g has length≤ C(a+ n) in H. Since the nj nj n −1 action of H on N by conjugation is unipotent, we deduce that the size in N of g r g is nj nj nj polynomially bounded with respect to a+ n,say (a+ n) (uniformly in j ). Therefore n n d+1 h has size (a+ n) . Since (h ) has exponential growth in N, we deduce that (a ) also n n n n grows exponentially. Let us emphasize that at this point, the proof of Theorem 11.A.1(3) is not yet complete. Indeed, given a real triangulable group G with H (g ) = 0, we need to check 2 0 that we can apply Proposition 11.C.1. This is the contents of the following theorem. If V is a G-module, V denotes the set of G-ﬁxed points. Also, see Deﬁnition 4.D.1 for the deﬁnition of g . ∞ G Theorem 11.C.2. —Let G be a triangulable Lie group. Suppose that H (g ) ={0}.Then there exists an extension of connected triangulable Lie groups 1 −→ N −→ H −→ G −→ 1 with N hypercentral in H and with N∩ H ={1}. The theorem will easily follow from the analogous (more general) result about solvable Lie algebras. By epimorphism of Lie algebras we mean a surjective homomorphism, and we de- note it by a two-headed arrow h g.Suchanepimorphismis k-hypercentral if its kernel z is contained in the kth term h of the ascending central series of h;when k= 1, that is, when z is central in h, it is simply called a central epimorphism. Also, if m is a g-module, we write m ={m∈ m :∀g∈ g, gm= 0}. Deﬁnition 11.C.3. — We say that a hypercentral epimorphism h g between Lie algebras ∞∞ has polynomial distortion if the induced epimorphism h g is bijective; otherwise we say it has non-polynomial distortion. The terminology is motivated by the fact that if G, H are triangulable Lie groups and H G is a hypercentral epimorphism, then its kernel is polynomially distorted in H if and only if the corresponding Lie algebra hypercentral epimorphism has polynomial distortion (in the above sense), and otherwise the kernel is exponentially distorted in H. ∞ g Theorem 11.C.4. —Let g be a K-triangulable Lie algebra. Assume that H (g ) ={0}. Then there exists a hypercentral epimorphism h g with non-polynomial distortion. ∞ g∞ The condition H (g ) ={0} can be interpreted as H (g ) ={0},where g is endowed 2 2 0 with a Cartan grading (see Section 4.D, especially Lemma 4.D.8). Theorem 11.C.4 will be proved in Section 11.D. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 203 Proof of Theorem 11.C.2 from Theorem 11.C.4.—Theorem 11.C.4 provides a hyper- central epimorphism h g, with kernel denoted by z, and with non-polynomial distor- tion, i.e. h∩ z ={0}. Since z is hypercentral, the action of g on z is nilpotent, hence triangulable, and since moreover g and z are triangulable, we deduce that h is triangu- lable. Let H→ G be the corresponding surjective homomorphism of triangulable Lie groups and Z its kernel, which is hypercentral. Then H∩ Z ={1}, because the Lie algebra counterpart holds. So the theorem is proved. 11.D. Proof of Theorem 11.C.4. Lemma 11.D.1. —Let g h l be epimorphisms of Lie algebras such that the composite homomorphism is a hypercentral epimorphism. If g l has polynomial distortion then so does h l. Proof. — This is trivial. Lemma 11.D.2. —Let g be a Lie algebra and (T ) be a one-parameter group of unipotent t∈K automorphisms of g.Let h g be a central epimorphism. Then there exists a Lie algebra k with an epimorphism ρ: k h such that the composite homomorphism k g is a central epimorphism, and such that the action of (T ) lifts to a unipotent action on k. Remark 11.D.3. — The conclusion of Lemma 11.D.2 cannot be simpliﬁed by the requirement that k= h, as we can see, for instance, by taking h to be the direct product of the 3-dimensional Heisenberg algebra and a 1-dimensional algebra and a suitable 1- parameter subgroup of unipotent automorphisms of g= h/[h, h]. Proof of Lemma 11.D.2. — Set z= Ker(h g) and denote the Hopf bracket (see Section 8.A.1)by [·,·]: g∧ g→ h. Pick a linear projection π: h→ z and deﬁne b(x∧ y)= π([x, y] ). This gives a linear identiﬁcation of h with g⊕ z, for which the law is given as x , z,x , z=[x , x], b(x∧ x ) 1 1 2 2 1 2 1 2 (in this proof, we write pairsx, z rather than (x, z) for the sake of readability). Observe that T naturally acts on g∧g, preserving Z (g) and B (g).If c∈ g∧g, deﬁne the function 2 2 α: K→ z u → b T c . If d is the dimension of g, let W denote the space of K-polynomial mappings of degree < 2d from K to z; the dimension of W is 2d dim(z).Now t → T is a polynomial 204 YVES CORNULIER, ROMAIN TESSERA of degree < d valued in the space of endomorphisms of g,soisalsopolynomialofdegree < 2d valued in the space of endomorphisms of g∧ g.So α is a polynomial of degree < 2d,from K to z. If c∈ g∧ g is a boundary then α= 0. So α deﬁnes a central extension k= g⊕ W (as a vector space) of g with kernel W, with law x ,ζ,x ,ζ=[x , x],α ,x ,ζ,x ,ζ ∈ g⊕ W. 1 1 2 2 1 2 x∧x 1 1 2 2 1 2 From now on, since elements of W are functions, it will be convenient to write elements of k asx,ζ(u),where u is thought of as an indeterminate. For t∈ K, the automorphism T lifts to an automorphism of k given by t t T x,ζ(u)= T x,ζ(u+ t) . This is obviously a one-parameter subgroup of linear automorphisms; let us check that these are Lie algebra automorphisms (in the computation, for readability we write the brackets as[·;·], with semicolons instead of commas). t t t t T x ,ζ (u); T x ,ζ (u)= T x ,ζ (u+ t); T x ,ζ (u+ t) 1 1 2 2 1 1 2 2 t t t t = T x; T x ,α (u) 1 2 T x∧T x 1 2 t t u t t = T x; T x , b T T x∧ T x 1 2 1 2 t t+u t+u = T[x; x], b T x∧ T x 1 2 1 2 = T[x; x],α (t+ u) 1 2 x∧x 1 2 = T[x; x],α (u) 1 2 x∧x 1 2 = T x ,ζ (u); x ,ζ (u) , 1 1 2 2 so these are Lie algebra automorphisms. Now the mapping k→ h ρ: x,ζ(u) → x,ζ(0) is clearly a surjective linear map; it is also a Lie algebra homomorphism: indeed ρ x ,ζ (u); ρ x ,ζ (u)= x ,ζ (0); x ,ζ (0) 1 1 2 2 1 1 2 2 =[x; x], b(x∧ x ) 1 2 1 2 =[x; x],α (0) 1 2 x∧x 1 2 = ρ[x; x],α (u) 1 2 x∧x 1 2 = ρ x ,ζ (u); x ,ζ (u) 1 1 2 2 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 205 Lemma 11.D.4. — The statement of Lemma 11.D.2 holds true if we replace, in both the hypotheses and the conclusion, central by k-hypercentral. Proof.—Thecase k= 1 was done in Lemma 11.D.2. Decompose h g as h h g,with h h central (with kernel z)and h g (k− 1)-hypercentral. By induction 1 1 1 hypothesis, there exists k with k h such that the composite epimorphism k g is (k− 1)-hypercentral and such that (T ) lifts to k. Consider the ﬁbered product h× k of the two epimorphisms h h and k h , so that the two lines in the diagram below are 1 1 central extensions and both squares commute. 0 z h h 0 0 z h× k k 0 Applying Lemma 11.D.2 again to h× k k,weobtain m h× l so that the h h 1 1 composite epimorphism m k is central and so that (T ) lifts to m. So the composite map m h is the desired homomorphism. We say that a Lie algebra g is spread if it can be written as g= n s where n is nilpotent, s is reductive and acts reductively on n.Itis spreadable if there exists such a decomposition. When g is solvable, s is abelian and a Cartan subalgebra of g is given by the centralizer h= C (s)= C (s)× s. In particular, the s-characteristic decomposition of g n g coincides with the h-characteristic decomposition, and the associated Cartan gradings are the same. This remark is useful when we have to deal with a homomorphism n s→ n s which is the identity on s: indeed such a homomorphism is graded for the Cartan gradings. Proof of Theorem 11.C.4. — We ﬁrst prove the result when g is spread, so g= u d. Let k be the dimension of u/g . We argue by induction on k. ∞∨ Suppose that k= 0. We have a grading of g , valued in d .Considerthe blow-up construction (Lemma 8.B.2): it gives a graded Lie algebra g and a graded surjective map ∞∞ g g with central kernel concentrated in degree zero and isomorphic to H (g ) . 2 0 206 YVES CORNULIER, ROMAIN TESSERA This grading, valued in d , deﬁnes a natural action of d on g and the epimorphism ∞∞ ∞∞ + + g d g d is central. By construction, the kernel H (g ) is contained in (g 2 0 d) so this (hyper)central epimorphism has non-polynomial distortion. Now suppose that k≥ 1. Let n be a codimension 1 ideal of g containing g d. Since n contains g⊕ d, the intersection of n with u is a hyperplane in u .Sothere 0 0 exists a one-dimensional subspace l⊂ u ,suchthat g= n l. Note that the grading of g, ∨∞∞∞ valued in d ,extends that of n and n= g and in particular, H (n ) ={0}. 2 0 By induction hypothesis, there exists a hypercentral epimorphism h n,with non-polynomial distortion. By Lemma 11.D.4, there exists a hypercentral epimorphism ad(l) m h (with kernel z) so that the action of e on n lifts to a unipotent action on m. This corresponds to a nilpotent action of l on m.Let z be the intersection of the ith term of the ascending central series of m with z,so z= z for some .Oneach z /z , the action of l i+1 i is nilpotent and the adjoint action of m is trivial. So the action of m l on each z /z , i+1 i hence on z, is nilpotent. That is, z is hypercentral in m l (this is where the argument would fail with “hypercentral” replaced by “central”). So m l g is the desired hy- percentral epimorphism: by Lemma 11.D.1, m n has non-polynomial distortion and therefore so does m l g. Now the result is proved when g is spread. In general, ﬁx a faithful linear represen- tation g→ gl ,and let h= u d be the splittable hull of g in gl (that is, the subalgebra n n generated by semisimple and nilpotent parts of elements of g for the additive Jordan de- composition, see [Bou, Chap. VII, §5]). If n is any Cartan subalgebra of h,then n= n∩g is a Cartan subalgebra of g [Bou,Chap. VII, §5,Ex. 8].Now n= (u∩ n)+ n ,but every n-weight of h vanishes on u∩n.Sothe n-grading of h extends the n -grading of g (in other ∞∞ words, the embedding g⊂ h is a graded map). In particular, since g= h , this equality is an isomorphism of graded algebras and we deduce that H (h ) ={0}.Soweobtain 2 0 ∞∞ a hypercentral extension m h, with non-polynomial distortion because g= h .By taking the inverse image of g in m, we obtain the desired hypercentral extension of g. 11.E. An example without central extensions. — We prove here that in the conclusion of Theorem 11.C.2, it is not always possible to replace, in the conclusion, hypercentral by central. We begin with the following useful general criterion. Proposition 11.E.1. —Let g be a ﬁnite-dimensional solvable Lie algebra with its Cartan grading. Then g has no central extension with non-polynomial distortion if and only if the image of (Ker(d )∩ (g∧ g )) in H (g) is zero (i.e. it is contained in Im(d )). 2 0 2 0 3 Proof. — Suppose that the image of the above map is nonzero. By the blow-up construction (Lemma 8.B.2), we obtain a central extension g˘ g with kernel H (g) 2 0 concentrated in degree zero. By the assumption, there exist x , y in g, of nonzero opposite i i weights±α ,suchthat x∧ y is a 2-cycle and is nonzero in H (g) . This means that in i i i 2 0 g˘ , the element z=[x , y] is a nonzero element of the central kernel H (g) .So z∈ g˘ i i 2 0 and the central epimorphism g˘ g does not have polynomial distortion. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 207 Conversely, suppose that there exists a central epimorphism g˘ g with non- polynomial distortion, with kernel z. Note that the Cartan grading lifts to g˘ , so that the kernel z is concentrated in degree zero. By assumption, z∩ g˘ contains a nonzero el- ement z. By Lemmas 8.A.3 and 8.A.4,wecan write,in g˘ , z=[x , y] with x , y of i i i i nonzero opposite weights. Thus in g, x∧ y is a nonzero element of H (g) . i i 2 0 Let G be the 15-dimensional R-group of 6× 6 upper triangular matrices of the form ⎛ ⎞ 1 x u u u u 12 13 14 15 16 ⎜ ⎟ 01 0 0 x x 25 26 ⎜ ⎟ ⎜ ⎟ 00 t u u u 3 34 35 36 ⎜ ⎟ (11.E.2) , ⎜ ⎟ 00 0 t u u 4 45 46 ⎜ ⎟ ⎝ ⎠ 00 0 0 1 x 0 0 0001 where t , t are nonzero. Its unipotent radical U consists of elements of the form (11.E.2) 3 4 ˜˜ with t= t= 1 and its exponential radical E consists of elements in U for which x= 3 4 12 x= x= x= 0. If D denotes the (two-dimensional) diagonal subgroup in G, the 25 26 56 ˜˜ quotient G/E is isomorphic to the direct product of D with a 4-dimensional unipotent group (corresponding to coefﬁcients x , x , x , x ). Note that the extension 1→ E→ 12 25 26 56 ˜˜˜ U→ U/E→ 1 is not split. Let Z the 2-dimensional subgroup of U consisting of matrices with all entries zero except u and x . Note that Z is hypercentral and has non-trivial intersection with the 16 26 exponential radical of G. ˜˜ Deﬁne G= G/Z; it is 13-dimensional. The weights of E= E/Z are arranged as follows (the principal weights are in boldface) 14 34 (11.E.3) 13 15 35 36 45 46 and the other basis elements of weight zero in g are 33, 44, 12, 25, 56. We see that e is 2-tame. Besides, H (e) = 0, as 13∧ 36 is a 2-cycle in degree 0 2 0 that is not a 2-boundary, as follows from the observation that e has an obvious nontrivial central extension in degree zero, given by at the level of groups by 1→ Z/Z→ E/Z→ E→ 1, 208 YVES CORNULIER, ROMAIN TESSERA where Z is the one-dimensional subgroup at position 26 (that is, the subgroup of Z con- ˜˜ sisting of matrices with u= 0), which is normalized by Ebut not by U. Since H (g ) ={0},byTheorem 11.C.4 there exists a hypercentral epimorphism 2 0 h→ g with non-polynomial distortion. By contrast, every central epimorphism h→ g has polynomial distortion. This follows from Proposition 11.E.1 and the following propo- sition. Proposition 11.E.4. —Let g be the above 13-dimensional triangulable Lie algebra. Then in H (g) ,the imageof (Ker(d )∩ (g∧ g )) is zero. 2 0 2 0 Proof. — To streamline the notation, we denote by ij the elementary matrix usually denoted by E ,with1at position (i, j) and zero everywhere else (including the diagonal). ij By considering each pair of nonzero opposite weights in (11.E.3), we can describe the map d on a basis of the 4-dimensional space (u∧ u ) . 2 0 13∧ 35 −→−15 13∧ 36 −→ 0, 14∧ 45 −→−15, 14∧ 46 −→ 0; accordingly a basis of (Ker(d )∩ (g∧ g )) is given by 2 0 13∧ 35− 14∧ 45, 13∧ 36, 14∧ 46; we have to check that these are all boundaries; let us snatch them one by one: 12∧ 25∧ 56 −→ 56∧ 15. 13∧ 34∧ 45 −→ 13∧ 45− 14∧ 45 13∧ 35∧ 56 −→ 56∧ 15+ 13∧ 36 14∧ 45∧ 56 −→ 56∧ 15+ 14∧ 46. Combining with Proposition 11.E.1,weget: Corollary 11.E.5. — We have H (g ) ={0}, but there is no central extension of Lie groups 2 0 1 −→ R −→ G −→ G −→ 1 with j(R) exponentially distorted in G. 12. G not 2-tame This section is totally independent from the previous ones. Its main result is Theo- rem 12.C.1, entailing Theorem E.2. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 209 Here we prove that any group satisfying the SOL obstruction has an at least expo- nential Dehn function. The method also provides the result that any group satisfying the non-Archimedean SOL obstruction is not compactly presented. Let us mention that, for instance, Theorem 12.C.1 reproves as particular case (us- ing an embedding as a cocompact lattice) the following theorem of Groves and Hermiller [GH01]: any strictly ascending HNN-extension of a ﬁnitely generated nilpotent group has an exponential Dehn function (the exponential upper bound being given by Corol- lary 3.B.6). 12.A. Combinatorial Stokes formula. Deﬁnition 12.A.1. —Let X be a set and let R be any commutative ring. We call a closed path a sequence c= (c ,..., c ) of points in X with c= c (so we can view it as indexed by Z/nZ). If 0 n 0 n α, β are functions X→ R,wedeﬁne β dα= β(c ) α(c )− α(c ) i i+1 i−1 i∈Z/nZ = β(c )α(c )− β(c )α(c ). i i+1 i+1 i i∈Z/nZ Clearly, this is invariant if we shift indices. The following properties are immediate consequences of the deﬁnition. • (Antisymmetry) We have , , β dα =− αdβ. c c • (Concatenation) If c= c= c and we write c= (c ,..., c ) and c= (c ,..., c ), 0 i n 0 i i n , , , β dα= β dα+ β dα. c c c • (Filiform vanishing) If n= 2 then the integral vanishes. More generally, the inte- gral vanishes when c is ﬁliform, i.e., n is even and c= c for all i. i n−i - - - Indeed, the difference β dα− β dα− β dα is equal to c c c β(c ) α(c )− α(c )+ α(c )− α(c ) 0 i+1 i−1 1 n−1 − α(c )− α(c )− α(c )− α(c )= 0. 1 i−1 i+1 n−1 The ﬁliform vanishing is immediate for n= 2 and follows in general by an induction based on the concatenation formula. 210 YVES CORNULIER, ROMAIN TESSERA Now let us deal with a Cayley graph of a group G with a generating set S, and we consider paths in the graph, that is sequences of vertices linked by edges. Thus any closed path based at 1 can be encoded by a unique element of the free group F , which is a relation (i.e. an element of the kernel of F→ G), and conversely, if r is a relation, we denote by[r] the corresponding closed path based at 1. Note that G acts by left translations on the set of closed paths. The above properties imply the following • (Product of relations) If r, r are relations, we have , , , β dα= β dα+ β dα. [rr][r][r] • (Conjugate of relations) If r is a relation and γ∈ F , , , β dα= β dα. −1 [γ rγ] γ·[r] • (Combinatorial Stokes formula) Suppose that a relation r is written as a product −1 r= γ r γ of conjugates of relations. Then i i i=1 i , , β dα= β dα. [r] γ·[r] i i i=1 The formula for products follows from concatenation if there is no simpliﬁcation in the product rr , and follows by also using the ﬁliform vanishing otherwise. The formula for conjugates also follows using the ﬁliform vanishing. The Stokes formula follows from the two previous by an immediate induction. Remark 12.A.2. — The above combinatorial Stokes formula is indeed analogous to the classical Stokes formula on a disc: here the left-hand term is thought of as an integral along the boundary, while the right-hand term is a discretized integral over the surface. 12.B. Loops in groups of SOL type. — Let K and K be two nondiscrete locally 1 2 compact normed ﬁelds. Consider the group G= (K× K ) −1 Z, 1 2 ( , ) where|| ≥|| > 1, with group law written so that the product depends afﬁnely on 2 K 1 K 2 1 the right term: n −n (x, y, n) x , y , n= x+ x , y+ y , n+ n . 1 2 Write|| =|| with 0 <μ≤ 1. 1 K 2 1 K Now we can also view x and y as the projections to the coordinates in the above description. In the next lemmas, we consider a normed ring K (whose norm is submul- tiplicative, not necessarily multiplicative), and functions A: K→ K,and B: K→ K, 1 2 yielding functions α, β: G→ K deﬁned by α= A◦ x and β= B◦ y. GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 211 Lemma 12.B.1. — Suppose that A is 1-Lipschitz and that B satisﬁes the Hölder-like condition |B(s)− B s |≤|s− s| ,∀s, s∈ K . Then β dα is bounded on triangles of bounded diameter, that is, when c ranges over triples (c ) i i∈Z/3Z −1 with c∈ G such all c c belong to some given compact subset of G. i j Proof. — Let us consider a triangle T of bounded diameter (viewed as a closed path of length three), i.e. three points (g , g h, g h ),where h and h are bounded (but not g !). 0 0 0 0 Note that as a consequence of the antisymmetry relation, β dα does not change if we add constants to both α and β . We can therefore assume that α(g )= β(g )= 0. Hence 0 0 β dα= β(g h)α g h− β g h α(g h). 0 0 0 0 In coordinates, suppose that g= (x , y , n ), h= (x, y, n) and h= (x , y , n ).Then g h= 0 0 0 0 0 n−n n−n 0 0 0 0 (x+ x, y+ y, n+ n),and g h= (x+ x , y+ y , n+ n ). Since A(x )= 0 0 0 0 0 0 0 0 1 2 1 2 B(x )= 0, we have −n n−n n 0 0 0 0 β dα= B y+ y A x+ x− B y+ y A x+ x . 0 0 0 0 2 1 2 1 By our assumptions on A and B, we have . . . . −n μ n −n μ n 0 0 0 0 . . β dα ≤| y|| x |+| y|| x| 2 1 2 1 . . μ μ =|y||x |+|y||x|, which is duly bounded when h, h are bounded. Fix n≥ 1. We consider the relation n−n n−1−n−1 γ= t xt yt x t y; 1,n this deﬁnes a closed path of length 4n+ 4, where x, y and t denote (by abuse of notation) the elements (1, 0, 0) , (0, 1, 0) and (0, 0, 1) of G= K× K Z. 1 2 Lemma 12.B.2. —Consider A, B,α,β as introduced before Lemma 12.B.1. Suppose that A(0)= B(1)= 0. Then we have β dα= 2B(0)A . 1,n 212 YVES CORNULIER, ROMAIN TESSERA Proof of Lemma 12.B.2. — To simplify the notation, let us denote by c the closed path of length 4n+ 4deﬁned by γ . In the integral β dα, only those points c for 1,n i which “β dα” is nonzero, i.e. both α(c ) = α(c ) and β(c ) = 0, do contribute. In this i+1 i−1 i example as well as the forthcoming ones, this will make most terms be equal to zero. The closed path c can be decomposed as c= (0, 0, 0), (0, 0, 1),...,(0, 0, n− 1), (0, 0, n)= c , 0 n n n n n c= , 0, n , , 0, n− 1 ,..., , 0, 1 , , 0, 0= c , n+1 2n+1 1 1 1 1 n n n n c= , 1, 0 , , 1, 1 ,..., , 1, n− 1 , , 1, n= c , 2n+2 3n+2 1 1 1 1 c= (0, 1, n), (0, 1, n− 1),...,(0, 1, 1), (0, 1, 0)= c . 3n+3 4n+3 We see that x(c ) = x(c ) only for i= n, n+ 1, 3n+ 2, 3n+ 3. Moreover, for i+1 i−1 i= 3n+ 2, 3n+ 3, y(c )= 1, so B(y(c ))= 0. Thus i i n+1 β dα= B y(c ) A x(c )− A x(c )= 2B(0) A − A(0); i i+1 i−1 i=n whence the result. To illustrate the interest of the notions developed here, note that this is enough to obtain the following result. Proposition 12.B.3. — Under the assumptions above, if 2 = 0 in K , the group G= (K× K ) −1 Z 1 2 ( , ) has at least exponential Dehn function, and if both K and K are ultrametric then G is not compactly 1 2 presented. Remark 12.B.4. — Actually the lower bound in Proposition 12.B.3 is optimal: if either K or K is Archimedean, then G has an exponential Dehn function. Indeed, the 1 2 upper bound follows from Corollary 3.B.6. We use the following convenient language: in a locally compact group G with a compact system of generators S, we say that a sequence of null-homotopic words (w ) in F has asymptotically inﬁnite area if for every R, there exists N(R) such that no w for S n n≥ N(R) is contained in the normal subgroup of F generated by null-homotopic words of length≤ R. By deﬁnition, the non-existence of such a sequence is equivalent to G being compactly presented. Proof of Proposition 12.B.3.—Set I= β dα. 1,n Deﬁne K= K× K . We need to use suitable functions A: K→ K and B: K→ 1 2 1 2 K satisfying the hypotheses of Lemmas 12.B.1 and 12.B.2.For x∈ K ,deﬁne o(x)∈ K 2 1 GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 213 to be equal to 1 if|x| < 1and to 0 if|x|≥ 1; also deﬁne (x)= max(1 −|x|, 0)∈ R; also keep in mind that any Archimedean locally compact ﬁeld naturally contains R as a closed subﬁeld. We deﬁne A and B according to whether K and K are Archimedean: 1 2 • K non-Archimedean (K arbitrary): A(x )= (x , 0) and B(x )= (o(x ), 0); 2 1 1 1 2 2 • K and K both Archimedean: A(x )= (x , 0) and B(x )= ((x ), 0); 1 2 1 1 2 2 • K Archimedean, K non-Archimedean: A(x )= (0,|x|) and B(x )= 2 1 1 1 2 (0,(x )). (Note that the cases K Archimedean and not K , and vice versa, cannot be treated 1 2 simultaneously because of the dissymmetry resulting from the condition μ≤ 1.) Since A(0)= B(1)= (0, 0), Lemma 12.B.2 implies that I= 2B(0)A( ),and thus n n I= (2 , 0) in the ﬁrst two cases, and I= (0, 2|| ) in the last case. It is clear that in each case, A is 1-Lipschitz. We have to check the Hölder-like condition for B, which is clear when K is non-Archimedean. If K is Archimedean, 2 2 we need to check the Hölder-like condition for f (x )= max(0, 1 −|x|)∈ R.Indeed,if 2 2 s, s∈ K ,ifboth|s|,|s |≥ 1, then f (s)= f (s );if|s|≤ 1 ≤|s|,then|f (s)− f (s )|= 1 −|s|; then since 1 ≤|s |≤|s− s|+|s|, we deduce 1 −|s|≤|s− s|.If|s− s |≤ 1, we deduce that |f (s)− f (s )|≤|s− s |≤|s− s| If|s− s |≥ 1, we directly see|f (s)− f (s )|= 1 −|s|≤ 1 ≤|s− s| . Finally if both|s|,|s |≤ 1, we have|f (s)− f (s )|=|s|−|s |≤ 1. Thus the inequality is again clear if|s− s |≥ 1, and otherwise|f (s)− f (s )|=|s|−|s |≤|s− s |≤ |s− s| . Therefore by Lemma 12.B.1, for each R there is a bound C(R) on the norm of β dα over any triangle of diameter≤ R. We now again discuss: and K both non-Archimedean, I= (2 , 0): by ultrametricity of K ,and • K 1 2 1 the combinatorial Stokes theorem, C(R) is a bound for the norm of β dα over an arbitrary loop that can be decomposed into triangles of diameter≤ R. Since (|2||| ) goes to inﬁnity, this shows that for every R there exists n such that K 1 0 1 K for every n≥ n , the loop γ cannot be decomposed into triangles of diameter 0 1,n ≤ R. Thus the sequence (γ ) has asymptotically inﬁnite area. This shows that 1,n G is not compactly presented. • K Archimedean, K arbitrary, I= (2 , 0). Fix R so that every combinatorial 1 2 loop in G can be decomposed into triangles of diameter≤ R. Suppose that γ can be decomposed into j triangles of diameter≤ R. By the combinatorial 1,n n Stokes formula (see Section 12.A) and Lemma 12.B.1,|I|≤ C(R)j . It follows that j≥ 2|| /C(R).Hence theareaof γ grows at least exponentially, so n 1 1,n the Dehn function of G grows at least exponentially. • K Archimedean, K non-Archimedean: I= (0, 2|| ). The argument is ex- 2 1 1 actly as in the previous case. Remark 12.B.5. — The assumption that K does not have characteristic two can be removed, but in that case we need to redeﬁne β dα as β(c )(α(c )− α(c )). i i+1 i The drawback of this deﬁnition is that the integral is not invariant under conjugation. 214 YVES CORNULIER, ROMAIN TESSERA However, with the help of Lemma 2.D.2, it is possible to conclude. Since we are not concerned with characteristic two here, we leave the details to the reader. However Proposition 12.B.3 is not enough for our purposes, because we do not only wish to bound below the Dehn function of the group G, but also of various groups H mapping onto G. In general, the loop γ does not lift to a loop in those groups, so we 1,n consider more complicated loops γ in G, which eventually lift to the groups we have in k,n mind. However, to estimate the area, we will go on working in G, because we know how to compute therein, and because obviously the area of a loop in H is bounded below by the area of its image in G. Deﬁne by induction −1−1 γ= γ g γ g . k,n k−1,n k k−1,n k Here, g denotes the element (0, y , 0) in the group G, where the sequence (y ) in K k k i 2 satisﬁes the following property: y= 1 and for any non-empty ﬁnite subset I of integers, . . . . . . y≥ 1. . . i∈I For instance, if K is ultrametric, this is satisﬁed by y= ;if K= R, the constant se- 2 i 2 quence y= 1 works. The sequence (y ) will be ﬁxed once and for all. i i Fix n. We wish to compute, more generally, β dα. Write the path γ as (c ). k,n i k,n Note that for given n, c does not depend on k (because γ is an initial segment of γ ). i k,n k+1,n Write the combinatorial length of γ as λ (λ= 4n+ 4, λ= 2λ+ 2). k,n k,n 1,n k+1,n k,n Lemma 12.B.6. — The number n being ﬁxed, we have (1) There exists a sequence ﬁnite subsets F of the set of positive integers, such that for all i, we have y(c )= y , and satisfying in addition: for all i <λ and all k≥ 1, we have i j k,n j∈F F ⊂{1,..., k}.Moreover y(c ) = 0 (and thus F =∅), unless either i i i – i≤ 2n+ 2,or – i= λ for some j . j,n (2) Assume that 1≤ i≤ n− 1,or n+ 2≤ i≤ 2n+ 2,or i= λ for some j . Then j,n x(c )= x(c ). i−1 i+1 Proof. (1) The sequence (F ) is constructed for i <λ , by induction on k.For k= 1, we i k,n set F =∅ if i≤ 2n− 1and F ={1} if 2n+ 2≤ i≤ 4n+ 3= λ− 1, and it i i 1,n satisﬁes the equality for y(c ) (see the proof of Lemma 12.B.2,where c is made i i explicit for all i≤ λ= 4n+ 4). 1,n GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 215 Now assume that k≥ 2and that F is constructed for i <λ= λ with the i k−1,n required properties. We set F =∅; since c= (0, 0, 0), the condition holds for λ λ i= λ. It remains to deal with i when λ< i <λ ; in this case c= g c ,so k,n i k 2λ−i y(c )= y+ y(c ). Thus if we set F ={k }∪ F , remembering by induction that i k i i 2λ−i F ⊂{1,..., k− 1}, we deduce that y(c )= y ; clearly F ⊂{1,..., k}. 2λ−i i j i j∈F (2) This was already checked for i≤ 2n+ 2 (see the proof of Lemma 12.B.2). In the case i= λ ,wehave c= g and c= g ,so x(c )= x(c ). j,n i−1 j i+1 j+1 i−1 i+1 Lemma 12.B.7. — Under the assumptions of Lemma 12.B.2, we have β dα =|2| . k,n Proof. — By Lemma 12.B.6,ifboth|y(c )| < 1and x(c ) = x(c ),then i= n or i i−1 i+1 i= n+ 1. It follows that the desired integral on γ is the same as the integral on γ k,n 1,n computed in Lemma 12.B.2. 12.C. Groups with the SOL obstruction. Theorem 12.C.1. —Let G be a locally compact, compactly generated group, and suppose there is a continuous surjective homomorphism G→ G= (K× K ) −1 Z, 1 1 2 ( , ) where 0 = 2 in K . Suppose that G has a nilpotent normal subgroup H whose image in G contains 1 1 K× K . Then 1 2 • the Dehn function of G is at least exponential. • if both K and K are ultrametric, then G is not compactly presented. 1 2 Proof.—Let k be the nilpotency length of H and ﬁx k≥ k . Using Lemma 12.B.7 0 0 and arguing as in the proof of Proposition 12.B.3 (using Lemma 12.B.7 instead if Lemma 12.B.2), we obtain that the loops γ , which have linear length with respect to k,n n, have at least exponential area, and asymptotically inﬁnite area in case K and K are 1 2 both ultrametric. n−n ˜˜˜ Lift x, y,and g to elements x˜, y˜, g˜ in H and t to an element t in G; set X= t x˜t ; k k n since H is normal, X∈ H. This lifts γ to a path γ+ based at 1; let v be its value at λ , n k,n k,n k,n k,n −1−1 −1−1 ˜˜ so v= X y˜X y˜ and v= v g˜ v g˜ . We see by an immediate induction that 1,n n k+1,n k,n k k n k,n v belongs to the (k+ 1)th term in the lower central series of H. Since k≥ k , we see that k,n 0 γ+ is a loop of G, of linear length with respect to n, mapping to γ . In particular, its area k,n k,n is at least the area of γ . So we deduce that γ+ has at least exponential area with respect k,n k,n to n, and has asymptotically inﬁnite area in case K and K are both ultrametric. 1 2 We will also need the following variant, in the real case. 216 YVES CORNULIER, ROMAIN TESSERA Theorem 12.C.2. —Let G be a locally compact, compactly generated group, and suppose there is a continuous homomorphism with dense image G→ G= (R× R) R, −t so that the element t∈ R acts by the diagonal matrix ( , ) (≥ > 0). Suppose that G has a 2 1 1 1 2 nilpotent normal subgroup H whose image in G contains R× R. Then the Dehn function of G is at least exponential. Proof. — Since the homomorphism has dense image containing R× R,there ex- ists some element t mapping to an element t of the form (0, 0,τ) with τ> 0. Changing the parameterization of G if necessary (replacing by for i= 1, 2), we can suppose that τ= 1. Then lift x and y and pursue the proof exactly as in the proof of Theo- rem 12.C.1. Remark 12.C.3. — To summarize the proof, the lower exponential bound is ob- tained by ﬁnding two functions α, β on G such that the integral β dα is bounded on triangles of bounded diameter, and a sequence (γ ) of combinatorial loops of linear di- ameter such that β dα grows exponentially. This approach actually also provides a lower bound on the homological Dehn function [Ger92, BaMS93, Ger99] as well. Let us recall the deﬁnition. Let G be a locally compact group with a generating set S and a subset R of the kernel of F→ G consist- ing of relations of bounded length, yielding a polygonal complex structure with oriented edges and 2-faces. Let A denote either Z or R.For i= 0, 1, 2, let C (G, A) be the A- module freely spanned by the set of vertices, resp. oriented edges, resp. oriented 2-faces. Endow each C (G, A) with the norm. There are usual boundary operators ∂ ∂ 2 1 C (G, A)→ C (G, A)→ C (G, A). 2 1 0 satisfying ∂◦ ∂= 0. If Z (G, A) is the kernel of ∂ , then it is easy to extend, by linearity, 1 2 1 1 the deﬁnition of β dα (from Section 12.A)to c∈ Z (G, A). Following [Ger99], deﬁne, for c∈ Z (G, A) HFill (c)= inf$P$: P∈ C (G, A), ∂ (P)= c . 1 2 2 G,S,R and A A Hδ (n)= sup HFill(z): z∈ Z (G, Z),$z$≤ n . 1 1 G,S,R G,S,R Clearly, if c is a basis element (so that its area makes sense) R Z G,S,R HFill(c)≤ HFill(c)≤ area (c) ≤∞; G,S,R G,S,R it follows that R Z Hδ (n)≤ Hδ (n)≤ δ (n), G,S,R G,S,R G,S,R GEOMETRIC PRESENTATIONS OF LIE GROUPS AND THEIR DEHN FUNCTIONS 217 where δ is the smallest superadditive function greater or equal to δ . (Examples of G,S,R G,S,R non-superadditive Dehn functions of ﬁnite presentations of groups are given in [GS99]; however the question, raised in [GS99], whether any Dehn function of a ﬁnitely pre- sented group is asymptotically equivalent to a superadditive function, is still open.) The function HFill(c) is called the A-homological Dehn function of (G, R, S) G,S,R (the function HFill(c) is called abelianized isoperimetric function in [BaMS93]). If G,S,R ﬁnite, it can be shown by routine arguments that its≈-asymptotic behavior only depends on G. Some Bestvina–Brady groups [BeBr97] provide examples of ﬁnitely generated groups with ﬁnite homological Dehn function but inﬁnite Dehn function. Until recently, no example of a compactly presented group was known for which the integral (or even real) homological Dehn function is not equivalent to the integral homological Dehn func- tion; the issue was raised, for ﬁnitely presented groups, both in [BaMS93, p. 536] and [Ger99, p. 1]; the ﬁrst examples have ﬁnally been obtained by Abrams, Brady, Dani and Young in [ABDY13]. Let us turn back to G (a group satisfying the hypotheses of Theorem 12.C.1 or 12.C.2): for this example, since R consists of relations of bounded length, it follows that the integral of β dα over the boundary of any polygon is bounded. Since β dα grows exponentially, it readily follows that HFill (γ ) grows at least exponentially and R Z hence Hδ (n) (and thus Hδ (n)) grows at least exponentially. G G 1 1 Acknowledgements We are grateful to Christophe Pittet for several fruitful discussions. We thank Cor- nelia Dru¸ tu for discussions and pointing out useful references. We thank Pierre de la Harpe for many useful corrections. We are grateful to the referee for a thorough work and various suggestions improving the presentation of the paper. REFERENCES [Ab72] H. ABELS, Kompakt deﬁnierbare topologische gruppen, Math. Ann., 197 (1972), 221–233. [Ab87] H. ABELS, Finite Presentability of S-Arithmetic Groups. 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[Y13a] R. YOUNG, Filling inequalities for nilpotent groups through approximations, Groups Geom Dyn., 7 (2013), 977–1011. [Y13b] R. YOUNG, The Dehn function of SL(n; Z), Ann. Math., 177 (2013), 969–1027. Y. C. Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France [email protected] R. T. Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France [email protected] Manuscrit reçu le 1 novembre 2013 Manuscrit accepté le 28 novembre 2016 publié en ligne le 20 décembre 2016.

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Geometric presentations of Lie groups and their Dehn functions

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