Finitely presented simple groups and measure equivalence
FFinitely presented simple groups and measureequivalence
Antonio López NeumannJanuary 2021
Abstract
We exhibit explicit infinite families of finitely presented, Kazhdan, simplegroups that are pairwise not measure equivalent. These groups are latticesacting on products of buildings. We obtain the result by studying vanishingand non-vanishing of their L -Betti numbers.2010 Mathematics Subject Classification: 20E32, 20E42, 20F05, 20F55, 20J05,20J06, 22E41, 28D15.Keywords and Phrases: Infinite simple groups, finite presentation, L -Bettinumbers, measure equivalence, Coxeter groups, buildings, cohomological finite-ness. Infinite finitely presented simple groups are rare in geometric group theory. To thisdate, few examples are known: Burger-Mozes groups acting on products of trees[BM], non-affine irreducible Kac-Moody lattices over a finite field acting on twinbuildings [CR1] and variants of Thompson groups [Röv].Naturally, we study these groups not up to isomorphism but up to equivalencerelations that are relevant to the theory, like quasi-isometry or measure equivalence.Some results have been obtained on the quasi-isometry side. All Burger-Mozesgroups are quasi-isometric since they act properly and cocompactly on productsof trees, which are bi-Lipschitz equivalent. It is shown in [CR2] that there is aninfinite family of non-affine irreducible Kac-Moody lattices that are pairwise notquasi-isometric. Using cohomological finiteness properties, the same result is shownin [SWZ] for Röver-Nekrashevych variants of Thompson groups.We obtain the same result as in [CR2] and in [SWZ] but for measure equivalence.Namely:
Theorem 1.1.
There are infinitely many measure equivalence classes of finitelypresented, Kazhdan, simple groups. These groups are Kac-Moody lattices over finitefields with well chosen non-affine Weyl groups.
It was shown by Gaboriau that L -Betti numbers of discrete countable groupsup to proportionality are invariant by measure equivalence [Gab, 6.3]. Petersenprovides a first family of finitely generated simple (Kac-Moody) groups having non1 a r X i v : . [ m a t h . G R ] J a n roportional sequences of L -Betti numbers [Pet, 6.8] using an example by Dymara-Januszkiewicz [DJ, 8.9]. Unfortunately, the groups in question are not Kazhdanand it is not known if they are finitely presented. A conjecture [AG] says that suchgroups should never be finitely presented, due to the presence of ∞ ’s in the Coxeterdiagram of the Weyl group.We follow the strategy outlined by Petersen but for families of Kac-Moody groupsenjoying better properties. Now we describe the strategy in two steps. First, weuse Dymara and Januszkiewicz’s formula for L -Betti numbers of locally compactgroups acting on some buildings ([DJ, Section 8], here Theorem 2.9). These groupsinclude complete Kac-Moody groups, which can be seen as the ambient spaces ofKac-Moody lattices. Second, we use a result by Petersen [Pet, 5.9] relating the L -Betti numbers of a topological group and its lattices to translate information fromthe topological group given by the formula at the level of the lattice.The formula for L -Betti numbers of complete Kac-Moody groups is explicitbut still hard to manipulate. Roughly, the formula splits into two parts: a topo-logical one and a representation theoretic one. As explained in [DJ], the behaviourof the representation theoretic part is well understood. Thus, the difficulty comesfrom understanding the topological part. More precisely, one has to compute thecohomology of some subcomplexes of the Davis chamber which encode the combi-natorial complexity of the groups we consider. The Davis chamber, as explainedlater, can be constructed entirely from the Weyl group of the building. This reducesour study of L -Betti numbers to a purely combinatorial study of Coxeter groups.We compute the cohomology of some of these spaces, precisely enough to give anon-vanishing criterion of an L -Betti number (in high degree) for groups acting onbuildings having a certain Weyl group. The condition we require for the Weyl groupis that the canonical generating set may be partitioned into two parts, such thatone part generates a finite Coxeter group and the other generates an affine Coxetergroup. This is enough to obtain Theorem 1.1.We give explicit families of groups as in Theorem 1.1 where we compute all ofthese cohomology spaces, thus simplifying the formula for L -Betti numbers of thecorresponding complete Kac-Moody groups and Kac-Moody lattices. This formulacan be stated explicitly by following the proof, we do not do it because what mattersto us is in which degrees the L -Betti numbers vanish and in which degrees they donot.Now we present the contents of the sections.Section 2 introduces the necessary background for the rest of the paper. Wefirst define the classes of simplicial complexes and groups discussed in [DJ], as wellas some combinatorial properties of the Davis chamber. We introduce Kac-Moodygroups and list the results making them interesting for us. We then state the maintheorems from [DJ], especially the formula for L -Betti numbers of groups actingon buildings.Section 3 deals with the cohomology of some of the subcomplexes D σ appearingin the formula of L -Betti numbers given in Section 2. We give a vanishing criterionfor this cohomology, slightly simplifying this formula. We then give a non-vanishingcriterion under a condition on the Weyl group, and apply it to Kac-Moody lattices.This proves Theorem 1.1.Section 4 gives concrete examples of Coxeter groups where results from Section2 compute all L -Betti numbers of the corresponding complete and discrete Kac-Moody groups.Section 5 addresses cohomological finiteness properties of the simple Kac-Moodylattices we study. Our arguments point out that it should not be possible to showthe results of [CR2] using cohomological finiteness criteria as in [SWZ]. Contents L -Betti numbers of groups acting on buildings 3 L -Betti numbers of groups acting on buildings . . 6 L -Betti numbers of groups acting on buildings This section is mainly a review of some parts of [DJ]. First, we introduce theclasses of simplicial complexes of [DJ, Section 1]. Let X be a purely n -dimensional countable simplicial complex (every simplex is a face of an n -dimensional simplex).The top dimensional simplices in X will be called alcoves . Let Aut( X ) be the groupof simplicial automorphisms equipped with the compact-open topology. Let G be aclosed subgroup of Aut( X ) . We consider the following properties on the pair ( X, G ) . B -dimensional links in X are finite. B Links of dimension ≥ in X are gallery connected: for any two alcoves insuch a link, there exists a path of alcoves connecting them, such that two consecutiveelements meet on a face of codimension 1. B All the links in X are either finite or contractible (including X itself).The three properties listed above deal with the space X only. The followingcondition is the only one that considers both the group G and the space X . Roughlyit requires the group to have the good size: it is big enough to act transitively onalcoves, and small enough to have a fundamental domain of maximal dimension. B The group G acts transitively on the set of alcoves of X and the quotientmap X → X/G restricts on an isomorphism on each alcove.The next two properties are spectral conditions on the Laplacian that are fun-damental in [DJ] to prove the results presented in Section 2.3.3 δ Links of dimension 1 are compact and the nonzero eigenvalues of the Laplacianon 1-dimensional links are ≥ − δ . B ∗ δ The spectrum of the Laplacian on 1-dimensional links in X is a subset of { } ∪ [1 − δ, + ∞ [ .We now begin a brief discussion on buildings, since they are the main source ofexamples of spaces satisfying (most of) these conditions. A standard reference forbuildings is [Br]. They are simplicial complexes obtained by gluing subcomplexescalled apartments under two incidence conditions: any two simplices lie in an apart-ment and their position is independent of the apartment. Each apartment is a copyof the same Coxeter complex, a purely dimensional simplicial complex with a simplytransitive action (on its set of alcoves) of a Coxeter group called the Weyl group ofthe building. The number of alcoves containing a given face of codimension 1 iscalled the thickness (of that face) and for buildings we want it to be ≥ for all suchfaces. The buildings we are interested in have finite thickness, they satisfy B . Let X be a simplicial complex satisfying B . Definition 2.1.
Let X (cid:48) be the first barycentric subdivision of X . The Davis complex X D of X is the subcomplex of X (cid:48) generated by the barycenters of simplices of X with compact links.This definition is interesting for two reasons. The first, is that X D is a deforma-tion retract of X [DJ, 1.4] which has the same automorphism group Aut( X ) , butthe action of the latter becomes proper over X D . The second reason stems frombuildings, and is summarized in the following proposition. Proposition 2.2.
Let X be a building. Then the link of every simplex is a building.Suppose X is a non-compact building. Then X D can be endowed with a CAT (0) metric. In particular, X and X D are contractible. (cid:3) The first assertion is [Br, IV.1 Prop 3] and the second one can be found in [Dav].In particular, this shows that buildings satisfy B and B .Property B is not always satisfied in this setting since one can consider buildingswithout automorphisms. However, if G is a group with BN -pair and X is thebuilding constructed from it, then the pair ( X, G ) satisfies B .If a pair ( X, G ) satisfies B and B , the intersections D = X D ∩ ∆ are simpliciallyisomorphic for any top dimensional simplex ∆ . We call such an intersection a Davischamber of X and we note it D . We can see D as a cone over D ∩ ∂ ∆ with apexthe barycenter of ∆ , thus D is contractible.In the case of buildings, the Davis chamber may also be constructed from itsWeyl group. The construction is equivalent to the previous one if the building comesfrom a BN-pair.Let ( W, S ) be a Coxeter system with | S | = n +1 et let ∆ be the standard simplexof dimension n . We associate to each codimension 1 face of ∆ a generator s ∈ S .This choice determines for each face σ in ∆ a parabolic subgroup W σ of W , where W σ = W J = (cid:104) J (cid:105) and J ⊆ S is the set of generators corresponding to codimension 1faces containing σ . 4 efinition 2.3. Let ∆ (cid:48) be the first barycentric subdivision of ∆ . We define the Davis chamber of W as the subcomplex D = D W of ∆ (cid:48) generated by the barycentersof faces σ in ∆ whose corresponding parabolic subgroup W σ is finite.The next definition measures in a certain sense how many simplices we mustremove from ∆ to obtain D . Definition 2.4.
Let ( W, S ) be a Coxeter system. We say ( W, S ) is k -spherical iffor all J ⊆ S with | J | ≤ k , the parabolic subgroup W J = (cid:104) J (cid:105) of W is finite.This definition will appear often as a hypothesis on the Weyl group for the resultswe will obtain. We will discuss it in more detail in Section 5.If σ and τ are faces of ∆ , then σ ⊆ τ is equivalent to W σ ⊇ W τ . To describe allfaces σ of ∆ whose corresponding subgroup W σ is finite, we have to identify thosecorresponding to maximal finite parabolic subgroups of W .If σ is a face of codimension k of ∆ , then the subcomplex of ∆ (cid:48) generated by thebarycenters of faces containing σ is a (simplicial subdivision of a) cube of dimension k . Each maximal finite parabolic subgroup W J ( J ⊆ S ) corresponds to a cube ofdimension | J | in the Davis chamber.The Davis chamber D is then obtained as follows. We start with the disjointunion of these cubes, and then we glue some of their faces: if W I and W J are twomaximal finite parabolic subgroups of W , the intersection of their correspondingcubes in ∆ (cid:48) is the cube corresponding to W I ∩ J .This gives an equivalent construction of the Davis chamber that is independentof ∆ . From each parabolic subgroup W J ( J ⊆ S ) of W we define the flag complexof parabolic subgroups { W J (cid:48) , J (cid:48) ⊆ J } contained in W J ordered by inclusion. Weconstruct D as the union of all flag complexes of maximal finite parabolic subgroupsof W , where we glue the complexes corresponding to W I and W J over the flagcomplex of W I ∩ J .For σ ⊆ ∆ , let ∆ σ be the union of faces not containing σ and D σ = D ∩ ∆ σ .Notice ∆ σ is always a union of ( n − -dimensional simplices of ∆ and that ∆ σ = ∂ ∆ if and only if σ = ∆ . More precisely, if σ ⊆ ∆ corresponds to the parabolic subgroup W J , then ∆ σ is exactly the union of codimension 1 faces of ∆ corresponding to thegenerators s j for j ∈ S \ J . The family of simple groups we want to exhibit in Theorem 1.1 are Kac-Moodygroups. We introduce them following the presentation of [DJ, Appendix TKM] andthen list the properties that make them interesting to us.
Definition 2.5. A Kac-Moody datum is the data ( I, Λ , ( α i ) i ∈ I , ( h i ) i ∈ I , A ) of:1. A finite set I .2. A finitely generated abelian free group Λ .3. Elements α i ∈ Λ , i ∈ I .4. Elements h i ∈ Λ ∨ = Hom(Λ , Z ) , i ∈ I .5. A generalized Cartan matrix ( A ij ) i,j ∈ I given by A ij = (cid:104) α i , h j (cid:105) , satisfying A ii = 2 , if i (cid:54) = j then A ij ≤ and A ij = 0 if and only if A ji = 0 . M = ( m ij ) i,j ∈ I as follows: m ii = 1 and for i (cid:54) = j, m ij = 2 , , , or ∞ as A ij A ji = 0 , , , or is ≥ , respectively.We consider the Coxeter group W associated to this matrix: W = (cid:104) r i | ( r i r j ) m ij = 1 , for m ij (cid:54) = ∞(cid:105) . If a Kac-Moody datum is fixed, Tits defines a group functor associating to eachfield (or commutative ring in general) k a group Λ( k ) [Ti]. The group Λ( k ) hastwo BN-pairs ( B + , N ) and ( B − , N ) such that their Weyl groups B ± / ( B ± ∩ N ) areisomorphic to the group W coming from the generalized Cartan matrix.These BN-pairs define two buildings X + and X − of thickness | k | + 1 and Weylgroup W (therefore the dimensions of these buildings is | I | − ), such that Λ( k ) acts transitively on their sets of chambers [DJ, Appendix TKM]. These buildingsare simplicially isomorphic, we denote them X when it is not necessary to distin-guish them. Denote by G ± the completion of Λ( k ) in Aut( X ± ) with respect to thecompact-open topology.The following theorem summarizes the properties that justify the study of Kac-Moody groups for our purposes. Theorem 2.6.
Let Λ be a Kac-Moody group over F q with Weyl group W . Then Λ is finitely generated. Moreover:1. The covolume of Λ in G + × G − (diagonally injected) is W ( q ) , where W ( t ) = (cid:88) w ∈ W t l ( w ) . In particular for q > | I | , the group Λ is a lattice in G + × G − .2. If W is non-affine, irreducible and Λ is a lattice in G + × G − , then Λ /Z (Λ) issimple, where Z (Λ) is the center of Λ .3. If q ≥ and all the entries of the Coxeter matrix are finite (ie the Weyl group is -spherical), then Λ is finitely presented.Proof. Assertions 1 and 2 are respectively Proposition 2 and Theorem 20 in [CR2].Assertion 3 is a simplified version of the main corollary in [AM].
Remark.
In what follows we will systematically consider center-free Kac-Moodygroups. This can always be done without further assumptions on the generalizedCartan matrix by choosing the adjoint root datum, where Λ is generated by the α i ’s. L -Betti numbers of groups acting on build-ings Dymara and Januszkiewicz state their results for classes B t and B + of pairs ( X, G ) groups acting on simplicial complexes. The class B t are pairs ( X, G ) satisfying B − and B ∗ n , the class B + are pairs ( X, G ) satisfying B − and B n . For groups witha BN-pair and their associated buildings, the class B t corresponds to large minimal6hickness and B + corresponds to large minimal thickness and finiteness of all entriesof the Coxeter matrix [DJ, 1.7], that is, 2-sphericity of its Weyl group.In particular, a complete Kac-Moody group over a finite field F q is in B t forlarge q , and is in B + if all the entries of its Coxeter matrix are finite.We now mention three important results of [DJ]. The initial motivation of[DJ] is to find examples of Kazhdan groups. The first theorem we state addressesthis question and gives a criterion for the vanishing of the first and also highercohomology groups for pairs in B + with a finiteness condition. Theorem 2.7. [DJ, 5.2] Let ( X, G ) be in B + . Suppose the links of X of dimensions , . . . , k are compact. Then for any unitary representation ( ρ, V ) of G , we have: H ict ( G, ρ ) = 0 for i = 1 , . . . , k. For buildings of finite thickness, compactness of all links of dimension ≤ k isequivalent to having a ( k + 1) -spherical Weyl group. This combinatorial conditionis necessary for cohomological vanishing: one can consider the group D ∞ actingsimplicially on a tree X with edge set E , the induced action on L ( E ) does not havea fixed point. Thus we have a space X not verifying the condition of the theorem(the only link of dimension 1 is X = Lk ( ∅ ) ) and H ct ( G, L ( E )) (cid:54) = 0 .The second theorem we mention is a formula for cohomology spaces of groupsin B + with values in unitary representations. Theorem 2.8. [DJ, 7.1] Suppose either thatthe pair ( X, G ) is in B + and ( ρ, V ) is a unitary representation of G , or that,the pair ( X, G ) is in B t and ( ρ, V ) is a subrepresentation of (cid:76) ∞ L ( G ) .Then H ∗ ct ( G, ρ ) = (cid:77) σ ⊆ ∆ (cid:101) H ∗− ( D σ ; V σ ) . We draw the attention on the right hand side depending on the classical coho-mology groups of the spaces D σ defined at the end of Section 2.1. Theorem 2.7 isproven in [DJ, Section 7] from this formula by noticing that the cohomology H ∗ ( D σ ) is the cohomology H ∗ ( U ) of a simple covering U of D σ . The combinatorial conditionof Theorem 2.7 on the Weyl group implies vanishing of H ∗ ( U ) in low degrees for all σ ⊆ ∆ . The proofs given in Section 3 are in the same spirit: under combinatorialconditions on the Weyl group, we compute H ∗ ( D σ ) for some σ ⊆ ∆ . The differenceis that instead of considering all simplices σ ⊆ ∆ and obtaining a partial descriptionof the cohomology of D σ , we pick particular σ ⊆ ∆ for which we can fully describethe spaces H ∗ ( D σ ) .The last theorem we mention is the starting point for this paper, though theexpression we present is obtained directly from the previous theorem. It is a formulafor L -Betti numbers, as defined in [Sau, 4.1], of groups acting on buildings of finitethickness in B t . As said before, groups in this class include complete Kac-Moodygroups over finite fields of large cardinality. Theorem 2.9. [DJ, 8.5] Let ( X, G ) be a building in B t of thickness q + 1 . Thenthe L -Betti numbers of G are given by β k ( G ) = (cid:88) σ ⊆ ∆ dim C (cid:101) H k − ( D σ ; C ) · dim G L ( G ) σ . oreover, the sum can be taken for σ with compact links. The fundamental observation by Dymara and Januszkiewicz is that for q > n , dim G L ( G ) σ > for all σ ⊆ ∆ with compact links [DJ, p. 612]. Thus, for q large enough, the problem of determining whether β k ( G ) is zero or not reduces todetermining if there exists σ ⊆ ∆ such that (cid:101) H k − ( D σ ; C ) (cid:54) = 0 or not. In the end,this reduces to the study of the combinatorics of the Weyl group. In the next twosections we study the topology of D σ via the combinatorics of the Weyl group ofthe building. Remark.
Dymara and Januszkiewicz apply their results to Kac-Moody groups whoseWeyl group W P is the right-angled Coxeter group defined by the intersection rela-tions of the faces of codimension 1 of a polytope P of dimension n [DJ, 8.9]. Mostof the associated Kac-Moody lattices are non-affine and irreducible, hence simple.When P is dual to a triangulation of a sphere (up to considering a barycentric sub-division), it is shown that all L -Betti numbers of the completions of such groupsvanish except in degree n . As stated by [Pet], this gives a first example of an infi-nite family of finitely generated simple groups with non-proportional sequences of L -Betti numbers.Petersen says such groups should often be finitely presented. Unfortunately, theCoxeter matrices of their Weyl groups have ∞ entries, so it is not known whetherthese groups are finitely presented or not, and a conjecture [AG] says that suchgroups should never be finitely presented. The examples we will give here are finitelypresented in view of Theorem 2.6, Assertion 3. In this section we elaborate on the formula of Theorem 2.9 for L -Betti numbersof groups acting on buildings. We study the contribution of the topological part ofthis formula to obtain first a vanishing criterion, slightly simplifying the formula,and then a non-vanishing criterion. We then apply this non-vanishing criterion toKac-Moody lattices and prove Theorem 1.1. We now start a brief discussion on the contractibility of D σ . Here contractibleunderstates non-empty. The following observation is fundamental for what follows. Lemma 3.1.
Let X be a finite simplicial complex covered by { X i , i ∈ I } a finitefamily of subcomplexes such that (cid:84) j ∈ J X j is contractible for all nonempty J ⊆ I .Then X = (cid:83) i ∈ I X i is contractible.Proof. Start with the case | I | = 2 . Write X = A ∪ B where A, B and A ∩ B arecontractible finite simplicial complexes. By [Hat, 0.16], given a continuous map f defined on X and a homotopy f t of f | Y on a subcomplex Y , we can extend f t to ahomotopy of f defined over X . This property implies, since Y = A ∩ B is contractible[Hat, 0.17], that the quotient map X → X/ ( A ∩ B ) is a homotopy equivalence. Thisquotient space is just the wedge sum of A/ ( A ∩ B ) and B/ ( A ∩ B ) . Both of these8paces are contractible because, by the same argument, they are homotopicallyequivalent to A and B respectively. Thus X is contractible.The general case is done by induction on | I | . Let I = { , . . . , n } and write X = (cid:83) ni =0 X i = A ∪ B , where A = X and B = (cid:83) ni =1 X i . By induction hypothesis, B is contractible. The same is true for A ∩ B = (cid:83) ni =1 ( X ∩ X i ) , since the spaces X ∩ X i are contractible with contractible intersections. Thus X is contractible bythe case | I | = 2 .In the previous section, we considered ( W, S ) a Coxeter system and ∆ a simplexof dimension | S | − . Let D be the Davis chamber of ( W, S ) . We defined for σ ⊆ ∆ ,the subset ∆ σ to be the union of the faces of ∆ not containing σ and D σ = D ∩ ∆ σ .Notice that ∆ σ is always a union of codimension 1 faces of ∆ . More precisely, if ∆ s isthe codimension 1 face of ∆ corresponding to the generator s ∈ S and σ = (cid:84) s ∈ J ∆ s then ∆ σ = (cid:83) s ∈ J c ∆ s . Remark. If σ is the simplex in ∆ corresponding to the parabolic subgroup W J , D ∩ σ is the geometric realization of the flag complex of finite parabolic subgroupscontaining W J . Thus, every cube of maximal dimension in this geometric realizationhas the barycenter of σ as a vertex. The intersections of these cubes are cubes oflower dimensions. Thus, if D ∩ σ is non-empty, Lemma 3.1 implies that D ∩ σ iscontractible (in particular this shows again that D is contractible). Notice that D ∩ σ is non-empty if and only if W J is finite. Proposition 3.2.
Let ∆ s denote the codimension 1 face of ∆ corresponding to s ∈ S , and let D s = D ∩ ∆ s . For σ ⊆ ∆ , we write σ = (cid:84) s ∈ J ∆ s for some J ⊆ S ,so that D σ = (cid:83) s ∈ J c D s . If W J c = (cid:104) J c (cid:105) is finite, or equivalently if (cid:84) s ∈ J c D s isnon-empty, then D σ is contractible.Proof. If (cid:84) s ∈ J c D s is non-empty, then every sub-intersection (cid:84) s ∈ J (cid:48) D s is non-emptyfor J (cid:48) ⊆ J c , hence contractible by the previous remark. Now the result follows fromLemma 3.1.We wish to compute spaces (cid:101) H ∗ ( D σ ) for σ ⊆ ∆ that appear in the formula ofTheorem 2.9. The sum ranges over σ with compact links, that is, over σ with finitecorresponding parabolic subgroups W σ . What we said allows us to focus on a smallerclass of simplices σ . Thus, the sum in the formula of Theorem 2.9 reduces to thefollowing. Proposition 3.3.
Let ( X, G ) be a building in B t of thickness q + 1 . Then the L -Betti numbers of G are given by β k ( G ) = (cid:88) σ ⊆ ∆ dim C (cid:101) H k − ( D σ ; C ) · dim G L ( G ) σ . Moreover, the sum can be taken over simplices σ corresponding to finite parabolicsubgroups W J such that W J c is infinite.Proof. The proof of Theorem 2.9 in [DJ, 8.5] already shows we can restrict ourselvesto those σ whose corresponding parabolic subgroup W J is finite (the argument isthat L ( G ) σ ⊆ L ( G ) G σ and if the link of σ is non-compact, then L ( G ) G σ = { } ).Now suppose σ ⊆ ∆ corresponds to a finite parabolic subgroup W J and that theparabolic subgroup W J c is also finite. Then by proposition 3.2, D σ is contractible,so its cohomology does not contribute to the sum in any degree.9 .2 Non-vanishing of cohomologies The previous section gives a vanishing criterion. To show the result stated in theintroduction we will use the following non-vanishing criterion.
Lemma 3.4.
Let X be a finite simplicial complex covered by { X i , i ∈ I } a finitefamily of subcomplexes such that (cid:84) j ∈ J X j is contractible for all proper nonemptysubsets J (cid:40) I . Suppose (cid:84) i ∈ I X i is empty. Then X = (cid:83) i ∈ I X i has the cohomologyof an ( | I | − -dimensional sphere.Proof. Put I = { , . . . , n } , I ∗ = { , . . . , n } and B = (cid:83) i ∈ I ∗ X i . We will show theresult by induction on | I | = n + 1 . For n = 1 , the space X is the disjoint union of X and X , so it has the cohomology of two points. For n ≥ , write X = X ∪ B. By Lemma 3.1, B is contractible. Since X is contractible, we look at the intersection X ∩ B = X ∩ (cid:91) i ∈ I ∗ X i = (cid:91) i ∈ I ∗ ( X ∩ X i ) . Each X ∩ X i is contractible with contractible intersections (except for the smallestone that is empty), thus X ∩ B is a union of | I |− n subspaces as in the assertion.By induction, X ∩ B has the cohomology of a ( n − -dimensional sphere. Thenthe Mayer-Vietoris exact sequence [Hat, p. 203] gives H ∗ ( X ) = H ∗− ( X ∩ B ) .Now we go back to the situation where ( W, S ) is an irreducible Coxeter systemwith Davis chamber D . We can restate the previous result for X = D σ as follows. Proposition 3.5.
Let ∆ s denote the codimension 1 face of ∆ corresponding to s ∈ S , set D s = D ∩ ∆ s and write D σ = (cid:83) s ∈ J D s for some J ⊆ S . If W J = (cid:104) J (cid:105) is infinite, but every proper parabolic subgroup of W J is finite, or equivalently if (cid:84) s ∈ J (cid:48) D s (cid:54) = ∅ for all nonempty J (cid:48) (cid:40) J but (cid:84) s ∈ J D s = ∅ , then D σ has the cohomologyof a ( | J | − -dimensional sphere. (cid:3) Remark.
Infinite irreducible Coxeter groups such that every proper parabolic sub-group is finite are classified: they are exactly affine and compact hyperbolic Coxetergroups [Bou, p.133, exercice 14]. Compact hyperbolic Coxeter groups have rank ≤ [Bou, p.133, exercice 15.c], so in higher rank the only examples are affine Coxetergroups [Bou, p.100, Proposition 10].The following result summarizes the main idea, that is, we have non-vanishingof an L -Betti number in high degree for groups acting on buildings whose Weylgroup is obtained as a perturbation of an affine Coxeter group by a finite Coxetergroup. Theorem 3.6.
Let ( X, G ) be a building in B t of thickness q +1 and irreducible Weylgroup ( W, S ) . Suppose S admits a partition S = J sph (cid:116) J aff such that W sph = (cid:104) J sph (cid:105) is finite and W aff = (cid:104) J aff (cid:105) is an infinite affine Coxeter group. Put | J aff | = n + 1 .Then, for q large enough, β n ( G ) > . roof. Let σ be the simplex corresponding to the subgroup W sph . Hence, D σ = (cid:83) s ∈ J aff D s and thus by 3.5, the space (cid:101) H ∗ ( D σ ) is non-zero in degree n − . Recallthat for q > | S | , we have dim G L ( G ) σ > [DJ, p. 612]. Therefore by Theorem 2.9we have β n ( G ) ≥ (cid:101) H n − ( D σ ) dim G L ( G ) σ > . We now apply the previous result to L -Betti numbers of Kac-Moody groups. Recallfrom Section 2.2 that we can suppose our Kac-Moody groups to be center-freewithout adding conditions on the generalized Cartan matrix. Corollary 3.7.
Let Λ be a center-free Kac-Moody group over a finite field F q withirreducible Weyl group ( W, S ) as in Theorem 3.6, we set S = J sph (cid:116) J aff with J sph (cid:54) = ∅ .Suppose also that its Coxeter matrix has finite entries. Put | J aff | = n + 1 . For q large enough, β k (Λ) (cid:26) = 0 if k ≥ | S | − > if k = 2 n Moreover, Λ is infinite finitely presented, Kazhdan and simple.Proof. Let G be the complete Kac-Moody group associated to Λ . Theorem 3.6applies to G , so β n ( G ) > for large q . Let X be the building coming from theBN-pair of G . The group G acts properly cocompactly on the Davis complex X D .This complex has dimension ≤ | S | − , thus [DJ, 3.4] gives H kct ( G, ρ ) = 0 for k ≥ | S | any quasi-complete representation ( ρ, V ) . Hence the Künneth formula for L -Bettinumbers [Pet, 6.5] gives for large qβ k ( G × G ) (cid:26) = 0 if k ≥ | S | − > if k = 2 n For q > n + 1 , the group Λ is a lattice in G × G by Theorem 2.6 Assertion 1. By[Pet, 5.9], the sequences of L -Betti numbers of Λ and G × G are proportional.Since the entries of the Coxeter matrix of ( W, S ) are finite, Theorem 2.7 tells us G has property ( T ) (for the same bound on q ), thus so does G × G and any latticein G × G . The group Λ is finitely presented and simple in view of Theorem 2.6assertions 2, 3 and because we assumed Z (Λ) = { } .The previous corollary gives a control on the vanishing of L -Betti numbersfor simple Kac-Moody lattices. We can now prove the theorem stated in the in-troduction using Gaboriau’s projective invariance of L -Betti numbers by measureequivalence [Gab, 6.3]. Proof of Theorem 1.1.
It suffices to take an affine diagram with n + 1 generators s , . . . , s n +1 , add a generator s , declare that it does not commute with at leastone s i , i ≥ and that the products s s i have order ≤ . Thus the Coxeter system ( W, S ) , where S sph = { s } , S aff = { s , . . . , s n +1 } , S = S sph (cid:116) S aff and W = (cid:104) S (cid:105) ,satisfies the conditions of the previous corollary with | S | = n + 2 .11et Λ n be a Kac-Moody group over F q with Weyl group ( W, S ) with q as in thecorollary (its Coxeter matrix comes from a generalized Cartan matrix because ofthe hypothesis on the order of the products s s i ). Then β k (Λ n ) (cid:26) = 0 if k ≥ n + 3 ,> if k = 2 n. Hence the groups (Λ n ) have non-proportional sequences of L -Betti numbers, hencethey are pairwise non measure equivalent in view of [Gab, 6.3]. In this section we deal with a concrete family of Coxeter diagrams having a decom-position as in Theorem 3.6. The first aim is to exhibit a concrete family as before.However, in this example we can say more: the previous arguments compute all L -Betti numbers of a pair ( X, G ) in B t with Weyl groups corresponding to theseparticular diagrams. Let ( W, S ) be the Coxeter system defined by the diagram (cid:101) A n, (with n + 1 generators and n ≥ ) as below: (cid:101) A n, Let s , . . . , s n be the generators corresponding to the affine subgroup of type (cid:101) A n − in W and s be the remaining generator, so that ( s s ) = 1 , ( s s ) = 1 , ( s s i ) = 1 for i = 3 , . . . , n .The subgroup generated by s , s and s is of type (cid:101) A , thus it is infinite. Toobtain finite parabolic subgroups one has to consider subsets J ⊂ S that do notcontain these three generators simultaneously.For simplicity, call W i = (cid:104) S \ { s i }(cid:105) and for i (cid:54) = j call W i,j = (cid:104) S \ { s i , s j }(cid:105) . Thesubgroups W and W are of type A n − , thus finite. Therefore, they are maximalfinite parabolic subgroups of W . The subgroup W is of type (cid:101) A n − , thus infinite. Itis affine, so every proper parabolic subgroup of W is finite. Therefore the parabolicsubgroups W ,i for ≤ i ≤ n are maximal finite. Remark.
We can proceed in the same way for Kac-Moody groups with Weyl groupscoming from the same alteration of an affine Coxeter diagram.4 44 12n each case, the structure of maximal finite parabolic subgroups is the same,therefore the following results for W remain the same. The following result completes the computation of the cohomology of D σ for all σ ⊆ ∆ . More precisely, the simplices σ appearing in the following theorem areexactly those whose corresponding parabolic subgroup W J of W is finite but with W J c infinite. Let ∆ i be the face of codimension 1 of ∆ corresponding to the generator s i of W and D i = D ∩ ∆ i . Theorem 4.1.
Let D be the Davis chamber of ( W, S ) of type (cid:101) A n, . . For σ = ∆ , the space D σ = (cid:83) j (cid:54) =0 D j has the cohomology of an ( n − -dimensional sphere. . For σ = ∆ , the space D σ = D ∩ ∂ ∆ has the cohomology of an ( n − -dimensionalsphere. . Let τ ⊂ ∆ be the simplex such that ∆ τ = ∆ ∪ ∆ ∪ ∆ . The space D τ has thecohomology of the circle. . For τ (cid:40) σ (cid:40) ∆ , the space D σ is contractible.Proof. Assertions 1. and 3. follow from 3.4 since W ∆ and (cid:104) s , s , s (cid:105) are infiniteaffine Coxeter groups of type (cid:101) A n − and (cid:101) A . Denote A k = D ∪ . . . ∪ D k , k ≥ , (whatmatters is that A k contains D and D , up to reordering the others, we can alwaysreduce to this case) and D ij = D i ∩ D j . We first show the intersection D ∩ A k iscontractible: D ∩ A k = k (cid:91) l =1 D ∩ D l = D , ∪ D , ∪ k (cid:91) l =3 D ,l . The union A (cid:48) k = (cid:83) kl =3 D ,l is contractible because of 3.2. The intersections A (cid:48) k ∩ D , and A (cid:48) k ∩ D , are contractible. Notice D , ∩ D , = (∆ ∩ ∆ ∩ ∆ ) ∩ D = ∅ . Thus,the intersection D ∩ A k is contractible.Therefore Mayer-Vietoris tells us D ∪ A k has the same cohomology as A k . If k = n , we have A n = D ∆ so from the first assertion we know A n has the cohomologyof an ( n − -dimensional sphere. Thus D ∆ = D ∩ ∂ ∆ = D ∪ A n has the cohomologyof an ( n − -dimensional sphere. If ≤ k < n , then A k is contractible because of3.2, hence the union D ∪ A k is contractible.We now recover the corresponding results for L -Betti numbers of groups in B t with Weyl group (cid:101) A n, . Corollary 4.2.
Let ( X, G ) be a building in B t of thickness q + 1 and Weyl groupof type (cid:101) A n, with n ≥ . Normalize the Haar measure µ on G so that the stabilizer G ∆ of an alcove ∆ has measure 1. Then we have: k ( G ) = (cid:40) dim G L ( G ) ∆ + dim G L ( G ) ∆ k = n − , dim G L ( G ) τ k = 2 , otherwise.Proof. The sum in the formula 2.9 runs over σ ⊆ ∆ such that its correspondingparabolic subgroup W J , J ⊆ S is finite and such that W J c is not finite. Such sim-plices are exactly the ones treated in the theorem. Their non-vanishing cohomologygroups give the result.Thus, by the same arguments as in the previous section, the Künneth formulagives the following more precise statement for Kac-Moody lattices Λ having Weylgroup of type (cid:101) A n, . Moreover, the theorems we use are quantitative: they give us anexplicit formula for β k (Λ) . We do not give the formulas since the only informationthat matters to us is in which degrees β k (Λ) vanishes and it which it does not vanishfor large q . Corollary 4.3.
Let Λ be a center-free Kac-Moody group over a finite field F q withWeyl group of type (cid:101) A n, . For q large enough, β k (Λ) (cid:26) > k ∈ { , n + 1 , n − } , = 0 otherwise . Moreover, Λ is infinite finitely presented, Kazhdan and simple. There is a link between n -sphericity and finiteness properties F n and F P n of aKac-Moody group (over a finite field). Theorem 2.6 3, says that for large q , the2-sphericity condition implies property F , that is, finite presentation. The converseis still a conjecture, but it has at least been proven in particular cases [AG]. Muchless is known for higher finiteness properties.Theorem 2.7 can be stated as ( k + 1) -sphericity implying a certain cohomologicalfiniteness (sometimes called k -Kazhdan).Discrete Kac-Moody groups with Weyl group of type (cid:101) A n, are finitely presented(at least for q > ) since they are -spherical, but they are not -spherical because ofthe subdiagram of type (cid:101) A that we introduced. We can ask if it is possible to obtainstronger finiteness properties for non-affine Kac-Moody groups. Here we present afamily of non-affine Coxeter diagrams that are -spherical but not -spherical. Wecall (cid:101) B n, the Coxeter diagram with n + 1 generators as below: (cid:101) B n, (cid:101) B n, is similar to that of (cid:101) A n, .It contains a subdiagram of type (cid:101) E when n ≥ . If we remove the generator s atthe left, we obtain an affine subdigram of type (cid:101) B n − with generators s , . . . s n . Ifwe take out any other generator s i ( i = 1 , . . . , ) of (cid:101) E , we obtain a maximal finite14arabolic subgroup. Hence the Davis chamber consists of an affine part ( n cubesof dimension n − ) and cubes of dimension n . The result for the diagram (cid:101) A n, remains valid for (cid:101) B n, . Theorem 5.1.
Let D be the Davis chamber of ( W, S ) of type (cid:101) B n, . . For σ = ∆ , D σ = (cid:83) j (cid:54) =0 D s j has the cohomology of an ( n − -dimensional sphere. . For σ = ∆ , D σ = D ∩ ∂ ∆ has the cohomology of an ( n − -dimensional sphere. . Let τ ⊂ ∆ be the simplex such that ∆ τ = (cid:83) i =0 ∆ i . The space D τ has thecohomology of a -dimensional sphere. . For τ (cid:40) σ (cid:40) ∆ , D σ is contractible.Proof. The proof is almost the same as for (cid:101) A n, : assertions 1. and 3. follow from3.4 while 2. and 4. need some adjustment. We keep the same notation as in theother proof. The only difference is that in order to show D ∪ A k has the samecohomology as A k for k ≥ , one has to iterate the contractibility argument 8 timeson D ∩ A k to show it is contractible.Thus we obtain the same corollaries for Kac-Moody groups as in the previoussection. The complete Kac-Moody groups with Weyl group of type (cid:101) B n, have non-vanishing L -Betti numbers (for large thickness) exactly in degrees n − and .At the level of discrete Kac-Moody groups, we obtain an infinite family of infi-nite finitely presented, Kazhdan, -spherical, simple groups that are pairwise non-measure equivalent. By the Künneth formula, their L -Betti numbers vanish exceptin degrees , n + 7 and n − .One cannot have better sphericity properties for a non-affine irreducible Coxeterdiagram. The following proposition has to be stated somewhere. I did not find areference so I give the proof here. Proposition 5.2. A -spherical irreducible Coxeter group is either finite or affine.Proof. Call W the Coxeter group in question, its Coxeter diagram is connected byirreducibility. The proof consists on ruling out all possibilities by looking at theclassification of finite and affine Coxeter groups. More precisely, we look at twofamilies of integers: the valencies of vertices of the Coxeter diagram as a graph andthe labelling of edges of the diagram.If the valency of every vertex of the diagram is ≤ , then the Coxeter diagram of W without labelling is of type A n or (cid:101) A n with n ≥ . If we label edges of a diagramof type A n by numbers ≥ , the -sphericity of W rules out all possibilities, excepthaving extremal edges labelled by : the possible diagrams for W are A n , B n = C n or (cid:101) C n . If we label an edge of a diagram of type (cid:101) A n by a number ≥ , the associatedgroup is not -spherical, hence the only possibility for W in this case is to have adiagram of type (cid:101) A n .If not, there exists a vertex y of valency ≥ . The valency of y cannot be ≥ since this would give directly an infinite subgroup of rank 5. Frow now on y hasvalency 3. Call y , y and y the three neighbors of y . None of these vertices canhave valency ≥ since this would give rise to an infinite subgroup of rank 6. Thethree neighbors cannot have valency ≥ simultaneously since the Coxeter graphwould contain a subgraph of type (cid:101) E that corresponds to an infinite subgroup ofrank 7. Thus we can suppose y has valency 1.15f both y and y have valency 2, then there is a subdiagram of type (cid:101) E or oftype (cid:101) E , which respectively correspond to infinite subgroups of rank 8 and 9. Again, -sphericity rules out these possibilities.We may assume y and y have valency 1, and y has valency ≥ . This implies W is of type (cid:101) B n or (cid:101) D n . Indeed, the graph of W has to contain (cid:101) B n or (cid:101) D n as asubgraph, with possible vertices of higher valency. If such a vertex exists, thenagain there is a subdiagram of type (cid:101) E or of type (cid:101) E . Remark.
In view of this proposition, Theorem 2.7 does not tell us anything aboutcohomological vanishing in rank ≥ for groups with BN-pair acting on a buildingof finite thickness with non-affine Weyl group. When the Weyl group is affine, thesame vanishing result was obtained by Garland in [Gar].If we believe the conjecture saying that property F n implies n -sphericity for aKac-Moody group is true, then the previous proposition proves that every non-affineKac-Moody group over a finite field is at most F . Remark.
In [SWZ], it is used that properties F n are invariant by quasi-isometries[Alo]. They construct finitely presented simple groups that are F n − but not F n foreach n . This gives an infinite family of infinite finitely presented simple groups thatare pairwise not quasi-isometric.This method could not work for non-affine Kac-Moody groups if the previousconjecture is true. Thus the only method known to this date to show there is aninfinite family of non-affine Kac-Moody groups that are pairwise not quasi-isometricis non-distortion as in [CR2]. References [AG] P. Abramenko, Z. Gates,
Finite presentability of Kac-Moody groups over finitefields , arXiv:1912.05611, 2019.[AM] P. Abramenko, B. Mühlherr,
Présentations de certaines BN-paires jumeléescomme sommes amalgamées , Comptes-Rendus de l’Academie des sciences deParis, 325 (1997) 701–706.[Alo] J. M. Alonso,
Finiteness conditions on groups and quasi-isometries , J. PureAppl. Algebra, 95(2):121–129, 1994.[Bou] N. Bourbaki,
Groupes et algèbres de Lie , Chapitres IV–VI, Hermann 1968.[Br] K. Brown,
Buildings , Springer, New York, 1989.[BM] M. Burger, S. Mozes,
Lattices in product of trees , Inst. Hautes Études Sci.Publ. Math., (92):151–194 (2001), 2000.[CR1] P. E. Caprace, B. Rémy,
Abstract simplicity of non-affine Kac–Moody groups ,C. R. Acad. Sci. Paris, Ser. I 342 (2006).[CR2] P. E. Caprace, B. Rémy,
Simplicity and superrigidity of twin building lattices ,Inventiones Math 176 (2009) 169-221.16Dav] M. W. Davis,
Buildings are CAT(0) , in: Geometry and Cohomology in GroupTheory, London Math. Society Lecture Notes 252, Cambridge Univ. Press, Cam-bridge, 1998, pp. 108–123.[DJ] J. Dymara, T. Januszkiewicz,
Cohomology of buildings and of their automor-phism groups , Invent. Math. 150 (2002), no. 3, p. 579–627. 3.[Gab] D. Gaboriau,
Invariants L de relations d’équivalence et de groupes , Publ.Math. Inst. Hautes Etudes Sci., 95:93–150, 2002.[Gar] H. Garland, p-adic curvature and the cohomology of discrete subgroups of p-adic groups , Ann. of Math. (2) 97 (1973), 375–423.[Hat] A. Hatcher, Algebraic topology , Cambridge University Press, 2002.[Pet] H. D. Petersen, L -Betti numbers of locally compact groups , Thesis, Universityof Copenhagen, 2013.[Sau] R. Sauer, (cid:96) -Betti numbers of discrete and non-discrete groups , 2017,arXiv:1509.05234v2.[Röv] C. E. Röver, Constructing finitely presented simple groups that contain Grig-orchuk groups.
J. Algebra, 220(1):284–313, 1999.[SWZ] R. Skipper, S. Witzel, M. Zaremsky,
Simple groups separated by finitenessproperties , Invent. Math. 215 (2019) 713-740.[Ti] J. Tits,