CAT(0) cube complexes and inner amenability
aa r X i v : . [ m a t h . G R ] M a r CAT(0) CUBE COMPLEXES AND INNER AMENABILITY
BRUNO DUCHESNE, ROBIN TUCKER-DROB, AND PHILLIP WESOLEK
Abstract.
We here consider inner amenability from a geometric andgroup theoretical perspective. We prove that for every non-elementaryaction of a group G on a finite dimensional irreducible CAT(0) cubecomplex, there is a nonempty G -invariant closed convex subset such thatevery conjugation invariant mean on G gives full measure to the stabi-lizer of each point of this subset. Specializing our result to trees leadsto a complete characterization of inner amenability for HNN-extensionsand amalgamated free products. One novelty of the proof is that itmakes use of the existence of certain idempotent conjugation-invariantmeans on G .We additionally obtain a complete characterization of inner amenabil-ity for permutational wreath product groups. One of the main ingredi-ents used for this is a general lemma which we call the location lemma,which allows us to “locate” conjugation invariant means on a group G relative to a given normal subgroup N of G . We give several furtherapplications of the location lemma beyond the aforementioned charac-terization of inner amenable wreath products. Contents
1. Introduction 12. Preliminaries 43. Location lemma and wreath products 74. Inner amenability and CAT(0) cube complexes 185. Trees, amalgams and inner amenability 32References 341.
Introduction
A discrete group G is said to be inner amenable if there exists anatomless mean on G which is invariant for the action of G on itself byconjugation. This notion was isolated by Effros in [16] in order to elucidate Date : March 2019. Our definition is slightly different from the definition given in [16], where the meanis not required to be atomless, but rather supported on G − { G } . However, the twodefinitions coincide for infinite conjugacy class (ICC) groups, which were the main concernof [16]. Murray and von Neumann’s proof that the group von Neumann algebra ofthe free group on two generators has no nontrivial asymptotically centralsequences [32]. Similar connections between inner amenability and centralsequences were later found by Choda [12] and Jones and Schmidt [21] inthe context of ergodic theory. These connections to operator algebras andergodic theory have continued to provide a rich context and motivation forthe study of inner amenability; see, e.g., [38, 23, 25, 24, 11, 26, 27, 37, 19,33, 15, 20, 3, 22, 28]. Perhaps because of this, inner amenability has beenstudied primarily by virtue of its relevance to these two fields (with a fewexceptions, e.g., [1, 2, 36, 18]). In this article, by contrast, we explore inneramenability from the perspectives of geometry and group theory.1.A.
Conjugation invariant means on groups acting on trees.
Wegive a complete characterization of inner amenability for groups built viaamalgamated free products and HNN-extensions.We say that an amalgamated free product G = A ∗ H B is nondegenerate if H = A , H = B , and the index of H in either A or B is at least three. Theorem 1.1 (Corollary 5.3) . Let G = A ∗ H B be a nondegenerate amal-gamated free product. Then:(1) Every conjugation invariant mean on G concentrates on H .(2) G is inner amenable if and only if there exist conjugation invariant,atomless means m A on A and m B on B with m A ( H ) = m B ( H ) = 1 ,and m A ( E ) = m B ( E ) for every E ⊆ H .In particular, if G is inner amenable, then so are each of the groups A , B ,and H . Let H be a subgroup of a group K and let φ : H → K be an injectivegroup homomorphism. The associated HNN-extensionHNN( K, H, φ ) := h K, t | tht − = ϕ ( h ) , ( h ∈ H ) i is said to be ascending if either H = K or φ ( H ) = K . Otherwise, it iscalled non-ascending . Theorem 1.2 (Corollary 5.3) . Let G = HNN( K, H, φ ) be a non-ascendingHNN extension. Then:(1) Every conjugation invariant mean on G concentrates on H .(2) G is inner amenable if and only if there exists a conjugation invari-ant, atomless mean m on K with m ( H ) = 1 , and m ( E ) = m ( φ ( E )) for every E ⊆ H .In particular, if G is inner amenable, then so are the groups K and H . Theorems 1.1 and 1.2 are group theoretical consequences of a more generalgeometric statement, regarding groups acting on trees. Given a group G anda G -set X , we denote by G x the stabilizer subgroup of G at x ∈ X . AT(0) CUBE COMPLEXES AND INNER AMENABILITY 3
Theorem 1.3.
Suppose that a group G acts by automorphisms on a tree T . Assume that G does not fix a vertex, an edge, an end, or a pair of ends.Then there is a nonempty G -invariant subtree T of T such that m ( G x ) = 1 for every conjugation invariant mean m on G and every vertex x of T . This theorem follows directly from Theorem 5.1 by considering the uniqueminimal G -invariant subtree T of T .1.B. Groups acting on CAT( ) cube complexes. Theorem 1.3 is itselfa special case of the following general theorem concerning groups acting onfinite dimensional CAT(0) cube complexes.
Theorem 1.4 (Theorem 4.8) . Let G be a group acting essentially and non-elementarily on an irreducible finite dimensional CAT(0) cube complex X .Then there exists a nonempty G -invariant closed convex subspace X of X such that m ( G x ) = 1 for every conjugation invariant mean m on G andevery x ∈ X . Theorem 1.3 corresponds precisely to the special case of Theorem 1.4where X is one dimensional.The essentiality and irreducibility assumptions in Theorem 1.4 can beremoved at the cost of passing to a finite index subgroup; see Corollary 4.9.This allows us, for instance, to characterize when a graph product of groupsis inner amenable in Theorem 4.14. Graph products generalize both directproducts and free products of groups; examples of graph products of groupsinclude right-angled Artin groups and right-angled Coxeter groups.These examples, along with Theorems 1.1 and 1.2, illustrate that themost interesting applications of Theorem 1.4 concern actions which are notnecessarily proper. While Theorem 1.4 easily implies that groups actingproperly and non-elementarily on finite dimensional CAT(0) cube complexesare not inner amenable (see Corollary 4.10), this result can also be deducedfrom other results in the literature; see Remarks 4.11 and 4.12.The proof of Theorem 1.4 requires substantially more work than the one-dimensional case covered by Theorem 1.3. One novelty of the proof in thehigher dimensional setting, which we now briefly describe, is that it makesuse of the existence of certain idempotent conjugation invariant means.The proof begins by observing that each conjugation invariant mean m on G must concentrate on the set of group elements which act elliptically(Proposition 4.3). The next step uses a transversality argument (Proposition4.5) to show that, for each half-space h of X , there is a m -conull set ofgroup elements which fix some point in h . At this point, the proof in theone dimensional setting is essentially complete, but the situation in higherdimensions becomes more complicated; after moving to a minimal convexsubspace X of X and fixing some x ∈ X , we adapt an argument of Capraceand Sageev [8] to our setting (Lemma 4.6) to show that for each x ∈ X theintegral ϕ ( x ) := R G d ( x, gx ) dm ( g ) is finite, and hence (using the CAT(0)inequality) the function x ϕ ( x ) achieves a unique minimum at some point B. DUCHESNE, R. TUCKER-DROB, AND P. WESOLEK z ∈ X (Lemma 4.7). If the mean m is idempotent under convolution, thenwe can easily deduce (using discreteness of G -orbits on X ) that this point z is fixed by a m -conull set of group elements, which would complete theproof. Fortunately, by using a simple stationarity argument (Lemma 2.2)combined with Ellis’s Lemma, we are able to reduce the proof of Theorem 1.4to the special case where the means m under consideration are additionallyassumed to be idempotent.1.C. Wreath products and the location lemma.
We obtain a completecharacterization of inner amenability for wreath products.
Theorem 1.5 (Theorem 3.9) . Let H = 1 and K be discrete groups, let X be a set on which K acts, and let G := H ≀ X K be the (restricted) wreathproduct. Then G is inner amenable if and only if one of the following holds(1) The action K y X admits an atomless K -invariant mean;(2) H is inner amenable and the action K y X has a finite orbit;(3) There is an atomless K -conjugation invariant mean m on K satis-fying m ( K x ) = 1 for all x ∈ X . One of the key ingredients to the proof of Theorem 1.5 is Lemma 3.5,which gives a way of locating various conjugation invariant means on agroup, relative to some normal subgroup. Lemma 3.5 also leads to a com-plete characterization of when the commutator subgroup of an inner amenablegroup is itself inner amenable (see Corollary 3.8 and the paragraph preced-ing it). To state a further consequence, we first make a definition. If m is amean on a group G , then we define ker( m ) := { g ∈ G : m ( C G ( g )) = 1 } . It iseasy to see that ker( m ) is a subgroup of G , and if m is conjugation-invariant,then ker( m ) is a normal subgroup of G . Theorem 1.6 (Corollary 3.6) . Let G be an inner amenable group with nonontrivial finite normal subgroups, and let N be a normal subgroup of G .Then either there exists an atomless conjugation invariant mean on G whichconcentrates on N , or else N ≤ ker( m ) for every conjugation invariant mean m on G .Moreover, there exists an atomless conjugation invariant mean m on G with either m ([ N, N ]) = 1 or N ≤ ker( m ) . Acknowledgments.
We would like to thank Yair Hartman for his manycontributions to this project. We would also like to thank Amine Marrakchifor explaining to us another proof of Theorem 1.1 and allowing us to includehis argument in Remark 5.4. RTD was supported in part by NSF grant DMS1600904. BD was supported in part by French projects ANR-14-CE25-0004GAMME and ANR-16-CE40-0022-01 AGIRA.2.
Preliminaries
For G a group, H ≤ G , and g ∈ G , we write g H := { hgh − | h ∈ H } . AT(0) CUBE COMPLEXES AND INNER AMENABILITY 5
Let m and n be means on a group G . The convolution of m and n , denoted m ∗ n , is the mean defined by( m ∗ n )( A ) := Z g ∈ G n ( g − A ) dm ( g )for A ⊆ G . We denote by ˇ m the mean on G defined by ˇ m ( A ) := m ( A − ) for A ⊆ G .If either m or n is atomless, then m ∗ n is atomless as well; this is adirect computation under the assumption that n is atomless, and it is ashort exercise under the assumption that m is atomless. Likewise, if m and n are both invariant under a subgroup H of Aut( G ) then so are m ∗ n andˇ m .It follows readily from the definition that m m ∗ n is continuous forthe weak- ∗ topology but, the continuity of n m ∗ n does not hold primafacie . Lemma 2.1.
Let m be a mean on G and let H be a subgroup of G . Then ( ˇ m ∗ m )( H ) ≥ P gH ∈ G/H m ( gH ) . In particular, if there is some g ∈ G suchthat m ( gH ) > , then ( ˇ m ∗ m )( H ) > .Proof. For any finite subset F ⊆ G/H , by finite additivity of m we have( ˇ m ∗ m )( H ) = Z G m ( xH ) dm ( x ) ≥ X gH ∈ F Z gH m ( xH ) dm ( x )= X gH ∈ F Z gH m ( gH ) dm ( x )= X gH ∈ F m ( gH ) . Taking the supremum over all such F proves the lemma. (cid:3) Lemma 2.2.
Let H be a subgroup of a group G and let m and n be meanson G . Assume that n ( H ) = 1 . If either n ∗ m = n or m ∗ n = n then m ( H ) = 1 as well.Proof. Suppose first that n ∗ m = n . Then1 = n ( H ) = ( n ∗ m )( H ) = Z G m ( g − H ) dn ( g )= Z H m ( g − H ) dn ( g ) = Z H m ( H ) dn ( g ) = m ( H ) , where the fourth and last equalities hold since n concentrates on H .Suppose now that m ∗ n = n . Then n ( g − H ) = 0 for all g H , hence1 = n ( H ) = ( m ∗ n )( H ) = Z G n ( g − H ) dm ( g )= Z H n ( H ) dm ( g ) = Z H dm ( g ) = m ( H ) . (cid:3) B. DUCHESNE, R. TUCKER-DROB, AND P. WESOLEK
Proposition 2.3.
Let G be a discrete group and let H be a subgroup of Aut( G ) . Let m be an atomless H -invariant mean on G . Then there existsanother atomless H -invariant mean n on G satisfying:(i) If K is a subgroup of G with m ( K ) = 1 then n ( K ) = 1 .(ii) If K is a subgroup of G with m ( K ) = 1 then n ( L ) = 1 for every H -invariant subgroup L of G with | K : K ∩ L | < ∞ .In particular, if G is inner amenable, then there is an atomless conjugationinvariant mean m on G , such that m ( G ) = 1 for every finite index subgroup G of G .Proof. Let C be the collection of all subgroup K of G with m ( K ) = 1,and let C be the collection of all H -invariant subgroups L of G with | K : K ∩ L | < ∞ for some K ∈ C . Observe that C is a directed set under reverseinclusion: if L , L ∈ C are H -invariant and K , K ∈ C are such that | K i : K i ∩ L i | < ∞ , then K ∩ K ∈ C and | K ∩ K : K ∩ K ∩ L ∩ L | < ∞ since both | K ∩ K : ( K ∩ L ) ∩ K | ≤ | K : K ∩ L | < ∞ and | ( K ∩ L ) ∩ K : ( K ∩ L ) ∩ ( K ∩ L ) | ≤ | K : K ∩ L | < ∞ .Since m is atomless and H -invariant, so is ˇ m ∗ m . If K ∈ C then wehave ( ˇ m ∗ m )( K ) = R K m ( kK ) dm ( k ) = R K m ( K ) dm ( k ) = 1. Moreover, if K ∈ C and L is any subgroup of G with | K : K ∩ L | < ∞ , then since m is finitely additive there must be some g ∈ G such that m ( g L ) >
0, andhence ( ˇ m ∗ m )( L ) > L is additionally H -invariant(i.e., if L ∈ C ), then the normalized restriction, n L , of ˇ m ∗ m to L , (definedby n L ( A ) := ( ˇ m ∗ m )( A ∩ L ) / ( ˇ m ∗ m )( L )) is an H -invariant atomless meanwith n L ( L ) = 1 and n L ( K ) = 1 for all K ∈ C . The assignment L n L is then a net from the directed set C to the compact space of all atomless H -invariant means on G . Any cluster point n of this net then satisfies (i)and (ii). (cid:3) The following proposition will be improved significantly in Lemma 3.3,although it is important enough that we state it now.
Proposition 2.4.
Suppose that G is inner amenable and let N be a normalsubgroup of G . Then either(1) There is an atomless G -conjugation invariant mean on N , or(2) G/N is inner amenable.In particular, either N is inner amenable or G/N is inner amenable.Proof.
Let m be an atomless conjugation invariant mean on G . Let p : G → G/N denote the natural projection map. Then p ∗ m is a conjugationinvariant mean on G/N . If p ∗ m is atomless, then G/N is inner amenable sowe are done. Otherwise, if p ∗ m has an atom, then there is some g ∈ G suchthat m ( gN ) >
0. Then ( ˇ m ∗ m )( N ) > m ∗ m to N is an atomless G -invariant mean on N . (cid:3) AT(0) CUBE COMPLEXES AND INNER AMENABILITY 7
Proposition 2.5.
Let N be a finite normal subgroup of a group G . Thenfor every conjugation invariant mean m on G/N , there is a conjugationinvariant mean m on G which projects to m .Proof. Let m be a conjugation invariant mean on G/N . Define the mean m on G by m ( A ) := R gN ∈ G/N | A ∩ gN | / | N | dm ( gN ). This clearly works. (cid:3) Proposition 2.6.
Let H be a subgroup of a group G . Assume that (a) There is an atomless H -conjugation invariant mean m H on G , (b) The action G y G/H is amenable with G -invariant mean m G/H .Then G is inner amenable, as witnessed by the atomless G -conjugation in-variant mean m := Z gH ∈ G/H gm H g − dm G/H ( gH ) In particular, if N is a normal subgroup of G which is inner amenable, andif G/N is amenable, then G is inner amenable and, moreover, there is anatomless G -conjugation invariant mean m on G with m ( N ) = 1 .Proof. This is a straightforward computation. (cid:3)
Proposition 2.7.
Let G be a group.(1) (Giordano, de la Harpe [18] ) Let H be a finite index subgroup of G .Then G is inner amenable if and only if H is inner amenable.(2) Let N be a finite normal subgroup of G . Then G is inner amenableif and only if G/N is inner amenable.Proof. (1) follows from Propositions 2.3 and 2.6, and (2) follows from Propo-sition 2.5. (cid:3) Location lemma and wreath products
Lifting almost invariant probability measures.
Let X and Y be G -sets and let π : X → Y be a G -map from X to Y . Let ˜ X := X ⊗ π X = { ( x , x ) ∈ X : π ( x ) = π ( x ) } , so that ˜ X is a G -invariant subset of X (for the diagonal G -action). Let p ∈ ℓ ( X ) be a probability vector on X (i.e., a nonnegative unit vector), and view p as a probability measure on X .For y ∈ Y let p y be the normalized restriction of p to π − ( y ) (put p y = 0if p ( π − ( y )) = 0). Define ˜ p := P y ∈ Y p ( π − ( y ))( p y ⊗ p y ), so that ˜ p is aprobability vector on ˜ X . Lemma 3.1.
For any g ∈ G we have k g ˜ p − ˜ p k ≤ k gp − p k .Proof. For each y ∈ Y let q ( y ) := p ( π − ( y )) so that ˜ p = P y ∈ Y q ( y )( p y ⊗ p y ).We write gp y for the translate of the function p y by g . We have k g ˜ p − ˜ p k = k X y ∈ Y (cid:0) q ( y )( gp y ⊗ gp y ) − q ( gy )( p gy ⊗ p gy ) (cid:1) k ≤ X y ∈ Y | q ( y ) − q ( gy ) |k gp y ⊗ gp y k + X y ∈ Y q ( gy ) k ( gp y ⊗ gp y ) − ( p gy ⊗ p gy ) k Since k gp y ⊗ gp y k ≤ p y = 0), the first sum is bounded by k gp − p k . Let Y := { y ∈ Y : q ( y ) > } . The second sum is bounded by X y ∈ Y q ( gy ) (cid:0) k ( gp y − p gy ) ⊗ gp y k + k p gy ⊗ ( gp y − p gy ) k (cid:1) ≤ X y ∈ Y q ( gy ) k gp y − p gy k = 2 X y ∈ Y X x ∈ π − ( gy ) | q ( gy ) p y ( g − x ) − p ( x ) |≤ k gp − p k + 2 X y ∈ Y X x ∈ π − ( gy ) | q ( gy ) p y ( g − x ) − p ( g − x ) | = 2 k gp − p k + 2 X y ∈ Y | q ( gy ) q ( y ) − | X x ∈ π − ( gy ) p ( g − x ) ≤ k gp − p k + 2 X y ∈ Y | q ( gy ) − q ( y ) |≤ k gp − p k . (cid:3) Conjugation invariant means on normal subgroups.
In whatfollows, for a subgroup M of a group G , and a nonempty subset S of G , wedefine C G/M ( S ) := { g ∈ G : g ( sM ) g − = sM for all s ∈ S } . For h ∈ G we write C G/M ( h ) for C G/M ( { h } ). If M is the trivial subgroup,we simply denote C G ( S ) for the centralizer of S in G . Proposition 3.2.
Let M be a subgroup of G , let S be a nonempty subset of G , and let h S i be the subgroup generated by S . Then:(1) C G/M ( S ) is a subgroup of G contained in the normalizer, N G ( M ) ,of M in G .(2) Suppose that S ⊆ N G ( M ) . Then C G/M ( h S i ) = C G/M ( S ) .(3) Suppose that M and S are finite and S ⊆ N G ( M ) . Then C G ( S ) ∩ C G ( M ) is a finite index subgroup of C G/M ( S ) .Proof. (1): C G/M ( S ) is clearly a group. To see that C G/M ( S ) is containedin N G ( M ), observe that for g ∈ C G/M ( S ) and s ∈ S we have (a) M = s − g − sM g , and (b) s − gsg − ∈ M . By (b) we have g − M = s − g − sM ,and applying this to (a) we see that M = s − g − sM g = g − M g , and hence g ∈ N G ( M ).(2): This is clear.(3): Since M is finite, for each s ∈ S the group C G ( M ) ∩ C G/M ( s ) hasfinite index in C G/M ( s ), and hence the group C G ( sM ) has finite index in C G/M ( s ), being the kernel of the homomorphism ( C G ( M ) ∩ C G/M ( s )) → M , g [ s, g ]. Therefore, since S is finite, C G ( S ) ∩ C G ( M ) = T s ∈ S C G ( sM ) hasfinite index in C G/M ( S ) = T s ∈ S C G/M ( s ). (cid:3) AT(0) CUBE COMPLEXES AND INNER AMENABILITY 9
Let m be a mean on a group G . We defineker( m ) := { g ∈ G : m ( C G ( g )) = 1 } . It is easy to see that ker( m ) is a subgroup of G . If m is invaraint underconjugation by a subgroup K of G , then ker( m ) is normalized by K . Observethat if H is any finitely generated subgroup of ker( m ), then m ( C G ( H )) = 1.Thus, if the mean m is atomless then C G ( H ) is infinite for every finitelygenerated subgroup H of ker( m ).Given two elements h and k in a group G , we denote their commutatorby [ h, k ] := hkh − k − . If H and K are subgroups of G then we define [ H, K ]to be the subgroup [
H, K ] := h{ [ h, k ] : h ∈ H, k ∈ K }i . Clearly [
H, K ] = [
K, H ]. Note that the group [
H, K ] is normalized by both H and K . To see this, observe that if [ h, k ] is any generator for [ H, K ],where h ∈ H and k ∈ K , then for any h ∈ H we have h [ h, k ] h − =[ h h, k ][ h , k ] − ∈ [ H, K ]. Hence [
H, K ] is normalized by H , and by sym-metry it is also normalized by K .The following lemma, along with the more general Lemma 3.5, is oneof our main tools for understanding the “location” of conjugation invariantmeans on a group. Lemma 3.3.
Let N , G and K be subgroups of a group G , with N nor-malized by K . Assume that there is no nontrivial finite subgroup of G ∩ [ G , K ∩ N ] which is normalized by K . Then at least one of the followingholds: (1) There is an atomless K -conjugation invariant mean on G which con-centrates on G ∩ [ G , K ∩ N ] , (2) For every K -conjugation invariant mean m on G which concentrateson G , we have K ∩ N ≤ ker( m ) . Remark 3.4.
The assumption that there is no nontrivial finite subgroup of G ∩ [ G , K ∩ N ] which is normalized by K can be removed at the expenseof making the conclusion of the lemma a bit messier to state. See Lemma3.5 below. Proof of Lemma 3.3.
Assume that (1) fails and we will show that (2) holds.Fix then a mean m on G with m ( G ) = 1 and which is invariant underconjugation by K . We must show that m ( C G ( h )) = 1 for each h ∈ K ∩ N .We first claim that there is no g ∈ G −{ G } with g K finite and contained in G ∩ [ G , K ∩ N ]. For suppose otherwise. If the group h g K i were finite, then itwould be a nontrivial finite subgroup of G ∩ [ G , K ∩ N ] which is normalizedby K , a contradiction. So the group h g K i would have to be infinite. Butevery element of this group has a finite orbit under conjugation by K , sosince h g K i is infinite we can find an atomless K -conjugation invariant meanwhich concentrates on h g K i ≤ G ∩ [ G , K ∩ N ], which contradicts thehypothesis that (1) fails. Let X = G and let α : K y X denote the conjugation action of K on X , α ( k ) x := kxk − . By the Hahn-Banach Theorem we may find a net ( p i ) i ∈ I of probability vectors on X , which weak ∗ -converges to m in ℓ ∞ ( X ) ∗ , andsatisfies k α ( k )( p i ) − p i k → k ∈ K . Since m concentrates on G , wemay additionally assume that each p i concentrates on G . Let Y denote theset of all orbits of α ( K ∩ N ) on X , and let π : X → Y denote the projectionmap, π ( x ) := α ( K ∩ N ) x . Since K normalizes K ∩ N , K naturally acts on Y so that π is a K -equivariant map. Let˜ X := X ⊗ π X = { ( x , x ) ∈ X × X : π ( x ) = π ( x ) } , and let ˜ α : K y ˜ X denote the diagonal action of K , i.e., ˜ α ( k )( x , x ) :=( α ( k )( x ) , α ( k )( x )) = ( kx k − , kx k − ). For each probability vector p on X and each y ∈ Y , let p y be the normalized restriction of p to π − ( y ) (wherewe put p y = 0 if p ( π − ( y )) = 0), and define the probability vector ˜ p on ˜ X by ˜ p := P y ∈ Y p ( π − ( y )) p y ⊗ p y . By Lemma 3.1, for each k ∈ K we have k ˜ α ( k )˜ p i − ˜ p i k →
0. Thus, after moving to a subnet of ( p i ) i ∈ I if necessary, wemay assume without loss of generality that the net (˜ p i ) i ∈ I weak ∗ -convergesin ℓ ∞ ( ˜ X ) ∗ to a mean ˜ m on ˜ X which is invariant under ˜ α ( K ). Then ˜ m concentrates on ˜ X ∩ ( G × G ) since each ˜ p i does.For each ( x , x ) ∈ ˜ X ∩ ( G × G ) we have x , x ∈ G and x = hx h − for some h ∈ K ∩ N , hence x x − = x hx − h − ∈ G ∩ [ G , K ∩ N ]. We let ϕ : ˜ X → X be the map ϕ ( x , x ) := x x − . The map ϕ is K -equivariant,i.e., ϕ ◦ ˜ α ( k ) = α ( k ) ◦ ϕ , hence the pushforward ϕ ∗ ˜ m , of ˜ m under ϕ , is a K -conjugation invariant mean on X = G satisfying ( ϕ ∗ ˜ m )( G ∩ [ G , K ∩ N ]) =1. By our assumption, the mean ϕ ∗ ˜ m must by purely atomic and, being K -invariant, must therefore concentrates on the collection of finite orbits of α ( K ) which are contained in G ∩ [ G , K ∩ N ]. As we saw above, the onlysuch orbit is the trivial orbit of the identity element. This means that ϕ ∗ ˜ m is the point mass at the identity element of G , and hence ˜ m ( △ X ) = 1, where △ X := { ( x, x ) : x ∈ X } ⊆ ˜ X .For each i ∈ I let q i = π ∗ p i . For i ∈ I and y ∈ Y , let s i ( y ) :=sup x ∈ π − ( y ) p yi ( x ) = max x ∈ π − ( y ) p yi ( x ). Since (˜ p i ) i ∈ I weak ∗ -converges to ˜ m ,we have 1 = ˜ m ( △ X ) = lim i ˜ p i ( △ X ), and so1 = lim i ˜ p i ( △ X ) = lim i X y ∈ Y q i ( y ) X x ∈ π − ( y ) p yi ( x ) ≤ lim i X y ∈ Y q i ( y ) s i ( y ) X x ∈ π − ( y ) p yi ( x )= lim i Z Y s i ( y ) dq i ( y ) . (3.1)For each i ∈ I and y ∈ Y choose some x yi ∈ π − ( y ) with p yi ( x yi ) = s i ( y ). Let r := (although any number strictly between and 1 will do), and define Y i := { y ∈ Y : s i ( y ) > r } and X i := { x yi : y ∈ Y } . Since 0 ≤ s i ( y ) ≤ AT(0) CUBE COMPLEXES AND INNER AMENABILITY 11 have Z s i dq i = Z Y i s i dq i + Z Y \ Y i s i dq i ≤ q i ( Y i ) + r (1 − q i ( Y i ))= r + (1 − r ) q i ( Y i ) , and hence, by (3.1),(3.2) lim i p i ( π − ( Y i )) = lim i q i ( Y i ) ≥ lim i [ Z s i dq i − r ] / (1 − r ) = 1 . In addition,(3.3) lim i p i ( X i ) = lim i X y ∈ Y p i ( x yi ) = lim i Z Y s i ( y ) dq i ( y ) = 1 . Fix now any h ∈ K ∩ N . For each x = x yi ∈ ( X i ∩ π − ( Y i )) \ C G ( h ), we have α ( h ) x = x , hence p yi ( α ( h ) x ) < − p yi ( x ) < − r , and p i ( x )(2 r − ≤ q i ( y )( r − (1 − r )) ≤ q i ( y )( p yi ( x ) − p yi ( α ( h ) x ))= | p i ( x ) − p i ( α ( h ) x ) | , hence p i (( X i ∩ π − ( Y i )) \ C G ( h )) ≤ k p i − α ( h ) p i k / (2 r − m ( X \ C G ( h )) = lim i p i ( X \ C G ( h )) = lim i p i (( X i ∩ π − ( Y i )) \ C G ( h )) ≤ lim i k p i − α ( h ) p i k r −
1= 0 . Therefore m ( C G ( h )) = 1 for all h ∈ K ∩ N , as was to be shown. This finishesthe proof (cid:3) Lemma 3.5 (Location lemma) . Let N , G and K be subgroups of a group G , with N normalized by K . Let P be a subgroup of G which is normalizedby G and contains G ∩ [ G , K ∩ N ] (e.g., P = G ∩ [ G , K ∩ N ] ), and let M be defined by M := { g ∈ G : g K is finite and contained in P } , so that M is a subgroup of P which is normalized by K . Then at least oneof the following holds: (1) There is an atomless K -conjugation invariant mean on G which con-centrates on P , (2) The group M is finite, and for every K -conjugation invariant mean m on G which concentrates on M G , we have m ( C G/M ( h )) = 1 forevery h ∈ M K ∩ N .Proof of Lemma 3.5. Assume (1) fails and we will show that (2) holds. Sup-pose toward a contradiction that M is infinite. Then we could finite a se-quence C , C , . . . , of distinct finite K -orbits for the conjugation action of K on M . If we let m C n denote normalized counting measure on C n , then any weak ∗ -cluster point of m C , m C . . . , is an atomless K -conjugation invariantmean on G which concentrates on P , a contradiction. Therefore, M mustbe finite. Let N G ( M ) denote the normalizer of M in G . We next establishthe following:( ∗ ) If m is any K -conjugation invariant mean on G which concentrates on M G , then m ( N G ( M )) = 1.To see this, let m be a K -conjugation invariant mean on G with m ( M G ) =1. Since P is normalized by G and M ≤ P , we obtain a mean n on G ,concentrating on P , defined by n := R g ∈ MG n g − Mg dm ( g ), where n g − Mg denotes normalized counting measure on the finite group g − M g ≤ P for g ∈ M G . Since m is invariant under conjugation by K and M is normalizedby K , the mean n is invariant under conjugation by K . Since (1) fails, themean n must be purely atomic and hence we must have n ( M ) = 1, whichimplies that m ( N G ( M )) = 1. This proves ( ∗ ).Let G ′ be the quotient group G ′ := N G ( M ) /M , and let proj : N G ( M ) → G ′ be the projection map. Define K ′ := proj( K ), G ′ := proj( G ∩ N G ( M )), N ′ := proj( N ∩ N G ( M )), and P ′ := proj( P ∩ N G ( M )) ≥ G ′ ∩ [ G ′ , K ′ ∩ N ′ ].Observe that the group M ′ := { x ∈ G ′ : x K ′ is finite and contained in P ′ } is trivial: every orbit of the conjugation action K ′ y G ′ , is the image underproj of an orbit of the conjugation action K y N G ( M ), and since M isfinite this means that finite orbits of K ′ y G ′ are images of finite orbitsof K y N G ( M ), hence M ′ is trivial. In particular, there is no nonidentity x ∈ G ′ − { G ′ } such that x K ′ is finite and contained in G ′ ∩ [ G ′ , K ′ ∩ N ′ ].This establishes all of the hypotheses of Lemma 3.3 for the groups G ′ , N ′ , G ′ ,and K ′ in place of G, N, G , and K respectively. Observe that (1) of Lemma3.3 fails for these groups: if m ′ were an atomless K ′ -conjugation invariantmean on G ′ which concentrates on G ′ ∩ [ G ′ , K ′ ∩ N ′ ], then we would ob-tain an atomless K -conjugation invariant mean m on G concentrating on P , defined by m ( A ) := R gM ∈ P ′ | A ∩ gM || M | dm ′ ( gM ), a contradiction. Thus,(2) of Lemma 3.3 must hold, i.e., for every K ′ -conjugation invariant mean m ′ on G ′ which concentrates on G ′ , we have m ′ ( C G ′ ( h ′ )) = 1 for every h ′ ∈ K ′ ∩ N ′ = proj( M K ∩ N ). In particular, by ( ∗ ), this applies to allmeans m ′ which are the projection, m ′ := proj ∗ m , of some K -conjugationinvariant mean m on G which concentrates on M G . Therefore, for all suchmeans m we have m ( C G/M ( h )) = 1, as was to be shown. (cid:3) As a Corollary to Lemma 3.3 we obtain:
Corollary 3.6.
Let N be a normal subgroup of a group G and assume thatthere is no nontrivial finite normal subgroup of G contained in [ G, N ] . Thenat least one of the following holds: ( A. There is an atomless G -conjugation invariant mean which concen-trates on [ N, N ] . AT(0) CUBE COMPLEXES AND INNER AMENABILITY 13 ( A. There is an atomless G -conjugation invariant mean which concen-trates on [ G, N ] , and for every G -conjugation invariant mean m on G which concentrates on N , we have that N ≤ ker( m ) . ( A. For every G -conjugation invariant mean m on G we have N ≤ ker( m ) .In particular, if G is inner amenable then there exists an atomless G -conjugation invariant mean m on G with either m ([ N, N ]) = 1 or N ≤ ker( m ) .Proof. Assume that ( A.
1) and ( A.
2) fail and we will prove that ( A.
3) holds.Apply Lemma 3.3 with G = N and K = G . The assumption that ( A. G -conjugation invariant mean m on G which con-centrates on N we have N ≤ ker( m ). But this means that the second partof ( A.
2) holds, so the assumption that ( A.
2) fails then implies that thereis no atomless G -conjugation invariant mean which concentrates on [ G, N ].Applying Lemma 3.3 again, but this time using G = K = G , this meansthat (1) of that lemma fails, and hence (2) must hold, i.e., for every G -conjugation invariant mean m on G we have N ≤ ker( m ), which is precisely( A. (cid:3) Corollary 3.7.
Let G be an inner amenable group and let N be a normalfinitely generated subgroup of G . Then there is an atomless G -conjugationinvariant mean m on G with either m ([ N, N ]) = 1 or m ( C G ( N )) = 1 .Proof. Note that if [
G, N ] contains no nontrivial finite normal subgroup of G then this follows immediately from Corollary 3.6. In the general case, wewill argue similarly, but apply Lemma 3.5 instead of Lemma 3.3. Supposethat there is no atomless G -conjugation invariant mean which concentrateson [ N, N ]. Then (1) of Lemma 3.5 fails (applied to K = G , G = N ,and P = [ N, N ]), so the group M , of all elements of [ N, N ] with finite G -conjugacy classes, is finite, and every G -conjugation invariant mean m which concentrates on N must satisfy m ( C G/M ( N )) = 1 (since N is finitelygenerated). We now consider cases.Case 1: There exists an atomless G -conjugation invariant mean m whichconcentrates on N . In this case, by what we already showed we have m ( C G/M ( N )) = 1. Since M is finite and N is finitely generated, the group C G ( N ) has finite index in C G/M ( N ). Therefore, by Proposition 2.3, we canfind an atomless G -conjugation invariant mean on G which concentrates on C G ( N ), as was to be shown.Case 2: There does not exist an atomless G -conjugation invariant mean m which concentrates on N . In this case, applying Lemma 3.5 again, butthis time with K = G = G and P = N , we see that (1) of that lemmafails, and hence (2) holds, i.e., the group M , of all elements of N with finite G -conjugacy class, is finite, and every G -conjugation invariant mean m on G satisfies m ( C G/M ( N )) = 1 (once again, since N is finitely generated). Since by assumption G is inner amenable, we can find an atomless G -conjugationinvariant mean m on G , which must necessarily satisfy m ( C G/M ( N )) = 1.We now argue as in Case 1. Since M is finite and N is finitely generated, thegroup C G ( N ) has finite index in C G/M ( N ), so we can apply Proposition 2.3to find an atomless G -conjugation invariant mean satisfying m ( C G ( N )) =1. (cid:3) Taking N = K = G = G and P = [ G, G ] in Lemma 3.5 shows that, if G is an inner amenable group, then either the commutator subgroup of G isinner amenable, or else there exists a finite normal subgroup M of G suchthat G/M admits an asymptotically central sequence (i.e., the centralizer ofevery finite subset of
G/M is infinite). If G is additionally finitely generatedand has no nontrivial finite normal subgroups, then we obtain: Corollary 3.8.
Let G be a group. Assume that G is finitely generated andcontains no nontrivial finite normal subgroups. If G is inner amenable thenexactly one of the following holds:(1) The commutator subgroup [ G, G ] of G is inner amenable,(2) The center, Z ( G ) , of G , is infinite, [ G, G ] ∩ Z ( G ) = 1 , and ev-ery [ G, G ] -conjugation invariant mean on G concentrates on Z ( G ) .Moreover, there is a finitely generated subgroup H of [ G, G ] such that C G ( H ) = Z ( G ) .Remark: In alternative (2), Z ( G ) must be isomorphic to Z n for some n ≥
1: since G has no nontrivial finite normal subgroups, it follows that Z ( G ) is torsion free, and since G is finitely generated with [ G, G ] ∩ Z ( G ) = 1,it follows that Z ( G ) is isomorphic to a torsionfree subgroup of the finitelygenerated abelian group G/ [ G, G ]. Proof.
The two alternatives are indeed mutually exclusive: if (1) holds, thenthe collection C , of atomless [ G, G ]-conjugation invariant means on [
G, G ]is a nonempty compact convex set on which G acts by conjugation, with[ G, G ] acting trivially. Since G/ [ G, G ] is abelian, by the Markov-Kakutanifixed point theorem, there is a fixed point m ∈ C , which corresponds to anatomless G -conjugation invariant mean with m ([ G, G ]) = 1. Therefore, (2)cannot hold since it would imply that m ( Z ( G )) = 1 and hence m ( { } ) = m ([ G, G ] ∩ Z ( G )) = 1, contradicting that m is atomless.Assume now (1) fails and we will show that (2) holds. Since G is finitelygenerated, applying Lemma 3.3 (with N = K = G = G ) shows that m ( Z ( G )) = 1 for every G -conjugation invariant mean m on G . Suppose to-ward a contradiction that there were some [ G, G ]-conjugation invariant mean n on G with n ( Z ( G )) = r <
1. Then the set D of all [ G, G ]-conjugationinvariant means n on G with n ( Z ( G )) = r is a nonempty compact convexset on which G acts by conjugation, with [ G, G ] acting trivially. Applyingthe Markov-Kakutani fixed point theorem once more yields a mean m ∈ D which is invariant under conjugation by G , a contradiction. AT(0) CUBE COMPLEXES AND INNER AMENABILITY 15
We claim that there is some finitely generated subgroup H of [ G, G ]such that [
G, G ] ∩ C G ( H ) = 1 (and hence [ G, G ] ∩ Z ( G ) = 1). Otherwise,for each finitely generated subgroup H of [ G, G ] we can find some x H ∈ [ G, G ] ∩ C G ( H ) with x H = 1. The collection of finitely generated subgroupsof [ G, G ] is a directed set under inclusion and, taking any weak ∗ clusterpoint of the net of point masses, ( δ x H ), in the space of means on [ G, G ],we obtain a [
G, G ]-conjugation invariant mean m on [ G, G ] which is not thepoint mass at the identity element. Since (1) does not hold, the group M , ofall elements of [ G, G ] with finite conjugacy class, must be finite and m mustconcentrate on M . Since M is characteristic in [ G, G ], it is normal in G , andby hypothesis G has no nontrivial finite normal subgroups, hence M = 1.This contradicts that m is not the point mass at the identity element.Similarly, we claim that there must be a finitely generated subgroup H of [ G, G ] such that C G ( H ) = Z ( G ). Otherwise, for each finitely generatedsubgroup H of [ G, G ] we can find some y H ∈ C G ( H ) with y H Z ( G ). Bytaking any weak ∗ cluster point of the net of point masses, ( δ y H ), in the spaceof means on G , we obtain a [ G, G ]-conjugation invariant mean n on G with n ( Z ( G )) = 0, which we already showed is impossible. (cid:3) Wreath products.Theorem 3.9.
Let K and H be groups, with H = 1 , let X be a set on which K acts, and let G := H ≀ X K = ( L X H ) ⋊ K be the (restricted) generalizedwreath product. Then G is inner amenable if and only if one of the followingholds(1) The action K y X admits an atomless K -invariant mean,(2) H is inner amenable and the action K y X has a finite orbit,(3) There is an atomless K -conjugation invariant mean m on K satis-fying m ( K x ) = 1 for all x ∈ X , where K x denotes the stabilizer of x in K .Remark: It follows from the proof that either (1) or (2) holds if and onlyif there is an atomless conjugation invariant mean on G which concentrateson L X H , and that (3) holds if and only if there is an atomless conjugationinvariant mean on G which concentrates on K . Proof.
Let N := L X H , so that elements of N are functions z : X → H whose support, supp( z ) := { x ∈ X : z ( x ) = 1 H } , is finite. Then K acts on N via ( k · z )( x ) := z ( k − · x ), and we identify N and K with subgroups of G so that G is the internal semidirect product G = N ⋊ K and kzk − = k · z for all k ∈ K , z ∈ N .Suppose first that (1) holds. Let m be an atomless K -invariant mean on X . Fix some h ∈ H , h = 1, and for each x ∈ X let z x ∈ N be the function z x ( x ) = h and z x ( y ) = 1 H if y = x . Define the mean ˜ m on N to be thepushforward of m under the map x z x . Since z k · x = kz x k − , the mean˜ m is invariant under conjugation by K . Since m is atomless, ˜ m is atomless,and for any z ∈ N we have ˜ m ( C N ( z )) ≥ m ( X \ supp( z )) = 1 since supp( z ) is finite. Therefore, ˜ m is invariant under conjugation by N and K , hence byall of G , so G is inner amenable.Suppose next that (2) holds. Let X ⊆ X be a finite orbit of the action K y X , and let K = T x ∈ X K x , so that K has finite index in K . Thenthe subgroup N := { z ∈ N : supp( z ) ⊆ X } is normal in G and isomorphicto the finite direct sum N ∼ = L X H . Since H is inner amenable, N isinner amenable, and hence N C G ( N ) is inner amenable. Since N C G ( N )contains N K , it is of finite index in G , hence G is inner amenable.Suppose that (3) holds. For any z ∈ N we have C K ( z ) = T x ∈ supp( z ) K x hence m ( C K ( z )) = 1. Therefore, in addition to being K -conjugation in-variant, the mean m is N -conjugation invariant, hence it is G -conjugationinvariant, hence G is inner amenable.Assume now that G is inner amenable and we will show that (1), (2),or (3) holds. Let M denote the group of all elements of N having finite G -conjugacy class, and let M H denote the group of elements of H havingfinite H -conjugacy class. Notice that if z ∈ M , then supp( z ) is contained ina finite K -invariant set, and z ( x ) ∈ M H for all x ∈ X . Therefore, if M isinfinite then either K has infinitely many finite orbits on X , in which case(1) holds, or K has a (nonzero) finite number of finite orbit on X and M H is infinite, in which case (2) holds. We may therefore assume without lossof generality that M is finite.Let π : G → N be the map π ( zk ) = z for z ∈ N , k ∈ K . This map isequivariant for the conjugation actions of K on G and N respectively. If m is any mean on G with m ( K ) < m denote the normalizedrestriction of m to G − K and we define the mean ϕ ( m ) on X by ϕ ( m )( A ) := Z z ∈ N | A ∩ supp( z ) || supp( z ) | dπ ∗ m ( z )for A ⊆ X . It is clear that if m is K -conjugation invariant, then ϕ ( m ) isinvariant under the action of K on X . We will break the rest of the proofinto three cases: C1 There exists a K -conjugation invariant mean m on G with m ( K ) < ϕ ( m ) is not supported on a finite subset of X . C2 There exists an atomless G -conjugation invariant mean m on G , with m ( M K ) < m ( G ) = 1 for every finite index subgroup G of G , and such that ϕ ( m ) is supported on a finite subset of X . C3 There exists an atomless G -conjugation invariant mean m on G suchthat m ( C G/M ( z )) = 1 for all z ∈ N , and m ( M C K ( M )) = 1.We will show that C1 implies (1), C2 implies (2), and C3 imples (3).Let us first assume that these three implications hold and finish the proof.Suppose first that there exists an atomless G -conjugation invariant mean m on G which concentrates on N . By Proposition 2.3 we may assume that m ( G ) = 1 for every finite index subgroup of G . Since m is atomless and M is finite we have m ( M K ) = m ( N ∩ M K ) = m ( M ) = 0. If ϕ ( m ) is not AT(0) CUBE COMPLEXES AND INNER AMENABILITY 17 supported on a finite subset of X then C1 holds, hence (1) holds, and if ϕ ( m )is supported on a finite subset of X then C2 holds, hence (2) holds. We maytherefore assume that there is no atomless G -conjugation invariant meanwhich concentrates on N . Then by Lemma 3.5, for every G -conjugationinvariant mean m on G we must have m ( C G/M ( z )) = 1 for all z ∈ N . Since G is inner amenable we may find an atomless G -conjugation invariant mean m on G , and by Proposition 2.3 we may assume that m ( G ) = 1 for everyfinite index subgroup G of G . If m ( M K ) < C1 or C2 holdonce again, so either (1) or (2) holds and we are done. We may thereforeassume that m ( M K ) = 1. Since M is finite and normal in G , C K ( M ) isa finite index normal subgroup of K , and N C K ( M ) has finite index in G ,hence m ( N C K ( M )) = 1. Thus, m ( M C K ( M )) = m ( N C K ( M ) ∩ M K ) = 1,so C3 holds, and hence (3) holds. It remains to prove the implications Cj ⇒ (j) for j = 1 , , C1 . Since ϕ ( m ) is K -invariant, if ϕ ( m ) is not purely atomicthen, after renormalizing ϕ ( m ) on its atomless part if necessary, we see that(1) holds, and if ϕ ( m ) is purely atomic then the action K y X has infinitelymany finite orbits, so (1) holds nonetheless.Assume C2 . There must be a finite K -invariant subset X ⊆ X such that ϕ ( m )( X ) = 1 (in particular, the action K y X has a finite orbit). Since ϕ ( m )( X ) = 1, we have that ( π ∗ m )( S ) = 1, where S := n z ∈ N \ { N } : | X ∩ supp( z ) || supp( z ) | > | X || X | + 1 o . Since the number | X ∩ supp( z ) || supp( z ) | belongs to the set { i/ | supp( z ) | : 0 ≤ i ≤ | X |} ,which (by considering the cases | supp( z ) | > | X | and | supp( z ) | ≤ | X | ) isseen to be disjoint from the open interval with endpoints | X | / ( | X | + 1)and 1, it follows that any z ∈ S satisfies | X ∩ supp( z ) || supp( z ) | = 1, i.e., supp( z ) ⊆ X . Therefore, ( π ∗ m )( N ) = 1, where N := { z ∈ N : supp( z ) ⊆ X } .This implies that m ( N K ) = 1. Since X is K -invariant, N is a normalsubgroup of G , and since X is additionally finite, the set K := C K ( N ) = T x ∈ X K x has finite index in K , and hence N K has finite index in G . Thus, m ( N K ) = 1 and therefore m ( N K ) = m ( N K ∩ N K ) = 1. Since N and K commute, the restriction of the map π to N K is equivariant for theconjugation action of N on N K and N respectively, and hence π ∗ m isinvariant under conjugation by N . Since N C G ( N ) has finite index in G ,the group M , of all elements of N with finite N -conjugacy class, is finite iscontained in M . Therefore, π ∗ m ( M ) ≤ m ( M K ) <
1, and hence π ∗ m mustnot be purely atomic. This shows that N is inner amenable, and since X isfinite and N ∼ = L X H , finitely many applications of Proposition 2.4 showthat H is inner amenable and hence (2) holds.Assume C3 . Since M ≤ N is finite and normal in G , the set X := S z ∈ M supp( z ) is a finite (possibly empty, if M is trivial) K -invariant subsetof X . Put K := T x ∈ X K x (and K := K if M is trivial). Then M = { z ∈ K : supp( z ) ⊆ X and z ( X ) ⊆ M H } , from which it follows that K = C K ( M ).Given z ∈ N , we claim that C G/M ( z ) ∩ z − M K z ∩ M K = M C K ( z ). Thecontainment ⊇ is clear since M is normal in G , so we must show the inclusion ⊆ . We can write z = z z = z z where supp( z ) ⊆ X and supp( z ) ⊆ X \ X . Let g ∈ C G/M ( z ) ∩ z − M K z ∩ M K . Then g = xk for some x ∈ M and k ∈ K , and zxkz − k − ∈ M . Since z commutes with M and z commuteswith K we have zxkz − k − = z xz − z kz − k − , and since z xz − ∈ M thisimplies that z kz − k − ∈ M . But then supp( z kz − k − ) ⊆ X , and alsosupp( z kz − k − ) ⊆ X \ X , since supp( z ) ⊆ X \ X and X is K -invariant.Therefore, z kz − k − = 1 and so k ∈ C K ( z ) ∩ C K ( z ) ≤ C K ( z ).It now follows that m ( M C K ( z )) = 1 for all z ∈ N . Define now themean m K on K by m K ( A ) := m ( M A ) for A ⊆ K . This is K -conjugationinvariant since M is normal in G . In addition, m K ( K x ) = 1 for all x ∈ X since m K ( C K ( z )) = m ( M C K ( z )) = 1 for all z ∈ N . This shows that (3)holds. (cid:3) Example:
Consider a Baumslag-Solitar group G = BS( p, q ) = h t, a | ta p t − = a q i with 1 < p < q . Let A be the cyclic subgroup generated by a . Let G acton X = G/A by translation and consider the wreath product W := H ≀ X G ,where H is a non-trivial group. Let S = L X H , and identify S and G withtheir images in W . By [36], we can find a G -conjugation invariant atomlessmean m which satisfies m ( A ) = 1 for every finite index subgroup A of A .We claim that m is in fact conjugation invariant for all of W . It suffices toshow it is invariant under conjugation by elements of S . Given s ∈ S let Q s ⊆ X be the support of s . Then the pointwise stabilizer A Q s = T x ∈ Q s A x of Q s in A is a finite index subgroup of A , so satisfies m ( A Q s ) = 1. Since A Q s ⊆ C G ( s ), we have m ( C G ( s )) = 1, and hence m is invariant underconjugation by s .4. Inner amenability and
CAT(0) cube complexes
We now study the structure of inner amenable groups acting on CAT(0)cube complexes.4.A.
Generalities.
A CAT(0) cube complex is, roughly speaking, a non-positively curved complex built from cubes - i.e. subspaces isometric to Eu-clidean cubes [0 , n for some n ∈ N - glued together via isometries along aface. For an introduction to CAT(0) cube complexes, we refer to [34]; wehere rely primarily on [8]. We emphasize that the metric statements in thissection are always for the CAT(0) metric.A CAT(0) cube complex X has finite dimension if there some d such thatevery cube of X has dimension at most d . The dimension of X is the leastsuch d .The link of a vertex x in a CAT(0) cube complex X is the simplicialcomplex whose vertex set is the set of edges of X incident to x . A set of AT(0) CUBE COMPLEXES AND INNER AMENABILITY 19 n + 1 vertices of this complex corresponds to a n -simplex if and only if thecorresponding edges lie in a common cube.A group G acting by simplicial automorphisms on an irreducible CAT(0)cube complex X acts elementarily if there is an invariant subset S ⊂ ∂X with at most 2 elements. Let us emphasize that this notion of elementarityis not the usual one used for groups acting on general CAT(0) spaces (whereelementarity refers to the existence either of an invariant Euclidean subspaceor a fixed point at infinity). This notion coincides with elementarity forgroups acting on Gromov hyperbolic spaces without bounded orbits. Theaction is essential if no G -orbit remains in a bounded neighborhood of somehalf-space of X .For a half-space h , we denote by ˆ h the corresponding hyperplane and by h ∗ the other half-space corresponding to ˆ h . Sometimes, for a given hyperplane b h , we let h ± denote its two corresponding half-spaces. Two half-spaces h and k are facing if h ∩ k = ∅ and they are not nested. A facing triple is atriple of pairwise facing halfspaces.For a group G acting by isometries on a CAT(0) space ( X, d ), the trans-lation length of an element g ∈ G is | g | := inf x ∈ X d ( g ( x ) , x ) . An element g ∈ G is called elliptic if there is a fixed point. That is to say, | g | = 0, and the infimum value is achieved. We say that g is hyperbolic if | g | > x ∈ X . Elements which areneither elliptic nor hyperbolic are called parabolic . For a finite dimensionalCAT(0) cube complex, it can be shown that there are no parabolic isometries[4].These three classes of elements are disjoint and invariant under conjuga-tion, because the action is by isometries. We denote these three subsets of G by Ell( G ), Hyp( G ), and Par( G ), respectively.A hyperbolic isometry g has axes which are isometrically embedded reallines on which g acts as a translation of length | g | . A hyperbolic isometryis contracting if there is an axis L and some R > L has a projection to L with diameter atmost R . A contracting isometry has exactly 2 fixed points at infinity; anattractive one and a repulsive one which are the end points of any of itsaxes.A CAT(0) cube complex X is said to be pseudo-Euclidean it has anAut( X )-invariant isometricaly embedded Euclidean subspace. If the spacehas dimension 1, the complex is said to be R - like .We will need two results from the work [6] of Caprace and Lytchack. Theyintroduced the telescopic dimension of CAT(0) spaces. A finite dimensionalCAT(0) cube complex has finite telescopic dimension. Proposition 4.1 (Caprace–Lytchack, [6, Theorem 1.1]) . Let X be a com-plete CAT(0) space of finite telescopic dimension and ( X i ) i ∈ N be a nested sequence of closed convex subspaces of X . If T X i = ∅ , then T ∂X i isnonempty and has radius at most π/ . In particular, the center of all thecircumcenters of T ∂X i is unique. Corollary 4.2 (Caprace–Lytchack, [6, Corollary 1.5]) . Let X be a complete CAT(0) space of finite telescopic dimension. For a parabolic element g ∈ Isom( X ) , the set of g -fixed points in ∂X is nonempty and has a canonicalpoint ξ ( g ) which is the center of the circumcenters of g -fxed points in ∂X . This implies that, for a parabolic element g , one has ξ ( hgh − ) = hξ ( g )for any h ∈ Isom( X ); that is, the point ξ ( g ) is equivariant in g . Fora hyperbolic element g , we denote by ξ ± ( g ) the attracting and repulsingfixed points of g . These are the boundary points such that for any x ∈ X , g ± n ( x ) → ξ ± ( g ). Again these points are equivariant with respect to g : forany h ∈ Isom( X ), ξ ± ( hgh − ) = hξ ± ( g ).4.B. Main theorem.
We first start with a general statement for groupsequipped with a conjugation invariant mean acting on CAT(0) spaces.
Proposition 4.3.
Let X be complete CAT(0) space of finite telescopic di-mension without Euclidean de Rham factor and let G be a group acting min-imally on X without fixed points at infinity. If m is a conjugation invariantmean on G , then m (Ell( G )) = 1 .Proof. Assume that m (Par( G )) > m (Hyp( G )) > m | Par( G ) and renormalizing it, we may assume that m (Par( G )) =1 (respectively m (Hyp( G )) = 1). Pushing forward m via ξ (respectively ξ + ), we get a G -invariant mean on ∂X . One can think of this mean asa positive linear functional µ : ℓ ∞ ( ∂X ) → R that is G -invariant. Let usfix x ∈ X . For ξ ∈ ∂X , let us denote by x β ξ ( x ) the Busemannfunction associated to ξ that vanishes at x . For x ∈ X and any ξ ∈ ∂X , | β ξ ( x ) | ≤ d ( x, x ). Hence, ξ β ξ ( x ) is a bounded function on ∂X . Inparticular, one can “integrate” this function with respect to µ to obtaina real number that we denote by R ∂X β ξ ( x ) dµ ( ξ ). Let us denote by f thefunction x R ∂X β ξ ( x ) dµ ( ξ ). For any ξ ∈ ∂X , the function x β ξ ( x )is 1-Lipschitz and convex, and furthermore, µ is linear and positive with µ ( ) = 1. It now follows that f is a convex 1-Lipschitz function on X that is quasi-invariant. Moreover it lies in the closed convex hull C (whichis compact) of Busemann functions vanishing at x . Actually, since R X isendowed with the pointwise convergence topology, to prove that f ∈ C , byHahn-Banach theorem, it suffices to check that for any x ∈ X ,inf ϕ ∈C ϕ ( x ) ≤ f ( x ) ≤ sup ϕ ∈C ϕ ( x ) . This follows from the definition of f and the positivity of m . If f has nominimum then the intersection of its sublevel sets yields a G -invariant pointat infinity. So f has a minimum m . The set { x ∈ X | f ( x ) = m } is closedconvex and G -invariant. So by minimality, this subset coincides with X , AT(0) CUBE COMPLEXES AND INNER AMENABILITY 21 that is f is actually constant. By [6, Proposition 4.8 and Lemma 4.10], X has a non-trivial Euclidean de Rham factor which is a contradiction. (cid:3) Corollary 4.4.
Let X be an irreducible CAT(0) cube complex of finite di-mension and let G be a group acting essentially and non-elementarily byautomorphims on X . If m is a conjugation invariant mean on G , then m (Ell( G )) = 1 .Proof. Since there is no fixed point at infinity, there is a minimal G -invariantCAT(0) subspace Y ⊂ X [6, Proposition 1.8(ii)].Thanks to Proposition 4.3, it suffices to show that Y has trivial Euclideande Rham factor. So, let us assume that Y has a splitting Y = Y ′ × E where E is the Euclidean de Rham factor of Y and dim( E ) ≥
1. By [8, Theorem6.3], G has a contracting isometry (with an axis in Y ) and thus dim( E ) ≤ E ) = 1. The boundary ∂E gives a G -invariant pair of points in ∂Y ⊂ ∂X and thus a contradiction with non-elementarity. (cid:3) Proposition 4.5.
Let G be a group acting essentially and non-elementarilyon a finite dimensional irreducible CAT(0) cube complex X . If m is a con-jugation invariant mean on G , then m ( { g ∈ G | Fix( g ) ∩ h } ) = 1 for every half-space h .Proof. We consider two collections of hyperplanes of X : W := n ˆ h | m (cid:0) { g | Fix( g ) ∩ h = ∅ and Fix( g ) ∩ h ∗ = ∅} (cid:1) = 1 o , and W := n ˆ h | m (cid:0) { g | Fix( g ) ⊂ h } (cid:1) > m (cid:0) { g | Fix( g ) ⊂ h ∗ } (cid:1) > o . By Proposition 4.4, we know that m (Ell( G )) = 1, hence W = W ⊔ W where W is the set of all hyperplanes of X . We aim to show that W = ∅ .Let us argue that W and W are transverse. That is, for any b h i ∈ W i , b h ∩ b h = ∅ . Suppose toward a contradiction that there are ˆ h ∈ W andˆ h ∈ W that are not transverse. We may assume that h and h arefacing. Since h ∗ ⊂ h , m ( { g | Fix( g ) ⊂ h ∗ } ) = 0. On the other hand, thedouble skewer lemma, [8, Section 1.2], ensures that there is δ ∈ G such that δ h ⊂ h ∗ . Hence, m ( { g | Fix( g ) ⊂ h } ) = 0 which is a contradiction. Weconclude that W and W are transverse.By [8, Lemma 2.5], X splits as a product of CAT(0) cube complexes X × X (with possibly a trivial factor) where W and W are respectivelythe hyperplane systems of X and X . By irreducibility, we know that oneof this factor is trivial. If W = ∅ then X = X . In this case, define W ′ := n ˆ h ∈ W | m ( { g | Fix( g ) ⊂ h } ) = m ( { g | Fix( g ) ⊂ h ∗ } ) o . For ˆ h ∈ W ′ , we may choose the half-space h such that m ( { g | Fix( g ) ⊂ h } ) > m ( { g | Fix( g ) ⊂ h ∗ } ) . Take ˆ h and ˆ k in W ′ and suppose that h ∩ k = ∅ . It is then the case that k ⊂ h ∗ . Hence, m ( { g | Fix( g ) ⊂ h } ) > m ( { g | Fix( g ) ⊂ h ∗ } ) ≥ m ( { g | Fix( g ) ⊂ k } ) . On the other hand, h ⊂ k ∗ , so the same computation with the roles re-versed gives that m ( { g | Fix( g ) ⊂ h } ) < m ( { g | Fix( g ) ⊂ k } ), which is acontradiction.It now follows, thanks to the Helly-type property for CAT(0) cube com-plexes, that T ˆ h ∈W ′ h is an intersection of nested closed convex sets. Theintersection T ˆ h ∈W ′ h is furthermore G -invariant, and since the action of G on X is essential, this intersection is empty. Thanks to Proposition 4.1,there is a global fixed point at infinity for the action of G contradicting thatthe action is non-elementary. We conclude that W ′ = ∅ .For any Hyperplane ˆ h ∈ W , it is thus the case that m ( { g | Fix( g ) ⊂ h } ) = m ( { g | Fix( g ) ⊂ h ∗ } ) , and this value is non-zero. Since X is irreducible, Aut( X ) has no invariantEuclidean subspace otherwise it would be R -like ([8, Lemma 7.1]) and thus,there would be an invariant pair at infinity corresponding to the ends ofthe invariant line. Let us now consider a facing triple h , h , h ∈ W ,which exists thanks to [8, Theorem 7.2]. That the triple is facing ensuresthat h ∗ j ⊆ h k for k = j . In particular, h ∗ i ∩ h ∗ j = ∅ for i = j . Setting α i := m ( { g | Fix( g ) ⊂ h ∗ i } ), we see that α ≥ α + α and α ≥ α + α .Hence, α = 0 which is a contradiction. (cid:3) Lemma 4.6.
Let G be a group acting essentially and non-elementary onan irreducible CAT(0) cube complex X . Let m be any conjugation invariantmean on G . Then there is some x ∈ X and some C > such that m ( { g ∈ G | Fix( g ) ∩ B ( x , C ) = ∅} ) = 1 . Moreover the set X C , of all such points, is convex and G -invariant. The lemma follows essentially from Proposition 4.5 and the ideas thatgives the existence of contracting isometries in [8]. We urge the reader tohave a copy of [8] at hand to follow the proof.
Proof.
By the first paragraph of the proof of [8, Theorem 6.3], there is g ∈ G that skewers some hyperplan b h , with g h + ⊂ h + , and such that b h and g b h arestrongly separated. Let x be the intersection of some axis of g and b h . Bythe proof of [8, Lemma 6.1], there is C > a ∈ g − h − and b ∈ g h + , the geodesic [ a, b ] meets B ( x , C ). By Proposition 4.5, there is ameasure 1 set of elements of G such that any g in this set, Fix( g ) ∩ g − h − = ∅ and Fix( g ) ∩ g h + = ∅ . Thus for any g in this set, Fix( g ) ∩ B ( x , C ) = ∅ .Since the mean m is conjugation invariant, it follows directly from thedefinition that X C is G -invariant. Now let x, y ∈ X C and let c : [0 , → X be a parametrization of [ x, y ] proportional to arc-length. There is a measure AT(0) CUBE COMPLEXES AND INNER AMENABILITY 23 G ⊂ G such that for all g ∈ G , Fix( g ) ∩ B ( x, C ) = ∅ and Fix( g ) ∩ B ( y, C ) = ∅ . Fix g ∈ G , x ′ ∈ Fix( g ) ∩ B ( x, C ) and y ′ ∈ Fix( g ) ∩ B ( y, C ).Let c ′ be the similar parametrization of [ x ′ , y ′ ]. By convexity of the metric[5, II.2.2], d ( c ( t ) , c ′ ( t )) ≤ C for any t ∈ [0 , c ′ ( t ) is fixed by g aswell, the result follows. (cid:3) A mean on a group G is said to be an idempotent if m ∗ m = m . Lemma 4.7.
Let G be a group acting minimally by isometries on somecomplete CAT(0) space X . Let m be a conjugation invariant idempotentmean on G . Let C > and suppose m ( { g ∈ G | Fix( g ) ∩ B ( x, C ) = ∅} ) = 1 for every x ∈ X .Then m ( { g ∈ G | d ( x, gx ) < ε } ) = 1 for every x ∈ X and every ε > .Proof. Let us fix x ∈ X . For g ∈ G such that Fix( g ) = ∅ and x ∈ X , d ( x, gx ) ≤ d ( x, Fix( g )) + d (Fix( g ) , x )and thus the following formula ϕ ( x ) = Z G d ( x, gx ) dm ( g )defines a function ϕ : X → R + . The continuity follows from the inequality (cid:12)(cid:12) d ( x, gx ) − d ( y, gx ) (cid:12)(cid:12) = | d ( x, gx ) − d ( y, gx ) | · ( d ( x, gx ) + d ( y, gx )) ≤ d ( x, y ) (2 d ( x, gx ) + d ( x, y ))which integrates into | ϕ ( x ) − ϕ ( y ) | ≤ d ( x, y ) (2 ϕ ( x ) + d ( x, y )) . It follows from linearity and the CAT(0) inequality that for any x, y ∈ X and c their midpoint that(4.4) ϕ ( c ) ≤
12 ( ϕ ( x ) + ϕ ( y )) − d ( x, y ) . We claim that ϕ has a unique minimum z and for any z ′ ∈ X , d ( z, z ′ ) ≤ ϕ ( z ′ ) − ϕ ( z )) . Only the existence of this minimum requires an argument, the inequalityand the uniqueness follow directly from Equation (4.4). So let us provethe existence. For g in a set of measure 1, d ( x , Fix( g )) ≤ C and thusfor any g in this set, d ( x , gx ) ≤ C . In particular, ϕ ( x ) ≤ C . Now,for a point y ∈ X \ B ( x , C ), the reverse triangle inequality, implies that d ( y, gx ) ≥ C and thus ϕ ( y ) ≥ C . Since any continuous convex functionon a bounded complete CAT(0) space has a minimum (this follows from [31,Theorem 14]), ϕ has a minimum which lies in B ( x , C ). By idempotence of m we have, ϕ ( z ) = Z G d ( z, gx ) dm ∗ m ( g )= Z h ∈ G Z k ∈ G d ( h − z, kx ) dm ( k ) dm ( h )= Z h ∈ G ϕ ( h − z ) dm ( h ) . In particular, for any ε > h ∈ G such that ϕ ( h − z ) < ϕ ( z ) + ε and thus d ( z, h − z ) ≤ ε . This gives that { x ∈ X | ∀ ε > , m ( { g | d ( x, gx ) < ε } ) = 1 } is nonempty. Since this set isclosed convex and G -invariant, it is X itself. (cid:3) Theorem 4.8.
Let G be a group acting essentially and non-elementary onan irreducible CAT(0) cube complex X . There is a nonempty closed convexsubspace X such that for any conjugation invariant mean m on G and any x ∈ X , m ( G x ) = 1 .Proof. Since the action is non-elementary, there are minimal invariant closedconvex subspaces. Moreover the union of all such minimal subspaces split asa product X × C [7, Theorem 4.3 (B.ii)] where X is one of these minimalsubspaces and the action is diagonal, being trivial on C . So, all minimalsubspaces are equivariantly isometric. Let us fix a minimal closed subspace X .Assume first that n is a conjugation invariant mean on G that is addi-tionally idempotent. Then we can apply Lemma 4.7 to some minimal closedconvex G -invariant subspace X of X C given by Lemma 4.6. Since the orbitof any point in X is discrete under the action of Aut( X ), we conclude that n ( G x ) = 1 for every x ∈ X . Since X and X are equivariantly isometric,for every x ∈ X , n ( G x ) = 1.Now let m be a conjugation invariant mean on G . Since the map n n ∗ m is affine, continuous, and the set of conjugation invariant means is a convexcompact subspace of a locally convex topological vector space, the Markov-Kakutani fixed point theorem gives the existence of an m -stationary conju-gation invariant mean, i.e., a mean n satisfying the equation n ∗ m = n . Theset of all such m -stationary conjugation invariant means is a compact lefttopological semigroup for convolution, hence Ellis’s Lemma [17, Lemma 1]gives the existence of an m -stationary conjugation invariant mean n whichis furthermore idempotent. By the paragraph above, we have n ( G x ) = 1for every x ∈ X . Therefore, by Lemma 2.2, since n is m -stationary, weconclude that m ( G x ) = 1 for every x ∈ X . (cid:3) A few applications.Corollary 4.9.
Let G be a group acting on a finite dimensional CAT(0) cube complex X with no fixed points and no finite orbit in ∂X . Then thereexists a finite index subgroup G and a closed convex G -invariant subspace AT(0) CUBE COMPLEXES AND INNER AMENABILITY 25 X such that for any conjugation invariant mean m on G and for any point x ∈ X , m (( G ) x ) = 1 .Proof. Since G has no fixed point in X , nor any fixed point at infinity, thereis a nonempty invariant subcomplex called the essential core, Y ⊂ X onwhich G acts essentially [8, Proposition 3.5]. This complex Y has a canonicalsplitting Y = Y × · · · × Y n which is preserved by Aut( Y ) [8, Proposition2.6]. Let us gather the pseudo-Euclidean factors as the k first ones. Thatis Y i is pseudo-Euclidean if and only if i ≤ k . In particular, if we denote by Y euc the product Y × · · · × Y k , then Y euc is pseudo-Euclidean and the actionAut( Y euc ) y Y euc is essential. Let us also denote by Y non-euc the productof the remaining factors. Observe that the splitting Y = Y euc × Y non-euc isAut( Y )-invariant and this gives an action of Aut( Y ) on Y euc . By [8, Lemma7.1], each factor Y i , for i ≤ k , is R -like. In particular, each factor hasa Aut( Y i )-invariant line which gives an Aut( Y i )-invariant pair of points in ∂Y i . So this gives a finite orbit in ∂Y euc for Aut( Y ) and thus for G as well.This is a contradiction and this implies that Y has no pseudo-Euclideanfactor.Let G be the finite index subgroup of G that preserves each factor of thedecomposition Y = Y × · · · × Y n . For any i ≤ k , the action G y Y i isessential and non-elementary (otherwise there would be a finite G -orbit in ∂Y ). By Theorem 4.8, there is a closed convex subspace X i ⊂ Y i such thatthe stabilizer of any point in X i has measure 1 for any conjugation invariantmean m . Let us denote by X = X × · · · × X n ⊂ Y . By intersecting finitelymeasure 1 sets, it follows that the stabilizer of any point in has measure1. (cid:3) Corollary 4.10.
Let G be a group acting on some finite dimensional CAT(0) cube complex properly and without finite orbit at infinity. Then G is not in-ner amenable.Proof. Let us assume toward a contradiction that G is inner amenable. Wecontinue with the same notations as in Corollary 4.9. Say that m is aconjugation invariant atomless mean on G (which exists since it has finiteindex in G ). By Corollary 4.9, the stabilizer of any point in X has measure1. However, that the action is proper ensures the stabilizer of any vertex isfinite. Therefore the mean m has atoms, which is a contradiction. (cid:3) Remark 4.11.
Corollary 4.C can also be deduced from previous results.From [35, Theorem 1.3] (see also [13, 2.23]), one can deduce the existence ofnon-degenerate hyperbolically embedded subgroup of G and thus the group G is not inner amenable [13, 2.35]. Remark 4.12.
Under the hypotheses of Corollary 4.C, we can also showthat G has a natural proper 1-cocycle into a non-amenable representationand thus G is properly proximal in the sense of [3] and hence not inneramenable. There is a well-known natural 1-cocycle for the quasi-regularrepresentation of G on ℓ ( H ), associated to the action of G on the set of halfspaces H , and this cocycle is proper if the action of G on X is proper [10, § G -invariant mean on H .Assume for the sake of contradiction that there is such a G -invariant mean m on H . We can push it forward via h b h to get a G -invariant mean m onthe set of hyperplanes W . Thanks to the same argument as in the proof ofCorollary 4.9, it suffices to consider the case where X is irreducible and theaction is essential. Analogous to the proof of Proposition 4.7, let us define W = { b h | m ( { b k | b k ⊥ b h } ) = 1 } , W = { b h | m ( { b k | b k ⊂ h } ) > m ( { b k | b k ⊂ h ∗ } ) > } . For similar reasons as in Proposition 4.7, W = W ⊔ W and these col-lections are transverse. By irreducibility, one is trivial and arguing with W as in the proof of Proposition 4.7, we show that W = W . Thus, for anytwo b h , b h , there is a measure 1 set of hyperplanes which simultaneouslycross them both. This contradicts the existence of strongly separated hy-perplanes which is guaranteed thanks to the irreducibility of the complex [8,Proposition 5.1]. Example 4.13.
The Higman group is not inner amenable. Let us recallthat the Higman group H is the group given by the presentation. H = h a , a , a , a | a i a i +1 a − i = a i +1 with i ∈ Z / Z i . The work [29] exhibits an action of H on an irreducible CAT(0) squarecomplex. From the description of the action, it follows that the convex hullof any orbit meets the interior of some square and the action is essentialand non-elementary. Moreover, the action on squares is regular So , byTheorem 4.8, for any conjugacy invariant mean on H m ( { } ) = 1, so H isnot inner amenable.4.D. Graph products of groups.
Let Γ be a finite simplicial graph withvertex set V Γ and edge set E Γ. The neighborhood N ( v ) of v ∈ V Γ is theset { w ∈ V Γ , w = v or { v, w } ∈ E Γ } . A clique in Γ is subset C ⊂ V Γsuch that the induced graph is complete. By a maximal clique , we meana clique which is maximal for inclusion. The flag simplicial complex F Γassociated to Γ is the simplicial complex with 1-skeleton Γ and simplicescorresponding to cliques.Assume that for each v ∈ V Γ, a non-trivial group G v is given. The groups G v are called the vertex groups . For any simplex σ of F Γ, that is a cliquein Γ, we define G σ = Q v ∈ σ G v . In particular, for two simplices τ ⊂ σ , wehave the natural inclusion ψ στ : G τ → G σ . The graph product G Γ is thedirect limit of the system given by the groups G σ and homomorphisms ψ στ .It can also been described as the quotient of the free product of all vertexgroups by the normal subgroup generated by the commutators [ a, b ] with a ∈ G v , b ∈ G w and { v, w } ∈ E Γ. AT(0) CUBE COMPLEXES AND INNER AMENABILITY 27
More generally, if S is a subset of V Γ, we denote by Γ S the subgraph ofΓ induced by S . The subgroup of G Γ generated by { G v } v ∈ S is denoted by G S and is isomorphic to the graph product G Γ S .The graph Γ is a join if there are two proper subsets V , V ⊂ V Γ, V Γ = V ⊔ V and such that for any v ∈ V and v ∈ V , { v , v } ∈ E Γ. In thiscase, if Γ i is the graph induced by V i then the graph product G Γ splits asthe direct product G Γ × G Γ . The complement graph Γ is the graphwith same vertex set V Γ = V Γ and { v, w } ∈ E Γ if and only if { v, w } / ∈ E Γ.Let Γ , . . . , Γ n be the subgraphs of Γ induced by the vertex sets of theconnected components of Γ. By the above remark, the group G Γ splits as adirect product G Γ × · · · × G Γ n . This canonical splitting is maximal in thesense that no Γ i is a join.The goal of this subsection is to prove the following characterization ofinner amenability of graph products of groups and to specialize this resultto the cases of right-angled Artin groups and right-angled Coxeter groups. Theorem 4.14.
The graph product G Γ is inner amenable if and only if • there is v ∈ V Γ such that N ( v ) = V Γ and G v is inner amenable or • there are v , v ∈ V such that N ( v ) = V \ { v } , N ( v ) = V \ { v } and G v ≃ G v ≃ Z / Z . In particular G Γ splits as direct productwith the infinite dihedral group D ∞ . To prove this theorem, we use a nice combinatorial action of G Γ on aCAT(0) cube complex X Γ due to Meier and Davis [30, 14]. This constructionis also described in [5, Example II.12.30.(2)]. We refer to these referencesfor an explicit construction. This action has a cubical complex C Γ as strictfundamental domain, which is the cubulation of the simplicial complex F Γ.Let us describe it. The complex C Γ is completely determined by its 1-skeleton C Γ (1) and thus we only describe this graph (this 1-skeleton is amedian graph see [9, § C Γ (1) is the set of cliques of Γ together with the emptyset (seen as the empty clique). Two cliques σ and τ are joined by an edgeif and only if their symmetric difference is a singleton. More generally, twocliques lie in a common cube if and only if their union is a clique. So allmaximal cubes of C Γ have a set of vertices given by the set of all subsets ofa maximal clique. In particular, the link of the vertex ∅ is F Γ.For a point x ∈ C Γ, we denote by σ ( x ) the smallest clique (for inclusion)appearing as vertex of the smallest cube containing x . The CAT(0) cubecomplex X Γ (which depends on the vertex groups whereas C Γ does not) isobtained as a quotient ( G Γ × C Γ) / ∼ where ( g, x ) ∼ ( h, y ) if x = y and g − h ∈ G σ ( x ) .This quotient is naturally endowed with the cubical structure coming from C Γ and G Γ acts by automorphisms via g · ( h, x ) = ( gh, x ). For example, thestabilizer of a vertex σ is exactly G σ (with the convention that G ∅ = { } ). The vertices of X Γ are in bijection with cosets gG σ ∈ G Γ /G σ , that is theunion indexed by the set of all cliques (possibly empty) X (0)Γ = G σ G Γ /G σ . Observe that C Γ embeds in X Γ by the map x (1 , x ). Under this embed-ding the link of ∅ in X Γ is the same as in C Γ, that is F Γ. This follows fromthe fact that the stabilizer of ∅ is G ∅ = { } . Example 4.15.
To explicit a bit this construction, we illustrate it on somesimple examples in Table 1. For the sake of simplicity, the vertex groups arecyclic but the construction is not restricted to this case.
Lemma 4.16.
Let S ⊂ V Γ . The CAT(0) cube complex X Γ S embeds as aconvex subcomplex of X Γ in a G S -equivariant way.Proof. By construction, the set of vertices of X Γ S is the union G σ ⊂ S G S /G σ over cliques included in S . This can be seen as a subset of G σ ⊂ V Γ G Γ /G σ and this gives the embedding at the level of vertices. It is clearly G S -equivariant. Moreover, by construction, two vertices of X Γ S lie in a commoncube of X Γ if and only if they lie in a common cube of X Γ S and thus theembedding is convex. (cid:3) Lemma 4.17.
The action of G Γ y ∂X Γ has no fixed point.Proof. Let σ be a maximal clique of the graph Γ. Assume that ξ ∈ ∂X Γ is a G Γ -fixed point. The geodesic ray L from the vertex σ to ξ is pointwise fixedby G σ . The maximal cubes having σ as vertex are images by some element g ∈ G σ of the cube C with vertex set { τ | τ ⊂ σ } . In particular, if g ∈ G σ does not lie in any G v for v ∈ σ then gC ∩ C = { σ } . So G σ acts transitivelyon maximal cubes attached to σ and the intersection of all these cubes isreduced to σ . We have a contradiction because there is some maximal cubeattached to σ such that the intersection of L and this cube is not reducedto { σ } . (cid:3) If Γ is a join, the group G Γ splits as direct product and the CAT(0) cubecomplex X Γ splits as a direct product with factors associated to the factorsof G Γ . The converse is also true. Lemma 4.18. If Γ is not a join then X Γ is irreducible.Proof. In the complex C Γ, vertices are in bijection with cliques S ⊂ V Γ(possibly empty). Two such vertices are connected by an edge if they differby one element. In particular, edges with one end ∅ have some singleton AT(0) CUBE COMPLEXES AND INNER AMENABILITY 29
Γ Vertex groups andtheir graph products C Γ X Γ • v G v = G Γ = Z / Z •∅ •{ v } • • • • • • { v }• v • v ′ G v = Z / Z , G v ′ = Z / Z G Γ = Z / Z ∗ Z / Z •∅ •{ v }•{ v ′ } • • •••• •••••• • ... • ... • ... • ... • ... • ... • ... • ... ∅ { v }{ v ′ } • v • v ′ G v = Z / Z , G v ′ = Z / Z G Γ = Z / Z × Z / Z •∅ • { v }•{ v ′ } • { v, v ′ } • ∅ • { v } • { v ′ } • { v, v ′ } • (0 , • (0 , • (1 , • (1 , • (1 , •• v • v ′ • v ′′ G v = G v ′ = G v ′′ = Z / Z G Γ = Z / Z ∗ ( Z / Z × Z / Z ) •∅ • { v }•{ v ′ } • { v, v ′ }• { v ′′ } •∅•• • • •• • •••• ............ Table 1.
Some examples of graph products and their asso-ciated CAT(0) cube complexes. { v } for the other end. Since any maximal cube in C Γ contains the vertex ∅ , to any hyperplane of C Γ, one can associate a unique v ∈ V Γ, which isthe unique v such that this hyperplane is the parallelism class of a uniqueedge with ends ∅ and { v } . We denote by h v this hyperplane (seen as anhyperplane of C Γ or X Γ ). By construction, for any v , v , h v crosses h v ifand only if { v , v } ∈ E Γ.Assume that X Γ splits as as product of CAT(0) cube complexes thenthere is a canonical non-trivial decomposition X Γ = X × · · · × X n andthis decomposition is stable under the action of the automorphism group(which possibly permutes the isomorphic factors). In that case, the set ofhyperplanes W is the (non-trivial) disjoint union W ⊔ · · · ⊔ W n where any b h i ∈ W i meets any b h j ∈ W j . See [8, Proposition 2.6]. This partition of W induces a partition of V Γ in the following way. If V i = n v, b h v ∈ W i o then V Γ = V ⊔ · · · ⊔ V n and for any i, j distinct, v i ∈ V i , v j ∈ V j , { v i , v j } ∈ E Γ.Moreover, this partition is non-trivial because C Γ is a fundamental domainfor the action of G Γ . (cid:3) Let us recall that an action by automorphisms of a group G on a CAT(0)cube complex is essential (or G - essential if we aim to emphasize the ac-tion) if all hyperplanes are essential, that is there is no orbit at a boundeddistance from an half-space. Lemma 4.19. If Γ is not a join and has at least two vertices then the action G Γ y X Γ is essential.Proof. For an hyperplane, to be G Γ -essential is a G Γ -invariant property. Soit suffices to show that hyperplanes corresponding to edges with ends ∅ and { v } (for some v ∈ V Γ) are essential. So let v ∈ V Γ. Since Γ has at leasttwo vertices and is not a join then there is v ′ ∈ V Γ such that { v, v ′ } is notan edge of Γ. Let S = { v, v ′ } , b h be the hyperplane corresponding to edgebetween ∅ and { v } and b h ′ the one between ∅ and { v ′ } . These hyperplanesdo not cross since { v, v ′ } is not an edge and none of their images under G S = G v ∗ G v ′ ≤ G Γ . The images of the above edges under G S span theinfinite tree without leaf X Γ S (which is convexly embedded by Lemma 4.16)and thus b h is essential. (cid:3) Lemma 4.20.
Let Γ be a graph that is not a join and such that | V Γ | ≥ .Then, there is S ⊂ V Γ such that | S | = 3 and | E Γ S | ≤ .Proof. Since Γ is not a join, Γ is not a complete graph and there are vertices v , v such that { v , v } / ∈ E Γ. Now, assume for a contradiction that for any S ⊂ V Γ of cardinal 3, one has | E Γ S | ≥
2. So, for any v ∈ V Γ \ { v , v } , { v , v } , { v , v } ∈ E Γ and thus Γ is the join of { v , v } and V Γ \ { v , v } ,and we have a contradiction. (cid:3) Proof of Theorem 4.14.
Thanks to Proposition 2.4, a direct product is inneramenable if and only at least of its factor is. So, it suffices to prove that if
AT(0) CUBE COMPLEXES AND INNER AMENABILITY 31
Γ is not a join and G Γ is inner amenable then Γ has a unique vertex v (andthus G Γ = G v is inner amenable), or Γ = Γ and the vertex groups haveorders 2 (that is G Γ ≃ D ∞ ).From now on, we assume that Γ is not a join, not reduced to a vertex norto a pair of edges with vertex groups Z and we show that in this case G Γ is not inner amenable. By Lemma 4.18, X Γ is irreducible.Let us show that X Γ is not pseudo-Euclidean (in our irreducible situation,this means R -like [8, Lemma 7.1]) and that implies that there is no invariantpair of points at infinity. That is, the action is non-elementary. So, for thesake of contradiction, let us assume, there is a Aut( X Γ )-invariant Euclideansubspace E ⊂ X Γ . Since the fundamental domain C Γ is compact, theprojection π : X Γ → E is a quasi-isometry. In particular, the subcomplexes X Γ S (for S ⊂ V Γ) can’t be hyperbolic without being quasi-isometric to areal interval.If | V Γ | = 2 then at least one vertex group has order greater than 2 andthen X Γ is tree with no leaf and at least one vertex with valency greaterthan 2. Thus it can’t be quasi-isometric to a Euclidean space. So it remainsto deal with the case where | V Γ | ≥
3. Thanks to Lemma 4.20, there is S = { v , v , v } ⊂ V Γ such that | E Γ S | ≤
1. If E Γ S = ∅ then X Γ S is a treewithout leaf and the vertex ∅ has valency 3, which gives a contradiction. If | E Γ S | = 1, we may assume that E Γ S = { v , v } . The complex X Γ { v ,v } isbounded (all squares are attached to the vertex corresponding to G { v ,v } )and G { v ,v } and has at least 4 elements (this is similar to the last examplein Table 1). So X Γ S is quasi-isometric to a tree without leaf and a vertexwith valency at least 4. Once again, this gives a contradiction.So, we know that X Γ is not pseudo-Euclidean. By [8, Theorem 7.2],contains a facing triple and each of this hyperplane b h i is skewered by somecontracting isometry g i . If h , h , h are the corresponding half-spaces,wemay assume that g i h i is properly contained in h i . Each contracting isometryhas exactly 2 fixed points. By the configuration of the triple of hyperplanes,the three attractive points of the isometries g i are distincts and this showsthat Γ has no invariant pair of points at infinity.By Lemmas 4.17 and 4.19, the action of G Γ has non-elementary and isessential. We can apply Theorem 4.5 and thus we know the existence of X ⊂ X such that for any x ∈ X and any conjugation invariant mean m (( G Γ ) x ) = 1. Let x ∈ X . Up to apply an element of G Γ , we may assumethat x belongs to some square containing the vertex ∅ . The stabilizer of x is then G σ ( x ) where is σ ( x ) is the minimal clique appearing as vertex inthe smallest cube containing x . We claim that there is γ ∈ G Γ such that γG σ ( x ) γ − ∩ G σ ( x ) = { } . It follows that m ( { } ) = 1 and thus G Γ cannotbe inner amenable. Since Γ is not a join, for any v ∈ σ ( x ), there is g v insome G v ′ such that g v does commute with G v . So if σ ( x ) = { v , . . . , v n } , itsuffices to take γ = g v · · · g v n . (cid:3) A right-angled Artin group is a graph product of groups where allvertex groups are infinite cyclic and a right-angled Coxeter group isa graph product where all vertex groups are Z / Z . We readily get thefollowing two consequences. Corollary 4.21.
A right-angled Artin group is inner amenable if and onlyif it splits as a direct product with Z . Remark 4.22.
A right-angled Artin group also acts on its Davis complexwhich is a different CAT(0) cube complex from the one we use here.
Corollary 4.23.
A right-angled Coxeter group is inner amenable if andonly if it splits as a direct product with the infinite dihedral group D ∞ . Trees, amalgams and inner amenability
In this section, we prove Theorem 1.3, and we also sketch a direct argu-ment for Theorem 1.3 which does not rely on the general results on CAT(0)cube complexes.Let us say that a group action on a tree is minimal if there is no properinvariant subtree.
Theorem 5.1.
Let G be a group acting non-elementarily and minimally ona tree T and let m be a conjugation invariant mean on G . Then m ( G x ) = 1 for every x ∈ T .Proof. If T is reduced to a point then the result is trivial. If it is not reducedto a point, then no orbit is bounded and thus the action is essential. Theconclusion therefore follows from Theorem 4.8. (cid:3) Remark 5.2.
Let us sketch a direct proof of Theorem 5.1 that does notrely on Theorem 4.8. Let m be some conjugation invariant mean on G .Then we first argue that m concentrates on the elliptic group elements.Otherwise, if m (Hyp( G )) >
0, then one can push forward the normalizationof m | Hyp( G ) to ∂T by associating to any hyperbolic element its attractivefixed point. This yields an invariant mean on ∂T . The removal of any oneedge partitions the tree into two half spaces, which in turn yields a partitionof the boundary of T into two pieces. One then considers the measure ofeach of these two boundary pieces. There can be no edge whose associatedboundary pieces each have measure 1/2, since otherwise the collection ofall such edges would necessarily be a G -invariant segment, half-line, or line,which would contradict non-elementarity of the action. Therefore, for eachedge, one of its associated boundary pieces has measure strictly greaterthan 1/2. By considering the intersection of all half-spaces corresponding toboundary pieces with measure strictly greater than 1 /
2, we obtain a pointin T or in ∂T which has to be fixed by G , which yields a contradiction onceagain. Thus we know that m (Ell( G )) = 1.We claim that for any edge e , the measure of the point wise stabilizer of e is 1. Observe that if this holds for one edge, then it in fact holds for every AT(0) CUBE COMPLEXES AND INNER AMENABILITY 33 edge by G -invariance, convexity and minimality. So let us assume toward acontradiction that it holds for no edge. That is, for every edge e of T , themeasure of the set of elliptic elements whose fixed point set is completelycontained in one of the two connected components of T \ e is positive. Bycomparing the measure of the set of group elements having a fixed point setin the first connected component of T \ e with the measure of the set of groupelements having fixed point set in the second connected component of T \ e ,we can argue as above (or, more concretely, as in the analysis of W in thethird paragraph of the proof of Proposition 4.5) to obtain a contradictionwith non-elementarity of the action.So, we know that for any edge, the measure of its pointwise stabilizer is1, and this implies that for any vertex its stabilizer has measure 1. Corollary 5.3.
Let G = A ∗ H B be a nondegenerate amalgamated freeproduct. Then every conjugation invariant mean on G concentrates on H .Thus, G is inner amenable if and only if there exist conjugation invariant,atomless means m A on A and m B on B with m A ( H ) = m B ( H ) = 1 , and m A ( E ) = m B ( E ) for every E ⊆ H .Let G = HN N ( K, H, ϕ ) be a non-ascending HNN extension. Then everyconjugation invariant mean on G concentrates on H . Thus, G is inneramenable if and only if there exists a conjugation invariant, atomless mean m on K with m ( H ) = 1 , and m ( E ) = m ( φ ( E )) for every E ⊆ H .Proof. It follows from Bass-Serre theory that in both cases (amalgams andHNN-extension), G has a minimal non-elementary action on a tree such that H is exactly the pointwise stabilizer of some edge. By Theorem 5.1, for anyconjugation invariant mean m on G , m ( H ) = 1. The respective characteriza-tions of inner amenability are a straightforward consequence (the sufficiencyof the conditions for inner amenability is obvious, and the necessity followsdirectly from the first part). (cid:3) Remark 5.4.
Amine Marrakchi, remarking on an earlier version of thisarticle, informed us that he found another proof of Corollary 5.3 for nonde-generate amalgams G = A ∗ H B , which he kindly allowed us to reproducehere.We may assume that | A : H | ≥ | B : H | ≥
3. Let G = G \ H , A = A \ H and B = B \ H . By the definition of an amalgamated freeproduct, the family of subsets A ( B A ) n , ( A B ) n +1 , B ( A B ) n , ( B A ) n +1 , n ≥ G . Let S = S n ≥ A ( B A ) n ∪ ( A B ) n +1 be the setof all elements starting with a letter in A . Take a ∈ A . Then we have S ∪ aSa − = G . Take b , b ∈ B such that b − b ∈ B . Then the sets S , b Sb − and b Sb − are disjoint in G . Now suppose that m is a conjugationinvariant mean on G . Then we must have 2 µ ( S ) ≥ m ( S ∪ aSa − ) = m ( G )and 3 m ( S ) = m ( S ⊔ b Sb − ⊔ b Sb − ) ≤ m ( G ). This shows that m ( G ) = 0as we wanted. References
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