Vanishing of $\ell^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence
aa r X i v : . [ m a t h . A T ] M a r VANISHING OF ℓ -BETTI NUMBERS OF LOCALLYCOMPACT GROUPS AS AN INVARIANT OF COARSEEQUIVALENCE ROMAN SAUER AND MICHAEL SCHRÖDL
Abstract.
We provide a proof that the vanishing of ℓ -Betti numbersof unimodular locally compact second countable groups is an invariantof coarse equivalence. To this end, we define coarse ℓ -cohomology forlocally compact groups and show that it is isomorphic to continuous co-homology for unimodular groups and invariant under coarse equivalence. Introduction
The insight that the vanishing of ℓ -Betti numbers provides a quasi-isometry invariant is due to Gromov (see [12, Chapter 8] for a statementwithout proof), and positive results around this insight have a long history.The most important contribution is by Pansu [18] whose work on asymptotic ℓ p -cohomology includes a proof that the vanishing of ℓ -Betti numbers ofdiscrete groups of type F ∞ , is a quasi-isometry invariant.There is a growing interest in the metric geometry of locally compactgroups [2, 3]. We thus think it is important to have the quasi-isometry andcoarse invariance of the vanishing of ℓ -Betti numbers available in the great-est generality. Following Pansu’s ideas and relying on more recent advancesin the theory of ℓ -Betti numbers, we provide a proof of the following result. Theorem 1.
Let G and H be unimodular locally compact second countablegroups. If G and H are coarsely equivalent then the n -th ℓ -Betti number of G vanishes if and only the n -th ℓ -Betti number of H vanishes. The coarse invariance for discrete groups was proven earlier in a paperof Mimura-Ozawa-Sako-Suzuki [16, Corollary 6.3].Every locally compact, second countable group G (hereafter abbrevi-ated by lcsc ) has a left-invariant proper continuous metric by a theoremof Struble [26]. As any two left-invariant proper continuous metrics on G are coarsely equivalent, every lcsc group has a well defined coarse geometry.Further, any coarse equivalence between compactly generated lcsc groups Mathematics Subject Classification.
Primary 20F65; Secondary 22D99.
Key words and phrases.
Coarse geometry, locally compact groups, ℓ -Betti numbers. is a quasi-isometry with respect to word metrics of compact symmetricgenerating sets and vice versa. In particular, a coarse equivalence betweenfinitely generated discrete groups is a quasi-isometry. See [3, Chapter 4] fora systematic discussion of these notions.To even state Theorem 1 in that generality, recent advances in the theoryof ℓ -Betti numbers were necessary. ℓ -Betti numbers of discrete groupsenjoy a long history but it was not until recently that ℓ -Betti numberswere defined for arbitrary unimodular lcsc groups by Petersen [19], and asystematic theory analogous to the discrete case emerged [13, 19, 20]. Earlierstudies of ℓ -Betti numbers of locally compact groups in specific cases canbe found in [4, 6, 10]. Previous results on coarse invariance.
Pansu [18] introduced asymptotic ℓ p -cohomology of discrete groups and proved its invariance under quasi-isometries. If a group Γ is of type F ∞ , then the ℓ p -cohomology of Γ coin-cides with its asymptotic ℓ p -cohomology [18, Théorème 1]. The geometricexplanation for the appearance of the type F ∞ condition is that the finite-dimensional skeleta of the universal covering of a classifying space of finitetype are uniformly contractible. As an immediate consequence of Pansu’s re-sult, the vanishing of ℓ -Betti numbers is a quasi-isometry invariant amongdiscrete groups of type F ∞ . The same arguments work for totally discon-nected groups admitting a topological model of finite type [23].Elek [7] investigated the relation between ℓ p -cohomology of discrete groupsand Roe’s coarse cohomology and proved similar results. Another indepen-dent treatment is due to Fan [8]. Genton [11] elaborated upon Pansu’s meth-ods in the case of metric measure spaces.Oguni [17] generalised the quasi-isometry invariance of the vanishing of ℓ -Betti numbers from discrete groups of type F ∞ to discrete groups whosecohomology with coefficients in the group von Neumann algebra satisfiesa certain technical condition. A similar technical condition appears in theproof of quasi-isometry invariance of Novikov-Shubin invariants of amenablegroups [25], and it is unclear how much this condition differs from the type F ∞ -condition. Oguni’s groupoid approach is inspired by [9, 25] and quitedifferent from the approaches by Elek, Fan, and Pansu.The coarse invariance of vanishing of ℓ -Betti numbers for discrete groupswas shown by Mimura-Ozawa-Sako-Suzuki [16, Corollary 6.3]. Li [14] re-cently reproved this using groupoid techniques as a consequence of moregeneral cohomological coarse invariance results. -BETTI NUMBERS AND COARSE EQUIVALENCE 3 Structure of the paper.
We review the necessary basics of ℓ -Betti numbersand continuous cohomology in Section 2. In Section 3 we define coarse ℓ -cohomology for lcsc groups and show that it is isomorphic to continuouscohomology. In Section 4 we conclude the proof of Theorem 1 and discusswhat fails for non-unimodular groups.2. Continuous cohomology and ℓ -Betti numbers of lcscgroups Let G be a unimodular lcsc group with Haar measure µ . Let X be alocally compact second countable space with Radon measure ν . Let E be aFréchet space.The space C ( X, E ) of continuous functions from X to E becomes aFréchet space when endowed with the topology of compact convergence. Let L loc ( X, E ) be the space of equivalence classes of measurable maps f : X → E up to ν -null sets such that || f | K || E is square-integrable for every compactsubset K ⊂ X . The L -norm of the function || f | K || E defines a semi-norm p K on L loc ( X, E ) . The family of semi-norms p K , K ⊂ E , turns L loc ( X, E ) into a Fréchet space.We call a Fréchet space E with a continuous (i.e. G → E , g gv , iscontinuous for every v ∈ E ) linear G -action a G -module . A continuous linear G -equivariant map between G -modules is a homomorphism of G -modules .If E is a G -module and G acts continuously and ν -preserving on X then C ( X, E ) and L loc ( X, E ) become G -modules via ( g · f )( x ) = gf ( g − x ) for x ∈ X and g ∈ G [1, Proposition 3.1.1]. The usual homogeneous coboundarymap(1) d n − f ( g , ..., g n ) = n X i =0 ( − i f ( g , ..., b g i , ..., g n ) defines cochain complexes C ( G ∗ +1 , E ) and L loc ( G ∗ +1 , E ) of G -modules (cf. [1,Proposition 3.2.1]). Here we take the diagonal G -action on G ∗ +1 . We recallthe following definition. Definition 2.
The (continuous) cohomology of G with coefficients in E isthe cohomology H n ( G, E ) = H n (cid:0) C ( G ∗ +1 , E ) G (cid:1) of the G -invariants of C ( G ∗ +1 , E ) . The reduced (continuous) cohomology H ∗ ( G, E ) is a quotient of H ∗ ( G, E ) obtained by taking the quotient withthe closure of im d ∗− instead of im d ∗− . R. SAUER AND M. SCHRÖDL
We have an obvious inclusion(2) I ∗ : C ( G ∗ +1 , E ) → L loc ( G ∗ +1 , E ) . The maps I ∗ form a cochain map of G -modules. Taking a positive function χ ∈ C c ( G ) there is a cochain map R ∗ : L loc ( G ∗ +1 , E ) → C ( G ∗ +1 , E ) of G -modules ( R n f )( g , ..., g n ) = Z G n +1 f ( h , ..., h n ) χ ( g − h ) · ... · χ ( g − n h n ) dµ ( h , ..., h n ) such that I ∗ ◦ R ∗ and R ∗ ◦ I ∗ are homotopic (as cochain maps of G -modules)to the identity [1, Proposition 4.8]. So we have the following useful fact: Theorem 3.
The cochain map I ∗ in (2) induces isomorphisms in cohomol-ogy and in reduced cohomology. Next we turn to the case where the coefficient module E = L ( G ) is theregular representation, relevant for the definition of ℓ -Betti numbers.Let L ( G ) be the von Neumann algebra of G ; the Haar measure µ defines asemifinite trace tr µ on L ( G ) . There are a natural left G -action and a naturalright L ( G ) -action on L ( G ) , and the two actions commute. Hence also the G -actions on C ( G ∗ +1 , L ( G )) and L loc ( G ∗ +1 , L ( G )) considered previously andthe L ( G ) -actions induced from the right L ( G ) -action on L ( G ) commute. Sothe (reduced and non-reduced) continuous cohomology of G with coefficientsin L ( G ) is naturally a L ( G ) -module . Obviously, the cochain map I ∗ aboveis compatible with the L ( G ) -module structures. The groups H ∗ ( G, L ( G )) are called the (continuous) ℓ -cohomology of G . Similarly for the reducedcohomology.Petersen [19] extended Lück’s dimension function from finite von Neu-mann algebras to semifinite von Neumann algebras. The dimension function dim µ with respect to ( G, µ ) is a non-trivial dimension for (algebraic) right L ( G ) -modules that is additive for short exact sequences of L ( G ) -modules.It scales as dim cµ = c − dim µ for c > . The fact that a L ( G ) -module hasdimension zero can be expressed without referring to the trace: it is an al-gebraic fact. The following criterion was shown by the first author for finitevon Neumann algebras [24, Theorem 2.4]; it was extended to the semifinitecase by Petersen [19, Lemma B.27]. Theorem 4. An L ( G ) -module M satisfies dim µ ( M ) = 0 if and only if forevery x ∈ M there is an increasing sequence ( p i ) of projections in L ( G ) with sup p i = 1 such that xp i = 0 for every i ∈ N . When talking about L ( G ) -modules we mean the algebraic module structure andignore topologies. -BETTI NUMBERS AND COARSE EQUIVALENCE 5 Definition 5.
The n -th ℓ - Betti number of G is the L ( G ) -dimension of itsreduced continuous cohomology with coefficients in L ( G ) , i.e. β n (2) ( G ) := dim µ H n ( G, L ( G )) ∈ [0 , ∞ ] . Remark . Equivalently, the n -th ℓ -Betti number can be defined as the L ( G ) -dimension of the non-reduced cohomology H n ( G, L ( G )) . This is anon-trivial fact (see [13, Theorem A]). For discrete G , our definition co-incides with Lück’s definition in [15]. Again, this is non-trivial and shownin [21, Theorem 2.2].The following lemma was observed in [19, Proposition 3.8]. Since it is adirect consequence of Theorem 4 we present the argument. Lemma 7. β n (2) ( G ) = 0 ⇔ H n ( G, L ( G )) = 0 . Proof.
Let β n (2) ( G ) = 0 . Let f : G n +1 → L ( G ) be a cocycle representing acohomology class [ f ] in H n ( G, L ( G )) . By Theorem 4 there is an increasingsequence of projections p j ∈ L ( G ) whose supremum is such that each f p j is a coboundary d n − b j . It is clear that f p j = d n − b j converges to f in thetopology of C ( G n +1 , L ( G )) , thus [ f ] = 0 . (cid:3) Coarse equivalence and coarse ℓ -cohomology Let G be a lcsc group. We fix a left-invariant proper continuous metric d on G . Let µ be a Haar measure on G . Let µ n be the n -fold product measureof µ on G n .For every R > and n ∈ N we consider the closed subset G nR := { ( g , ..., g n − ) ∈ G n | d ( g i , g j ) ≤ R for all ≤ i, j ≤ n − } and a family of semi-norms for measurable maps α : G n +1 → C defined by k α k R = Z G n +1 R | α ( g , ..., g n ) | dµ n +1 ∈ [0 , ∞ ] . Let CX n (2) ( G ) be the space of equivalence classes (up to µ n +1 -null sets)of measurable maps α : G n +1 → C such that k α k R < ∞ for every R > . The semi-norms k _ k R , R > , turn CX n (2) ( G ) into a Fréchet space. Itis straightforward to verify that the homogeneous differential (1) yields awell-defined, continuous homomorphism CX n (2) ( G ) → CX n +1(2) ( G ) (cf. [11,Proposition 2.3.3]). Thus we obtain a cochain complex of Fréchet spaces. Definition 8.
The coarse ℓ -cohomology of G is defined as HX n (2) ( G ) = H n (cid:0) CX ∗ (2) ( G ) (cid:1) . R. SAUER AND M. SCHRÖDL
By taking the quotients by the closure of the differentials, one defines sim-ilarly the reduced coarse ℓ -cohomology HX n (2) ( G ) . Remark . The previous definition is the continuous analog of Elek’s defini-tion [7, Definition 1.3] in the discrete case (Elek gives credits to Roe [22]).It is very much related to Pansu’s asymptotic ℓ -cohomology [18], which wasconsidered in the generality of metric measure spaces by Genton [11]. Thedifference of our definition to the one in Genton [11] is as follows: CX ∗ (2) ( G ) is an inverse limit of spaces L ( G ∗ +1 R ) . Unlike us, Genton takes first thecohomology of L ( G ∗ +1 R ) and then the inverse limit. Under some uniformcontractibility assumptions the two definitions coincide but likely not ingeneral. Theorem 10.
Let G be a unimodular lcsc group. For every n ≥ , the n -th continuous cohomology of G with coefficients in the left regular repre-sentation L ( G ) is isomorphic to the n -th coarse ℓ -cohomology of G , andlikewise for reduced cohomology.Proof. We have the obvious embedding L loc ( G n +1 , L ( G )) ⊂ L loc ( G n +1 , L loc ( G )) and the exponential law (see [1, Lemme 1.4] for a proof but beware of thetypo in the statement) L loc ( G n +1 , L loc ( G )) ∼ = L loc ( G n +1 × G ) . Thus an element in L loc ( G n +1 , L ( G )) G is represented by a measurable com-plex function in ( n + 2) -variables. For α ∈ L loc ( G n +1 , L ( G )) G we define µ n +2 -almost everywhere F n ( α )( x , . . . , x n , x ) = α ( x − x , . . . , x − x n )( x ) . The measurable function F n ( α ) is invariant by translation in the ( n + 2) -th variable. By Fubini’s theorem we may regard F n ( α ) as a measurablefunction E n ( α ) : G n +1 → C in the first ( n + 1) -variables. We may think of E n ( α ) as an evaluation of α at e . Let B ( R ) denote the R -ball around e ∈ G .Next we show that k E n ( α ) k R < ∞ for every R > , thus E n ( α ) ∈ CX n (2) ( G ) .Since α ∈ L loc ( G n +1 , L ( G )) G we have ∞ > Z B (2 R ) n +1 Z G | α ( x , x , ..., x n )( x ) | dµdµ n +1 = Z B (2 R ) n +1 Z G | α ( x, xx − x , ..., xx − x n )( x ) | dµdµ n +1 . -BETTI NUMBERS AND COARSE EQUIVALENCE 7 The map m : G n +2 → G n +2 , ( x , . . . , x n , x ) ( x, xx − x , . . . , xx − x n , x ) is measure preserving since it is the composition of taking inverses in the lastcoordinate, left multiplication by xx − , conjugation by x and taking inversesin the last coordinate. Note that this requires unimodularity. Further, wehave m (cid:0) G n +1 R × B ( R ) (cid:1) ⊂ B (2 R ) n +1 × G. This implies the first inequality below. The first equality follows from thefact that ( x , . . . , x n , x ) ( x − x , . . . , x − x n , x ) is a measure preservingmeasurable automorphism of G n +1 R × B ( R ) . Z B (2 R ) n +1 Z G | α ( x, xx − x , ..., xx − x n )( x ) | dµdµ n +1 ≥ Z G n +1 R Z B ( R ) | α ( x , . . . , x n )( x ) | dµdµ n +1 = Z G n +1 R Z B ( R ) | α ( x − x , . . . , x − x n )( x ) | dµdµ n +1 = µ ( B ( R )) k E n ( α ) k R . Hence k E n ( α ) k R is finite for every R > . That E ∗ : L loc ( G ∗ +1 , L ( G )) G → CX ∗ (2) ( G ) defines a cochain map is obvious. The above computation also implies that E ∗ is continuous with respect to the Fréchet topologies.Given β ∈ CX n (2) ( G ) we define M n ( β )( g , . . . , g n )( g ) = β ( g − g , . . . , g − g n ) for µ n +2 -almost every ( g , . . . , g n , g ) . The function M n ( β ) defines an elementin L loc ( G n +1 , L ( G )) G . The G -invariance of M n ( β ) is obvious. We have toshow that k M n ( β ) | B ( R ) n +1 k is square-integrable for every R > . This followsfrom the following computations which is based on the arguments above inreversed order. µ ( B ( R )) Z G n +12 R | β ( g , ..., g n ) | dµ n +1 = Z G n +12 R Z B ( R ) | β ( g , ..., g n ) | dµdµ n +1 ≥ Z B ( R ) n +1 Z G | β ( g − g , . . . , g − g n ) | dµdµ n +1 R. SAUER AND M. SCHRÖDL
Obviously, M ∗ is a chain map. Continuity follows from the previous com-putation. It is clear that M ∗ and E ∗ are mutual inverses. Using Theorem 3,this concludes the proof. (cid:3) Coarse invariance
We recall the notion of coarse equivalence. A map f : ( X, d X ) → ( Y, d Y ) between metric spaces is coarse Lipschitz if there is a non-decreasing func-tion a : [0 , ∞ ) → [0 , ∞ ) with lim t →∞ a ( t ) = ∞ such that d Y ( f ( x ) , f ( x ′ )) ≤ a ( d ( x, x ′ )) for all x, x ′ ∈ X . We say that two such maps f, g are close if sup x ∈ X d Y ( f ( x ) , g ( x )) < ∞ . A coarse Lipschitz map f : X → Y is a coarse equivalence if there is a coarseLipschitz map g : Y → X such that f g and gf are close to the identity. Wesay g is a coarse inverse of f . Lemma 11.
Coarsely equivalent lcsc groups are measurably coarse equiv-alent, i.e. if G and H are coarse equivalent lcsc groups then there are mea-surable coarse Lipschitz maps f : G → H and g : H → G such that f g and gf are close to the identity. Proof.
We choose left-invariant continuous proper metrics d G and d H on G and H , respectively. Let f : G → H be a coarse Lipschitz map with d H ( f ( x ) , f ( x ′ )) ≤ a ( d G ( x, x ′ )) . Let t > . We pick a countable measurablepartition U of G whose elements have diameter ≤ t and choose an element x U ∈ U for every U ∈ U .By setting ˜ f ( x ) = f ( x U ) for x ∈ U we obtain a coarse Lipschitz map ˜ f : G → H which satisfies d ( ˜ f ( x ) , ˜ f ( x ′ )) ≤ a ( d ( x, x ′ ) + 2 t ) and is close to f with d ( ˜ f ( x ) , f ( x )) ≤ a (2 t ) . Analogously, we construct a measurable coarseLipschitz map ˜ g , constructed from a coarse Lipschitz map g : H → G whichis a coarse inverse to f . It is obvious that ˜ g is a coarse inverse to ˜ f . (cid:3) Theorem 12.
Coarsely equivalent lcsc groups have isomorphic reduced andnon-reduced coarse ℓ -cohomology groups.Proof. Let G and H lcsc groups with Haar measures µ and ν , respectively.Let f : G → H be a coarse equivalence with coarse inverse g . Because oflemma 11 we can further assume that f and g are measurable. We define amap χ : G × G → R by χ ( x, y ) = B x ( c ) ( y ) µ ( B ( c )) -BETTI NUMBERS AND COARSE EQUIVALENCE 9 where we choose c such that µ ( B ( c )) ≥ . Then χ is a measurable func-tion with χ ( x, y ) = χ ( y, x ) and R G χ ( x, y ) dµ ( y ) = 1 . We use the followingnotation: χ : G n +1 × G n +1 → R , χ (( x , ..., x n ) , ( y , ..., y n )) = χ ( x , y ) · ... · χ ( x n , y n ) . Analogously, we define χ ′ : H n +1 × H n +1 → R with some radius c ′ . Nowwe can define the maps f ∗ : HX ∗ (2) ( H ) → HX ∗ (2) ( G ) and g ∗ : HX ∗ (2) ( G ) → HX ∗ (2) ( H ) as follows where we use x i for elements in G and y i for elementsof H : f ∗ α ( x , ..., x n ) = Z H n +1 α ( y , ..., y n ) χ ′ (( f ( x ) , ..., f ( x n )) , ( y , ..., y n )) dν n +1 g ∗ β ( y , ..., y n ) = Z G n +1 β ( x , ..., x n ) χ (( g ( y ) , ..., g ( y n )) , ( x , ..., x n )) dµ n +1 . The idea of averaging over a function like χ goes back to Pansu; it is nec-essary in our context since the maps f and g do not preserve the measureclasses, in general.First of all, we check that these are well-defined continuous cochain maps. ∞ > k α k a ( R )+ c ′ = Z H n +1 | α ( y , ..., y n ) | · H na ( R )+ c ′ dν n +1 ≥ Z H n +1 | α ( y , ..., y n ) | Z G n +1 R χ ′ (( f ( x ) , ..., f ( x n )) , ( y , ..., y n )) dµ n +1 dν n +1 = Z G n +1 R Z H n +1 | α ( y , ..., y n ) | χ ′ (( f ( x ) , ..., f ( x n )) , ( y , ..., y n )) dν n +1 dµ n +1 ≥ Z G n +1 R (cid:12)(cid:12)(cid:12)(cid:12)Z H n +1 α ( y , ..., y n ) χ ′ (( f ( x ) , ..., f ( x n )) , ( y , ..., y n )) dν n +1 (cid:12)(cid:12)(cid:12)(cid:12) dµ n +1 = Z G n +1 R | f n α ( x , ..., x n ) | dµ n +1 = k f n α k R It is a direct computation that d n ◦ f n = f n +1 ◦ d n .It remains to show that there is a cochain homotopy h : CX ∗ (2) ( H ) → CX ∗− ( H ) such that Id − g ∗ f ∗ = hd + dh . We define h n +1 i : CX n +1(2) ( H ) → CX n (2) ( H ) by h n +1 i α ( y , ..., y n )= Z H n +1 α (˜ y , ..., ˜ y i , y i , ..., y n ) χ ′ (( y , ..., y n ) , (˜ y , ..., ˜ y n )) dν n +1 (˜ y ) and set h n +1 = n X i =0 ( − i h n +1 i . That h ∗ is well-defined is a similar consideration as to show that f ∗ and g ∗ are well-defined. Now let us denote the i-th term of the coboundary map by d ni , i.e. d ni α ( y , ..., y n +1 ) = α ( y , ..., b y i , ..., y n +1 ) . It is straightforward toverify that we have the following relations: h n +1 n ◦ d nn +1 = g n ◦ f n ,h n +10 ◦ d n = Id CX n (2) ( H ) ,h n +1 j ◦ d ni = d n − i ◦ h nj − for ≤ j ≤ n and i ≤ j,h n +1 j ◦ d ni = d n − i − ◦ h nj for ≤ i ≤ n and i > j. We get h n +1 d n + d n − h n = Id CX n (2) ( H ) − g n ◦ f n . The same construction appliesto f ∗ g ∗ which completes the proof. (cid:3) Proof of Theorem 1.
Let G and H be unimodular lcsc groups. Let G and H be coarsely equivalent. Then we have the following equivalences: β ( G ) = 0 ⇔ H n ( G, L ( G )) = 0 ( Lemma ⇔ HX n (2) ( G ) = 0 ( Theorem ⇔ HX n (2) ( H ) = 0 ( Theorem
Going the same steps backwards for the group H finishes the proof. (cid:3) Remark . Since the Borel subgroup
B < SL ( R ) of upper triangularmatrices is cocompact, the solvable Lie groups B and SL ( R ) are quasi-isometric. So B belongs to the class of amenable hyperbolic lcsc groups ofwhich a systematic study was undertaken in [2].The group B is not unimodular and thus its ℓ -Betti number are notdefined. Nevertheless, one may ask what exactly breaks down in the proofabove which can be formulated to a large part without the notion of ℓ -Bettinumbers. By a result of Delorme [5, Corollaire V.3], we have H ( B, L ( B ))) =0 . Since Theorem 12 does not require unimodularity, we have HX ( B ) ∼ = HX n (2) (SL ( R )) = 0 since β (SL ( R )) = 0 . So it is Theorem 10 that failsfor the non-unimodular group B . Acknowledgements.
We acknowledge support by the German ScienceFoundation via the Research Training Group 2229.
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Institute for Algebra and Geometry, Karlsruhe Institute of Technol-ogy, Englerstr. 2, 76128 Karlsruhe, Germany
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