Vanishing of $\ell^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

TL;DR

This paper proves that the property of having zero $ ext{l}^2$-Betti numbers for unimodular locally compact second countable groups remains unchanged under coarse equivalence, establishing it as a coarse invariant.

Contribution

It demonstrates that the vanishing of $ ext{l}^2$-Betti numbers is preserved under coarse equivalence for a broad class of groups, extending previous invariance results.

Findings

Vanishing of $ ext{l}^2$-Betti numbers is a coarse invariant.

The result applies to unimodular locally compact second countable groups.

Provides a new tool for classifying groups via coarse geometry.

Abstract

We provide a proof that the vanishing of $\ell^2$-Betti numbers of unimodular locally compact second countable groups is an invariant of coarse equivalence.

Equations

$$d^{n-1}f(g_0,...,g_n)=\sum _{i=0}^n(-1{)}^{i}f(g_0,...,\widehat{g_i},...,g_n)$$

$$H^{n}(G,E)=H^{n}\bigl{(}C(G^{*+1},E)^{G}\bigr{)}$$

$$I^{\ast }:C(G^{\ast +1},E)\to L_{loc}^2(G^{\ast +1},E).$$

$$(R^{n}f)(g_0,...,g_n)={\int }_{G^{n+1}}f(h_0,...,h_n)\chi (g_0^{-1}{h}_0)\cdot ...\cdot \chi (g_n^{-1}{h}_n)d\mu (h_0,...,h_n)$$

$${\beta }_{(2)}^n(G):={\dim }_{\mu }{\underline{H}}^n(G,L^2(G))\in [0,\infty ].$$

$$G_R^n:=\{(g_0,...,g_{n-1})\in G^n|d(g_i,g_j)\le R\text{ for all }0\le i,j\le n-1\}$$

$$\Vert \alpha {\Vert }_R^2={\int }_{G_R^{n+1}}|\alpha (g_0,...,g_n){|}^{2}d{\mu }_{n+1}\in [0,\infty ].$$

$$HX_{(2)}^{n}(G)=H^{n}\bigl{(}CX_{(2)}^{\ast}(G)\bigr{)}.$$

$$L_{loc}^2(G^{n+1},L^2(G))\subset L_{loc}^2(G^{n+1},L_{loc}^2(G))$$

$$L_{loc}^2(G^{n+1},L_{loc}^2(G))\cong L_{loc}^2(G^{n+1}\times G).$$

$$F^n(\alpha )(x_0,\dots ,x_n,x)=\alpha (x^{-1}{x}_0,\dots ,x^{-1}{x}_n)(x).$$

$$\infty$$

$$={\int }_{B(2R{)}^{n+1}}{\int }_G|\alpha (x,x{x}_0^{-1}{x}_1,...,x{x}_0^{-1}{x}_n)(x_0){|}^{2}d\mu d{\mu }_{n+1}.$$

$$m:G^{n+2}\to G^{n+2},(x_0,\dots ,x_n,x)\mapsto (x,x{x}_0^{-1}{x}_1,\dots ,x{x}_0^{-1}{x}_n,x_0)$$

$$m\bigl{(}G_{R}^{n+1}\times B(R)\bigr{)}\subset B(2R)^{n+1}\times G.$$

$${\int }_{B(2R{)}^{n+1}}{\int }_G|\alpha (x,x{x}_0^{-1}{x}_1$$

$$\ge {\int }_{G_R^{n+1}}{\int }_{B(R)}|\alpha (x_0,\dots ,x_n)(x){|}^{2}d\mu d{\mu }_{n+1}$$

$$={\int }_{G_R^{n+1}}{\int }_{B(R)}|\alpha (x^{-1}{x}_0,\dots ,x^{-1}{x}_n)(x){|}^{2}d\mu d{\mu }_{n+1}$$

$$=\mu (B(R))\Vert E^n(\alpha ){\Vert }_R.$$

$$E^{\ast }:L_{loc}^2(G^{\ast +1},L^2(G){)}^G\to C{X}_{(2)}^{\ast }(G)$$

$$M^n(\beta )(g_0,\dots ,g_n)(g)=\beta (g^{-1}{g}_0,\dots ,g^{-1}{g}_n)$$

$$\mu (B(R)){\int }_{G_{2R}^{n+1}}|$$

$$={\int }_{G_{2R}^{n+1}}{\int }_{B(R)}|\beta (g_0,...,g_n){|}^{2}d\mu d{\mu }_{n+1}$$

$$\ge {\int }_{B(R{)}^{n+1}}{\int }_G|\beta (g^{-1}{g}_0,\dots ,g^{-1}{g}_n)|d\mu d{\mu }_{n+1}$$

$$d_Y(f(x),f(x^{\prime }))\le a(d(x,x^{\prime }))$$

$$\underset{x\in X}{\sup }{d}_Y(f(x),g(x))<\infty .$$

$$\chi (x,y)=\frac{\text{mathds}{1}_{B_x(c)}(y)}{\mu (B(c))}$$

$$\chi :G^{n+1}\times G^{n+1}\to \mathbb{R},\chi ((x_0,...,x_n),(y_0,...,y_n))=\chi (x_0,y_0)\cdot ...\cdot \chi (x_n,y_n).$$

$$f^{\ast }\alpha (x_0,...,x_n)$$

$$g^{\ast }\beta (y_0,...,y_n)$$

$$\infty >\Vert \alpha {\Vert }_{a(R)+c^{\prime }}^2={\int }_{H^{n+1}}|\alpha (y_0,...,y_n){|}^2\cdot \text{mathds}{1}_{H_{a(R)+c^{\prime }}^n}d{\nu }_{n+1}$$

$$\ge {\int }_{H^{n+1}}|\alpha (y_0,...,y_n){|}^2{\int }_{G_R^{n+1}}{\chi }^{\prime }\left((f(x_0),...,f(x_n)),(y_0,...,y_n)\right)d{\mu }_{n+1}d{\nu }_{n+1}$$

$$={\int }_{G_R^{n+1}}{\int }_{H^{n+1}}{\left|\alpha (y_0,...,y_n)\right|}^2{\chi }^{\prime }\left((f(x_0),...,f(x_n)),(y_0,...,y_n)\right)d{\nu }_{n+1}d{\mu }_{n+1}$$

$$\ge {\int }_{G_R^{n+1}}{\left|{\int }_{H^{n+1}}\alpha (y_0,...,y_n){\chi }^{\prime }\left((f(x_0),...,f(x_n)),(y_0,...,y_n)\right)d{\nu }_{n+1}\right|}^{2}d{\mu }_{n+1}$$

$$={\int }_{G_R^{n+1}}|f^n\alpha (x_0,...,x_n){|}^{2}d{\mu }_{n+1}=\Vert f^n\alpha {\Vert }_R^2$$

$$h_i^{n+1}\alpha (y_0,...,y_n)$$

Full text

1. Introduction

The insight that the vanishing of $\ell^{2}$-Betti numbers provides a quasi-isometry invariant is due to Gromov (see [12]*Chapter 8 for a statement without proof), and positive results around this insight have a long history. The most important contribution is by Pansu [18] whose work on asymptotic $\ell^{p}$-cohomology includes a proof that the vanishing of $\ell^{2}$-Betti numbers of discrete groups of type $F_{\infty}$, is a quasi-isometry invariant.

There is a growing interest in the metric geometry of locally compact groups [3, 2]. We thus think it is important to have the quasi-isometry and coarse invariance of the vanishing of $\ell^{2}$-Betti numbers available in the greatest generality. Following Pansu’s ideas and relying on more recent advances in the theory of $\ell^{2}$-Betti numbers, we provide a proof of the following result.

Theorem 1**.**

Let $G$ and $H$ be unimodular locally compact second countable groups. If $G$ and $H$ are coarsely equivalent then the $n$-th $\ell^{2}$-Betti number of $G$ vanishes if and only the $n$-th $\ell^{2}$-Betti number of $H$ vanishes.

The coarse invariance for discrete groups was proven earlier in a paper of Mimura-Ozawa-Sako-Suzuki [16]*Corollary 6.3.

Every locally compact, second countable group $G$ (hereafter abbreviated by lcsc) has a left-invariant proper continuous metric by a theorem of 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 is a quasi-isometry with respect to word metrics of compact symmetric generating sets and vice versa. In particular, a coarse equivalence between finitely generated discrete groups is a quasi-isometry. See [3]*Chapter 4 for a systematic discussion of these notions.

To even state Theorem 1 in that generality, recent advances in the theory of $\ell^{2}$-Betti numbers were necessary. $\ell^{2}$-Betti numbers of discrete groups enjoy a long history but it was not until recently that $\ell^{2}$-Betti numbers were defined for arbitrary unimodular lcsc groups by Petersen [19], and a systematic theory analogous to the discrete case emerged \citelist[19][13][20]. Earlier studies of $\ell^{2}$-Betti numbers of locally compact groups in specific cases can be found in [10, 6, 4].

Previous results on coarse invariance

Pansu [18] introduced asymptotic $\ell^{p}$-cohomology of discrete groups and proved its invariance under quasi-isometries. If a group $\Gamma$ is of type $F_{\infty}$, then the $\ell^{p}$-cohomology of $\Gamma$ coincides with its asymptotic $\ell^{p}$-cohomology [18]*Théorème 1. The geometric explanation for the appearance of the type $F_{\infty}$ condition is that the finite-dimensional skeleta of the universal covering of a classifying space of finite type are uniformly contractible. As an immediate consequence of Pansu’s result, the vanishing of $\ell^{2}$-Betti numbers is a quasi-isometry invariant among discrete groups of type $F_{\infty}$. The same arguments work for totally disconnected groups admitting a topological model of finite type [23].

Elek [7] investigated the relation between $\ell^{p}$-cohomology of discrete groups and Roe’s coarse cohomology and proved similar results. Another independent treatment is due to Fan [8]. Genton [11] elaborated upon Pansu’s methods in the case of metric measure spaces.

Oguni [17] generalised the quasi-isometry invariance of the vanishing of $\ell^{2}$-Betti numbers from discrete groups of type $F_{\infty}$ to discrete groups whose cohomology with coefficients in the group von Neumann algebra satisfies a certain technical condition. A similar technical condition appears in the proof of quasi-isometry invariance of Novikov-Shubin invariants of amenable groups [25]*, and it is unclear how much this condition differs from the type $F_{\infty}$-condition. Oguni’s groupoid approach is inspired by [9, 25] and quite different from the approaches by Elek, Fan, and Pansu.

The coarse invariance of vanishing of $\ell^{2}$-Betti numbers for discrete groups was shown by Mimura-Ozawa-Sako-Suzuki [16]*Corollary 6.3. Li [14] recently reproved this using groupoid techniques as a consequence of more general cohomological coarse invariance results.

Structure of the paper

We review the necessary basics of $\ell^{2}$-Betti numbers and continuous cohomology in Section 2. In Section 3 we define coarse $\ell^{2}$-cohomology for lcsc groups and show that it is isomorphic to continuous cohomology. In Section 4 we conclude the proof of Theorem 1 and discuss what fails for non-unimodular groups.

2. Continuous cohomology and $\ell^{2}$-Betti numbers of lcsc groups

Let $G$ be a unimodular lcsc group with Haar measure $\mu$. Let $X$ be a locally compact second countable space with Radon measure $\nu$. Let $E$ be a Fréchet space.

The space $C(X,E)$ of continuous functions from $X$ to $E$ becomes a Fréchet space when endowed with the topology of compact convergence. Let $L^{2}_{loc}(X,E)$ be the space of equivalence classes of measurable maps $f\colon X\to E$ up to $\nu$-null sets such that $||f|_{K}||_{E}$ is square-integrable for every compact subset $K\subset X$. The $L^{2}$-norm of the function $||f|_{K}||_{E}$ defines a semi-norm $p_{K}$ on $L^{2}_{loc}(X,E)$. The family of semi-norms $p_{K}$, $K\subset E$, turns $L^{2}_{loc}(X,E)$ into a Fréchet space.

We call a Fréchet space $E$ with a continuous (i.e. $G\to E$, $g\mapsto gv$, is continuous for every $v\in E$) 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 $\nu$-preserving on $X$ then $C(X,E)$ and $L^{2}_{loc}(X,E)$ become $G$-modules via $(g\cdot f)(x)=gf(g^{-1}x)$ for $x\in X$ and $g\in G$ [1]*Proposition 3.1.1. The usual homogeneous coboundary map

[TABLE]

defines cochain complexes $C(G^{*+1},E)$ and $L^{2}_{loc}(G^{\ast+1},E)$ of $G$-modules (cf. [1]*Proposition 3.2.1). Here we take the diagonal $G$-action on $G^{\ast+1}$. We recall the following definition.

Definition 2**.**

The (continuous) cohomology of $G$ with coefficients in $E$ is the cohomology

[TABLE]

of the $G$-invariants of $C(G^{*+1},E)$. The reduced (continuous) cohomology $\underline{H}^{*}(G,E)$ is a quotient of $H^{\ast}(G,E)$ obtained by taking the quotient with the closure of $\operatorname{im}d^{\ast-1}$ instead of $\operatorname{im}d^{\ast-1}$.

We have an obvious inclusion

[TABLE]

The maps $I^{\ast}$ form a cochain map of $G$-modules. Taking a positive function $\chi\in C_{c}(G)$ there is a cochain map $R^{*}:L_{loc}^{2}(G^{*+1},E)\to C(G^{*+1},E)$ of $G$-modules

[TABLE]

such that $I^{*}\circ R^{*}$ and $R^{*}\circ I^{*}$ 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 $I^{\ast}$ in (2) induces isomorphisms in cohomology and in reduced cohomology.

Next we turn to the case where the coefficient module $E=L^{2}(G)$ is the regular representation, relevant for the definition of $\ell^{2}$-Betti numbers.

Let $L(G)$ be the von Neumann algebra of $G$; the Haar measure $\mu$ defines a semifinite trace $\operatorname{tr}_{\mu}$ on $L(G)$. There are a natural left $G$-action and a natural right $L(G)$-action on $L^{2}(G)$, and the two actions commute. Hence also the $G$-actions on $C(G^{\ast+1},L^{2}(G))$ and $L^{2}_{loc}(G^{\ast+1},L^{2}(G))$ considered previously and the $L(G)$-actions induced from the right $L(G)$-action on $L^{2}(G)$ commute. So the (reduced and non-reduced) continuous cohomology of $G$ with coefficients in $L^{2}(G)$ is naturally a $L(G)$-module111When talking about $L(G)$-modules we mean the algebraic module structure and ignore topologies.. Obviously, the cochain map $I^{\ast}$ above is compatible with the $L(G)$-module structures. The groups $H^{\ast}(G,L^{2}(G))$ are called the (continuous) $\ell^{2}$-cohomology of $G$. Similarly for the reduced cohomology.

Petersen [19] extended Lück’s dimension function from finite von Neumann algebras to semifinite von Neumann algebras. The dimension function $\operatorname{dim}_{\mu}$ with respect to $(G,\mu)$ 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 $\operatorname{dim}_{c\mu}=c^{-1}\operatorname{dim}_{\mu}$ for $c>0$. The fact that a $L(G)$-module has dimension zero can be expressed without referring to the trace: it is an algebraic fact. The following criterion was shown by the first author for finite von Neumann algebras [24]*Theorem 2.4; it was extended to the semifinite case by Petersen [19]*Lemma B.27.

Theorem 4**.**

An $L(G)$-module $M$ satisfies $\operatorname{dim}_{\mu}(M)=0$ if and only if for every $x\in M$ there is an increasing sequence $(p_{i})$ of projections in $L(G)$ with $\sup p_{i}=1$ such that $xp_{i}=0$ for every $i\in{\mathbb{N}}$.

Definition 5**.**

The $n$-th $\ell^{2}$-Betti number of $G$ is the $L(G)$-dimension of its reduced continuous cohomology with coefficients in $L^{2}(G)$, i.e.

[TABLE]

Remark 6*.*

Equivalently, the $n$-th $\ell^{2}$-Betti number can be defined as the $L(G)$-dimension of the non-reduced cohomology $H^{n}(G,L^{2}(G))$. This is a non-trivial fact (see [13]*Theorem A). For discrete $G$, our definition coincides with Lück’s definition in [15]. Again, this is non-trivial and shown in [21]*Theorem 2.2.

The following lemma was observed in [19]*Proposition 3.8. Since it is a direct consequence of Theorem 4 we present the argument.

Lemma 7**.**

$\beta_{(2)}^{n}(G)=0\Leftrightarrow\underline{H}^{n}(G,L^{2}(G))=0.$

Proof.

Let $\beta_{(2)}^{n}(G)=0$. Let $f\colon G^{n+1}\to L^{2}(G)$ be a cocycle representing a cohomology class $[f]$ in $\underline{H}^{n}(G,L^{2}(G))$. By Theorem 4 there is an increasing sequence of projections $p_{j}\in L(G)$ whose supremum is $1$ such that each $fp_{j}$ is a coboundary $d^{n-1}b_{j}$. It is clear that $fp_{j}=d^{n-1}b_{j}$ converges to $f$ in the topology of $C(G^{n+1},L^{2}(G))$, thus $[f]=0$. ∎

3. Coarse equivalence and coarse $\ell^{2}$-cohomology

Let $G$ be a lcsc group. We fix a left-invariant proper continuous metric $d$ on $G$. Let $\mu$ be a Haar measure on $G$. Let $\mu_{n}$ be the $n$-fold product measure of $\mu$ on $G^{n}$.

For every $R>0$ and $n\in\mathbb{N}_{0}$ we consider the closed subset

[TABLE]

and a family of semi-norms for measurable maps $\alpha\colon G^{n+1}\to\mathbb{C}$ defined by

[TABLE]

Let $CX_{(2)}^{n}(G)$ be the space of equivalence classes (up to $\mu_{n+1}$-null sets) of measurable maps $\alpha\colon G^{n+1}\to\mathbb{C}$ such that $\|\alpha\|_{R}<\infty$ for every $R>0$. The semi-norms $\|\_\|_{R}$, $R>0$, turn $CX_{(2)}^{n}(G)$ into a Fréchet space. It is straightforward to verify that the homogeneous differential (1) yields a well-defined, continuous homomorphism $CX_{(2)}^{n}(G)\to CX_{(2)}^{n+1}(G)$ (cf. [11]*Proposition 2.3.3). Thus we obtain a cochain complex of Fréchet spaces.

Definition 8**.**

The coarse $\ell^{2}$-cohomology of $G$ is defined as

[TABLE]

By taking the quotients by the closure of the differentials, one defines similarly the reduced coarse $\ell^{2}$-cohomology $\underline{H}X_{(2)}^{n}(G)$.

Remark 9*.*

The previous definition is the continuous analog of Elek’s definition [7]*Definition 1.3 in the discrete case (Elek gives credits to Roe [22]). It is very much related to Pansu’s asymptotic $\ell^{2}$-cohomology [18], which was considered in the generality of metric measure spaces by Genton [11]. The difference of our definition to the one in Genton [11] is as follows: $CX_{(2)}^{\ast}(G)$ is an inverse limit of spaces $L^{2}(G^{\ast+1}_{R})$. Unlike us, Genton takes first the cohomology of $L^{2}(G^{\ast+1}_{R})$ and then the inverse limit. Under some uniform contractibility assumptions the two definitions coincide but likely not in general.

Theorem 10**.**

Let $G$ be a unimodular lcsc group. For every $n\geq 0$, the $n$-th continuous cohomology of $G$ with coefficients in the left regular representation $L^{2}(G)$ is isomorphic to the $n$-th coarse $\ell^{2}$-cohomology of $G$, and likewise for reduced cohomology.

Proof.

We have the obvious embedding

[TABLE]

and the exponential law (see [1]*Lemme 1.4 for a proof but beware of the typo in the statement)

[TABLE]

Thus an element in $L^{2}_{loc}(G^{n+1},L^{2}(G))^{G}$ is represented by a measurable complex function in $(n+2)$-variables. For $\alpha\in L^{2}_{loc}(G^{n+1},L^{2}(G))^{G}$ we define $\mu_{n+2}$-almost everywhere

[TABLE]

The measurable function $F^{n}(\alpha)$ is invariant by translation in the $(n+2)$-th variable. By Fubini’s theorem we may regard $F^{n}(\alpha)$ as a measurable function $E^{n}(\alpha)\colon G^{n+1}\to{\mathbb{C}}$ in the first $(n+1)$-variables. We may think of $E^{n}(\alpha)$ as an evaluation of $\alpha$ at $e$. Let $B(R)$ denote the $R$-ball around $e\in G$. Next we show that $\|E^{n}(\alpha)\|_{R}<\infty$ for every $R>0$, thus $E^{n}(\alpha)\in CX_{(2)}^{n}(G)$.

Since $\alpha\in L^{2}_{loc}(G^{n+1},L^{2}(G))^{G}$ we have

[TABLE]

The map

[TABLE]

is measure preserving since it is the composition of taking inverses in the last coordinate, left multiplication by $xx_{0}^{-1}$, conjugation by $x$ and taking inverses in the last coordinate. Note that this requires unimodularity. Further, we have

[TABLE]

This implies the first inequality below. The first equality follows from the fact that $(x_{0},\dots,x_{n},x)\mapsto(x^{-1}x_{0},\dots,x^{-1}x_{n},x)$ is a measure preserving measurable automorphism of $G_{R}^{n+1}\times B(R)$.

[TABLE]

Hence $\|E^{n}(\alpha)\|_{R}$ is finite for every $R>0$. That

[TABLE]

defines a cochain map is obvious. The above computation also implies that $E^{\ast}$ is continuous with respect to the Fréchet topologies.

Given $\beta\in CX_{(2)}^{n}(G)$ we define

[TABLE]

for $\mu_{n+2}$-almost every $(g_{0},\ldots,g_{n},g)$. The function $M^{n}(\beta)$ defines an element in $L^{2}_{loc}(G^{n+1},L^{2}(G))^{G}$. The $G$-invariance of $M^{n}(\beta)$ is obvious. We have to show that $\|M^{n}(\beta)|_{B(R)^{n+1}}\|$ is square-integrable for every $R>0$. This follows from the following computations which is based on the arguments above in reversed order.

[TABLE]

Obviously, $M^{\ast}$ is a chain map. Continuity follows from the previous computation. It is clear that $M^{\ast}$ and $E^{\ast}$ are mutual inverses. Using Theorem 3, this concludes the proof. ∎

4. Coarse invariance

We recall the notion of coarse equivalence. A map $f\colon(X,d_{X})\to(Y,d_{Y})$ between metric spaces is coarse Lipschitz if there is a non-decreasing function $a\colon[0,\infty)\to[0,\infty)$ with $\lim_{t\to\infty}a(t)=\infty$ such that

[TABLE]

for all $x,x^{\prime}\in X$. We say that two such maps $f,g$ are close if

[TABLE]

A coarse Lipschitz map $f\colon X\to Y$ is a coarse equivalence if there is a coarse Lipschitz map $g\colon Y\to X$ such that $fg$ and $gf$ are close to the identity. We say $g$ is a coarse inverse of $f$.

Lemma 11**.**

Coarsely equivalent lcsc groups are measurably coarse equivalent, i.e. if $G$ and $H$ are coarse equivalent lcsc groups then there are measurable coarse Lipschitz maps $f\colon G\to H$ and $g\colon H\to G$ such that $fg$ 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 $f\colon G\to H$ be a coarse Lipschitz map with $d_{H}(f(x),f(x^{\prime}))\leq a(d_{G}(x,x^{\prime}))$. Let $t>0$. We pick a countable measurable partition $\mathcal{U}$ of $G$ whose elements have diameter $\leq t$ and choose an element $x_{U}\in U$ for every $U\in\mathcal{U}$.

By setting $\tilde{f}(x)=f(x_{U})$ for $x\in U$ we obtain a coarse Lipschitz map $\tilde{f}\colon G\to H$ which satisfies $d(\tilde{f}(x),\tilde{f}(x^{\prime}))\leq a(d(x,x^{\prime})+2t)$ and is close to $f$ with $d(\tilde{f}(x),f(x))\leq a(2t)$. Analogously, we construct a measurable coarse Lipschitz map $\tilde{g}$, constructed from a coarse Lipschitz map $g\colon H\to G$ which is a coarse inverse to $f$. It is obvious that $\tilde{g}$ is a coarse inverse to $\tilde{f}$. ∎

Theorem 12**.**

Coarsely equivalent lcsc groups have isomorphic reduced and non-reduced coarse $\ell^{2}$-cohomology groups.

Proof.

Let $G$ and $H$ lcsc groups with Haar measures $\mu$ and $\nu$, respectively. Let $f\colon G\to H$ be a coarse equivalence with coarse inverse $g$. Because of lemma 11 we can further assume that $f$ and $g$ are measurable. We define a map $\chi\colon G\times G\to\mathbb{R}$ by

[TABLE]

where we choose $c$ such that $\mu(B(c))\geq 1$. Then $\chi$ is a measurable function with $\chi(x,y)=\chi(y,x)$ and $\int_{G}\chi(x,y)d\mu(y)=1$. We use the following notation:

[TABLE]

Analogously, we define $\chi^{\prime}\colon H^{n+1}\times H^{n+1}\to\mathbb{R}$ with some radius $c^{\prime}$. Now we can define the maps $f^{*}\colon HX_{(2)}^{*}(H)\to HX_{(2)}^{*}(G)$ and $g^{*}\colon HX_{(2)}^{*}(G)\to HX_{(2)}^{*}(H)$ as follows where we use $x_{i}$ for elements in $G$ and $y_{i}$ for elements of $H$:

[TABLE]

The idea of averaging over a function like $\chi$ goes back to Pansu; it is necessary in our context since the maps $f$ and $g$ do not preserve the measure classes, in general.

First of all, we check that these are well-defined continuous cochain maps.

[TABLE]

It is a direct computation that $d^{n}\circ f^{n}=f^{n+1}\circ d^{n}$.

It remains to show that there is a cochain homotopy $h\colon CX_{(2)}^{*}(H)\to CX_{(2)}^{*-1}(H)$ such that $\operatorname{Id}-g^{*}f^{*}=hd+dh$. We define $h_{i}^{n+1}\colon CX_{(2)}^{n+1}(H)\to CX_{(2)}^{n}(H)$ by

[TABLE]

and set

[TABLE]

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 $d^{n}_{i}$, i.e. $d^{n}_{i}\alpha(y_{0},...,y_{n+1})=\alpha(y_{0},...,\widehat{y_{i}},...,y_{n+1})$. It is straightforward to verify that we have the following relations:

[TABLE]

We get $h^{n+1}d^{n}+d^{n-1}h^{n}=\operatorname{Id}_{CX_{(2)}^{n}(H)}-g^{n}\circ f^{n}$. The same construction applies to $f^{*}g^{*}$ which completes the proof. ∎

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:

[TABLE]

Going the same steps backwards for the group $H$ finishes the proof. ∎

Remark 13*.*

Since the Borel subgroup $B<\operatorname{SL}_{2}({\mathbb{R}})$ of upper triangular matrices is cocompact, the solvable Lie groups $B$ and $\operatorname{SL}_{2}({\mathbb{R}})$ are quasi-isometric. So $B$ belongs to the class of amenable hyperbolic lcsc groups of which a systematic study was undertaken in [2].

The group $B$ is not unimodular and thus its $\ell^{2}$-Betti number are not defined. Nevertheless, one may ask what exactly breaks down in the proof above which can be formulated to a large part without the notion of $\ell^{2}$-Betti numbers. By a result of Delorme [5]*Corollaire V.3, we have $\underline{H}^{1}(B,L^{2}(B)))=0$. Since Theorem 12 does not require unimodularity, we have $\underline{H}X_{(2)}^{1}(B)\cong\underline{H}X_{(2)}^{n}(\operatorname{SL}_{2}({\mathbb{R}}))\neq 0$ since $\beta_{(2)}^{1}(\operatorname{SL}_{2}({\mathbb{R}}))\neq 0$. So it is Theorem 10 that fails for the non-unimodular group $B$.

Acknowledgements

We acknowledge support by the German Science Foundation via the Research Training Group 2229.