1. Introduction
Finiteness conditions of discrete groups are higher-dimensional generalisations of the notions of being finitely generated and finitely presented. If a group satisfies suitable finiteness conditions, one can expect the group homology to enjoy nice properties, like, for example, being finitely generated. The appeal of studying finiteness conditions stems from the interaction between topology (specifically, the topology of classifying spaces) and algebra (notably, homological algebra involving chain complexes over the group ring). A similar theory for total disconnected locally compact Hausdorff groups, which we refer to as tdlc groups, was still in its infancy a few years ago. What is different from the discrete case? On the topological side, tdlc groups often admit nice actions on CW-complexes or simplicial complexes but these actions are never free. On the algebraic side, smooth modules constitute a suitable abelian category; however, it possesses enough projectives only over the rationals.
If one is content with studying finiteness conditions of a tdlc group G modulo the family of its compact-open subgroups, an elegant framework encompassing both topological and algebraic aspects becomes available. The class of G-equivariant CW-complexes with compact-open stabilisers enjoys a well-developed equivariant homotopy theory, similar to the discrete case [
Reference Tom Dieck16
,
Reference Lück18
]. Its algebraic counterpart, the category of chain complexes over the orbit category of G with respect to the family of compact-open subgroups, is an abelian category, even when considered with integral coefficients [
Reference Lück18
]. Generally, however, it is unsatisfactory to be restricted to working modulo the family of compact-open subgroups. Thompson’s group V, for example, satisfies the finiteness condition
$F_\infty$
in the usual sense – an important early result of Geoghegan and Brown [
Reference Brown and Geoghegan8
] – but not in the category of chain complexes of the orbit category with respect to the family of finite subgroups. There is a similar situation for Neretin’s group, which is a totally disconnected analog of Thompson’s group V [
Reference Sauer and Thumann21
].
The recent work of Castellano and Corob Cook [ Reference Castellano and Corob Cook10 ] drops all these limitations and establishes a convenient and elegant algebraic theory of finiteness conditions for tdlc groups, which works also with integral coefficients. Many of the fundamental properties of the discrete theory, as presented in Brown’s foundational book [ Reference Brown5 ], now find analogs in the study of tdlc groups.
An important construction of tdlc groups from discrete groups is the Schlichting completion of a discrete group relative to a commensurated subgroup (see Section 2). The contribution of this paper is to prove finiteness properties of the Schlichting completions and to relate finiteness properties and cohomology of the Schlichting completion to the ones of its defining discrete group.
If the commensurated subgroup is normal, then the Schlichting completion is just the quotient, in particular, it is discrete. One should read Theorems1·1, 1·2 and 1·5 below with this in mind; the results for quotient groups are well known.
The definitions of properties
$\operatorname{FP}^R_n$
and
$\operatorname{F}_n$
are recalled in Section 3.
Theorem 1·1 Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda\lt\Gamma$
. Let R be a commutative ring. Then the following holds:
-
(i) if
$\Lambda$
and G have type
$\operatorname{FP}^R_n$
, then
$\Gamma$
has type
$\operatorname{FP}^R_n$
; -
(ii) if
$\Lambda$
and G have type
$\operatorname{F}_n$
, then
$\Gamma$
has type
$\operatorname{F}_n$
.
Theorem 1·2 Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda\lt\Gamma$
. Let R be a commutative ring. Then the following holds:
-
(i) if
$\Lambda$
has type
$\operatorname{FP}^R_{n-1}$
and
$\Gamma$
has type
$\operatorname{FP}^R_n$
, then G has type
$\operatorname{FP}^R_n$
; -
(ii) if
$\Lambda$
has type
$\operatorname{F}_{n-1}$
and
$\Gamma$
has type
$\operatorname{F}_n$
, then G has type
$\operatorname{F}_n$
.
Theorems 1·1 and 1·2 are proved in Section 3.
It is an interesting question when the restriction map from the continuous cohomology of a locally compact group to the cohomology of a dense subgroup is an isomorphism. For the inclusion
$\operatorname{SL}_n({\mathbb Q})\hookrightarrow \operatorname{SL}_n({\mathbb R})$
this was proved by Borel-Yang [
Reference Borel and Yang3
] in order to solve the rank conjecture in algebraic K-theory. In the next result, which is proved in Section 4, we consider the easier situation of the inclusion of a discrete group into its Schlichting completion.
Theorem 1·3 Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to a locally finite commensurated subgroup
$\Lambda\lt\Gamma$
. Then the restriction map
$H^\ast_c(G,{\mathbb R})\to H^\ast(\Gamma, {\mathbb R})$
is an isomorphism in all degrees.
Neretin’s group
$N_d$
, which is the group of almost automorphisms of a non-rooted
$(d+1)$
-regular tree, is the Schlichting completion of the Higman-Thompson’s group
$V_{d,2}$
relative to a locally finite commensurated subgroup [
Reference Le Boudec9
] example 6·7. Brown [
Reference Brown7
] showed the rational acyclicityFootnote
1
of
$V_{d,2}$
.
We obtain the following consequence.
Corollary 1·4. Let
$d\ge 2$
. The continuous cohomology
$H^i_c(N_d,{\mathbb R})$
of Neretin’s group
$N_d$
vanishes for every
$i\gt0$
.
In the next result,
$\chi^{(2)}(G, \mu)$
denotes the Euler characteristic of an unimodular tdlc group. This invariant is discussed in Section 5. If G is a discrete group with a finite model of its classifying space and
$\mu$
is the counting measure, then
$\chi^{(2)}(G, \mu)$
is the usual Euler characteristic. If G is discrete and has torsion, it is the
$\ell^2$
-Euler characteristic whenever it is defined.
Theorem 1·5. Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda\lt\Gamma$
. Suppose that G is unimodular and that
$\Lambda$
and G have type
$\operatorname{FP}^{\mathbb Q}$
. Then
$\Gamma$
has type
$\operatorname{FP}^{\mathbb Q}$
and we have
\[ \chi^{(2)}(\Lambda)\cdot\chi^{(2)}(G,\mu)=\chi^{(2)}(\Gamma)\]
for the Haar measure
$\mu$
with
$\mu(U)=1$
where
$U\lt G$
is the closure of
$\Lambda$
.
2. The Schlichting completion
The Schlichting completion of a discrete group
$\Gamma$
relative to the commensurated subgroup
$\Lambda$
is a tdlc group which we denote by
$G=\Gamma /\!/\Lambda$
. This construction was introduced in [
Reference Tzanev25
], following an earlier idea of Schlichting [
Reference Schlichting22
].
A nice background reference is the work of Shalom and Willis [
Reference Shalom and Willis23
] who call the Schlichting completion the relative profinite completion of
$\Gamma$
with respect to
$\Lambda$
.
Let
$\Gamma$
be a discrete group and
$\Lambda \lt \Gamma$
be a commensurated subgroup. Then
$\Gamma$
acts by left multiplication on
$\Gamma /\Lambda$
and thus defines a homomorphism
\[\alpha \colon \Gamma \to \operatorname{Sym}(\Gamma/\Lambda).\]
We equip
$\operatorname{Sym}(\Gamma/\Lambda)$
with the topology of pointwise convergence. The closure
\[G=\Gamma /\!/ \Lambda= \overline{\alpha(\Gamma)}\]
is the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda$
. Strictly speaking, the Schlichting completion is not a completion of
$\Gamma$
since
$\alpha$
might not be injective.
In the following, we collect some properties of this construction.
Proposition 2·1 ([
Reference Shalom and Willis23
, Section 3]). Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda\lt\Gamma$
.
-
(i) If
$\Lambda$
is normal in
$\Gamma$
then
$G=\Gamma/\Lambda$
. -
(ii) G is a tdlc group.
-
(iii) The map
$\alpha \colon \Gamma \to G$
has a dense image. Its kernel is the largest subgroup that is normal in
$\Gamma$
and contained in
$\Lambda$
. -
(iv) The closure of the image
$\overline{\alpha(\Lambda)}$
is a compact open subgroup of G. In particular, it is commensurated in G.
For the proof of Theorem 2·3 we use the following easy fact about the Schlichting completion.
Lemma 2·2 ([ Reference Le Boudec9 , Lemmas 6.3 and 6.4]). The following holds for the Schlichting completion:
-
(i)
$\Gamma/\!/\Lambda=\overline{\alpha(\Lambda)}\alpha(\Gamma)$
; -
(ii)
$\overline{\alpha(\Lambda)}\cap \alpha(\Gamma)=\alpha(\Lambda)$
.
An easy consequence is that
$\alpha$
induces an isomorphism
$\Gamma/\Lambda\xrightarrow{\cong} G/U$
where
$G=\Gamma/\!/\Lambda$
and
$U=\overline{\alpha(\Lambda)}$
. This fact will be used frequently.
Theorem 2·3. Let
$G=\Gamma /\!/\Lambda$
be the Schlichting completion of
$\Gamma $
relative to the commensurated subgroup
$\Lambda \lt \Gamma$
.
-
(i) If G is compactly generated and
$\Lambda$
is finitely generated, then
$\Gamma$
is finitely generated. -
(ii) If G is compactly presented and
$\Lambda$
is finitely presented, then
$\Gamma$
is finitely presented. -
(iii) If
$\Gamma$
is finitely generated, then G is compactly generated. -
(iv) If
$\Gamma$
is finitely presented and
$\Lambda$
is finitely generated, then G is compactly presented [
Reference Le Boudec9
] Theorem 6·1.
Before we prove this, we consider the following two propositions which show that Theorem2·3 implies Theorem1·1 part (ii) and Theorem 1·2 part (ii) for the cases
$n=1,2$
. For the notion of type
$F_n$
see Definition 3·1.
Proposition 2·4 ([ Reference Geoghegan14 , Proposition 7.2.1]). Let G be a discrete group. Then the following equivalences hold:
-
(i) G is of type
$F_1$
if and only if G is finitely generated; -
(ii) G is of type
$F_2$
if and only if G is finitely presented.
The analogous proposition can be formulated for tdlc groups.
Proposition 2·5 ([ Reference Castellano and Corob Cook10 , Proposition 3.4]). Let G be a tdlc group. Then the following equivalences hold:
-
(i) G is of type
$F_1$
if and only if G is compactly generated; -
(ii) G is of type
$F_2$
if and only if G is compactly presented.
Proof of Theorem
2·3. The proof of (iii) immediate since the union of a generating set of
$\Gamma$
and a compact-open subgroup of G is a generating set of G. As indicated in the statement, (iv) is proved by Le Boudec.
Next we prove (i) and (ii). If G is compactly generated or compactly presented, then G is of type
$F_1$
or
$F_2$
, respectively. Being of type
$F_1$
or
$F_2$
is witnessed by a contractible proper smooth G-CW complex X with cocompact 1-skeleton or 2-skeleton. Since the stabilizers of G are compact-open, they are commensurable with
$\overline{\alpha(\Lambda)}$
. It follows from Lemma 2·2 that the stabilizers of the restricted
$\Gamma$
-action on X are commensurable with
$\Lambda$
. In the first case the stabilizers of the
$\Gamma$
-action are finitely generated, in the second case they are finitely presented. Therefore,
$\Gamma$
is finitely generated by a special case of the Schwarz-Milnor lemma or finitely presented by a theorem of Brown [
Reference Brown4
, Proposition 3.1], respectively.
3. Finiteness properties of the Schlichting completion
Finiteness properties of tdlc groups over
${\mathbb Q}$
were introduced and studied by Castellano-Weigel [
Reference Castellano and Weigel12
]. Castellano-Corob Cook developed a theory of finiteness properties of tdlc groups over an arbitrary commutative ground ring [
Reference Castellano and Corob Cook10
] which we briefly review first.
A natural setting for the homological algebra of tdlc groups is the category
${}_{R[G]}{\mathbf{dis}}$
of discrete R[G]-modules, that is, of R-modules equipped with a left action of G such that the stabilizer of each element is open. A discrete R[G]-module of the form
$R[\Omega]$
where
$\Omega$
is a discrete set with a continuous G-action is called a discrete permutation R[G]-module. This means that the continuous G-action on
$\Omega$
has open stabilizers. If the stabilizers are also compact, then
$R[\Omega]$
is proper.
The category
${}_{R[G]}{\mathbf{dis}}$
is an abelian category that has enough injectives. If
${\mathbb Q}\subset R$
, then
${}_{R[G]}{\mathbf{dis}}$
also has enough projectives, and every proper discrete permutation R[G]-module is projective. For
$R={\mathbb Z}$
this is no longer true in general. For any R, the category
${}_{R[G]}{\mathbf{dis}}$
embeds into a quasi-abelian category
${}_{R[G]}{\mathbf{top}}$
that has enough projectives. Although proper discrete permutation R[G]-modules are not necessarily projective for arbitrary rings R we still have the equivalence (ii) in Theorem 3·2. As a consequence, a reader of this paper does not really have to know what
${}_{R[G]}{\mathbf{top}}$
and
$\operatorname{KP}_n^R$
are and can just work with the more intuitive notions
${}_{R[G]}{\mathbf{dis}}$
and
$\operatorname{FP}_n^R$
. However, the reason that the notion of
$\operatorname{FP}_n^R$
works well, as in e.g. Proposition 3·5, is that there is the quasi-abelian category
${}_{R[G]}{\mathbf{top}}$
in the background.
Definition 3·1. Let R be a commutative ring and
$n\in{\mathbb N}\cup\{\infty\}$
. We say that a tdlc group G has:
-
(1) type
$\operatorname{F}_n$
if there is a contractible proper smooth G-CW-complex with cocompact n-skeleton; -
(2) type
$\operatorname{FP}_n^R$
if the trivial R[G]-module R has a resolution
$P_\ast\to R$
by proper discrete permutation R[G]-modules
$P_\ast$
such that
$P_0,\dots,P_n$
are finitely generated; -
(3) type
$\operatorname{KP}_n^R$
if the trivial R[G]-module R has a projective resolution
$P_\ast$
in the category
${}_{R[G]}{\mathbf{top}}$
such that
$P_0,\dots, P_n$
are compactly generated.
Furthermore, a tdlc group G that admits a finite resolution by finitely generated proper discrete permutation R[G]-modules is said to have type
$\operatorname{FP}^R$
.
For the definition of G-CW-complexes see [ Reference Tom Dieck16 ]. A G-CW-complex is proper or smooth if all its stabilizers are compact or open subgroups, respectively.
Theorem 3·2 (Castellano-Corob Cook). Let G be a tdlc group. Let R be a commutative ring and
$n\in{\mathbb N}\cup\{\infty\}$
. Then the following holds:
-
(i) if G is compactly presented and G has type
$\operatorname{FP}_n^{\mathbb Z}$
then G has type
$\operatorname{F}_n$
[
Reference Castellano and Corob Cook10
, Proposition 3.13]; -
(ii) the group G has type
$\operatorname{FP}_n^R$
if and only if G has type
$\operatorname{KP}^R_n$
[
Reference Castellano and Corob Cook10
, Theorem 3.10].
Lemma 3·3. Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda\lt\Gamma$
. Let
$M=R[\Omega]$
be a finitely generated proper discrete permutation G-module over R. If
$\Lambda$
is of type
$\operatorname{FP}^R_n$
, then
$\operatorname{res}^G_\Gamma(M)$
has a projective
$R[\Gamma]$
-resolution
$P_\ast\twoheadrightarrow M$
such that
$P_0, \dots, P_n$
are finitely generated. If
$\Lambda$
is locally finite and
${\mathbb Q}\subset R$
, then
$\operatorname{res}^G_\Gamma(M)$
is a flat
$R[\Gamma]$
-module.
Proof. If
$\Lambda$
has type
$\operatorname{FP}^R_n$
then so does any subgroup of
$\Gamma$
that is commensurated with
$\Lambda$
by [
Reference Brown5
, (5.1) Proposition on p. 197]. Let
$\Lambda'\lt\Gamma$
commensurated with
$\Lambda$
. Let
$Q_\ast\to R$
be a projective
$R[\Lambda']$
-resolution of the trivial module such that
$Q_0, \dots, Q_n$
are finitely generated. Then
$R[\Gamma]\otimes_{R[\Lambda']} Q_\ast$
is a projective resolution of
$R[\Gamma/\Lambda']$
that is finitely generated in degrees
$0,\dots, n$
. Hence the
$R[\Gamma]$
-module
$R[\Gamma/\Lambda']$
has type
$\operatorname{FP}^R_n$
. The finitely generated proper discrete permutation G-module M is a finite sum of modules of the type
$R[G/U]$
where
$U\lt G$
is a compact-open subgroup. By Lemma 2·2 we have
$G/U\cong \Gamma/\alpha^{-1}(U)$
, and
$\alpha^{-1}(U)$
is commensurable with
$\alpha^{-1}(\overline{\alpha(\Lambda)})=\Lambda$
. Therefore
$\operatorname{res}^G_\Gamma R[G/U]$
is of type
$\operatorname{FP}^R_n$
.
If
$\Lambda$
and thus
$\Lambda'$
are locally finite and
${\mathbb Q}\subset R$
, then R is a flat
$R[\Lambda']$
-module [
Reference Bieri2
, Proposition 4·12 on p· 63]. Therefore
$R[\Gamma]\otimes_{R[\Lambda']} R=R[\Gamma/\Lambda']$
is a flat
$R[\Gamma]$
-module.
Lemma 3·4 ([
Reference Brown6
, Lemma 1.5]). Let
$C_\ast$
be a chain complex over a ring. Let
$P_\ast^{(i)}$
be a projective resolution of
$C_i$
. Then there is a chain complex
$Q_\ast$
with
$Q_n=\bigoplus_{i+j=n} P_i^{(j)}$
and a weak equivalence
$Q_\ast\to C_\ast$
.
Proof of Theorem 1·1. We only need to prove part (i), because part (ii) follows directly from part (i), Theorem3·2 and Theorem 2·3.
Let
\[\dots\to P_{n+1}\to P_n\to\dots \to P_0\to R\to 0\]
be a resolution of the trivial G-module by proper discrete permutation modules such that
$P_0,\dots, P_n$
are finitely generated. By Lemma3·3 each
$R[\Gamma]$
-module
$\operatorname{res}^G_\Gamma(P_j)$
,
$j\le n$
, has a projective resolution
$Q_\ast^{(j)}$
such that
$Q_i^{(j)}$
is finitely generated for
$i\in\{0,\dots,n\}$
. For
$j\gt n$
let be
$Q_\ast^{(j)}$
any projective resolution of
$\operatorname{res}^G_\Gamma(P_j)$
. By Lemma 3·4 there is a projective resolution
$Q_\ast$
of the trivial
$R[\Gamma]$
-module R such that
\[ Q_k=\bigoplus_{i+j=k}Q_i^{(j)},\]
which concludes the proof.
The following proposition follows from combining Proposition 3·9 and Theorem 3·10 in [ Reference Castellano and Corob Cook10 ].
Proposition 3·5 (Castellano-Corob Cook). Let G be a tdlc group and R be a commutative ring. Let
$0\to A'\to A\to A''\to 0$
a short exact sequence of discrete R[G]-modules. Then the following statements hold true:
-
(a) if A’ has type
$\operatorname{FP}^R_{n-1}$
and A has type
$\operatorname{FP}^R_n$
, then A” has type
$\operatorname{FP}^R_n$
; -
(b) if A has type
$\operatorname{FP}^R_{n-1}$
and A” has type
$\operatorname{FP}^R_n$
, then A’ has type
$\operatorname{FP}^R_{n-1}$
; -
(c) if A’ and A” have type
$\operatorname{FP}^R_n$
, then so does A.
Proposition 3·6. Let
$G=\Gamma/\!/\Lambda$
be the Schlichting completion of
$\Gamma$
relative to the commensurated subgroup
$\Lambda\lt\Gamma$
. Let R be a commutative ring. Let M be a discrete R[G]-module. If
$\Lambda$
has type
$\operatorname{FP}^R_m$
and
$\operatorname{res}^G_\Gamma(M)$
has type
$\operatorname{FP}^R_n$
then M has type
$\operatorname{FP}^R_{\min\{m+1,n\}}$
.
Proof. If
$\operatorname{res}_\Gamma^G(M)$
is not finitely generated, we are done. If
$\operatorname{res}_\Gamma^G(M)$
is finitely generated, then M is clearly finitely generated. In particular, there is a short exact sequence
\begin{equation}0\to K\to P\to M\to 0,\end{equation}
where P is a finitely generated proper discrete permutation module.
We show the statement by induction over n. The case
$n=0$
just means finite generation, and there is nothing more to do. Suppose the statement holds true for every restriction of an R[G]-module of type
$\operatorname{FP}^R_{n-1}$
. Let
$\operatorname{res}_\Gamma^G(M)$
be of type
$\operatorname{FP}^R_n$
and choose a sequence as in (3·1). We apply Proposition 3·5 to the short exact sequence (1) for the tdlc group G and to the short exact sequence
\begin{equation} 0\to \operatorname{res}^G_\Gamma(K)\to \operatorname{res}^G_\Gamma(P)\to \operatorname{res}^G_\Gamma(M)\to 0\end{equation}
for the discrete group
$\Gamma$
. By Lemma 3·3 the module
$\operatorname{res}^G_\Gamma(P)$
has type
$\operatorname{FP}^R_m$
. By part (b) of the above proposition the kernel
$\operatorname{res}_\Gamma^G(K)$
has type
$\operatorname{FP}^R_{\min\{m,n-1\}}$
. By induction hypothesis, K has type
$\operatorname{FP}^R_{\min\{m,n-1\}}$
. By part (a) of the above proposition, applied to (1), we obtain that M has type
$\operatorname{FP}^R_{\min\{m,n-1\}+1}=\operatorname{FP}^R_{\min\{m+1,n\}}$
. This concludes the proof.
Proof of Theorem 1·2. The first part of the theorem follows by applying Proposition 3·6 to the trivial G-module R. If
$\Lambda$
is compactly generated and
$\Gamma$
is compactly presented, then G is compactly presented by Theorem 2·3. By [
Reference Castellano and Corob Cook10
Proposition 3.13] being compactly presented and having type
$\operatorname{FP}^{\mathbb Z}_n$
is equivalent to having type
$\operatorname{F}_n$
. Therefore the second part of the theorem follows from the first one.
Example 3·7
(The Abels-Brown group) Let R be a commutative ring. Let
$\Gamma_n(R)$
denote the subgroup of
$\operatorname{GL}_{n+1}(R)$
that consists of upper triangular matrices
$(g_{i,j})$
such that
$g_{1,1}=g_{n+1,n+1}=1$
. For example,
$\Gamma_2(R)$
consists of matrices of the form
\[\left(\begin{array}{c@{\quad}c@{\quad}c} 1 & \ast & \ast\\ 0 & \ast & \ast\\ 0 & 0 & 1\end{array}\right).\]
This group was studied by Abels and Brown [
Reference Abels and Brown1
]. They showed that
$\Gamma_n({\mathbb Z}[1/p])$
is of type
$\operatorname{FP}^{\mathbb Z}_{n-1}$
but not of type
$\operatorname{FP}^{\mathbb Z}_n$
. Moreover, for
$n\ge 3$
it is finitely presented. The subgroup
$\Lambda_n=\Gamma_n({\mathbb Z})$
has entries
$\pm 1$
on the diagonal. Therefore,
$\Lambda_n$
is finitely generated nilpotent, hence of type
$\operatorname{FP}^{\mathbb Z}_\infty$
. Let
$G_n$
be the Schlichting completion of
$\Gamma_n({\mathbb Z}[1/p])$
relative to
$\Lambda_n$
. By Theorem 1·2,
$G_n$
is of type
$\operatorname{FP}^{\mathbb Z}_{n-1}$
. By Theorem 1·1,
$G_n$
is not of type
$\operatorname{FP}^{\mathbb Z}_n$
. By Theorem 2·3,
$G_n$
is compactly presented for
$n\ge 3$
.
4. Continuous cohomology vs. cohomology of the dense subgroup
The continuous cohomology of a locally compact group is defined by the complex of continuous cochains in the standard resolution. In Proposition 4·1 below, we compare the continuous cohomology to the discrete cohomology both with real coefficients. The discrete cohomology can be computed from a resolution by proper discrete permutation modules and was introduced as a derived functor in [ Reference Castellano and Weigel12 , Section 2.5].
We do not claim originality for Proposition 4·1. It can be deduced from the results in Guichardet’s book [
Reference Guichardet15
] but we give a proof because we need the specific chain map
$\phi$
used in the proof later. There is a similar statement in [
Reference Fust13
] but it seems to assume a discrete topology on the coefficients. Note that we consider the reals
${\mathbb R}$
with the usual topology.
Proposition 4·1. The continuous cohomology
$H_c^\ast(G,{\mathbb R})$
of a tdlc group G is isomorphic to the real discrete cohomology
${\operatorname{dH}}^\ast(G,{\mathbb R})$
.
Proof. Let
$U\lt G$
be a compact-open subgroup. Let
$\mu$
be the left-invariant Haar measure on G with
$\mu(U)=1$
. Then
${\mathbb R}[(G/U)^{\ast+1}]$
with the usual differentials of the bar resolution is a resolution of the trivial G-module
${\mathbb R}$
by proper discrete permutation modules. It suffices to show that the projection
$G\to G/U$
induces a homotopy equivalence
\begin{equation} \phi\colon \hom_{{\mathbb R}[G]}\bigl({\mathbb R}[(G/U)^{\ast+1}],{\mathbb R}\bigr)^G\to C\bigl(G^{\ast+1},{\mathbb R}\bigr)^G.\end{equation}
A homotopy inverse
$\rho$
is defined as follows. For a cochain
$f\colon G^{n+1}\to {\mathbb R}$
let
$\rho(f)\colon (G/U)^{n+1}\to{\mathbb R}$
be the map
\[ \rho(f)(g_0U,\dots,g_nU)=\int_{U^{n+1}} f\bigl( g_0u_0,\dots, g_nu_n \bigr)d\mu(u_0)\dots d\mu(u_n). \]
Since
$U^{n+1}$
is compact and f is continuous the integral exists. The definition is independent of the choice of representatives
$g_0,\dots, g_n$
of the U-coset classes by the left-invariance of
$\mu$
. Clearly,
$\rho$
is a cochain map and
$\rho\circ\phi=\operatorname{id}$
.
The chain homotopy
$\phi\circ\rho\simeq \operatorname{id}$
is defined as follows. Let
\[ S^n_i(f)(g_0,\dots, g_{n-1})\mathrel{\mathop:}=\int_{U^i} f\bigl(g_0u_0,\dots,g_{i-1}u_{i-1},g_i,\dots,g_{n-1}\bigr)d\mu(u_0)\dots d\mu(u_i)\]
for every
$n\ge 1$
and every
$0\le i\le n-1$
. Similarly as above, this formula defines a homomorphism
$S^n_i\colon C(G^{n+1}, {\mathbb R})^G\to C(G^n, {\mathbb R})^G$
. Then
$H^n=\sum_{i=0}^{n-1} (-1)^iS^n_i$
is the chain homotopy
$\phi\circ\rho\simeq \operatorname{id}$
.
Now we are able to quickly conclude the proof of Theorem 1·3.
Proof of Theorem 1·3. Let
$G=\Gamma/\!/\Lambda$
and U be the closure of
$\Lambda$
in G. Let
$P_\ast={\mathbb R}[(G/U)^{\ast+1}]$
be the resolution of the trivial G-module
${\mathbb R}$
appearing in the proof of Proposition 4·1. Each
$P_n$
is a proper discrete permutation module. By Lemma 3·3, the restricted resolution
$\operatorname{res}^G_\Gamma P_\ast={\mathbb R}[(\Gamma/\Lambda)^{\ast+1}]$
is a flat
${\mathbb R}[\Gamma]$
-resolution of
${\mathbb R}$
.
The map
$\psi$
in the following commutative square is induced by the projection
$\Gamma\to \Gamma/\Lambda$
. The map
$\phi$
is the one in (4·1).
The statement of Theorem 1·3 is that the upper horizontal restriction is a weak isomorphism. The map
$\phi$
is a weak isomorphism by the proof of Proposition 2·4. The forgetful lower horizontal map is obviously an isomorphism. So it suffices to show that
$\psi$
is a weak isomorphism.
By [
Reference Weibel26
, Lemma 3.2·8, p. 71] the projection from the projective resolution
${\mathbb R}[\Gamma^{\ast+1}]$
to the flat resolution
$\operatorname{res}^G_\Gamma P_\ast$
induces a weak isomorphism
\[ {\mathbb R}[\Gamma^{\ast+1}]\otimes_{{\mathbb R}[\Gamma]}{\mathbb R}\xrightarrow{\sim} \operatorname{res}^G_\Gamma P_\ast\otimes_{{\mathbb R}[\Gamma]}{\mathbb R}.\]
Its dual map
\[\hom_{\mathbb R}\bigl(\operatorname{res}^G_\Gamma P_\ast\otimes_{{\mathbb R}[\Gamma]}{\mathbb R},{\mathbb R}\bigr)\xrightarrow{\sim} \hom_{\mathbb R}\bigl({\mathbb R}[\Gamma^{\ast+1}]\otimes_{{\mathbb R}[\Gamma]}{\mathbb R},{\mathbb R} \bigr)\]
is isomorphic to
$\psi$
. The dual map is a weak isomorphism by the universal coefficient theorem over
${\mathbb R}$
[
Reference Weibel26
Theorem 3·6·5, p. 89].
5. The Euler characteristic of the Schlichting completion
The Euler characteristic
$\chi^{(2)}(G,\mu)\in {\mathbb R}$
of a unimodular tdlc group G with Haar measure
$\mu$
that admits a contractible smooth proper G-CW-complex or a finite resolution by proper discrete permutation R[G]-modules,
$R\subset{\mathbb C}$
, was introduced in [
Reference Petersen, Sauer and Thom20
]. A more general approach can be found in [
Reference Castellano, Chinello and Weigel11
].
If
\begin{equation} 0\to \bigoplus_{i\in I_n} {\mathbb C}[G/U_i^{(n)}]\to \dots\to \bigoplus_{i\in I_0}{\mathbb C}[G/U_i^{(0)}]\to {\mathbb C}\to 0\end{equation}
is a resolution by proper discrete permutation modules, then
\begin{equation} \chi^{(2)}(G,\mu)=\sum_{p=0}^n (-1)^p \sum_{i\in I_p} \mu\bigl(U_i^{(p)}\bigr)^{-1}.\end{equation}
If G is discrete, then one usually takes the counting measure as Haar measure and omits the Haar measure in the notation. In this case,
$\chi^{(2)}(G,\mu)$
coincides with the
$\ell^2$
-Euler characteristic of the group [
Reference Lück18
, Section 7.2]. Moreover, if G is discrete and of type F, then
$\chi^{(2)}(G,\mu)$
coincides with the classical Euler characteristic of the group.
By [
Reference Petersen, Sauer and Thom20
, Theorem 4.9] the Euler characteristic of a unimodular tdlc group is the alternating sum of its
$\ell^2$
-Betti numbers, which were introduced by Petersen [
Reference Petersen19
].
\begin{equation} \chi^{(2)}(G,\mu)=\sum_{p\ge 0}(-1)^p \beta^{(2)}_p(G,\mu)\end{equation}
The terms in (5·2) can be interpreted in terms of the von Neumann dimensions
$\dim_G$
of modules of the type
$L(G,\mu)\otimes_{\mathcal{H}(G)}{\mathbb C}[G/U]=L(G,\mu)p_U$
, that is,
\[ \dim_G L(G,\mu)\otimes_{\mathcal{H}(G)}{\mathbb C}[G/U]=\mu(U)^{-1},\]
where
$L(G,\mu)$
is the von Neumann algebra of G relative to
$\mu$
, and
$\mathcal{H}(G)$
is the Hecke algebra of complex-valued locally constant functions, and
$L(G,\mu)p_U$
is the projection onto the U-invariant vectors in
$L^2(G,\mu)$
. The proof of the second formula of
$\chi^{(2)}(G, \mu)$
is then just a matter of additivity of the von Neumann dimension. We refer to [
Reference Petersen, Sauer and Thom20
] for more details.
An important consequence of (5·3) and the corresponding property for
$\ell^2$
-Betti numbers [
Reference Kyed, Petersen and Vaes17
] is the equality
\begin{equation}\chi^{(2)}(\Gamma)=\operatorname{covol}(\Gamma,\mu)\cdot \chi^{(2)}(G,\mu)\end{equation}
for every lattice
$\Gamma$
in a locally compact group G with Haar measure
$\mu$
. In particular, if
$\Lambda\lt\Gamma$
is a subgroup of finite index in a discrete group
$\Gamma$
of type
$\operatorname{FP}^{\mathbb Q}$
, then
\begin{equation}\chi^{(2)}(\Lambda)=[\Gamma:\Lambda]\cdot \chi^{(2)}(\Gamma).\end{equation}
Proof of Theorem 1·5. As before, we denote the canonical map of
$\Gamma$
into the Schlichting completion
$G=\Gamma/\!/\Lambda$
by
$\alpha$
. Further, U is the closure of
$\alpha(\Lambda)$
. Consider a projective resolution of the trivial G-module by proper discrete permutation modules as in (5·1). For every
$j\in\{0,\dots,n\}$
and every
$i\in I_j$
we choose a finite projective
${\mathbb C}[\alpha^{-1}(U_i^{(j)})]$
-resolution
$\tilde Q(i,j)_\ast$
of the trivial
${\mathbb C}[\alpha^{-1}(U_i^{(j)})]$
-module
${\mathbb C}$
. Since
$\alpha^{-1}(U_i^{(j)})$
and
$\Lambda=\alpha^{-1}(U)$
are commensurable, the group
$\alpha^{-1}(U_i^{(j)})$
is of type
$\operatorname{FP}^{\mathbb Q}$
(thus,
$\operatorname{FP}^{\mathbb C}$
). This follows from the combination of [
Reference Bieri2
, Theorem 5.11, p. 78] and [
Reference Brown5
, Proposition 5.1, p. 197 and Proposition 6·1, p. 199]. Tensoring this resolution with
${\mathbb C}[\Gamma]$
we obtain a finite projective
${\mathbb C}[\Gamma]$
-resolution of
${\mathbb C}[\Gamma/\alpha^{-1}(U_i^{(j)})]$
which we denote by
$Q(i,j)_\ast$
. For every
$j\in\{0,\dots,n\}$
, the sum
$\bigoplus_{i\in I_j} Q(i,j)_\ast$
is a finite projective
${\mathbb C}[\Gamma]$
-resolution of
\[\operatorname{res}_\Gamma^G\bigl(\bigoplus_{i\in I_j} {\mathbb C}\bigl[G/U_i^{(j)}\bigr]\bigr)\cong \bigoplus_{i\in I_j} {\mathbb C}\bigl[\Gamma/\alpha^{-1}(U_i^{(j)})\bigr].\]
Similarly as in the proof of Theorem 1·1, we find a projective resolution
$Q_\ast$
of the trivial
${\mathbb C}[\Gamma]$
-module
${\mathbb C}$
such that
\[Q_n\cong\bigoplus_{k+j=n} \bigoplus_{i\in I_j}Q(i, j)_k.\]
Using the compatibility of the von Neumann dimension under induction, we conclude that
\begin{align*} \chi(\Gamma)&=\sum_{n\ge 0}(-1)^n\dim_{L(\Gamma)}\bigl(L(\Gamma)\otimes_{{\mathbb C}[\Gamma]}Q_n\bigr)\\ &=\sum_{j\ge 0}(-1)^j \sum_{k\ge 0}(-1)^k \sum_{i\in I_j} \dim_{L(\Gamma)}\bigl(L(\Gamma)\otimes_{{\mathbb C}[\Gamma]}Q(i,j)_k\bigr)\\ &=\sum_{j\ge 0}(-1)^j \sum_{i\in I_j}\sum_{k\ge 0}(-1)^k \dim_{L(\alpha^{-1}(U_i^{(j)}))}\bigl(L(\alpha^{-1}(U_i^{(j)}))\otimes_{{\mathbb C}[\alpha^{-1}(U_i^{(j)})]}\tilde Q(i,j)_k\bigr)\\ &=\sum_{j\ge 0}(-1)^j\sum_{i\in I_j}\chi^{(2)}\bigl(\alpha^{-1}(U_i^{(j)})\bigr)\\ &=\sum_{j\ge 0}(-1)^j\sum_{i\in I_j}\mu\bigl(U_i^{(j)}\bigr)^{-1}\chi^{(2)}(\Lambda)\qquad\text{(use (5$\cdot$5) and $\mu(U)=1$ and $\Lambda=\alpha^{-1}(U)$)}\\ &=\chi^{(2)}(G,\mu)\cdot\chi^{(2)}(\Lambda).\end{align*}
Example 5·1. The group
$\Gamma=\operatorname{SL}_n({\mathbb Z}[1/p])$
is a lattice in
$G=\operatorname{SL}_n({\mathbb R})\times \operatorname{SL}_n({\mathbb Q}_p)$
. Let
$\Lambda=\operatorname{SL}_n({\mathbb Z})\lt\Gamma$
. We compare Theorem 1·5 for
$\Gamma/\!/\Lambda$
to computations we obtain from the theory of
$\ell^2$
-Betti numbers of locally compact groups via (5·3). Let
$\mu$
and
$\nu$
be Haar measures of the left and right factor of G, respectively. Then
\[\chi^{(2)}\bigl( \Gamma\bigr) = \operatorname{covol}(\Gamma, \mu\times\nu)\cdot \chi^{(2)}\bigl( \operatorname{SL}_n({\mathbb R}),\mu)\cdot\chi^{(2)}\bigl(\operatorname{SL}_n({\mathbb Q}_p),\nu\bigr).\]
Similarly, since
$\operatorname{SL}_n({\mathbb Z})$
is a lattice of
$\operatorname{SL}_n({\mathbb R})$
we obtain that
\[ \chi^{(2)}\bigl( \operatorname{SL}_n({\mathbb Z})\bigr) = \operatorname{covol}\bigl(\operatorname{SL}_n({\mathbb Z}), \mu\bigr)\cdot \chi^{(2)}\bigl( \operatorname{SL}_n({\mathbb R}),\mu\bigr).\]
We normalize
$\nu$
so that
$\nu(\operatorname{SL}_n({\mathbb Z}_p))=1$
. The push-forward measure
$\xi$
on
$\operatorname{PSL}_n({\mathbb Q}_p)$
under the projection
$\operatorname{SL}_n({\mathbb Q}_p)\to \operatorname{PSL}_n({\mathbb Q}_p)$
satisfies
$\xi(\operatorname{PSL}_n({\mathbb Z}_p))=1$
. By [
Reference Petersen19
] and (5.3) we have
\[ \chi^{(2)}\bigl( \operatorname{SL}_n({\mathbb Q}_p),\nu\bigr)=\chi^{(2)}\bigl( \operatorname{PSL}_n({\mathbb Q}_p),\xi\bigr).\]
Therefore,
\begin{equation} \chi^{(2)}(\Gamma)=\frac{\operatorname{covol}(\Gamma,\mu\times\nu)}{\operatorname{covol}(\operatorname{SL}_n({\mathbb Z}),\mu)}\cdot \chi^{(2)}\bigl(\operatorname{SL}_n({\mathbb Z})\bigr)\cdot \chi^{(2)}\bigl(\operatorname{PSL}_n({\mathbb Q}_p),\xi\bigr).\end{equation}
There is an isomorphism
$\operatorname{SL}_n({\mathbb Z}[1/p])/\!/ \operatorname{SL}_n({\mathbb Z})\cong \operatorname{PSL}_n({\mathbb Q}_p)$
under which the closure of
$\operatorname{SL}_n({\mathbb Z})$
is mapped onto
$\operatorname{PSL}_n({\mathbb Z}_p)$
. See [
Reference Shalom and Willis23
, example 3.10]. By Theorem 1·5,
\[\chi^{(2)}(\Gamma)= \chi^{(2)}\bigl(\operatorname{SL}_n({\mathbb Z})\bigr)\cdot \chi^{(2)}\bigl(\operatorname{PSL}_n({\mathbb Q}_p),\xi\bigr).\]
As a consequence, the ratio of covolumes in (5·6) is 1.
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