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Register a new userRational discrete cohomology for totally disconnected locally compact groups
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Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $mathrm{Hom}$-$otimes$ identities associated to the rational discrete bimodule $mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretins group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $mathrm{FP}$ it is possible to define an Euler-Poincare characteristic $chi(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.
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It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $mathrm{dH}^1(G,mathrm{Bi}(G))$, where $mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $mathrm{dH}^bullet(G,_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).
In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings $R$, in particular for $R = mathbb{Z}$ and $R= mathbb{Q}$. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieris and Browns criteria for finiteness properties and deduce that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.
A connected, locally finite graph $Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $Gamma$. Define the minimal degree of $G$ as the minimal degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${rm Aut}(T_d)$ is $d$ for all $dgeq 2$.
We show that every abstract homomorphism $varphi$ from a locally compact group $L$ to a graph product $G_Gamma$, endowed with the discrete topology, is either continuous or $varphi(L)$ lies in a small parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not small is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If $L$ is a locally compact group and if $G$ is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism $varphi:Lto G$ is either continuous, or $varphi(L)$ is contained in the normalizer of a finite nontrivial subgroup of $G$. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.
Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $bdfine X$ which is a refinement of the visual boundary $bd X$. For each $x in bdfine X$, the stabilizer $G_x$ is amenable.
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