Rational discrete cohomology for totally disconnected locally compact groups

Abstract

Rational discrete cohomology and homology for a totally disconnected locally compact group G is introduced and studied. The Hom-⊗ identities associated to the rational discrete bimodule 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. Neretin's 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 FP it is possible to define an Euler-Poincaré characteristic χ(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.

Finiteness properties of totally disconnected locally compact groups

“…group G is defined to have type FP n over R (n ∈ N ∪ {∞}) if the trivial discrete R[G]-module R admits a resolution P → R formed by finitely generated proper discrete permutation R[G]-modules. This definition coincides with the one in [13] when R = Q and proper discrete permutation modules are projective. On the other hand, G will be said to have type KP n if there is a projective resolution P → R in R[G] top with P i a free k-R[G]-module on a compact space for i ≤ n. Since R [G] dis is an abelian subcategory of R[G] top, we may compare these notions: having type FP n turns out to be equivalent to having type KP n by Theorem 3.10.…”

“…In the context of discrete Q[G]-modules, there is an important discrete Q[G]-bimodules which can play some of the role of the group algebra: the so-called rational discrete standard G-bimodule Bi(G). In the case that G is unimodular, Bi(G) turns out to be isomorphic but not canonically isomorphic to the Q-vector space C c (G, Q) of continuous functions from G to Q with compact support; see [13] for the results. Here we recall the construction of Bi(G), which will play a crucial role throughout this paper.…”

“…Notice that Hom G (P, Bi(G)) is a finitely generated discrete right Q[G]-module with compact stabiliser whenever P is a left one. Since M is flat, one has the following commutative diagram [13,Prop. 4.6]).…”

“…Discrete G-CW-complexes. Following [13], let F be a non-empty set of open subgroups of G satisfying (F1) for A ∈ F and g ∈ G one has g A = gAg −1 ∈ F;…”

“…In [16] Cornulier and de la Harpe revisited σ-compactness, finite presentability and compact presentability of locally compact groups from the point of view of coarse geometry. A homological approach to finiteness conditions of totally disconnected locally compact groups over the rationals was introduced in [13]. Nevertheless, the theory of finiteness conditions for totally disconnected locally compact (t.d.l.c.)…”