Rational discrete first degree cohomology for totally disconnected locally compact groups

Abstract

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).

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References in corpus (2)

• Rational discrete cohomology for totally disconnected locally compact groups
• $L^2$-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices

Cited by in corpus (1)

• Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups