Rational discrete first degree cohomology for totally disconnected locally compact groups

TL;DR

This paper links the number of ends of totally disconnected locally compact groups to their rational discrete cohomology, extending classical results from discrete groups to a broader topological setting.

Contribution

It demonstrates that the number of ends of such groups can be detected via rational discrete cohomology, generalizing Stallings' and Abels' results.

Findings

Ends of groups correspond to cohomology groups.

Classifies certain groups via cohomological dimension.

Connects fundamental groups of graphs of profinite groups to cohomology.

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).