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

TL;DR

This paper characterizes hyperbolic totally disconnected locally compact groups using homological inequalities, showing hyperbolicity inheritance in certain subgroups, and explores examples including automorphism groups of negatively curved buildings.

Contribution

It introduces a homological characterization of hyperbolic TDLC-groups and proves hyperbolicity inheritance for subgroups with low cohomological dimension, extending understanding of their structure.

Findings

Hyperbolic TDLC-groups characterized by homological isoperimetric inequalities

Hyperbolicity inherited by certain closed subgroups with cohomological dimension ≤ 2

Examples include automorphism groups of negatively curved buildings and small cancellation quotients

Abstract

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of the article: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\leq 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group $\mathrm{Aut}(X)$ of a negatively curved locally finite $2$-dimensional building $X$ is a hyperbolic TDLC-group, whenever $\mathrm{Aut}(X)$ acts with finitely many orbits on $X$. Examples where this result applies include hyperbolic Bourdon's buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension $2$ when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.