Topological groups with a compact open subgroup, Relative hyperbolicity and Coherence

TL;DR

This paper develops geometric techniques to analyze topological groups with compact open subgroups, focusing on relative hyperbolicity, coherence, and extensions of discrete group results, providing new characterizations and invariance properties.

Contribution

It introduces a framework for studying relative hyperbolicity and coherence in topological groups with compact open subgroups, extending known discrete group results to the topological setting.

Findings

G is compactly generated relative to H iff certain associated graphs are quasi-isometric.

If G is compactly presented relative to H, then subgroups in H are also compactly generated or presented.

G being hyperbolic relative to H implies G is compactly presented relative to H.

Abstract

The main objects of study in this article are pairs $(G, \mathcal{H})$ where $G$ is a topological group with a compact open subgroup, and $\mathcal{H}$ is a finite collection of open subgroups. We develop geometric techniques to study the notions of $G$ being compactly generated and compactly presented relative to $\mathcal H$. This includes topological characterizations in terms of discrete actions of $G$ on complexes, quasi-isometry invariance of certain graphs associated to the pairs $(G,\mathcal H)$ when $G$ is compactly generated relative to $\mathcal H$, and extensions of known results for the discrete case. For example, generalizing results of Osin for discrete groups, we show that in the case that $G$ is compactly presented relative to $\mathcal H$: $\bullet$ if $G$ is compactly generated, then each subgroup $H\in \mathcal H$ is compactly generated; $\bullet$ if each subgroup $H\in \mathcal H$ is compactly presented, then $G$ is compactly presented. The article also introduces an approach to relative hyperbolicity for pairs $(G, \mathcal H)$ based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, we prove that if $G$ is hyperbolic relative to $\mathcal H$ then $G$ is compactly presented relative to $\mathcal H$. As applications of the results of the article we prove combination results for coherent topological groups with a compact open subgroup, and extend McCammond-Wise perimeter method to this general framework.