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

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.