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 $\mathcal H$.