  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.
