Abstract
For a pair $(G,\mathcal{P})$ consisting of a finitely generated group and
finite collection of subgroups, we introduce a simplicial $G$-complex
$\mathcal{K}(G,\mathcal{P})$ called the coset intersection complex. We prove
that the quasi-isometry type and the homotopy type of
$\mathcal{K}(G,\mathcal{P})$ are quasi-isometric invariants of the group pair
$(G,\mathcal{P})$. Classical properties of $\mathcal{P}$ in $G$ correspond to
topological or geometric properties of $\mathcal{K}(G,\mathcal{P})$, such as
having finite height, having finite width, being almost malnormal, admiting a
malnormal core, or having thickness of order one. As applications, we obtain
that a number of algebraic properties of $\mathcal{P}$ in $G$ are
quasi-isometry invariants of the pair $(G,\mathcal{P})$. For a certain class of
right-angled Artin groups and their maximal parabolic subgroups, we show that
the complex $\mathcal{K}(G,\mathcal{P})$ is quasi-isometric to the Deligne
complex; in particular, it is hyperbolic.