  For a pair $(G,\mathcal{P})$ consisting of a 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 Extension graph; in
particular, it is quasi-isometric to a tree.
