The quasi-isometry invariance of the Coset Intersection Complex

For a pair (G,P) consisting of a finitely generated group and finite collection of subgroups, we introduce a simplicial G-complex K(G,P) called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of K(G,P) are quasi-isometric invariants of the group pair (G,P). Classical properties of P in G correspond to topological or geometric properties of K(G,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 P in G are quasi-isometry invariants of the pair (G,P). For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex K(G,P) is quasi-isometric to the Deligne complex; in particular, it is hyperbolic.