Coarse geometry of expanders from rigidity of warped cones

We study quasi-isometry types of expanders that come from a warped cone construction over group actions on homogeneous spaces. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if the group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are conjugate in finite covers. As a consequence, we produce a continuum of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental group.