Quasi-isometries of warped cones and expanders

arped cones for action of a group on a metric space was first defined by Roe. Its metric records information about both group action and space. We show that in cases the spaces and actions are nice, quasi-isometry ofwarped cones implies actions are conjugate. As a application, when actions have spectral gap, we construct uncountably many non-quasi-isometricfamilies of expanders. This is a joint work with David Fisher and Woutervan Limbeek.