From the coarse geometry of warped cones to the measured coupling of groups

Abstract: In this article, we prove that if two warped cones corresponding to two finitely generated groups with free, isometric, measure-preserving, actions on two compact metric spaces with probability measures are level-wise quasi-isometric (with some extra natural assumptions), then the corresponding groups are uniformly measured equivalent (UME). It was earlier known from the works of de Laat-Vigolo and Sawicki that if two such warped cones are level-wise quasi-isometric, then their stable products are quasi-isometric. We strengthen this result and go further to prove UME of the groups. We also discuss many applications of our main result. We give countably infinite examples of groups and associated Warped cones such that the groups are mutually quasi-isometric, but the Warped cones are not mutually quasi-isometric in the sense of our main theorem. We also provide examples of two Warped cones (which are quasi-isometric to two different expander families) such that one of them does not quasi-isometrically embed into the other one in the sense of our main theorem.