A dynamical relation between groups from quasi-isometric warped cones

While studying large scale geometry of amenable groups, Yehuada Shalom proves the following beautiful result: if two finitely generated (f.g.) amenable groups are quasi-isometric (QI), then the groups are uniformly measured equivalent (UME), where UME is a dynamical relation between two groups with some geometric constraints. Later, this result was generalized by the author to residually finite groups or for “box spaces”: if two “box spaces” are QI, then the corresponding residually finite groups are UME. It was asked to the author by Piotr Nowak whether this result can be generalized to “warped cones”, where Warped cone is a geometric object associated to a free, isometric, probability measure-preserving action of a finitely generated group on a compact metric space. In this paper, we affirmatively answers this question and prove that if two warped cones are level-wise QI (with some extra natural assumptions), then the corresponding groups are UME. It also generalizes the following result of de Laat-Vigolo and Sawicki: if two warped cones are level-wise quasi-isometric, then their stable products are quasi-isometric. 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.