2020/04/21 by Kajal Das, Das, Kajal
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2004.10000
In this article, we prove that if two warped cones corresponding to two\nfinitely generated groups with free, isometric, measure-preserving, actions on\ntwo compact metric spaces with probability measures are level-wise\nquasi-isometric (with some extra natural assumptions), then the corresponding\ngroups are uniformly measured equivalent (UME). It was earlier known from the\nworks of de Laat-Vigolo and Sawicki that if two such warped cones are\nlevel-wise quasi-isometric, then their stable products are quasi-isometric. We\nstrengthen this result and go further to prove UME of the groups. We also\ndiscuss many applications of our main result. We give countably infinite\nexamples of groups and associated Warped cones such that the groups are\nmutually quasi-isometric, but the Warped cones are not mutually quasi-isometric\nin the sense of our main theorem. We also provide examples of two Warped cones\n(which are quasi-isometric to two different expander families) such that one of\nthem does not quasi-isometrically embed into the other one in the sense of our\nmain theorem.\n