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In this article, we prove that if two warped cones corresponding to twofinitely generated groups with free, isometric, measure-preserving, actions ontwo compact metric spaces with probability measures are level-wisequasi-isometric (with some extra natural assumptions), then the correspondinggroups are uniformly measured equivalent (UME). It was earlier known from theworks of de Laat-Vigolo and Sawicki that if two such warped cones arelevel-wise quasi-isometric, then their stable products are quasi-isometric. Westrengthen this result and go further to prove UME of the groups. We alsodiscuss many applications of our main result. We give countably infiniteexamples of groups and associated Warped cones such that the groups aremutually quasi-isometric, but the Warped cones are not mutually quasi-isometricin 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 ofthem does not quasi-isometrically embed into the other one in the sense of ourmain theorem.

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