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

In this article, we prove that if two warped cones corresponding to two groups with free, isometric, measure-preserving, ergodic actions on two manifolds are quasi-isometric, then the corresponding groups are uniformly measured equivalent (UME). It was earlier known from the work of de Laat-Vigolo that if two warped cones are QI, then their stable products are QI. Our result strengthens this result and go further to prove UME of the groups. However, Fisher-Nguyen-Limbeek proves that if the warped cones corresponding to two finitely presented groups with no free abelian factors are QI, then there is an affine commensuration of the two actions. Our result can be seen as an extension of their result in the setting of infinite presentability under some extra assumptions.Comment: Comments welcome. arXiv admin note: text overlap with arXiv:1512.08828. Some typos have been corrected and some minor gaps of Proposition 4.2 have been filled u