From the geometry of box spaces to the geometry and measured couplings of groups

In this paper, we prove that if two box spaces' of two residually finite groups are coarsely equivalent, then the two groups are uniform measured equivalent’ (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a `uniform measured equivalent embedding’ (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp.\ a coarse embedding) between the box spaces gives rise to a coarse equivalence (resp.\ a coarse embedding) between the groups. We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from $S{L}_n(\mathbb{Z})$ can not be coarsely embedded inside the expanders of $S{L}_m(\mathbb{Z})$, where $n>m$ and $n,m\ge 3$. Moreover, we obtain a countable class of residually groups which are mutually coarse-equivalent but any of their box spaces are not coarse-equivalent.