From the geometry of box spaces to the geometry and measured couplings of groups
Kajal Das
TL;DR
This work analyzes how coarse geometric relations between box spaces of residually finite groups reflect uniform measured equivalence (UME) between the groups themselves. It develops a translation from coarse relations of box spaces to ME-type couplings by constructing a topological coupling and a G-invariant measure via Gromov-Hausdorff convergence, thereby proving that coarse equivalence of box spaces implies G and H are UME, and that a coarse embedding induces a UME-embedding. The paper derives invariants such as $cd_R$ and $\beta_i^{(2)}$ that must be preserved up to proportionality under UME, and shows non-embeddability results for expanders like those from $SL_n(\mathbb{Z})$ into $SL_m(\mathbb{Z})$ for $n>m$, along with a countable class of residually finite groups with mutually coarse-equivalent group structures but non-coarse-equivalent box spaces. These results bridge coarse geometry of box spaces with measurable group equivalence, yielding new obstructions and refined comparisons between box spaces and their underlying groups, with implications for expanders and cohomological invariants.
Abstract
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 $SL_n(\mathbb{Z})$ can not be coarsely embedded inside the expanders of $SL_m(\mathbb{Z})$, where $n>m$ and $n,m\geq 3$. Moreover, we obtain a countable class of residually groups which are mutually coarse-equivalent but any of their box spaces are not coarse-equivalent.