Coarse equivalence versus bijective coarse equivalence of expander graphs

TL;DR

This paper characterizes when coarse equivalences between disjoint unions of expander graphs are close to bijective, and shows that isomorphic uniform Roe algebras imply bijective coarse equivalence of the underlying spaces.

Contribution

It provides a criterion for when coarse equivalences are close to bijective and links algebraic isomorphisms to geometric coarse equivalences of expander graph unions.

Findings

Coarse equivalences close to bijective under certain conditions.

Isomorphism of uniform Roe algebras implies bijective coarse equivalence.

Characterization of coarse disjoint unions of expanders.

Abstract

We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras of coarse disjoint unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent.