Coarse equivalence and topological couplings of locally compact groups

TL;DR

This paper extends Gromov's result by showing that any two compactly generated locally compact second countable groups are quasi-isometric if and only if they admit a topological coupling, broadening the class of groups for which this equivalence holds.

Contribution

The paper generalizes Gromov's theorem from finitely generated groups to all compactly generated locally compact second countable groups.

Findings

Establishes the equivalence between quasi-isometry and topological couplings for a broader class of groups.

Extends the concept of topological couplings to locally compact second countable groups.

Provides a unified framework linking geometric and topological group properties.

Abstract

A result due to M. Gromov states that any two finitely generated groups {\Gamma} and {\Lambda} are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions $\Gamma\curvearrowright X\curvearrowleft \Lambda$ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second countable groups.