Topological matchings and amenability

TL;DR

This paper characterizes amenability in topological groups using matchings related to uniform coverings, showing that certain classes of groups satisfy perfect matching properties, and linking automorphism groups of Fra"issé limits to Ramsey-type properties.

Contribution

It introduces a new characterization of amenability via matchings with finite uniform coverings and connects automorphism groups of Fra"issé limits to Ramsey-type matching properties.

Findings

Amenability characterized by matchings with finite uniform coverings.

Extremely amenable and compactly approximable groups satisfy perfect matching properties.

Automorphism groups of Fra"issé limits are amenable iff they have a Ramsey-type matching property.

Abstract

We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition -- the latter even with regard to arbitrary uniform coverings. Finally, we prove that the automorphism group of a Fra\"iss\'e limit of a metric Fra\"iss\'e class is amenable if and only if the considered metric Fra\"iss\'e class has a certain Ramsey-type matching property.