Topological matchings and amenability
Volume 238 / 2017
Fundamenta Mathematicae 238 (2017), 167-200
MSC: Primary 43A07, 28C10; Secondary 22A10, 05D10, 05C55.
DOI: 10.4064/fm248-10-2016
Published online: 27 February 2017
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 (i.e., possibly infinite) uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the class has a certain Ramsey-type matching property.
