A new characterization of topologically amenable groups

Abstract

A countable group G is called topologically amenable if there exist a compact Hausdorff space X on which G acts by homeomorphisms and weak*-continuous maps b n from X to the space, prob (G), of probability measures on G such that for every g ∈ G, limn→infinity supx∈X gbnx-bn gx1=0. For example, every amenable group is topologically amenable but not vice versa: The free group F2 is topologically amenable without being amenable. Inspired by a characterization of amenable groups due to Giordano and de la Harpe (a countable group G is amenable if and only if every continuous action of G on the Cantor set C admits an invariant probability measure), we give a new characterization of topologically amenable groups: A countable group G is topologically amenable if and only if it admits an amenable action on the Cantor set C.