On Følner sets in topological groups

Abstract

We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group G is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set G. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

Metric groups, unitary representations and continuous logic

“…Thus the formula in the formulation belongs to L ω1ω . By Theorem 4.5 of [14] and the discussion after the formulation of that theorem above we see that all amenable groups satisfy the statements in the formulation.…”

“…In Section 3 we apply the recent paper [14] for L ω1ω -axiomatization of (non-) amenability of metric groups. The case of property (T) looks slightly more complicated, because unbounded metric spaces are involved in the definition.…”

“…Thom in [14] in order to axiomatize in L ω1ω amenability for metric groups which are continuous structures. In fact we will see that this property is sup inf in terms of [7] 1 .…”

“…By Hall's matching theorem this value is the matching number of the graph (F 1 , F 2 , R U ). Theorem 4.5 of [14] gives the following description of amenable topological groups.…”

“…Let (G, d) satisfy the axioms from the formulation. We want to apply condition (2) of Theorem 4.5 of [14] in the case of balls B q . Fix θ, q and E as in the formulation so that…”