Strong amenability and the infinite conjugacy class property

TL;DR

This paper characterizes strongly amenable groups, showing finitely generated groups are strongly amenable if and only if they are virtually nilpotent, and more generally, relates strong amenability to the absence of ICC quotients.

Contribution

It provides a complete characterization of strongly amenable groups in terms of virtually nilpotent groups and ICC quotients, linking topological dynamics with group properties.

Findings

Finitely generated strongly amenable groups are exactly the virtually nilpotent groups.

Countable groups are strongly amenable iff none of their quotients have ICC.

Establishes a connection between strong amenability and the infinite conjugacy class property.

Abstract

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.