On extremely amenable groups of homeomorphisms

TL;DR

This paper characterizes extremely amenable groups of homeomorphisms on compact spaces through invariant subsets and provides an alternative proof of Pestov's theorem regarding the extreme amenability of order-preserving homeomorphism groups.

Contribution

It offers a new characterization of extremely amenable groups of homeomorphisms and presents an alternative proof of Pestov's theorem.

Findings

Equivalence of conditions for extreme amenability of groups of homeomorphisms.

New characterization involving minimal closed invariant subsets.

Alternative proof of Pestov's theorem on order-preserving homeomorphisms.

Abstract

A topological group $G$ is {\em extremely amenable} if every compact $G$-space has a $G$-fixed point. Let $X$ be compact and $G\subset{\mathrm{Homeo}} (X)$. We prove that the following are equivalent: (1) $G$ is extremely amenable; (2) every minimal closed $G$-invariant subset of $\exp R$ is a singleton, where $R$ is the closure of the set of all graphs of $g\in G$ in the space $\exp (X^2)$ ($\exp$ stands for the space of closed subsets); (3) for each $n=1,2,...$ there is a closed $G$-invariant subset $Y_n$ of $(\exp X)^n$ such that $\cup_{n=1}^\infty Y_n$ contains arbitrarily fine covers of $X$ and for every $n\ge 1$ every minimal closed $G$-invariant subset of $\exp Y_n$ is a singleton. This yields an alternative proof of Pestov's theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set (or of the interval $[0,1]$) is extremely amenable.