# On extremely amenable groups of homeomorphisms

Vladimir Uspenskij

Arxiv ID: 0710.5785•Last updated: 8/27/2021

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.

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