Extreme amenability of abelian $L_0$ groups

TL;DR

This paper proves that abelian $L_0$ groups are extremely amenable under certain conditions, extending previous results and answering open questions about their fixed point properties and characterizations.

Contribution

It generalizes fixed point results for abelian $L_0$ groups with diffused submeasures, establishing their extreme amenability and characterizing when $L_0( u,R)$ groups are extremely amenable.

Findings

Every continuous action of $L_0( u,G)$ on a compact space has a fixed point.

$L_0( u,R)$ is extremely amenable iff it has no nontrivial characters.

The proof involves chromatic number estimates and algebraic topology tools.

Abstract

We show that for any abelian topological group $G$ and arbitrary diffused submeasure $\mu$, every continuous action of $L_0(\mu,G)$ on a compact space has a fixed point. This generalizes earlier results of Herer and Christensen, Glasner, Furstenberg and Weiss, and Farah and Solecki. This also answers a question posed by Farah and Solecki. In particular, it implies that if $H$ is of the form $L_0(\mu,\mathbb{R})$, then $H$ is extremely amenable if and only if $H$ has no nontrivial characters, which gives an evidence for an affirmative answer to a question of Pestov. The proof is based on estimates of chromatic numbers of certain graphs on $\mathbb{Z}^n$. It uses tools from algebraic topology and builds on the work of Farah and Solecki.