# Marcin Sabok: Extreme amenability of abelian $L_0$ groups

07/Sep/2012, 13:30–15:00
Fields institute,Room 230

Speaker: Marcin Sabok

Title: Extreme amenability of abelian $L_0$ groups

Abstract: Answering a question of Farah, Solecki and Pestov, we show that the group $L_0(\mu)$ is extremely amenable if and only if $\mu$ is a diffused submeasure. This provides a witness for a positive answer to the Glasner–Pestov problem, as it implies that $L_0(\mu)$ is extremely amenable if and only if it has no nontrivial characters.

Given a submeasure $\mu$ on an algebra of subsets of a set $X$, the group $L_0(\mu)$ is the set of all real-valued measurable functions on $X$ with the pointwise addition. $L_0(\mu)$ is endowed with the topology of convergence in submeasure $\mu$. A topological group is extremely amenable if every its continuous action on a compact space has a fixed point.