Concentration of measure and whirly actions of Polish groups

TL;DR

This paper explores the concept of whirly actions of Polish groups on measure spaces, showing that certain non-Lévy groups can have whirly actions, unlike locally compact groups, and provides new examples of such groups.

Contribution

It demonstrates that non-Lévy Polish groups can admit whirly actions, answering an open question and expanding understanding of group actions in measure theory.

Findings

Locally compact groups do not admit whirly actions.

Ergodic near-actions by Polish Lévy groups are whirly.

Constructed examples of non-Lévy groups with whirly actions.

Abstract

A weakly continuous near-action of a Polish group $G$ on a standard Lebesgue measure space $(X,\mu)$ is whirly if for every $A\subseteq X$ of strictly positive measure and every neighbourhood $V$ of identity in $G$ the set $VA$ has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish L\'evy group in the sense of Gromov and Milman, such as $U(\ell^2)$, is whirly (Glasner--Tsirelson--Weiss). We give examples of closed subgroups of the group $\Aut(X,\mu)$ of measure preserving automorphisms of a standard Lebesgue measure space (with the weak topology) whose tautological action on $(X,\mu)$ is whirly, and which are not L\'evy groups, thus answering a question of Glasner and Weiss.