Concentration of measure and whirly actions of Polish groups

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.