## Advanced Studies in Pure Mathematics

### 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 Lévy 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 Lévy groups, thus answering a question of Glasner and Weiss.

#### Article information

Dates
Revised: 25 March 2009
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/ euclid.aspm/1543086329

Digital Object Identifier
doi:10.2969/aspm/05710383

Mathematical Reviews number (MathSciNet)
MR2648270

Zentralblatt MATH identifier
1213.37011

#### Citation

Pestov, Vladimir. Concentration of measure and whirly actions of Polish groups. Probabilistic Approach to Geometry, 383--403, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710383. https://projecteuclid.org/euclid.aspm/1543086329