Advanced Studies in Pure Mathematics

Concentration of measure and whirly actions of Polish groups

Vladimir Pestov

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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

Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 383-403

Received: 2 March 2009
Revised: 25 March 2009
First available in Project Euclid: 24 November 2018

Permanent link to this document euclid.aspm/1543086329

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A15: General groups of measure-preserving transformations [See mainly 22Fxx] 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 43A05: Measures on groups and semigroups, etc.

Concentration of measure on high-dimensional structures Polish groups greatest ambit Lévy groups invariant measures whirly actions groups of measure-preserving transformations


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.

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