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Israel Journal of Mathematics

, Volume 127, Issue 1, pp 317–357 | Cite as

Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups

  • Vladimir Pestov
Article

Abstract

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso\(\left( \mathbb{U} \right)\) of isometries of Urysohn’s universal complete separable metric space\(\mathbb{U}\), equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if\(\mathbb{U}\) is replaced with any generalized Urysohn metric spaceU that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.

Keywords

Topological Group Uniform Space Isometric Embedding Strong Operator Topology Compacta Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Hebrew University Magnes Press 2002

Authors and Affiliations

  • Vladimir Pestov
    • 1
  1. 1.School of Mathematical and Computing SciencesVictoria University of WellingtonWellingtonNew Zealand

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