The isometry group of the Urysohn space as a Levy group

Details The isometry group of the Urysohn space as a Levy group The isometry group of the Urysohn space as a Levy group

2005-09-18

arxiv.org/abs/math/0509402v3

Proceedings of the 6-th Iberoamerican Conference on Topology and its Applications (Puebla, Mexico, 4-7 July 2005). A special i

Mathematics

General Topology

20 pages, LaTeX 2e with Elsevier macros, final version, to appear in Proc. 6-th Iberoamerican Conf. on Topology and its Applic

Scientific paper

We prove that the isometry group $\Iso(\Ur)$ of the universal Urysohn metric space $\Ur$ equipped with the natural Polish topology is a L\'evy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that $\Iso(\Ur)$ has a dense locally finite subgroup.

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