The isometry group of the Urysohn space as a Levy group
General Topology
2007-09-03 v3
Abstract
We prove that the isometry group of the universal Urysohn metric space 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 has a dense locally finite subgroup.
Cite
@article{arxiv.math/0509402,
title = {The isometry group of the Urysohn space as a Levy group},
author = {Vladimir Pestov},
journal= {arXiv preprint arXiv:math/0509402},
year = {2007}
}
Comments
20 pages, LaTeX 2e with Elsevier macros, final version, to appear in Proc. 6-th Iberoamerican Conf. on Topology and its Applications (Puebla, Mexico, 4-7 July 2005). Section 3 is removed to become a part of a joint paper with V.V. Uspenskij ``Representations of residually finite groups by isometries of the Urysohn space'' (math.RT/0601700)