The isometry group of the Urysohn space as a Levy group
Abstract
Description
We prove that the isometry group $\Iso(\Ur)$ of the universal Urysohn metric space $\Ur$ equipped with the natural Polish topology is a Lévy 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.
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)
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)