The isometry group of the Urysohn space as a Levy group

We prove that the isometry group Iso(U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Levy 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(U) has a dense locally finite subgroup. (C) 2007 Elsevier B.V. All rights reserved.