[math/0601700] Representations of residually finite groups by isometries of the Urysohn space

[math/0601700] Representations of residually finite groups by isometries of the Urysohn space: "Representations of residually finite groups by isometries of the Urysohn space
Authors: Vladimir G. Pestov, Vladimir V. Uspenskij
Comments: 12 pages, LaTeX 2e
Subj-class: Representation Theory
MSC-class: 43A65; 20C99; 22A05; 22F05; 22F50; 54E50

As a consequence of Kirchberg's work, Connes Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise approximated by homomorphisms with a precompact range. In this form, the property (which we call Kirchberg's property) makes sense for an arbitrary topological group. We establish the validity of the Kirchberg property for the isometry group $Iso(U)$ of the universal Urysohn metric space $U$ as a consequence of a stronger result: every representation of a residually finite group by isometries of $U$ can be pointwise approximated by representations with a finite range. This brings up the natural question of which other concrete infinite-dimensional groups satisfy the Kirchberg property."