Representations of residually finite groups by isometries of the Urysohn space

Details Representations of residually finite groups by isometries of the Urysohn space Representations of residually finite groups by isometries of the Urysohn space

2006-01-28

arxiv.org/abs/math/0601700v2

Journal of the Ramanujan Mathematical Society 21 (2006), 189-203.

Mathematics

Representation Theory

12 pages, LaTeX 2e, minor changes (bibliography updates etc.)

Scientific paper

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.

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