Representations of residually finite groups by isometries of the Urysohn space

TL;DR

This paper proves that the isometry group of the universal Urysohn space can approximate representations of residually finite groups, linking geometric group actions with operator algebra conjectures.

Contribution

It establishes the Kirchberg property for the isometry group of the Urysohn space by showing all residually finite group representations can be approximated by finite-range representations.

Findings

The isometry group of the Urysohn space satisfies Kirchberg's property.

Representations of residually finite groups can be approximated by finite-range isometries.

The result connects geometric group actions with operator algebra conjectures.

Abstract

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.