C*-stability of discrete groups

Abstract

A group may be considered $C^{\ast }$-stable if almost representations of the group in a $C^{\ast }$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^{\ast }$-stable or only stable with respect to some subclass of $C^{\ast }$-algebras, e.g. finite dimensional $C^{\ast }$-algebras. We provide criteria and invariants for stability of groups and this allows us to completely determine stability/non-stability of crystallographic groups, finitely generated torsion-free step-2 nilpotent groups, surface groups, virtually free groups and certain Baumslag-Solitar groups.

The results in section 4.2 (finitely generated torsion-free 2-step nilpotent groups) have been strengthened slightly. Clarified in the introduction that we only consider unitary group representations. A few misprints fixed. 39 pages