C*-stability of discrete groups

A group may be considered C*-stable if almost representations of the group in a C*-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are C*-stable or only stable with respect to some subclass of C*-algebras, e.g. finite dimensional C*-algebras. We provide criteria and invariants for stability of groups and this allows us to completely determine stability/non-stability of crystallographic groups, surface groups, virtually free groups, and certain Baumslag-Solitar groups. We also show that among the non-trivial finitely generated torsion-free 2-step nilpotent groups the only C*-stable group is Z. (C) 2020 Elsevier Inc. All rights reserved.