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.