C*-stability of discrete groups

Journal article, 2020

A group may be considered ⁎-stable if almost representations of the group in a ⁎-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are ⁎-stable or only stable with respect to some subclass of ⁎-algebras, e.g. finite dimensional ⁎-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 ⁎-stable group is .