C*-stability of discrete groups

TL;DR

This paper investigates which discrete groups are $C^*$-stable, meaning almost representations are close to true representations, providing criteria and classifying stability for various group classes.

Contribution

It introduces criteria and invariants for $C^*$-stability, enabling complete classification of stability for several important classes of groups.

Findings

Crystallographic groups are classified as stable or not.

Finitely generated torsion-free step-2 nilpotent groups are analyzed for stability.

Surface groups and certain Baumslag-Solitar groups are characterized regarding $C^*$-stability.

Abstract

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, finitely generated torsion-free step-2 nilpotent groups, surface groups, virtually free groups and certain Baumslag-Solitar groups.