On uniform Hilbert Schmidt stability of groups

TL;DR

This paper classifies finitely generated residually finite groups that are uniformly Hilbert Schmidt stable, exploring the necessity of residual finiteness and extending results to amenable groups.

Contribution

It provides a complete classification of uniformly HS stable groups among finitely generated residually finite groups and discusses the role of residual finiteness and amenability.

Findings

Complete classification of uniformly HS stable finitely generated residually finite groups.

Necessity of residual finiteness for HS stability discussed.

Extension of results to groups assuming only amenability.

Abstract

A group $\Gamma$ is said to be uniformly HS stable if any map $\varphi : \Gamma \to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability.