Stability of group relations under small Hilbert-Schmidt perturbations

TL;DR

This paper investigates the stability of group relations under small Hilbert-Schmidt perturbations, establishing new results for 1-relator groups with non-trivial centers and characterizing matricial stability for amenable groups.

Contribution

It introduces the concept of $II_1$-factor stability for groups, proves stability for certain 1-relator groups, and characterizes matricial stability for amenable groups via approximation properties.

Findings

All 1-relator groups with non-trivial center are $II_1$-factor stable.

Matricial stability is characterized for amenable groups through approximation of characters.

Matricial stability is proved for the discrete Heisenberg group and virtually abelian groups.

Abstract

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on $II_1$-factors we similarly define $II_1$-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are $II_{1}$-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group $\mathbb H_3$ and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it.