The Local Lifting Property, Property FD, and stability of approximate representations

TL;DR

This paper proves key properties like the Local Lifting Property and Property FD for important classes of groups, showing they are very flexibly stable under certain norms.

Contribution

It establishes these properties for groups central to geometric and combinatorial group theory, linking them to stability in operator algebra contexts.

Findings

Groups like 3-manifold groups and right-angled Artin groups have Property FD and are very flexibly stable.

These groups also possess Kechris's property (E)MD, implying stability in finite actions.

The results are accessible to both operator algebraists and group theorists.

Abstract

We establish Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for classes of finitely generated groups of central importance in geometric and combinatorial group theory: $3$-manifold groups, limit groups, and certain one-relator groups and right-angled Artin groups. We deduce that such groups are very flexibly stable, with respect to normalized unitarily invariant norms. In the appendix, we show that these groups also have Kechris's property (E)MD, and hence are stable in finite actions, in the selse of Gohla--Thom. The exposition is made accessible to operator algebraists and group theorists alike.