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

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.