Stability of approximate group actions: uniform and probabilistic

TL;DR

This paper proves that amenable groups exhibit uniform flexible stability in permutations, showing approximate homomorphisms are close to true homomorphisms, and explores stability properties of various groups including $ ext{SL}_r(bZ)$ and free groups.

Contribution

It establishes uniform flexible stability for amenable groups in permutations and introduces a probabilistic stability variant with applications to property testing.

Findings

Amenable groups are uniformly flexibly stable in permutations.

The group $bZ$ is not uniformly strictly stable.

$ ext{SL}_r(bZ)$ for $r extgreater 2$ is uniformly flexibly stable, but free groups are not.

Abstract

We prove that every uniform approximate homomorphism from a discrete amenable group into a symmetric group is uniformly close to a homomorphism into a slightly larger symmetric group. That is, amenable groups are uniformly flexibly stable in permutations. This answers affirmatively a question of Kun and Thom and a slight variation of a question of Lubotzky. We also give a negative answer to Lubotzky's original question by showing that the group $\mathbb{Z}$ is not uniformly strictly stable. Furthermore, we show that $\text{SL}_{r}(\mathbb{Z})$, $r\geq3$, is uniformly flexibly stable, but the free group $F_{r}$, $r\geq 2$, is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.