Stability of approximate group actions: uniform and probabilistic

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 ${\text{SL}}_r(\mathbb{Z})$, $r\ge 3$, is uniformly flexibly stable, but the free group $F_r$, $r\ge 2$, is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.