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 \operatorname{SL}_{r}(\mathbb{Z}) , r\geq3 , is uniformly flexibly stable, but the free group F_{r} , r\geq2 , is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.