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 Z is not uniformly strictly stable. Furthermore, we show that SLᵣ(Z) , r≥3 , is uniformly flexibly stable, but the free group Fᵣ , r≥2 , is not. We define and investigate a probabilistic variant of uniform stability that has an application to property testing.