近似群作用的稳定性：一致性和概率性
Stability of approximate group actions: uniform and probabilistic
摘要：
证明了离散顺从群到对称群的一致近似同态一致接近于到稍大的对称群的同态。也就是说，顺从群在置换中是一致灵活稳定的。这肯定地回答了昆和托姆的一个问题，以及卢博茨基的一个问题的一个轻微变化。我们也给出了一个否定的答案Lubotzky的原始问题，表明该集团$\mathbb{Z}$是不一致严格稳定。此外，我们证明了$\text{SL}_{r}(\mathbb{Z})$，$r\geq3$，是一致柔性稳定的，但自由群$F_{r}$，$r\geq 2$，不是。我们定义和调查的概率变量的一致稳定性，有一个应用程序的属性测试。
英文摘要：
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.
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