论文标题
近似小组动作的稳定性：统一和概率
Stability of approximate group actions: uniform and probabilistic
论文作者
论文摘要
我们证明，从离散的正态组到对称组的每个均匀的近似同态均匀地均接近同构形成稍大的对称组。也就是说，在排列中，可正态的组均匀地稳定。这是肯定地回答了Kun和Thom的问题，以及Lubotzky问题的略有变化。我们还向Lubotzky的原始问题提供了负面答案，表明该组$ \ mathbb {z} $并非严格稳定。此外，我们表明$ \ text {sl} _ {r}（\ mathbb {z}）$，$ r \ geq3 $，均匀地稳定，但免费组$ f_ {r} $，$ r \ geq 2 $不是。我们定义和研究具有统一稳定性的概率变体，该变体应用于财产测试。
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.
