Stability of approximate group actions: uniform and probabilistic

2020/05/13 by Oren M. Becker, Michael Chapman, Becker, Oren +1 · 2 citations
Medicine · #FOS: Mathematics #Group Theory (math.GR) #Neurological and metabolic disorders

paper · pdf · doi:10.48550/arxiv.2005.06652

openalex publication_date 2020/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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