On uniform Hilbert Schmidt stability of groups

@article{Akhtiamov2020OnUH,
  title={On uniform Hilbert Schmidt stability of groups},
  author={Danil Akhtiamov and Alon Dogon},
  journal={arXiv: Group Theory},
  year={2020}
}
A group $\Gamma$ is said to be uniformly HS stable if any map $\varphi : \Gamma \to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability. 
Ultrametric analogues of Ulam stability of groups
We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a p-adic analogue of Ulam stability, where we take GLn(Zp)

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