On uniform Hilbert Schmidt stability of groups

Akhtiamov, Danil; Dogon, Alon

doi:10.1090/proc/15772

On uniform Hilbert Schmidt stability of groups
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by Danil Akhtiamov and Alon Dogon

Proc. Amer. Math. Soc. 150 (2022), 1799-1809

DOI: https://doi.org/10.1090/proc/15772

Published electronically: January 13, 2022

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Abstract:

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.

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