Local Hilbert–Schmidt stability

We introduce a notion of local Hilbert–Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert–Schmidt stable but not Hilbert–Schmidt stable. For amenable groups, we provide a criterion for local Hilbert–Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert–Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert–Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert–Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.