On amenable Hilbert-Schmidt stable groups

TL;DR

This paper explores Hilbert-Schmidt stability in amenable groups, providing new characterizations, stability preservation results, and the first example of an amenable HS-stable group that is not permutation stable.

Contribution

It offers new proofs of HS-stability for nilpotent groups, characterizes stability in semidirect products, and constructs the first non-permutation stable amenable HS-stable group.

Findings

Finitely generated nilpotent groups are HS-stable.

HS-stability is preserved under finite central quotients.

Constructed the first amenable HS-stable group not permutation stable.

Abstract

We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. We give a short, elementary proof that finitely generated nilpotent groups are HS-stable. We investigate the permanence of HS-stability under central extensions by showing HS-stability is preserved by finite central quotients, but is not preserved in general. We give a characterization of HS-stability for semidirect products $G\rtimes_\gamma \mathbb{Z}$ with $G$ abelian. We use it to construct the first example of a finitely generated amenable HS-stable group which is not permutation stable. Finally, it is proved that for amenable groups flexible HS-stability is equivalent to HS-stability, and very flexible HS stability is equivalent to maximal almost periodicity. There is some overlap of our work with the very recent and very nice preprint of Levit and Vigdorovich. We detail this overlap in the introduction. Where our work overlaps it appears that we take different approaches to the proofs and we feel the two works compliment each other.