On amenable Hilbert-Schmidt stable groups

We examine Hilbert-Schmidt stability (HS-stability) of dis-crete 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 un-der central quotients 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 )4 gamma Z with G abelian. We use it to construct the first exam-ple 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 pe-riodicity. There is some overlap of our work with the very recent and very nice preprint [27] 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 complement each other.(c) 2023 Published by Elsevier Inc.