Residually finite amenable groups that are not Hilbert-Schmidt stable

TL;DR

This paper constructs the first examples of residually finite amenable groups that are not Hilbert-Schmidt stable, revealing new distinctions in group stability properties and providing explicit examples with complex algebraic structures.

Contribution

It introduces the first known residually finite amenable groups that are not Hilbert-Schmidt stable, including finitely generated nilpotent-by-cyclic and solvable linear examples.

Findings

Constructed finitely generated, class 3 nilpotent-by-cyclic groups not HS-stable.

Provided examples of amenable groups that are flexibly HS-stable but not very flexibly HS-stable.

Demonstrated these groups are not operator-HS-stable, with almost homomorphisms that cannot be perturbed to true homomorphisms.

Abstract

We construct the first examples of residually finite amenable groups that are not Hilbert-Schmidt (HS) stable. We construct finitely generated, class 3 nilpotent by cyclic examples and solvable linear finitely presented examples. This also provides the first examples of amenable groups that are very flexibly HS-stable but not flexibly HS-stable and the first examples of residually finite amenable groups that are not locally HS-stable. Along the way we exhibit (necessarily not-finitely-generated) class 2 nilpotent groups $G = A\rtimes \Z$ with $A$ abelian such that the periodic points of the dual action are dense but it does not admit dense periodic measures. Finally we use the Tikuisis-White-Winter theorem to show all of the examples are not even operator-HS-stable; they admit operator norm almost homomorphisms that can not be HS-perturbed to true homomorphisms.