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Residually finite amenable groups that are not Hilbert-Schmidt stable

Caleb Eckhardt

TL;DR

The paper addresses whether residual finiteness implies Hilbert-Schmidt (HS) stability for amenable groups, and demonstrates a negative answer by constructing explicit residually finite amenable groups that are not HS-stable. Employing Hadwin’s trace-approximation framework and Abels-type embeddings, the authors build three tiers of counterexamples: an infinite-generated group $G_p$, a finitely generated group $K_p$, and a finitely presented group $\tilde{G}_p$, each manifesting obstructions in the finite-dimensional traces arising from dual dynamical systems. They further exhibit explicit non-perturbable approximate homomorphisms and apply the Tikuisis–White–Winter theorem to obtain operator-norm counterexamples, strengthening the negation to operator-HS stability. The work also clarifies the landscape of HS notions by showing amenable residually finite groups can be very flexibly HS-stable but not flexibly HS-stable or locally HS-stable, and discusses density of periodic points versus periodic measures in associated dual dynamics. Overall, this provides the first family of amenable, residually finite groups that fail HS-stability and related strong forms, widening the understanding of stability phenomena in group C*-algebras and representations.

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.

Residually finite amenable groups that are not Hilbert-Schmidt stable

TL;DR

The paper addresses whether residual finiteness implies Hilbert-Schmidt (HS) stability for amenable groups, and demonstrates a negative answer by constructing explicit residually finite amenable groups that are not HS-stable. Employing Hadwin’s trace-approximation framework and Abels-type embeddings, the authors build three tiers of counterexamples: an infinite-generated group , a finitely generated group , and a finitely presented group , each manifesting obstructions in the finite-dimensional traces arising from dual dynamical systems. They further exhibit explicit non-perturbable approximate homomorphisms and apply the Tikuisis–White–Winter theorem to obtain operator-norm counterexamples, strengthening the negation to operator-HS stability. The work also clarifies the landscape of HS notions by showing amenable residually finite groups can be very flexibly HS-stable but not flexibly HS-stable or locally HS-stable, and discusses density of periodic points versus periodic measures in associated dual dynamics. Overall, this provides the first family of amenable, residually finite groups that fail HS-stability and related strong forms, widening the understanding of stability phenomena in group C*-algebras and representations.

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 with 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.
Paper Structure (7 sections, 13 theorems, 33 equations)

This paper contains 7 sections, 13 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Residually finite, infinitely generated non-HS stable groups
  4. Residually finite, finitely generated non HS-stable groups
  5. Residually finite, finitely presented non (locally) HS-stable groups
  6. Explicit non-perturbable approximate homomorphisms
  7. A strong negation of HS-stability

Key Result

Theorem 1

There are infinitely many non-isomorphic finitely generated, residually finite, non HS-stable groups of the form $N\rtimes \mathbb{Z}$ where $N$ is class 3 nilpotent. Moreover these groups are not operator-HS-stable (See Theorem thm:sform).

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 19 more