Residually finite amenable groups that are not Hilbert-Schmidt stable
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916623782248448 |
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| author | Eckhardt, Caleb |
| author_facet | Eckhardt, Caleb |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Residually finite amenable groups that are not Hilbert-Schmidt stable Eckhardt, Caleb Group Theory Operator Algebras 43A35, 43A07, 20E26 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. |
| title | Residually finite amenable groups that are not Hilbert-Schmidt stable |
| topic | Group Theory Operator Algebras 43A35, 43A07, 20E26 |
| url | https://arxiv.org/abs/2501.07791 |