A Cup Product Obstruction to Frobenius Stability
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909097184460800 |
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| author | Glebe, Forrest |
| author_facet | Glebe, Forrest |
| contents | A countable discrete group $Γ$ is said to be Frobenius stable if a function from the group that is "almost multiplicative" in the point Frobenius norm topology is "close" to a genuine unitary representation in the same topology. The purpose of this paper is to show that if $Γ$ is finitely generated and a non-torsion element of $H^2(Γ;\mathbb{Z})$ can be written as a cup product of two elements in $H^1(Γ;\mathbb{Z})$ then $Γ$ is not Frobenius stable. In general, 2-cohomology does not obstruct Frobenius stability. Some examples are discussed, including Thompson's group $F$ and Houghton's group $H_3$. The argument is sufficiently general to show that the same condition implies non-stability in unnormalized Schatten $p$-norms for $1<p\le\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01533 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Cup Product Obstruction to Frobenius Stability Glebe, Forrest Operator Algebras Group Theory K-Theory and Homology A countable discrete group $Γ$ is said to be Frobenius stable if a function from the group that is "almost multiplicative" in the point Frobenius norm topology is "close" to a genuine unitary representation in the same topology. The purpose of this paper is to show that if $Γ$ is finitely generated and a non-torsion element of $H^2(Γ;\mathbb{Z})$ can be written as a cup product of two elements in $H^1(Γ;\mathbb{Z})$ then $Γ$ is not Frobenius stable. In general, 2-cohomology does not obstruct Frobenius stability. Some examples are discussed, including Thompson's group $F$ and Houghton's group $H_3$. The argument is sufficiently general to show that the same condition implies non-stability in unnormalized Schatten $p$-norms for $1<p\le\infty$. |
| title | A Cup Product Obstruction to Frobenius Stability |
| topic | Operator Algebras Group Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2312.01533 |