Nuclear Dimension and Rigidity Results for Virtually Abelian Groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917527324459008 |
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| author | Chan, Frankie Lippert, S. Joseph Moutzouris, Iason Weld, Ellen |
| author_facet | Chan, Frankie Lippert, S. Joseph Moutzouris, Iason Weld, Ellen |
| contents | Let $G$ be a finitely generated virtually abelian group. We show that the Hirsch length, $h(G)$, is equal to the nuclear dimension of its group $C^*$-algebra, $\dim_{nuc}(C^*(G))$. We then specialize our attention to a generalization of crystallographic groups dubbed \textit{crystal-like}. We demonstrate that in this scenario a \textit{point group} is well defined and the order of this point group is preserved by $C^*$-isomorphism. We close by using these tools to demonstrate that crystallographic (as a group property) is preserved by $C^*$-isomorphism. These three tools combine to prove that $2D$ crystallographic groups are $C^*$-superrigid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20850 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nuclear Dimension and Rigidity Results for Virtually Abelian Groups Chan, Frankie Lippert, S. Joseph Moutzouris, Iason Weld, Ellen Operator Algebras Group Theory Representation Theory 2D10, 22D25, 46L05 Let $G$ be a finitely generated virtually abelian group. We show that the Hirsch length, $h(G)$, is equal to the nuclear dimension of its group $C^*$-algebra, $\dim_{nuc}(C^*(G))$. We then specialize our attention to a generalization of crystallographic groups dubbed \textit{crystal-like}. We demonstrate that in this scenario a \textit{point group} is well defined and the order of this point group is preserved by $C^*$-isomorphism. We close by using these tools to demonstrate that crystallographic (as a group property) is preserved by $C^*$-isomorphism. These three tools combine to prove that $2D$ crystallographic groups are $C^*$-superrigid. |
| title | Nuclear Dimension and Rigidity Results for Virtually Abelian Groups |
| topic | Operator Algebras Group Theory Representation Theory 2D10, 22D25, 46L05 |
| url | https://arxiv.org/abs/2504.20850 |