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arXiv · 2504.20850

Nuclear Dimension and Rigidity Results for Virtually Abelian Groups

Frankie Chan·S. Joseph Lippert·Iason Moutzouris·Ellen Weld
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Abstract

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.

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Frankie Chan, S. Joseph Lippert, Iason Moutzouris, Ellen Weld. 2025-04-29. Nuclear Dimension and Rigidity Results for Virtually Abelian Groups. https://arxiv.org/abs/2504.20850

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