arXiv subjects

S. Joseph Lippert

Publications and source records attributed to S. Joseph Lippert.

4 recordsLinked to original sources

arXiv2026

Nuclear Dimension and Rigidity Results for Virtually Abelian Groups

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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arXiv2025

Products, crossed products, and Zappa--Szép products for $k$-graphs

We use the lens of Zappa--Szép decomposition to examine the relationship between directed graph products and $k$-graph products. There are many examples of higher-rank graphs, or $k$-graphs, whose underlying directed graph may be factored as a product, but the $k$-graph itself is not a product. In such examples, we establish that the Zappa--Szép structure of the $k$-graph gives rise to "actions'' of the underlying directed factors on each other. Although these "actions'' are in general poorly behaved, if one of them is trivial (or trivial up to isomorphism), we obtain a crossed-product-like structure on the $k$-graph. We provide examples where this crossed-product structure is visible in the associated $C^*$-algebra, and we characterize those $k$-graphs whose Zappa--Szép induced actions are trivial up to isomorphism.

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arXiv2024

On The Evans Chain Complex

We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialized case where the higher-rank graph consists of one vertex, we are able to use the Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

math.OA↗

arXiv2024

Refinement of Higher-Rank Graph Reduction

Given a row-finite, source-free, graph of rank k, we extend the definition of reduction introduced by Eckhardt et al. This constitutes a large step forward in the extension of the geometric classification of finite directed graph $C^*$-algebras presented by Eilers et al. to higher-rank graph $C^*$-algebras. This new move acts as an inverse to delay, directly extends the previous version, and provides previously undocumented Morita classes of k-graphs. In pursuit of this extension, we formalize what constitutes a higher-rank graph move. Specifically, we use this formalization as a bridge between the new geometric reasoning and the classical category theoretic construction.

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