Higher-dimensional digraphs from cube complexes and their spectral theory

We define k -dimensional digraphs and initiate a study of their spectral theory. The k -dimensional digraphs can be viewed as generating graphs for small categories called k -graphs. Guided by geometric insight, we obtain several new series of k -graphs using cube complexes covered by Cartesian prod...

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Bibliographic Details
Published in:Groups, geometry and dynamics geometry and dynamics, 2024-12, Vol.18 (4), p.1427-1467
Main Authors: Larsen, Nadia S., Vdovina, Alina
Format: Article
Language:English
Subjects:
Graph theory
Group theory
Online Access:Get full text
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Summary:We define k -dimensional digraphs and initiate a study of their spectral theory. The k -dimensional digraphs can be viewed as generating graphs for small categories called k -graphs. Guided by geometric insight, we obtain several new series of k -graphs using cube complexes covered by Cartesian products of trees, for k\geq 2 . These k -graphs can not be presented as virtual products and constitute novel models of such small categories. The constructions yield rank- k Cuntz–Krieger algebras for all k\geq 2 . We introduce Ramanujan k -graphs satisfying optimal spectral gap property and show explicitly how to construct the underlying k -digraphs.
ISSN:1661-7207
1661-7215
DOI:10.4171/ggd/787