Higher dimensional digraphs from cube complexes and their spectral theory
Abstract
We define -dimensional digraphs and initiate a study of their spectral theory. The -dimensional digraphs can be viewed as generating graphs for small categories called -graphs. Guided by geometric insight, we obtain several new series of -graphs using cube complexes covered by Cartesian products of trees, for . These -graphs can not be presented as virtual products, and constitute novel models of such small categories. The constructions yield rank- Cuntz-Krieger algebras for all . We introduce Ramanujan -graphs satisfying optimal spectral gap property, and show explicitly how to construct the underlying -digraphs.
Cite
@article{arxiv.2111.09120,
title = {Higher dimensional digraphs from cube complexes and their spectral theory},
author = {Nadia S. Larsen and Alina Vdovina},
journal= {arXiv preprint arXiv:2111.09120},
year = {2022}
}
Comments
33 pages, many figures. This revised version of the paper features a new title and several changes. Most notably the old Proposition 3.3 is now Theorem 3.1 and has a new proof, and the old Theorem 3.4 is now stated in the more general form of Theorem 3.3. The old Proposition 4.7 is now Theorem 4.7 and is slightly revised. The introduction has been revised accordingly