Higher dimensional digraphs from cube complexes and their spectral theory

TL;DR

This paper introduces higher-dimensional digraphs derived from cube complexes, explores their spectral properties, and constructs new models of small categories with applications to Cuntz-Krieger algebras and Ramanujan graphs.

Contribution

It defines $k$-dimensional digraphs, develops their spectral theory, and constructs novel $k$-graphs from cube complexes, leading to new models and spectral gap results.

Findings

Constructed new series of $k$-graphs from cube complexes.

Established spectral properties and optimal spectral gaps for Ramanujan $k$-graphs.

Connected $k$-graphs to rank-$k$ Cuntz-Krieger algebras.

Abstract

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.