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 products of trees, for k 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 2. We introduce Ramanujan k -graphs satisfying optimal spectral gap property and show explicitly how to construct the underlying k -digraphs.