Higher rank graphs from cube complexes and their spectral theory

We propose constructions of $k$-graphs from combinatorial input and initiate a study of their spectral theory. Guided by geometric insight, we obtain several new series of $k$-graphs using cube complexes covered by Cartesian products of trees, for $k \geq 3$. 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 such $k$-graphs.