C^-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory

We unite elements of category theory, K-theory, and geometric group theory, by defining a class of groups called k-cube groups, which act freely and transitively on the product of k trees, for arbitrary k.The quotient of this action on the product of trees defines a k-dimensional cube complex, which induces a higher-rank graph.We make deductions about the K-theory of the corresponding rank-k graph C -algebras, and give examples of k-cube groups and their K-theory.These are among the first explicit computations of K-theory for an infinite family of rank-k graphs for k ≥ 3, which is not a direct consequence of the Künneth theorem for tensor products.