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 Ku center dot nneth theorem for tensor products.