C*-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory
Abstract
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 k-rank graph C*-algebras, and give explicit examples of k-cube groups and their K-theory. We give explicit computations of K-theory for an infinite family of k-rank graphs for k≥ 3, which is not a direct consequence of the K\"unneth Theorem for tensor products.
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