$C^{\ast }$-algebras associated to graphs of groups

Nathan Brownlowe, Alexander Mundey, David Pask, Jack Spielberg and Anne Thomas

Abstract

To a large class of graphs of groups we associate a $C^{\ast }$-algebra universal for generators and relations. We show that this $C^{\ast }$-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple $C^{\ast }$-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.