## $$C^*$$-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 $$C^*$$-algebra universal for generators and relations. We show that this $$C^*$$-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 $$C^*$$-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.

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 Monday, October 31, 2016