C*-algebras associated to graphs of groups

TL;DR

This paper constructs C*-algebras from graphs of groups, explores their properties via group actions on trees, and classifies the resulting algebras as either Kirchberg or Bunce-Deddens types.

Contribution

It introduces a universal C*-algebra for graphs of groups and characterizes its properties through boundary actions and minimality conditions.

Findings

C*-algebra is stably isomorphic to a crossed product from the fundamental group's action

Characterization of minimal and locally contractive actions

Dichotomy: simple C*-algebras are either Kirchberg or Bunce-Deddens

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.