$C^*$-algebras associated with the fundamental groups of graphs of groups

Abstract

We construct a nuclear $C^*$-algebra associated with the fundamental group of a graph of groups of finite type. It is well-known that every word-hyperbolic group with zero-dimensional boundary, in other words, every group acting trees with finite stabilizers is given by the fundamental group of such a graph of groups. We show that our $C^*$-algebra is $*$-isomorphic to the crossed product arising from the associated boundary action and is also given by a Cuntz-Pimsner algebra. We also compute the K-groups and determine the ideal structures of our $C^*$-algebras.