$C^*$-algebras associated to directed graphs of groups, and models of Kirchberg algebras

Abstract

We introduce $C^*$-algebras associated to directed graphs of groups. In particular, we associate a combinatorial $C^*$-algebra to each row-finite directed graph of groups with no sources, and show that this $C^*$-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these $C^*$-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.