C*-algebras associated to group actions on trees, and connections to classification

Abstract

We study C*-algebras arising from actions of countable, discrete groups on directed trees. Such an action gives rise to a C*-algebra in two ways: the action induces an action of the group on the boundary of the tree, from which one can construct a crossed product C*-algebra; but also the action gives rise to a (directed) graph of groups as a quotient object, and one can also associate a C*- algebra to this directed graph of groups. After exploring these two C*-algebras as part of separate, more general classes (crossed products arising from group actions on the boundaries of multitrees, and C*-algebras of groupoid-embeddable categories), we show that the crossed product C*-algebra is Morita equivalent to the directed graph-of-groups C*-algebra. Finally, we show that directed graphof- groups C*-algebras (and their corresponding crossed products) contain all stable UCT Kirchberg algebras (a class of C*-algebras classified completely by K-theory).