A dichotomy for groupoid C*-algebras

We study the finite versus infinite nature of C -algebras arising from étale groupoids. For an ample groupoid, we relate infiniteness of the reduced C -algebra to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C -algebra of in the sense that if is ample, minimal, topologically principal, and is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph -algebras as well.