  We study the finite versus infinite nature of C*-algebras arising from etale
groupoids. For an ample groupoid G, we relate infiniteness of the reduced
C*-algebra of G to notions of paradoxicality of a K-theoretic flavor. We
construct a pre-ordered abelian monoid S(G) which generalizes the type
semigroup introduced by R{\o}rdam and Sierakowski for totally disconnected
discrete transformation groups. This monoid reflects the finite/infinite nature
of the reduced groupoid C*-algebra of G. If G is ample, minimal, and
topologically principal, and S(G) is almost unperforated we obtain a dichotomy
between stable finiteness and pure infiniteness for the reduced C*-algebra of
G.
