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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
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
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.
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