We study conditions that will ensure that a crossed product
of a C-algebra by a discrete exact group is purely innite (simple or
non-simple). We are particularly interested in the case of a discrete nonamenable
exact group acting on a commutative C-algebra, where our
sucient conditions can be phrased in terms of paradoxicality of subsets
of the spectrum of the abelian C-algebra.
As an application of our results we show that every discrete countable
non-amenable exact group admits a free amenable minimal action on the
Cantor set such that the corresponding crossed product C-algebra is a
Kirchberg algebra in the UCT class.