  We study conditions that will ensure that a crossed product of a C*-algebra
by a discrete exact group is purely infinite (simple or non-simple). We are
particularly interested in the case of a discrete non-amenable exact group
acting on a commutative C*-algebra, where our sufficient 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.
