Simple Purely Infinite C*-Algebras and n-Filling Actions

Let n be a positive integer. We introduce a concept, which we call the n-filling property, for an action of a group on a separable unital C*-algebra A. If A=C(Ω) is a commutative unital C*-algebra and the action is induced by a group of homeomorphisms of Ω then the n-filling property reduces to a weak version of hyperbolicity. The n-filling property is used to prove that certain crossed product C*-algebras are purely infinite and simple. A variety of group actions on boundaries of symmetric spaces and buildings have the n-filling property. An explicit example is the action of Γ=SLn(Z) on the projective n-space.