Simple purely infinite C∗-algebras and nfilling actions

Abstract. 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 homeo-morphisms of Ω then the n-filling property reduces to a weak ver-sion 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