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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 $Γ=SL_n({\bf Z})$ on the projective $n$-space.