  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 $A=C(\Omega)$ is a commutative unital $C^*$-algebra and
the action is induced by a group of homeomorphisms of $\Omega$ 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 $\Gamma=SL_n({\bf Z})$ on the projective
$n$-space.
