  For n>6, we show that if G is a torsion-free hyperbolic group whose visual
boundary is an (n-2)-dimensional Sierpinski space, then G=\pi_1(W) for some
aspherical n-manifold W with nonempty boundary. Concerning the converse, we
construct, for each n>3, examples of aspherical manifolds with boundary, whose
fundamental group G is hyperbolic, but with visual boundary not homeomorphic to
an (n-2)-dimensional Sierpinski space.
