  We prove that simple, thick hyperbolic P-manifolds of dimension >2 exhibit
Mostow rigidity. We also prove a quasi-isometry rigidity result for the
fundamental groups of simple, thick hyperbolic P-manifolds of dimension >2. The
key tool in the proofs of these rigidity results is a strong form of the Jordan
separation theorem, for maps from S^n to S^{n+1} which are not necessarily
injective.
