Strong Jordan separation and applications to rigidity
  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. 
J.-F. Lafont