STRONG JORDAN SEPARATION AND APPLICATIONS TO RIGIDITY

We prove that simple, thick hyperbolic P-manifolds of dimension at least three exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension at least three. The key tool in the proof of these rigidity results is a strong form of the Jordan separation theorem, for maps from Sn → Sn+1 which are not necessarily injective.