Strong Jordan Separation and Applications to Rigidity

We prove that simple, thick hyperbolic P-manifolds of dimensionat least three exhibit Mostow rigidity. We also prove a quasi-isometryrigidity result for the fundamental groups of simple, thickhyperbolic P-manifolds of dimension at least three. The keytool in the proof of these rigidity results is a strong formof the Jordan separation theorem, for maps from Sⁿ Sⁿ⁺¹ whichare not necessarily injective.