In this survey paper, we outline the proofs of rigidity results (Mostow type, quasi-isometric, and Diagram rigidity) for simple, thick, hyperbolic P-manifolds. These are stratified spaces, and are in some sense the simplest non-manifold locally CAT(−1) spaces one can create, having codimension one singularities along embedded totally geodesic submanifolds. All the proofs depend on the highly non-homogenous structure of the boundary at infinity of the (universal covers of the) spaces in question, and in particular, on an understanding of the local topology of the boundary at infinity. We emphasize the similarities and differences in the proofs of the various rigidity results. These results should be viewed as a first step towards understanding stratified locally CAT(−1) spaces.
Mostow rigidity Quasi-isometry Diagram of groups Singular spaces CAT(−1) Hyperbolic groups
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