Mostow Rigidity

1 The Proof Modulo some Analysis 1.1 Statement of the Result Given any metric space (X, d), a map H : X → X is BL (bi-lipshitz) if there is some constant K > 0 such that d(H(x), H(y)) ∈ [K, K] d(x, y), ∀x, y ∈ X. (1) Here is a limited form of Mostow Rigidity: Theorem 1.1 (Mostow) Suppose that M1 and M2 are both compact hyperbolic 3-manifolds. If there is a BL map f : M1 → M2 then there is an isometry g : M1 → M2. One can press on the proof to yield the stronger statement that f and g are homotopic maps. Also, if one is willing to work with quasi-isometries in place of BL maps, one can just assume that the map f is a homotopy equivalence. I’ll leave these matters to the interested reader. Note that if f is a diffeomorphism then f is automatically BL. Hence Corollary 1.2 If two closed hyperbolic 3 manifolds are diffeomorphic then they are isometric. Hence, the hyperbolic structure on a compact hyperbolic 3-manifold is unique. My proof is, in a certain sense, the standard one, but I figured out a good way to do it which relies on a lot less real analysis. The rest of this chapter assembles the ingredients of the proof and the last section of the chapter puts them together. The second and third chapters do the needed analysis.