Outline of the Proof of the Hyperbolization Theorem
Finally, we can present an outline of the proof of Thurston's Hyperbolization Theorem. Our discussion mainly follows [Mor84]. By Theorem 8.36, it is enough to prove the Hyperbolization Theorem for a finite covering of M. Thus we will consider only compact orientable pared manifolds (M, P), which contain orientable superincompressible surfaces and where \(P=\partial M\) is the designated parabolic locus. Thurston's theorem is proven by induction on levels of the Haken hierarchy of M. It turns out that the main problem in proving Theorem 1.42 is the last step of the induction.
KeywordsHyperbolic Manifold Kleinian Group Hyperbolic Structure Teichmiiller Space Hyperbolic Orbifold
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