565 research outputs found

    Geometric inflexibility and 3-manifolds that fiber over the circle

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    We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the thick part. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves. We use this inflexibility to give a new proof of the convergence of pseudo-Anosov double-iteration on the quasi-Fuchsian space of a closed surface, and the resulting hyperbolization theorem for closed 3-manifolds that fiber over the circle with pseudo-Anosov monodromy.Comment: 52 pages. Final version. Appeared in Journal of Topology. Material in original version on inflexibility of hyperbolic cone-manifolds has been rewritten into a new paper with identifier arXiv:1412.463

    Cone-manifolds and the density conjecture

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    We give an expository account of our proof that each cusp-free hyperbolic 3-manifold M with finitely generated fundamental group and incompressible ends is an algebraic limit of geometrically finite hyperbolic 3-manifolds.Comment: 19 Pages, 2 figures; to appear, proceedings of the Warwick Conference: Kleinian Groups and Hyperbolic 3-Manfiolds, September 200