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Geometric inflexibility and 3-manifolds that fiber over the circle. (English) Zbl 1220.30057
Summary: We prove that 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 from 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.

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds
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