Geometric inflexibility and 3-manifolds that fiber over the circle

TL;DR

This paper establishes the geometric inflexibility of hyperbolic 3-manifolds, providing new proofs for convergence in quasi-Fuchsian space and hyperbolization of fibered 3-manifolds with pseudo-Anosov monodromy.

Contribution

It introduces a novel approach to understanding hyperbolic 3-manifolds' rigidity and applies it to prove key results in 3-manifold topology.

Findings

Hyperbolic 3-manifolds are geometrically inflexible with exponential decay of bi-Lipschitz constants.

New proof of convergence of pseudo-Anosov double-iteration on quasi-Fuchsian space.

Hyperbolization theorem for closed 3-manifolds fibered over the circle with pseudo-Anosov monodromy.

Abstract

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.