  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.
