On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds

Abstract

Thurston’s ending lamination conjecture states that a hyperbolic manifold is uniquely determined by a collection of Riemann surfaces and geodesic laminations that describe the asymptotic geometry of its ends. We prove this conjecture for the case of manifolds whose fundamental group is freely indecomposable, and which admit a positive lower bound on injectivity radii. The techniques of the proof apply to show that a Kleinian surface group admitting a positive lower bound on injectivity radii is continuously semiconjugate to a Fuchsian group. This extends results of Cannon and Thurston. A further consequence is a rigidity theorem for surface groups satisfying the injectivity radius condition, namely that two such groups whose actions on the sphere are conjugate by a homeomorphism that is conformal on the domains of discontinuity must be conjugate by a Möbius transformation.