Quasi-rigidity of hyperbolic 3-manifolds and scattering theory

Abstract

Take two isomorphic convex co-compact co-infinite volume Kleinian groups, whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by these groups gives two hyperbolic 3-manifolds whose scattering operators may be compared. We prove that the operator norm of the difference between the scattering operators is small, then the groups are related by a coorespondingly small quasi-conformal deformation. This in turn implies that the two hyperbolic 3-manifolds are quasi-isometric.