1996/03/19 by David Borthwick, Borthwick, David, Alan McRae +5 · 1 citation
Mathematics · #Conformal map #Diffeomorphism #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic group #Hyperbolic manifold #Hyperbolic space #Kleinian group #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quotient #Regular polygon #Relatively hyperbolic group #Rigidity (electromagnetism) #dg-ga #math.DG
paper · pdf · doi:10.48550/arxiv.dg-ga/9603010
published in arXiv (Cornell University) (Cornell University) · LaTeX, 11 pages
arxiv created 1996/03/19 · openalex publication_date 1996/03/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.