Hyperbolic Manifolds and Kleinian Groups
A Kleinian group is a discrete subgroup of the isometry group of hyperbolic 3-space, which is also regarded as a subgroup of Möbius transformations in the complex plane. The present book is a comprehensive guide to theories of Kleinian groups from the viewpoints of hyperbolic geometry and complex analysis. After 1960, Ahlfors and Bers were the leading researchers of Kleinian groups and helped it to become an active area of complex analysis as a branch of Teichmüller theory. Later, Thurston brought a revolution to this area with his profound investigation of hyperbolic manifolds, and at the same time complex dynamical approach was strongly developed by Sullivan. This book provides fundamental results and important theorems which are needed for access to the frontiers of the theory from a modern viewpoint.
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3-manifold Abelian Ahlfors algebraic limit assume automorphism B-group Beltrami Bers slice compact complete hyperbolic condition conjugate contains converges geometrically convex core Corollary corresponding cusp neighborhoods define denote disk ending lamination equivalent exists finite Kleinian group finite number finitely generated Kleinian Fuchsian group Fuchsian model fundamental group geodesic lamination geometrically finite Kleinian group F Hence holomorphic homeomorphic homotopy hyperbolic manifold hyperbolic space hyperbolic structure hyperbolic surface implies incompressible induces infinite NP-end injective intersection isometric isomorphism Kleinian group Kleinian manifold Lemma limit set loxodromic loxodromic element measured lamination Mobius transformation Moreover neighborhood non-elementary Kleinian group non-trivial normal subgroup parabolic element parabolic fixed point pleated surfaces Proposition prove QH(T QHom(r quasi-isometric quasiconformal automorphism quasiconformal deformation quasiconformal map quasifuchsian group Riemann surface satisfies Schottky group second kind Section sequence simple closed curves simple closed geodesic subset Teichmuller space Thurston topological torus totally degenerate