Hyperbolic Manifolds and Discrete Groups

2009/08/03 by Michael Kapovich · 434 citations
Mathematics · #Geology #Geometric and Algebraic Topology #Mathematics #Pure mathematics

paper · doi:10.1007/978-0-8176-4913-5

published in Birkhäuser Boston eBooks

openalex publication_date 2009/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02

Abstract

Preface.-Three-dimensional Topology.-Thurston Norm.-Geometry of the Hyperbolic Space.-Kleinian Groups.-Teichmuller Theory of Riemann Surfaces.-Introduction to the Orbifold Theory.-Complex Projective Structures.-Sociology of Kleinian Groups.-Ultralimits of Metric Spaces.-Introduction to Group Actions on Trees.-Laminations, Foliations and Trees.-Rips Theory.-Brooks' Theorem and Circle Packings.-Pleated Surfaces and Ends of Hyperbolic Manifolds.-Outline of the Proof of the Hyperbolization Theorem.-Reduction to The Bounded Image Theorem.-The Bounded Image Theorem.- Hyperbolization of Fibrations.-The Orbifold Trick.-Beyond the Hyperbolization Theorem References.-Index.

Cited by

• Deforming Euclidean cone 3–manifolds
• Spectral and geometric bounds on 2-orbifold diffeomorphism type
• An extension of the Maskit slice for 4-dimensional Kleinian groups
• Non-realizability and ending laminations: Proof of the density conjecture
• On manifolds with quadratic curvature decay
• The bumping set and the characteristic submanifold
• Hyperbolization theorem [wikipedia]
• Kleinian group [wikipedia]

Related

• Hyperbolic boundaries vs. hyperbolic groups
• Lectures on complex hyperbolic Kleinian groups
• Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds
• The Geometry and Arithmetic of Kleinian Groups
• Hyperbolic Geometry and Distance Functions on Discrete Groups