Three dimensional manifolds, Kleinian groups and hyperbolic geometry

1982/05/01 by William P. Thurston · 20 citations
Mathematics · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory #Kleinian group #Hyperbolic 3-manifold #Mathematics #Hyperbolic geometry #Geometry #Hyperbolic triangle #Hyperbolic group #Hyperbolic manifold #Pure mathematics #Mathematical analysis #Algebraic geometry #Hyperbolic function

paper · pdf · doi:10.1090/s0273-0979-1982-15003-0

openalex publication_date 1982/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

Citations

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• A Unique Decomposition Theorem for 3-Manifolds
• Strong rigidity ofQ-rank 1 lattices

Cited by

• Geometrization of 3-dimensional orbifolds
• The Ricci Flow: An Introduction
• Perspectives on geometric analysis
• Reducing Dehn fillings and small surfaces
• The classification of Toroidal Dehn surgeries on Montesinos knots
• On growth rate and contact homology
• Topological Symmetry Groups of Graphs in 3-Manifolds
• Random Heegaard splittings
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Notes on Perelman’s papers
• Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry
• Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms
• Hyperbolic reflection groups
• Recognizing constant curvature discrete groups in dimension 3
• 3-manifold [wikipedia]
• Geometrization conjecture [wikipedia]
• Hyperbolic 3-manifold [wikipedia]
• Kleinian group [wikipedia]
• Orbifold [wikipedia]
• Virtually fibered conjecture [wikipedia]