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In the tradition of Thurston II : geometry and groups
- Available as
- Online
- Summary
-
The purpose of this volume and of the other volumes in the same series is to provide a collection of surveys that allows the reader to learn the important aspects of William Thurstons heritage. Thu...
The purpose of this volume and of the other volumes in the same series is to provide a collection of surveys that allows the reader to learn the important aspects of William Thurstons heritage. Thurstons ideas have altered the course of twentieth century mathematics, and they continue to have a significant influence on succeeding generations of mathematicians. The topics covered in the present volume include com-plex hyperbolic Kleinian groups, Mobius structures, hyperbolic ends, cone 3-manifolds, Thurstons norm, surgeries in representation varieties, triangulations, spaces of polygo-nal decompositions and of singular flat structures on surfaces, combination theorems in the theories of Kleinian groups, hyperbolic groups and holomorphic dynamics, the dynamics and iteration of rational maps, automatic groups, and the combinatorics of right-angled Artin groups.
Details
- Creator
- Ken'ichi Ohshika, Athanase Papadopoulos, editors
- Format
- Books
- Language
- English
- Contributors
- Publication
-
- Cham : Springer, 2022
- Physical Details
-
- 1 online resource (525 pages)
- data file
- ISBNs
- 3030975606, 3030975592, 9783030975616, 9783030975593, 9783030975609, 9783030975623
- OCLC
- on1338838242
- Includes bibliographical references and index.
- Intro -- Preface -- Contents -- 1 Introduction -- 2 A Survey of Complex Hyperbolic Kleinian Groups -- 2.1 Introduction -- 2.2 Complex Hyperbolic Space -- 2.3 Basics of Discrete Subgroups of PU(n,1) -- 2.4 Margulis Lemma and Thick-Thin Decomposition -- 2.5 Geometrically Finite Groups -- 2.6 Ends of Negatively Curved Manifolds -- 2.7 Critical Exponent -- 2.8 Examples -- 2.9 Complex Hyperbolic Kleinian Groups and Function Theory on Complex Hyperbolic Manifolds -- 2.10 Conjectures and Questions -- Appendix A: Horofunction Compactification -- Appendix B: Two Classical Peano Continua
- Appendix C: Gromov-Hyperbolic Spaces and Groups -- Appendix D: Orbifolds -- Appendix E: Ends of Spaces -- Appendix F: Generalities on Function Theory on Complex Manifolds -- Appendix G (by Mohan Ramachandran): Proof of Theorem 2.19 -- References -- 3 Möbius Structures, Hyperbolic Ends and k-Surfaces in Hyperbolic Space -- 3.1 Overview -- 3.1.1 Hyperbolic Ends and Möbius Structures -- 3.1.2 Infinitesimal Strict Convexity, Quasicompleteness and the Asymptotic Plateau Problem -- 3.1.3 Schwarzian Derivatives -- 3.1.4 Closing Remarks and Acknowledgements -- 3.2 Möbius Structures
- 3.2.1 Möbius Structures -- 3.2.2 The Möbius Disk Decomposition and the Join Relation -- 3.2.3 Geodesic Arcs and Convexity -- 3.2.4 The Kulkarni-Pinkall Form -- 3.2.5 Analytic Properties of the Kulkarni-Pinkall Form -- 3.3 Hyperbolic Ends -- 3.3.1 Hyperbolic Ends -- 3.3.2 The Half-Space Decomposition -- 3.3.3 Geodesic Arcs and Convexity -- 3.3.4 Ideal Boundaries -- 3.3.5 Extensions of Möbius Surfaces -- 3.3.6 Left Inverses and Applications -- 3.4 Infinitesimally Strictly Convex Immersions -- 3.4.1 Infinitesimally Strictly Convex Immersions -- 3.4.2 A Priori Estimates
- 3.4.3 Cheeger-Gromov Convergence -- 3.4.4 Labourie's Theorems and Their Applications -- 3.4.5 Uniqueness and Existence -- Appendix A: A Non-complete k-Surface -- Appendix B: Category Theory -- References -- 4 Cone 3-Manifolds -- 4.1 Introduction -- 4.2 Cone Manifolds -- 4.3 Hyperbolic Dehn Filling -- 4.4 Local Rigidity -- 4.5 Sequences of Cone Manifolds -- 4.5.1 Compactness Theorem -- 4.5.2 Cone-Thin Part -- 4.5.3 Decreasing Cone Angles: Global Rigidity -- 4.5.4 Increasing Cone Angles -- 4.6 Examples -- 4.6.1 Hyperbolic Two-Bridge Knots and Links -- 4.6.2 Montesinos Links -- 4.6.3 A Cusp Opening
- 4.6.4 Borromean Rings -- 4.6.5 Borromean Rings Revisited: Spherical Structures -- References -- 5 A Survey of the Thurston Norm -- 5.1 Introduction -- Organization -- Conventions and Notation -- 5.2 Foundations of the Thurston Norm -- 5.2.1 Thurston Norm -- 5.2.2 Norm Balls and Fibrations Over a Circle -- 5.2.3 Norm-Minimizing Surfaces and Codimension-1 Foliations -- 5.2.4 Singular and Gromov Norms -- 5.3 Alexander and Teichmüller Polynomials -- 5.3.1 Alexander Polynomial -- 5.3.2 Abelian Torsion -- 5.3.3 Teichmüller Polynomial -- 5.4 Seiberg-Witten Invariant -- 5.4.1 Seiberg-Witten Theory -- 5.4.2 Seiberg-Witten Invariant of a 3-Manifold