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