In the Tradition of Thurston: Geometry and Topology

Preface
Introduction
Contents
1 A Glimpse into Thurston's Work
1.1 Introduction
1.2 On Thurston's Works
1.2.1 Foliations and Groups of Homeomorphisms
1.2.2 Contact and Symplectic Geometry
1.2.3 One-Dimensional Dynamics
1.2.4 The Topology of 3-Manifolds
1.2.5 (G,X)-Structures and Geometric Structures
1.2.6 Geometrization of Cone-Manifolds
1.2.7 Dehn Surgery
1.2.8 Kleinian Groups
1.2.9 The Thurston Norm, the Gromov Norm and the Gromov Invariant
1.2.10 Conformal Geometry and Holomorphic Dynamics
1.2.11 Complex Projective Geometry
1.2.12 Circle Packings and Discrete Conformal Geometry
1.2.13 Word Processing in Groups, Automata and Tilings
1.2.14 Computers
1.2.15 Surfaces, Mapping Class Groups and Teichmüller Spaces
1.2.16 Fashion Design
1.3 On Thurston's Impact
1.3.1 The Proof of the Smith Conjecture
1.3.2 The Proofs of Ahlfors' Conjecture, Marden's Tameness Conjecture, the Ending Lamination Conjecture, and the Density Conjecture
1.3.3 The Proof of the Geometrization Conjecture
1.3.4 The Waldhausen Conjectures and the Virtual Fibering Conjecture
1.3.5 The Ehrenpreis Conjecture
1.3.6 The Cannon–Thurston Maps
1.3.7 Anti-de Sitter Geometry and Transitional Geometry
1.3.8 Linkages
1.3.9 Higher Teichmüller Theory
1.3.10 The Grothendieck–Thurston Theory
1.4 In Guise of a Conclusion
References
2 Thurston's Influence on Japanese Topologists up to the 1980s
2.1 Introduction
2.2 Foliations
2.3 Hyperbolic Manifolds
2.4 Conclusion
References
3 A Survey of the Impact of Thurston's Work on Knot Theory
3.1 Introduction
3.2 Knot Theory Before Thurston
3.2.1 The Fundamental Problem in Knot Theory
3.2.2 Seifert Surface
3.2.3 The Unique Prime Decomposition of a Knot
3.2.4 Knot Complements and Knot Groups
3.2.5 Fibered Knots
3.2.6 Alexander Invariants
3.2.7 Representations of Knot Groups onto Finite Groups
3.3 The Geometric Decomposition of Knot Exteriors
3.3.1 Prime Decomposition of 3-Manifolds
3.3.2 Torus Decomposition of Irreducible 3-Manifolds
3.3.3 The Geometrization Conjecture of Thurston
3.3.4 Geometric Decompositions of Knot Exteriors
3.4 The Orbifold Theorem and the Bonahon–Siebenmann Decomposition of Links
3.4.1 The Bonahon–Siebenmann Decompositions for Simple Links
3.4.2 2-Bridge Links
3.4.3 Bonahon–Siebenmann Decompositions and π-Orbifolds
3.4.4 The Orbifold Theorem and the Smith Conjecture
3.4.5 Branched Cyclic Coverings of Knots
3.5 Hyperbolic Manifolds and the Rigidity Theorem
3.5.1 Hyperbolic Space
3.5.2 Basic Facts for Hyperbolic Manifolds
3.5.3 Rigidity Theorem for Complete Hyperbolic Manifolds of Finite Volume
3.6 Computation of Hyperbolic Structures and Canonical Decompositions of Cusped Hyperbolic Manifolds
3.6.1 The Canonical Decompositions of Cusped Hyperbolic Manifolds
3.6.2 Ideal Triangulations and Computations of the Hyperbolic Structures
3.6.3 Other Geometric Invariants for Hyperbolic Knots and Effective Geometrization
3.7 Flexibility of Incomplete Hyperbolic Structures and the Hyperbolic Dehn Filling Theorem
3.7.1 Hyperbolic Dehn Filling Theorem
3.7.2 Outline of a Proof and Generalized Dehn Filling Coefficients
3.7.3 Geometry of the Hyperbolic Manifolds Obtained by Dehn Fillings
3.7.4 Exceptional Surgeries
3.8 Volumes of Hyperbolic 3-Manifolds
3.8.1 Calculation of Hyperbolic Volumes
3.8.2 Jørgensen–Thurston Theory for the Volumes of Hyperbolic 3-Manifolds
3.8.3 Small Volume Hyperbolic Manifolds
3.8.4 Gromov Norm
3.8.5 The Volume Conjecture
3.9 Commensurability and Arithmetic Invariants of Hyperbolic Manifolds
3.9.1 Commensurability Classes and Invariant Trace Fields
3.9.2 Commensurators and Hidden Symmetries
3.9.3 Arithmetic Versus Non-arithmetic
3.9.4 Siegel's Problem and Arithmetic Manifolds
3.10 Flexibility of Complete Hyperbolic Manifolds: Deformation Theory of Hyperbolic Structures
3.10.1 Convex Cores and Conformal Boundaries of Hyperbolic Manifolds
3.10.2 Deformation Space
3.10.3 Nielsen–Thurston Classification of Surface Homeomorphisms and Geometrization of Surface Bundles
3.10.4 Cannon–Thurston Maps and Veering Triangulations
3.11 Representations of 3-Manifold Groups
3.11.1 Character Variety
3.11.2 Hyperbolic Dehn Filling Theorem and Character Variety
3.11.3 The Culler–Shalen Theory and the Cyclic Surgery Theorem
3.11.4 A-Polynomials
3.12 Knot Genus and Thurston Norm
3.12.1 Thurston Norm
3.12.2 Evaluation of Thurston Norms in Terms of Twisted Alexander Polynomials
3.12.3 Harmonic Norm and Thurston Norm
3.13 Finite-Index Subgroups of Knot Groups and 3-Manifold Groups
3.13.1 Universal Knots/Links and Universal Groups
3.13.2 Virtual Fibering Conjecture
3.13.3 Profinite Completions of Knot Groups and 3-Manifold Groups
3.13.4 Homology Growth
References
4 Thurston's Theory of 3-Manifolds
4.1 Prologue
4.2 Pre-Thurston Era
4.3 Thurston Era
4.4 Post-Thurston Era
4.4.1 Geometrization Conjecture
4.4.2 Virtual Fiber Conjecture
4.5 Epilogue
References
5 Combinatorics Encoding Geometry: The Legacy of Bill Thurston in the Story of One Theorem
5.1 Introduction
5.1.1 An Introductory Overview
5.1.2 Dedication and Appreciation
5.2 The Koebe–Andre'ev–Thurston Theorem, Part I
5.2.1 Koebe Uniformization and Circle Packing
5.2.2 Koebe–Andre'ev–Thurston, or KAT for Short
5.2.3 A Proof of the Koebe Circle Packing Theorem
5.2.4 Maximal Packings and the Boundary Value Problem
5.3 The Koebe–Andre'ev–Thurston Theorem, Part II
5.3.1 Circle Packings of Compact Surfaces
5.3.2 KAT for Compact Surfaces
5.3.3 A Branched KAT Theorem and Polynomial Branching
5.3.4 Cusps and Cone Type Singularities
5.4 Infinite Packings of Non-compact Surfaces
5.4.1 The Discrete Uniformization Theorem
5.4.2 Types of Type
5.4.3 Koebe Uniformization for Countably-Connected Domains
5.5 Some Theoretical Applications
5.5.1 Approximating the Riemann Mapping
5.5.2 Uniformizing Equilateral Surfaces
5.6 Inversive Distance Circle Packings
5.6.1 A Quick Introduction to Inversive Distance
5.6.2 Some Advances on the Rigidity Question
5.6.3 Circle Frameworks and Möbius Rigidity
5.7 Polyhedra—From Steiner (1832) to Rivin (1996), and Beyond
5.7.1 Caging Eggs—Thurston and Schramm
5.7.2 Compact and Convex Hyperbolic Polyhedra—Hodgson and Rivin
5.7.3 Convex Ideal Hyperbolic Polyhedra—Rivin
5.7.4 New Millennium Excavations
5.7.4.1 Hyperideal Polyhedra—Bao and Bonahon
5.7.4.2 Weakly Inscribed Polyhedra—Chen and Schlenker
5.7.5 Addendum: Cauchy's Toolbox
5.8 In Closing, an Open Invitation
References
6 On Thurston's Parameterization of CP1-Structures
6.1 Introduction
6.2 CP1-Structures on Surfaces
6.3 Grafting
6.4 The Construction of Thurston's Parameters
6.4.1 The Construction of CP1-Structures from Measured Laminations on Hyperbolic Surfaces
6.4.2 The Construction of Measured Laminations on Hyperbolic Surfaces from CP1-Structures
6.5 Goldman's Theorem on Projective Structures with Fuchsian Holonomy
6.6 The Path Lifting Property in the Domain of Discontinuity
References
7 A Short Proof of an Assertion of Thurston ConcerningConvex Hulls
7.1 Introduction
7.2 Convex Subsets Viewed Extrinsically
7.3 Convex Subsets Viewed Intrinsically
References
8 The Double Limit Theorem and Its Legacy
8.1 Introduction
8.2 Compactifications of Deformation Spaces
8.2.1 Definitions
8.2.1.1 Deformation Spaces
8.2.1.2 Ahlfors–Bers Coordinates
8.2.2 Thurston's Compactification of Teichmüller Space
8.2.3 Culler–Morgan–Shalen's Compactification
8.3 The Double Limit Theorem
8.3.1 Thurston's Arguments: Efficiency of Pleated Surfaces
8.3.2 Otal's Proof: Real Trees and δ-Realization of Train Tracks
8.4 Manifolds with Incompressible Boundary
8.4.1 Thurston's Proof and Generalizations: Degenerating Simplices and Broken Windows
8.4.2 Morgan and Shalen's Arguments: Trees and Codimension-1 Laminations
8.4.3 Mixing the Arguments
8.5 Manifolds with Compressible Boundary
8.6 Necessary Conditions
8.7 Some Applications
References
9 Geometry and Topology of Geometric Limits I
9.1 Introduction
9.2 Main Results
9.3 Preliminaries
9.3.1 The Curve Graph and Tight Geodesics
9.3.2 Hyperbolic 3-manifolds and Geometric Limits
9.3.3 Least-Area Surfaces
9.4 Brick Manifolds
9.4.1 Embeddings of Brick Manifolds with Infinite Bricks
9.4.2 Conditions on Labelled Brick Manifolds
9.4.3 Tight Tube Unions
9.4.4 Block Decompositions of Labelled Brick Manifolds
9.4.5 Model Metrics on Brick Manifolds
9.4.6 Meridian Coefficients
9.5 The Bi-Lipschitz Model Theorem for Brick Manifolds
9.5.1 Minsky's Arguments
9.5.2 Length Bound
9.5.3 Homotoping f to a Lipschitz Map Preserving the Thin Part
9.5.4 Preliminary Steps to Homotope f3 to a Bi-Lipschitz Map
9.5.5 Homotoping f3 to a Homeomorphism
9.5.6 Topological Ordering of Joints
9.5.7 Deformation to a Bi-Lipschitz Map
9.6 Proofs of Theorems
9.6.1 Geometric Limits of Geometrically Finite Bricks
9.6.2 Proofs of Theorem A and Corollary B
9.6.3 Proof of Theorem C
9.6.4 Proof of Theorem D
References
10 Laminar Groups and 3-Manifolds
10.1 Introduction
10.2 S1-Bundle over the Leaf Space
10.3 Leaf Pocket Theorem and the Special Sections
10.4 The Case of Quasi-Geodesic and Pseudo-Anosov Flows
10.5 Invariant Laminations for the Universal Circles and Laminar Groups
10.6 Basic Notions and Notation to Study the Group Action on the Circle
10.7 Lamination Systems on S1 and Laminar Groups
10.8 Not Virtually Abelian Laminar Groups
10.9 Existence of a Non-abelian Free Subgroup in the Tight Pairs
10.10 Loose Laminations
10.11 Future Directions
References
11 Length Functions on Currents and Applications to Dynamics and Counting
11.1 Introduction
11.2 Background
11.2.1 Curves on Surfaces
11.2.2 Teichmüller Space and the Mapping Class Group
11.2.3 Measured Laminations
11.2.4 Geodesic Currents
11.2.5 Nielsen–Thurston Classification
11.2.6 Length Functions and the Intersection Number
11.3 Length Functions on Space of Currents
11.3.1 Length of Currents Through Liouville Currents
11.3.2 Stable Length of Currents
11.3.3 Stable Length as a Generalization of Intersection Length
11.4 Applications to Counting Curves
11.4.1 Thurston Measure
11.4.2 Counting with Respect to Length Functions
11.4.3 Orbits of Currents
11.5 Dynamics of Pseudo-Anosov Homeomorphisms
References
12 Big Mapping Class Groups: An Overview
12.1 Introduction
12.2 Preliminaries
12.2.1 Surfaces and Their Classification
12.2.1.1 Some Important Examples
12.2.2 Arcs and Curves
12.2.3 Mapping Class Group
12.2.4 Several Natural Subgroups
12.2.4.1 Pure Mapping Class Group
12.2.4.2 Compactly Supported Mapping Class Group
12.2.4.3 Torelli Group
12.2.5 Modular Groups
12.3 Two Important Results
12.3.1 Alexander Method
12.3.2 Automorphisms of the Curve Graph
12.4 Topological Aspects
12.4.1 The Permutation Topology
12.4.2 Basic Properties
12.4.3 Topological Generation
12.4.3.1 Torelli Group
12.4.4 Coarse Boundedness
12.4.5 Automatic Continuity
12.5 Algebraic Aspects
12.5.1 Algebraic Rigidity
12.5.1.1 Injective and Surjective Homomorphisms
12.5.1.2 General Homomorphisms
12.5.1.3 Rigidity of Subgroups
12.5.2 Abelianization
12.5.3 Quantifying Rigidity
12.5.4 Homology Representation
12.5.5 Nielsen Realization
12.5.6 The Relation with Thompson Groups
12.5.6.1 Thompson's Groups
12.5.6.2 Asymptotic Mapping Class Groups
12.5.6.3 The Case of the Cantor Tree Surface
12.5.6.4 Other Compact Surfaces with a Cantor Set Removed
12.5.6.5 The Case of the Blooming Cantor Tree
12.5.6.6 A Dense Asymptotic Mapping Class Group
12.6 Geometric Aspects
12.6.1 Complexes for Infinite-Type Surfaces
12.6.2 Weak Proper Discontinuity and Acylindricity
References
13 Teichmüller Theory, Thurston Theory, Extremal Length Geometry and Complex Analysis
13.1 Introduction
13.1.1 Background
13.1.2 Aim of This Chapter
13.2 Teichmüller Theory
13.2.1 Teichmüller Space
13.2.2 Complex Structure
13.2.3 Toy Model: The Case of Tori
13.3 Thurston's Theory on Surface Topology
13.3.1 Measured Foliations
13.3.2 Measured Laminations
13.3.3 Thurston's Measure
13.3.4 Toy Model: The Case of Tori
13.4 Thurston's Theory on Kleinian Surface Groups
13.4.1 Kleinian Groups
13.4.2 Quasiconformal Deformations
13.4.3 Classification of Marked Kleinian Surface Groups
13.4.3.1 End Invariants
13.4.3.2 Ending Lamination Theorem
13.4.4 Bers Slice
13.4.5 Structure of the Bers Boundary
13.4.5.1 Complex of Curves and the Gromov Boundary
13.4.5.2 Boundary Groups Without APTs
13.4.6 The Case of Once-Punctured Tori
13.5 Extremal Length and Thurston Measures on PMF
13.5.1 Hubbard–Masur Differentials and Extremal Length
13.5.2 Thurston Measures on PMF
13.5.3 Toy Model: The Case of Tori
13.6 Thurston Theory with Extremal Length
13.6.1 Gardiner–Masur Compactfication
13.6.2 Thurston's Theory with Extremal Length
13.6.3 Toy Model: The Case of Tori
13.7 Complex Analysis on Teichmüller Space
13.7.1 Complex Analysis
13.7.2 The Complex Structure on Teichmüller Space Revisited
13.7.3 Complex Analysis with Extremal Length
13.7.4 Pluricomplex Green Function
13.7.5 Pluriharmonic Measures
13.7.6 Toy Model: The Case of Tori
13.8 Toward Complex Analysis with Thurston Theory
13.8.1 Pluriharmonic Measures
13.8.2 Trace Functions
13.8.3 Holomorphic Functions
References
14 Signatures of Monic Polynomials
14.1 Introduction
14.2 Bi-regular Polynomials
14.3 Counting Bi-regular and Sub Bi-regular Signatures
14.4 Proofs
14.5 Pictures of Meromorphic Functions
14.6 Face Operations, Remarks, Questions
14.7 Sage and Pari Scripts
References
15 Anti-de Sitter Geometry and Teichmüller Theory
Introduction
Scope and Organization
Other Research Directions
15.1 Part 1: Anti-de Sitter Space
15.1.1 Preliminaries on Lorentzian Geometry
15.1.1.1 Basic Definitions
15.1.1.2 Maximal Isometry Groups and Geodesic Completeness
15.1.1.3 A Classification Result
15.1.1.4 Models of Anti de Sitter (n+1)-Space
15.1.1.5 The Quadric Model
15.1.1.6 The ``Klein Model'' and Its Boundary
15.1.1.7 The ``Poincaré Model'' for the Universal Cover
15.1.1.8 Geodesics
15.1.1.9 Polarity in Anti-de Sitter Space
15.1.2 Anti de Sitter Space in Dimension (2+1)
15.1.2.1 The PSL(2,R)-Model
15.1.2.2 The Boundary of PSL(2,R)
15.1.2.3 Levi-Civita Connection
15.1.2.4 Lorentzian Cross-Product
15.1.2.5 Geodesics in PSL(2,R)
15.2 Part 2: The Seminal Work of Mess
15.2.1 Causality and Convexity Properties
15.2.1.1 Achronal and Acausal Sets
15.2.1.2 Invisible Domains
15.2.1.3 Achronal Meridians in AdS2,1
15.2.1.4 Domains of Dependence
15.2.1.5 Properly Achronal Sets in AdS2,1
15.2.1.6 Convexity Notions
15.2.2 Globally Hyperbolic Three-Manifolds
15.2.2.1 General Facts
15.2.2.2 Genus r=1: Examples
15.2.2.3 Genus r=1: Classification
15.2.2.4 Genus r≥2: Examples
15.2.2.5 Genus r≥2: Classification
15.2.3 Gauss Map of Spacelike Surfaces
15.2.3.1 Spacelike Surfaces and Immersion Data
15.2.3.2 Germs of Spacelike Immersions in AdS Manifolds
15.2.3.3 Gauss Map and Projections
15.2.3.4 Consequences and Comments
15.2.3.5 Future Unit Tangent Bundle Perspective
15.2.3.6 Non-smooth Surfaces
15.2.3.7 The Fundamental Example
15.3 Part 3: Further Results
15.3.1 More on MGH Cauchy Compact AdS Manifolds
15.3.1.1 Foliations
15.3.1.2 Minimal Lagrangian Maps and Landslides
15.3.1.3 Cotangent Bundle of Teichmüller Space
15.3.1.4 Volume
15.3.1.5 Realization of Metrics and Laminations
15.3.2 Non-closed Surfaces
15.3.2.1 Foliations with Asymptotic Boundary
15.3.2.2 Extensions and Universal Teichmüller Space
15.3.2.3 Related Results
15.3.2.4 Cone Singularities and Manifolds with Particles
15.3.2.5 Boundary Components and Multi-Black Holes
References
16 Quasi-Fuchsian Co-Minkowski Manifolds
16.1 Introduction
16.2 Co-Minkowski Geometry
16.2.1 Definition of Co-Minkowski Space
16.2.1.1 Space of Spacelike Hyperplanes
16.2.1.2 Isometries
16.2.1.3 Connection, Geodesics
16.2.1.4 Co-Minkowski Space
16.2.2 Cylindrical Model
16.2.2.1 Klein Ball Model of the Hyperbolic Space
16.2.2.2 Affine Representation of Co-Minkowski Space
16.2.2.3 Duality
16.2.2.4 Isometries in Cylindrical Coordinates
16.2.2.5 Connection in Cylindrical Coordinates
16.2.2.6 Volume Form
16.2.3 Extrinsic Geometry of Graphs
16.2.3.1 Second Fundamental Form and Mean Curvature
16.2.3.2 Mean Surfaces
16.2.3.3 Convex Hull
16.2.3.4 The Mean Curvature Measure
16.2.3.5 The Fundamental Example of a Wedge
16.3 Action of Cocompact Hyperbolic Isometry Groups
16.3.1 Translation Parts as Cocycles
16.3.2 Equivariant Maps
16.3.3 Volume of the Convex Core and Asymmetric Norm
16.3.3.1 Convex Core
16.3.3.2 Asymmetric Norm
16.3.3.3 Mean Curvature Measure
16.3.3.4 Simplicial Measured Geodesic Laminations
16.3.4 The Case of Dimension 2+1
16.3.4.1 Goldman Isomorphism
16.3.4.2 Mess Homeomorphism
16.3.4.3 Length of Measured Geodesic Laminations
16.3.4.4 Thurston Earthquake Norm
16.3.4.5 Thurston Length Norm
16.4 Anosov Representations
References
Index