In the Tradition of Thurston III: Geometry and Dynamics (In the Tradition of Thurston, 3)

William Thurston’s ideas have altered the course of twentieth century mathematics, and they continue to have a significant influence on succeeding generations of mathematicians. The purpose of the present volume and of the other volumes in the same series is to provide a collection of articles that allows the reader to learn the important aspects of Thurston’s heritage. The topics covered in this volume include Kleinian groups, holomorphic motions, earthquakes from the Anti-de Sitter point of view, the Thurston and Weil–Petersson metrics on Teichmüller space, 3-manifolds, geometric structures, dynamics on surfaces, homeomorphism groups of 2-manifolds and the theory of orbifolds. Preface Contents 1 Introduction 2 The Geometry of the Thurston Metric: A Survey 2.1 Introduction 2.2 Background 2.2.1 Teichmüller Space 2.2.2 Geodesic Laminations and Measured Geodesic Laminations 2.2.3 The Thurston Metric and the Maximally Stretched Lamination 2.2.4 The Thurston Norm 2.3 Extremal Lipschitz Maps and Geodesic Rays 2.3.1 Deforming Polygons via Explicit Homeomorphisms 2.3.1.1 Thurston's Construction 2.3.1.2 The Construction of Papadapoulos-Théret 2.3.1.3 The Construction of Papadopoulos-Yamada 2.3.1.4 The Construction of Huang-Papadopoulos 2.3.2 Deforming Crowned Surfaces via Harmonic Diffeomorphisms 2.3.3 The Constructions of Guéritaud-Kassel and Alessandrini-Disarlo 2.3.4 The Construction of Daskalopoulos-Uhlenbeck 2.3.5 Concatenations of Geodesic Segments 2.4 Harmonic Stretch Lines 2.4.1 Harmonic Stretch Lines 2.4.2 Two Versions of the Geodesic Flow for the Thurston metric 2.5 Thurston Boundary 2.6 Isometry Rigidity 2.6.1 Global Rigidity 2.6.2 Infinitesimal Rigidity 2.7 Coarse Geometry 2.7.1 Short Markings 2.7.2 Curve Graphs 2.7.3 Subsurface Projection 2.7.4 Combinatorial Model 2.7.5 Short Curves 2.7.6 Length Spectrum Metric and the Product Theorem 2.8 Counting Lattice Points 2.9 Shearing Coordinates 2.10 Generalization of the Thurston Metric References 3 Thurston's Metric on the Teichmüller Space of Flat Tori 3.1 Introduction 3.1.1 Thurston's Original Work 3.1.2 Thurston's Metric on the Teichmüller Space of the Once-Punctured Torus 3.1.3 Generalizations of Thurston's Metric Theory 3.2 Teichmüller Space of Flat Tori 3.3 Affine Maps between Flat Tori 3.4 Geodesic Foliation on a Flat Torus 3.5 Lipschitz Constants of Homeomorphisms between Flat Tori 3.6 Curve Length Ratios for Two Flat Structures on S 3.7 Thurston's Metric on T(S) 3.8 Further Discussions on Extremal Lipschitz Constants and Singular Values of Affine Maps References 4 The Anti-de Sitter Proof of Thurston's Earthquake Theorem 4.1 Introduction 4.1.1 Mess' Groundbreaking Work and Later Developments 4.1.2 A Quick Comparison of the Two Proofs 4.1.3 Main Elements of the Anti-de Sitter Proof 4.2 Earthquake Maps 4.3 Anti-de Sitter Geometry 4.3.1 First Definitions 4.3.2 Boundary at Infinity 4.3.3 Spacelike Planes 4.3.4 Timelike Planes 4.3.5 Lightlike Planes 4.4 Convexity Notions 4.4.1 Affine Charts 4.4.2 Convex Hulls 4.4.3 Support Planes 4.4.4 Left and Right Projections 4.5 The Case of Two Spacelike Planes 4.5.1 The Fundamental Example 4.5.2 Simple Earthquake 4.5.3 The Example Is Prototypical 4.6 Proof of the Earthquake Theorem 4.6.1 Extension to the Boundary 4.6.2 Invertibility of the Projections 4.6.3 Earthquake Properties 4.6.4 Recovering Earthquakes of Closed Surfaces Appendix: Two Lemmas in the Hyperbolic Plane References 5 Homeomorphism Groups of Self-Similar 2-Manifolds 5.1 Introduction 5.1.1 The Main Object of Study: Self-Similar 2-Manifolds 5.1.2 Goals and Outline 5.2 Overview of Results 5.2.1 Normal Generation and Purity 5.2.2 Strong Distortion 5.2.3 Coarse Boundedness 5.2.4 Rokhlin Property 5.2.5 Automatic Continuity 5.2.6 Commutator Subgroups 5.3 Topology of Surfaces 5.4 Stable Sets 5.4.1 Definitions, Notations, and Conventions 5.4.2 Structure of Stable Sets 5.5 Freudenthal Subsurfaces and Anderson's Method 5.5.1 Definitions, Notations, Conventions 5.5.2 Anderson's Method 5.6 Topology of Homeomorphism Groups 5.6.1 The Compact-Open Topology 5.6.2 Homeomorphism Groups of 2-Manifolds 5.6.3 Defining the Mapping Class Group 5.7 Equivalent Notions of Self-Similarity 5.8 Normal Generation and Purity 5.9 Strong Distortion 5.10 Coarse Boundedness 5.11 Rokhlin Property 5.12 Automatic Continuity 5.13 Commutator Subgroups References 6 Weil–Petersson Teichmüller Theory of Surfaces of Infinite Conformal Type 6.1 Introduction 6.1.1 History and Motivation 6.1.2 Weil–Petersson Geometry of Surfaces of Infinite Conformal Type 6.1.3 Outline 6.2 Preliminaries 6.2.1 Surfaces, Borders, and Lifts 6.2.2 Differentials 6.2.3 Deformations: Quasiconformal Maps and Quasisymmetries 6.3 Three Models of Teichmüller Space 6.3.1 Definition of Teichmüller Space 6.3.2 Bers Embedding Model 6.3.3 Caps Fiber Model 6.3.4 Right Translation and Change of Base Point 6.4 Complex Manifold and Tangent Space Structure 6.4.1 Summary of the L∞ Theory 6.4.1.1 Bers Embedding 6.4.1.2 Deformation Model 6.4.1.3 Fiber Model 6.4.2 The Tangent Space in the Three Models 6.4.3 Weil–Petersson Teichmüller Space in a Nutshell 6.5 The Weil–Petersson Universal Teichmüller Space 6.5.1 History and Overview 6.5.2 Weil–Petersson Universal Teichmüller Space 6.5.2.1 Bers Embedding Model 6.5.2.2 Deformation Model 6.5.3 Some Further Characterizations of Weil–Petersson Quasisymmetries 6.6 The Period Mapping 6.6.1 Polarizations and the Siegel Disk 6.6.2 Interpretation of the Period Map 6.7 A Brief Overview of Other Refinements of TeichmüllerSpace 6.8 Weil–Petersson Teichmüller Spaces of General Surfaces 6.8.1 Overview 6.8.2 Manifold and Tangent Space Structure 6.8.2.1 Fiber Model 6.8.2.2 Bers Embedding Model 6.8.2.3 Deformation Model 6.9 Kähler Geometry and Global Analysis of the Weil–Petersson Teichmüller Space 6.9.1 Kählericity, Curvatures and Kähler Potentials 6.9.2 Chern Classes, Quillen Metric and Zeta Functions 6.10 Weil–Petersson Beyond Teichmüller Theory 6.10.1 Conformal Field Theory and String Theory 6.10.2 Fluid Mechanics 6.10.3 Loewner Energy References 7 Kleinian Groups and Geometric Function Theory 7.1 Introduction and Preliminaries 7.1.1 Hyperbolic Geometry 7.1.2 Quasiconformal Mappings 7.1.3 Kleinian Groups 7.2 Function Theory on the Components of Kleinian Groups 7.2.1 Cannon–Thurston Maps and the Geometric Function Theory 7.2.2 Hölder and John Domains 7.2.3 Distortion Estimates of Riemann Maps 7.2.4 Uniform Perfectness of the Limit Sets 7.3 Teichmüller Spaces and Univalent Functions 7.3.1 Schwarzian Derivative 7.3.2 The Teichmüller Space of a Fuchsian Group 7.3.3 Bers Embedding of Teichmüller Spaces 7.3.4 Function Theory on the Unit Disk 7.3.5 The Bers Conjecture for Fuchsian Groups 7.4 Holomorphic Motions, Quasiconformal Motions and Kleinian Groups References 8 Thurston's Broken Windows Only Theorem Revisited 8.1 Introduction 8.2 Preliminaries 8.2.1 JSJ Decompositon 8.2.2 Homotopy Equivalences and Books of I-Bundles 8.2.3 Pared Manifolds 8.2.4 Deformation Spaces 8.2.5 Morgan–Shalen Compactification 8.2.6 Skora's Theorem 8.3 Counter-Example 8.4 A Weaker Version of Thurston's Theorem 8.5 The Bounded Image Theorem References 9 Geometric Structures in Topology, Geometry, Global Analysis and Dynamics 9.1 Introduction 9.2 Domination, Monotonicity and Anosov Maps 9.2.1 The Domination Relation 9.2.2 Monotone Invariants 9.2.3 Anosov Diffeomorphisms 9.3 The Gromov Order for Thurston Geometries in Dimensions ≤4 9.3.1 Classification of Thurston's Geometries 9.3.2 Wang's Ordering 9.3.3 Ordering the 4-Dimensional Geometries 9.3.3.1 Manifolds Covered by Products 9.3.3.2 Finishing the Proof of Theorem 9.3.9 9.4 Geometric Kodaira Dimension, Monotonicity, and Simplicial Volume 9.4.1 Kodaira Dimension 9.4.1.1 Axiomatic Definition of κg 9.4.1.2 Classification in Dimensions ≤5 9.4.2 Monotonicity of the Kodaira Dimension 9.4.3 Kodaira Dimension Beyond Geometries and the Simplicial Volume 9.5 Anosov Diffeomorphisms 9.5.1 The Main Result 9.5.2 Proof of Theorem 9.5.1 9.5.2.1 Hyperbolic Geometries 9.5.2.2 Product Geometries 9.5.2.3 Non-product, Solvable or Compact Geometries References 10 Counting Problems for Invariant Point Processes 10.1 Introduction 10.2 SL(2,R)-Invariant Point Processes on C 10.3 Examples 10.3.1 Poisson Point Processes 10.3.2 Lattice Points 10.3.3 Saddle Connection Holonomies 10.4 Counting Asymptotics 10.4.1 Circles and Sectors 10.4.2 Hyperbolas 10.4.3 Triangles 10.5 Counting for Poisson Processes 10.5.1 Further Limit Theorems 10.6 Ergodic Theory of SL(2,R) 10.6.1 Moore Ergodicity 10.6.2 SL(2,R)-Ergodic Theorems 10.7 Circles and Sectors 10.7.1 Circle Averages and Counting 10.7.2 Applying Nevo's Ergodic Theorem 10.7.3 Technical Details 10.7.4 Counting Pairs 10.7.5 Central Limit Theorems 10.8 Hyperbolas and Geodesic Flow 10.8.1 Proving Theorem 10.4.2 10.8.2 Counting in Hyperbolas and Diophantine Approximation 10.8.3 Probabilistic Diophantine Approximation 10.8.4 Central Limit Theorems 10.9 Triangles and Horocycles 10.9.1 Slopes and Horocycles 10.9.2 BCZ-Type Maps 10.10 Further Questions/Directions 10.10.1 Shrinking Sectors 10.10.2 Siegel Measures on Rn 10.10.3 General G-Invariant Point Processes 10.10.4 Classification Questions 10.10.5 Representation Theory References 11 Orbifolds and the Modular Curve 11.1 Introduction 11.2 Complex Elliptic Curves 11.2.1 Complex Tori 11.2.2 Framed Elliptic Curves 11.2.3 The Modular Curve 11.3 Stacks 11.3.1 Prestacks 11.3.2 Stacks 11.3.3 Analytic Stacks 11.4 Orbifolds 11.5 Moduli Spaces 11.5.1 Coarse Moduli Spaces and Fine Moduli Spaces 11.5.2 GIT Quotients 11.6 The Moduli Stack of Elliptic Curves 11.6.1 Families of Elliptic Curves 11.6.2 Families of Framed Elliptic Curves 11.6.3 Orbifold Structure and Coarse Moduli Space References 12 Some Footnotes on Thurston's Notes The Geometry and Topology of 3-Manifolds 12.1 Introduction 12.2 Non-Euclidean Geometry 12.3 Excursus: Dante 12.4 Geometric Structures: From Ehresmann and Haefliger to Thurston 12.5 Spheres and Horospheres in Hyperbolic Space: Lobachevsky's Insight 12.6 Polyhedra: Andreev's Theorem 12.7 Volumes of Polyhedra: Lobachevsky and Milnor References Index