In the Tradition of Thurston II: Geometry and Groups

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
		5.4.3 Complexity of Surfaces in a 4-Manifold
		5.4.4 Harmonic Norm
	5.5 Floer Homology
		5.5.1 Heegaard Floer Homology
		5.5.2 Knot Floer Homology
	5.6 Torsion Invariants
		5.6.1 Reidemeister Torsion
		5.6.2 Twisted Alexander Polynomials
		5.6.3 Higher-Order Alexander Polynomials
		5.6.4 L2-Alexander Torsion
	5.7 Triangulations
		5.7.1 Thurston Norm Via Normal Surfaces
		5.7.2 Z / 2 Z-Thurston Norm and Complexity of 3-Manifolds
	5.8 Profinite Rigidity
	5.9 Conjectures and Questions
		5.9.1 Realization Problem
		5.9.2 Complexity Functions for Circle Bundles
		5.9.3 Twisted Alexander Polynomials for Hyperbolic Knots
		5.9.4 Higher-Order Alexander Polynomials and the Knot Genus
		5.9.5 Lower Bounds on Complexity of 3-Manifolds
		5.9.6 Thurston Norm Balls of Finite Covers
	References
6 From Hyperbolic Dehn Filling to Surgeries inRepresentation Varieties
	6.1 Introduction
	6.2 Hyperbolic Dehn Surgery
		6.2.1 Dehn Surgery
		6.2.2 Hyperbolic Dehn Surgery
		6.2.3 Haken Manifolds and Thurston's Uniformization
	6.3 Deformations of Hyperbolic Structures by Bending
	6.4 Higher Teichmüller Theory
		6.4.1 The Teichmüller Space
		6.4.2 Higher Teichmüller Spaces
		6.4.3 -Positive Representations
	6.5 Non-abelian Hodge Theory
		6.5.1 Moduli Spaces of G-Higgs Bundles
		6.5.2 G-Hitchin Equations
		6.5.3 The Non-abelian Hodge Correspondence
	6.6 Surgeries in Representation Varieties-General Theory
		6.6.1 Topological Gluing Construction
		6.6.2 Gluing in Exceptional Components of the Moduli Space
			6.6.2.1 Parabolic GL( n,C )-Higgs Bundles
		6.6.3 Complex Connected Sum of Riemann Surfaces
		6.6.4 Gluing at the Level of Solutions to Hitchin's Equations
			6.6.4.1 The Local Model
			6.6.4.2 Approximate Solutions of the SL(2,R)-Hitchin Equations
		6.6.5 Approximate Solutions to the G-Hitchin Equations
		6.6.6 The Contraction Mapping Argument
		6.6.7 Correcting an Approximate Solution to an Exact Solution
		6.6.8 Topological Invariants
	6.7 Examples: Model Higgs Bundles in Exceptional Components of Orthogonal Groups
		6.7.1 SO( p,q )-Higgs Bundle Data
		6.7.2 Hitchin Equations for Orthogonal Groups
		6.7.3 Model Parabolic SL( 2,R )-Higgs Bundles
		6.7.4 Parabolic SO( p,p+1 )-Models
			6.7.4.1 Models via the Irreducible Representation SL( 2,R )-3mu→SO( p,p+1 )
			6.7.4.2 Models via the General Map
		6.7.5 Gauge-Theoretic Gluing of Parabolic SO( p,p+1 )-Higgs Bundles
		6.7.6 Model Representations in the Exceptional Components of R( SO( p,p+1 ) )
		6.7.7 Model Representations and Positivity
	References
7 Acute Geodesic Triangulations of Manifolds
	7.1 Introduction
	7.2 In Dimension Three and Higher
		7.2.1 Polytopes and Dehn–Sommerville Equations
		7.2.2 Spherical Complexes
		7.2.3 Dimension Four and Five
		7.2.4 R3, S3 and More
	7.3 Dimension Two: General Riemannian and Flat Cone Metrics
		7.3.1 Riemannian Surfaces
		7.3.2 Euclidean and Flat Cone Surfaces
		7.3.3 Parametrizing Equilateral Triangulations
		7.3.4 Aperiodic Tilings
	7.4 Round Spheres
		7.4.1 Acute Triangulations from Right-Angled Hyperbolic 3-Polytopes
		7.4.2 The Koebe–Andreev–Thurston Theorem and Its Generalizations
		7.4.3 CAT(κ) Spaces
	References
8 Signature Calculation of the Area Hermitian Form on Some Spaces of Polygons
	8.1 Introduction
	8.2 Basic Facts on Hermitian Forms
	8.3 Spaces of Polygons and Signature Calculation
		8.3.1 The Area Hermitian Form and the Formula for Its Signature
		8.3.2 The Case n=2
		8.3.3 A Special Family of Polygons
		8.3.4 Cutting-Gluing Operations
		8.3.5 Signature Calculation
	References
9 Equilateral Convex Triangulations of R P2 with Three Conical Points of Equal Defect
	9.1 Introduction
	9.2 Triangulations of R P2 with Three Marked Points with Defects 2π/3
	9.3 Moduli Space of Flat Metrics on S2 with Six Pair-Wise Centrally Symmetric Conical Points of Equal Defect
	9.4 A Parametrization of Equilateral Triangulations of S2 with Six Centrally-Symmetric Points with Defects 2π/3
	9.5 Examples and Computer Computations
	References
10 Combination Theorems in Groups, Geometry and Dynamics
	10.1 Introduction
	10.2 Klein–Maskit Combination for Kleinian Groups
	10.3 Simultaneous Uniformization and Quasi-Fuchsian Groups
		10.3.1 Topologies on Space of Representations
		10.3.2 Simultaneous Uniformization
		10.3.3 Geodesic Laminations
	10.4 Thurston's Combination Theorem for Haken Manifolds
		10.4.1 Non-fibered Haken 3-Manifolds
		10.4.2 The Double Limit Theorem
	10.5 Combination Theorems in Geometric Group Theory:Hyperbolic Groups
		10.5.1 Trees of Spaces
		10.5.2 Metric Bundles
			10.5.2.1 Ladders
			10.5.2.2 Idea Behind the Proof of Theorem 10.18
		10.5.3 Relatively Hyperbolic Combination Theorems
			10.5.3.1 Relatively Hyperbolic Combination Theorem Using Acylindricity
			10.5.3.2 Relatively Hyperbolic Combination Theorem Using Flaring
		10.5.4 Effective Quasiconvexity and Flaring
	10.6 Combination Theorems in Geometric Group Theory:Cubulations
	10.7 Holomorphic Dynamics and Polynomial Mating
		10.7.1 Historical Comments
		10.7.2 Mating of Polynomials
	10.8 Combining Rational Maps and Kleinian Groups
		10.8.1 Mating Anti-polynomials with Reflection Groups
			10.8.1.1 Schwarz Reflection Maps and Motivating Examples
			10.8.1.2 Necklace Reflection Groups
			10.8.1.3 Conformal Mating of Anti-polynomials and Necklace Groups
			10.8.1.4 Examples of the Mating Phenomenon
			10.8.1.5 The General Theorem
		10.8.2 Mating Polynomials with Kleinian Groups
			10.8.2.1 The Fuchsian Case
			10.8.2.2 The Case of Bers Boundary Groups
	References
11 On the Pullback Relation on Curves Induced by a Thurston Map
	11.1 Introduction
	11.2 Conventions and Notation
	11.3 Non-dynamical Properties of f
		11.3.1 Known General Results
		11.3.2 Mechanisms for Triviality of f
		11.3.3 Computation of f
		11.3.4 When Each Curve Has a Nontrivial Preimage
	11.4 Dynamical Properties
		11.4.1 General Properties
		11.4.2 Bounds on the Size of the Attractor
		11.4.3 Examples with Symmetries
		11.4.4 Maps with the Same Fundamental Invariants
		11.4.5 Expanding vs. Nonexpanding Maps
	References
12 The Pullback Map on Teichmüller Space Induced from a Thurston Map
	12.1 Thurston's Characterization Theorem
		12.1.1 Levy Cycles
		12.1.2 An Application to Matings
	12.2 Further Developments
		12.2.1 Canonical Obstructions
		12.2.2 The Extension to a Boundary
		12.2.3 The g-Map and the Hurwitz Space
		12.2.4 The Pullback Map Near a Fixed Point
		12.2.5 Other Pullback Invariants
		12.2.6 Examples
		12.2.7 Eigenvalues of the Pullback Map
	References
13 A Classification of Postcritically Finite Newton Maps
	13.1 Introduction
	13.2 Thurston Theory on Branched Covers
	13.3 Extending Maps on Finite Graphs
	13.4 Abstract Extended Newton Graph
	13.5 Equivalence of Abstract Extended Newton Graphs
		13.5.1 Making Ray Endpoints and Accesses Coincide
		13.5.2 Equivalence on Newton Ray Grand Orbits
		13.5.3 Equivalence on Abstract Extended Newton Graphs
	13.6 Newton Maps from Abstract Extended Newton Graphs
		13.6.1 Contradiction for the Case ·≠0
		13.6.2 Contradiction for the Case ·= 0
	13.7 Proof of the Classification Theorem
	References
14 The Development of the Theory of Automatic Groups
	14.1 Introduction
		14.1.1 Historical Background
		14.1.2 Mathematical Background and Notation
	14.2 Automatic Groups
		14.2.1 Definition of an Automatic Group
		14.2.2 Basic Properties of Automatic Groups
		14.2.3 Basic Examples and Non-examples
			14.2.3.1 Virtually Abelian Groups, Soluble Groups
			14.2.3.2 Hyperbolic Groups
			14.2.3.3 Fundamental Groups of Compact 3-Manifolds
	14.3 Computing with Automatic Groups
		14.3.1 Building Automatic Structures
		14.3.2 Calculation Using the Automatic Structure
	14.4 Group Actions and Negative Curvature
	14.5 Some Automatic and Biautomatic Families
		14.5.1 Braid Groups, Artin Groups and Mapping Class Groups
		14.5.2 Coxeter Groups
	14.6 Open Problems
	References
15 Geometry and Combinatorics via Right-Angled Artin Groups
	15.1 Introduction
		15.1.1 Scope of This Survey
		15.1.2 Notation and Terminology
		15.1.3 A Remark About Generators
	15.2 The Cohomology Ring of a Right-Angled Artin Group
		15.2.1 The Topology of the Salvetti Complex
		15.2.2 Vector Spaces with a Vector-Space Valued Pairing
		15.2.3 The Cohomology Ring of A() Determines
	15.3 Translating Between Group Theory and Combinatorics
		15.3.1 Elementary Properties
		15.3.2 k-Colorability
		15.3.3 Hamiltonicity
		15.3.4 Graph Expanders
		15.3.5 Graph Automorphisms
		15.3.6 Some Further Entries in the Combinatorics–Algebra Dictionary
		15.3.7 Usefulness Beyond Group Theory and Combinatorics
			15.3.7.1 Complexity of Problems in Combinatorial Group Theory
			15.3.7.2 Hamiltonicity Testing
			15.3.7.3 Linear Algebraic Detection of Graph Expanders
			15.3.7.4 Interactive Proof Systems
			15.3.7.5 Group-Based Cryptosystems
	15.4 The Extension Graph and Its Properties
		15.4.1 Basic Properties of the Extension Graph
		15.4.2 The Extension Graph and Subgroups
		15.4.3 A Characterization of Cographs via Right-Angled Artin Groups and the Geometry of the Extension Graph
		15.4.4 More on the Geometry of the Extension Graph
		15.4.5 The Extension Graph as a Quasi-Isometry and Commensurability Invariant
	15.5 Further Directions
	References
Index