Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions

Contents
List of Illustrations
Preface
1. Hyperbolic space and its isometries
	1.1 Möbius transformations
	1.2 Hyperbolic geometry
	1.3 The circle or sphere at infinity
	1.4 Gaussian curvature
	1.5 Further properties of Möbius transformations
	1.6 Exercises and explorations
		1-3. Liouville measure.
		1-4. Euclidean and hyperbolic circle centers.
		1-6. Areas and side lengths of triangles.
		1-9. Möbius transformaions
		1-12. Formulas for the ball model.
		1-13.
		1-15. Iwasawa decomposition.
		1-17. The horizon in ℍ².
		1-21. Tangent bundle of a closed surface.
		1-22. Ideal tetrahedra.
		1-23. Volume of tetrahedra.
		1-24. Thinness of tetrahedra.
		1-26. Minkowski 3-space.
		1-27. Minkowski space, dimension-n.
		1-28. The quaternion description.
		1-31. Anti-Möbius transformations.
		1-33. Hyperbolic curvature of arcs.
		1-34. Conjugation by involution: Wada’s Lemma 2003.
		1-37. Two-jets of locally injective analytic functions.
		1-38. The hyperbolic Gauss map.
		1-40. Finer properties of isometric circles.
2. Discrete groups
	2.1 Convergence of Möbius transformations
	2.2 Discreteness
	2.3 Elementary discrete groups
	2.4 Kleinian groups
	2.5 Quotient manifolds and orbifolds
	2.6 Two fundamental algebraic theorems
	2.7 Introduction to Riemann surfaces and their uniformization
	2.8 Fuchsian and Schottky groups
	2.9 Riemannian metrics and quasiconformal mappings
	2.10 Teichmüller spaces of Riemann surfaces
	2.11 The mapping class group MCG(R)
	2.12 Exercises and explorations
		2-2.
		2-5. The modular group.
		2-6. A crash course on tori.
		2-7. Two tori.
		2-10. Punctured tori and the Farey sequence.
		2-11. Volume and area computations.
		2-12. A gaussian integer
		2-14. Genus-two surfaces.
		2-15. No tangents at loxodromic fixed points.
		2-18. Geometric group theory.
		2-19. Residual finiteness.
		2-21. Groups which are LERF.
		2-22. More on Schottky groups.
		2-24. Rational billiards.
		2-26. Homology and simple loops.
		2-27. Belyĭ functions on Riemann surfaces.
		2-28. Fuchsian subgroups of finite index.
		2-30. Euclidean orbifolds.
		2-31. Variation of length under a Dehn twist.
3. Properties of hyperbolic manifolds
	3.1 The Ahlfors Finiteness Theorem
	3.2 Tubes and horoballs
	3.3 Universal properties in hyperbolic 3-manifolds and orbifolds
	3.4 The thick/thin decomposition of a manifold
	3.5 Fundamental polyhedra
	3.6 Geometric finiteness
	3.7 Three-manifold surgery
	3.8 Quasifuchsian groups
	3.9 Geodesic and measured geodesic laminations
		Summary
	3.10 The convex hull of the limit set
	3.11 The convex core
	3.12 The compact and relative compact core
	3.13 Rigidity of hyperbolic 3-manifolds
	3.14 Exercises and explorations
		3-5. Figure-8 knot.
		3-6. Volume of maximal solid cusp tori [Adams 1987].
		3-7. Subgroups of geometrically finite groups [Thurston 1986b].
		3-8. Klein–Maskit combination theory.
		3-9. Extended quasifuchsian groups.
		3-12. Diameters of ordinary set components.
		3-14. Commensurability.
		3-15. Finiteness theorems.
		3-17. Cylindrical manifolds.
		3-19. Quasiisometries.
		3-20. Hausdorff dimension.
		3-21. Ergodic theory I.
		3-22. Ergodic theory II.
		3-23. The Patterson-Sullivan measure on limit sets.
		3-24. Poincaré series.
		3-26. Homotopic isometries.
		3-29. Generic Dirichlet regions: the Jørgenson-Marden Conjecture.
		3-31. A tower of covers.
		3-33. Intersections.
		3-35. Earthquakes.
		3-36. The Nielsen kernel.
		3-37. Extension from Ω(G) to ????².
		3-38. Geometric intersection number estimates. [Fathi et al. 1979, pp. 58–59].
		3-40. Interval exchange transformations [Bonahon 2001; Masur 1982].
		3-42. Rigidity of points in Teich(R).
		3-44. Ford polygon generalization.
4. Algebraic and geometric convergence
	4.1 Algebraic convergence
	4.2 Geometric convergence
	4.3 Polyhedral convergence
	4.4 The geometric limit
	4.5 Sequences of limit sets and regions of discontinuity
	4.6 New parabolics
	4.7 Acylindrical manifolds
	4.8 Dehn filling and Dehn surgery
	4.9 The prototypical example
	4.10 Manifolds of finite volume
	4.11 Exercises and explorations
		4-6. Convergence of limit sets. [McMullen 1996, Prop. 2.4]
		4-8. Geometric limits of fuchsian groups.
		4-9. Isomorphisms determining homeomorphisms.
		4-11. ℝ-trees.
		4-12. ℝ-trees and the degeneration of manifolds.
		4-13. The isoparametric inequality for ℍ³.
		4-14. The Gromov norm.
		4-15. The space of geodesics.
		4-17. Circle packings II.
		4-18. Right–angled hyperbolic polyhedra.
		4-19. Ideal triangulations; spinning.
		4-21. Simple loops in M(G); primitive curves.
		4-24. Every surface has a decomposition by pants of medium size.
5. Deformation spaces and the ends of manifolds
5.1 The representation variety
5.2 Homotopy equivalence
5.3 The quasiconformal deformation space boundary
5.4 The three conjectures for geometrically infinite manifolds
5.5 Ends of hyperbolic manifolds
5.6 Tame manifolds
5.7 The Ending Lamination Theorem
5.8 The Double Limit Theorem
5.9 The Density Theorem
5.10 Bers slices
5.11 The quasifuchsian space boundary
5.12 Examples of geometric limits at the Bers boundary
5.13 Classification of the geometric limits
5.14 Cannon-Thurston mappings
5.15 Exercises and explorations
	5-2. The bottom of the spectrum of eigenvalues.
	5-3. The eigenvalue λ₁(M).
	5-4. Nielsen transformations.
	5-5. The pinching estimate [Bers 1970a; McMullen 1999].
	5-6. Pinching.
	5-7. Nondensity of maximal cusps: an example of Curt McMullen.
	5-9. More on collapsing mappings (Thurston; see Minsky [1994b]).
	5-11. Anosov mappings of a torus.
	5-12. Surface automorphisms.
	5-13. Twists and traces.
	5-14. The orbit of a point of Teich(R) under iterated Dehn twists:
	5-15. Piling up double cusps.
	5-16. Shuffling a rolodex.
	5-17. Constructing a rolodex.
	5-18. The innards of an interesting manifold.
	5-21. The pants complex/graph.
	5-22. Outer space.
	5-23. Coset graph [Lyndon and Schupp 1977].
	5-26. Diskbusting curves; Canary’s trick.
	5-27. Pinched negative curvature manifolds.
	5-28. Representation varieties of fuchsian groups.
	5-30. Holomorphic motions.
	5-32. Two-generator groups.
	5-33. Quadratic differentials and measured laminations.
	5-34. Train tracks.
	5-35. Extension of boundary deformations to M(G).
	5-36. Representations into SL(3,ℝ) [Long et al. 2011]
	5-38.
6. Hyperbolization
	6.1 Hyperbolic manifolds that fiber over a circle
	6.2 Hyperbolic gluing boundary components
	6.3 Hyperbolization of 3-manifolds
	6.4 The three big conjectures, now theorems, for closed manifolds
	6.5 Geometrization
	6.6 Hyperbolic knots and links
	6.7 Computation of hyperbolic manifolds
	6.8 The Orbifold Theorem
	6.9 Exercises and explorations
		6-3. Drilling out simple geodesics.
		6-4. A nongeometrically finite limit on ∂T(G).
		6-6. Knottedness.
		6-7. Constructing a cone manifold on an unknotted geodesic: Bromberg’s construction.
		6-8. Grafting.
		6-9. Complex projective structures.
		6-10. Local connectedness of T(G).
		6-12. Ito’s expansion of Theorem 6.9.11.
		6-13. Meromorphic functions and laminations.
		6-14. Hyperbolic manifolds with corners.
		6-15. Infinitely generated kleinian groups.
		6-16.
		6-17. Splitting distance.
		6-19. Twisted face parings
7. Line geometry
	7.1 Half-rotations
	7.2 The Lie product
	7.3 Square roots
	7.4 Complex distance
	7.5 Complex distance and line geometry
	7.6 Exercises and explorations
		7-4. [Jørgensen 2000]
		7-7. A group G is determined by its traces.
		7-8. The length spectrum of a closed surface.
		7-13. A parametrization of two-generator groups.
		7-15. The McShane identity and Mirzakhani’s asymptotic formula for geodesic lengths.
		7-18. Extension of once-punctured torus groups [Wada 2003].
		7-20. [Brooks and Matelski 1981]
		7-22. Symmetry lines.
		7-23.
8. Right hexagons and hyperbolic trigonometry
	8.1 Generic right hexagons
	8.2 The sine and cosine laws
	8.3 Degenerate right hexagons
	8.4 Formulas for triangles, quadrilaterals, and pentagons
	8.5 Exercises and explorations
		8-3. Right-angled triangles.
		8-6. Planar right hexagons.
		8-8. Collars.
		8-9. Polar, cylindrical, and horocyclic coordinates.
		8-10. Orthogonal projection strictly reduces distances.
		8-13. Cutting a Riemann surface into euclidean polygons.
		8-14. Ptolemy relation for ideal quadrilaterals in ℍ².
		8-16.
		8-17. [Minsky 1999]
		8-20. The Parker–Series bending formula [Parker and Series 1995].
		8-21. Real traces.
		8-23. The single bend formula [Parker and Series 1995].
Bibliography
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Index
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