Hyperbolic Manifolds and Kleinian Groups

A Kleinian group is a discrete subgroup of the isometry group of hyperbolic 3-space, which is also regarded as a subgroup of Mobius transformations in the complex plane. The present book is a comprehensive guide to theories of Kleinian groups from the viewpoints of hyperbolic geometry andcomplex analysis. After 1960, Ahlfors and Bers were the leading researchers of Kleinian groups and helped it to become an active area of complex analysis as a branch of Teichmuller theory. Later, Thurston brought a revolution to this area with his profound investigation of hyperbolic manifolds,and at the same time complex dynamical approach was strongly developed by Sullivan. This book provides fundamental results and important theorems which are needed for access to the frontiers of the theory from a modern viewpoint.

0 Hyperbolic surfaces and Fuchsian groups: Summary

1

(14)

0.1 The hyperbolic plane and hyperbolic geometry

1

(3)

0.2 Hyperbolic surfaces

4

(1)

0.3 Aspects of finiteness of hyperbolic surfaces

5

(2)

0.4 Flexibility and geometrical boundedness

7

(2)

0.5 Action and deformation at infinity

9

(3)

0.6 The geometric limit

12

(3)

1 Hyperbolic 3-manifolds

15

(26)

1.1 The 3-dimensional hyperbolic space

15

(8)

1.1.1 The unit ball model and Mobius transformations

15

(4)

1.1.2 The upper half-space model and the projective model

19

(4)

1.2 Hyperbolic 3-manifolds and Kleinian models

23

(7)

1.2.1 Hyperbolic manifolds and developing maps

23

(2)

1.2.2 Kleinian groups

25

(5)

1.3 Examples of hyperbolic manifolds

30

(11)

1.3.1 Products and handlebodies

30

(2)

1.3.2 Pasting polyhedra

32

(5)

1.3.3 How to construct closed 3-manifolds

37

(4)

2 The basis of Kleinian group theory

41

(32)

2.1 Fundamental facts

41

(11)

2.1.1 Limit sets and non-elementary groups

41

(3)

2.1.2 Fundamental polyhedra

44

(2)

2.1.3 The action on the sphere at infinity

46

(3)

2.1.4 Fundamental sets on the sphere at infinity

49

(3)

2.2 The Margulis constant and cuspidal parts

52

(7)

2.2.1 The Jorgensen inequality

52

(4)

2.2.2 The thin parts of hyperbolic manifolds

56

(3)

2.3 The Selberg lemma

59

(3)

2.4 Construction of Kleinian groups

62

(11)

2.4.1 Combination and decomposition theorems

62

(5)

2.4.2 Examples of Kleinian groups

67

(6)

3 Geometrically finite Kleinian groups

73

(29)

3.1 Definitions of geometric finiteness

73

(7)

3.1.1 The definition using the convex core

73

(2)

3.1.2 The conical limit set

75

(3)

3.1.3 Equivalent conditions to geometric finiteness

78

(2)

3.2 Basic properties

80

(7)

3.2.1 Inheritance of geometric finiteness by subgroups

80

(4)

3.2.2 The Ahlfors measure zero theorem

84

(3)

3.3 Deformation and rigidity

87

(15)

3.3.1 Quasiconformal maps and quasi-isometries

87

(4)

3.3.2 Isomorphism theorems

91

(6)

3.3.3 The Mostow rigidity theorem

97

(5)

4 Finitely generated Kleinian groups

102

(29)

4.1 Finiteness theorems for Kleinian manifolds

102

(6)

4.1.1 The relative core theorem

102

(3)

4.1.2 Topological finiteness theorems

105

(3)

4.2 The Ahlfors finiteness theorem

108

(6)

4.2.1 Preparation from the theory of Teichmuller spaces

109

(3)

4.2.2 Proof of the Ahlfors finiteness theorem

112

(2)

4.3 Algebraic representation of qc deformation

114

(10)

4.3.1 PSL(2)(C)-representation space

114

(4)

4.3.2 Schottky space

118

(2)

4.3.3 Quasifuchsian space and Bers slice

120

(2)

4.3.4 Projective structures

122

(2)

4.4 B-groups

124

(7)

4.4.1 Kleinian manifolds of B-groups

124

(3)

4.4.2 Boundary groups

127

(4)

5 The sphere at infinity

131

(37)

5.1 Conservative Kleinian groups

131

(8)

5.1.1 Action on the limit set

131

(2)

5.1.2 Dirichlet sets and horospherical limit sets

133

(4)

5.1.3 Sufficient conditions for conservativity

137

(2)

5.2 The Sullivan rigidity theorem

139

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5.2.1 Divergent groups and ergodicity

139

(3)

5.2.2 Extension of the rigidity theorem

142

(2)

5.2.3 Proof of the Sullivan rigidity theorem

144

(8)

5.3 The Teichmuller space of the hyperbolic structures

152

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5.3.1 Quasiconformal deformation and Teichmuller spaces

152

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5.3.2 Quasi-isometric deformation space

155

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5.4 Stability

160

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5.4.1 Quasiconformal stability

160

(4)

5.4.2 Structural stability

164

(4)

6 Infinite ends of hyperbolic manifolds

168

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6.1 Measured laminations and pleated surfaces

168

(12)

6.1.1 Measured laminations

168

(5)

6.1.2 The Thurston compactification

173

(3)

6.1.3 Realizable laminations in pleated surfaces

176

(4)

6.2 A step to the uniformization theorem

180

(7)

6.2.1 The double limit theorem

180

(3)

6.2.2 Hyperbolic structure of a mapping torus

183

(4)

6.3 Geometrically tame Kleinian groups

187

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6.3.1 Tameness of infinite ends

187

(2)

6.3.2 The ending lamination

189

(5)

6.3.3 The Bonahon theorem

194

(5)

6.3.4 Thurston's theorem on the Ahlfors conjecture

199

(5)

7 Algebraic and geometric convergence

204

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7.1 Geometric convergence

204

(6)

7.1.1 Definitions of geometric convergence

204

(4)

7.1.2 Geometric convergence of representations

208

(2)

7.2 Difference between algebraic and geometric limits

210

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7.2.1 Dehn surgery

210

(5)

7.2.2 Iteration of modular transformations

215

(4)

7.2.3 The algebraic limit in the geometric limit

219

(2)

7.3 Strong convergence

221

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7.3.1 Caratheodory convergence

221

(3)

7.3.2 The covering theorem

224

(2)

7.3.3 Conditions for strong convergence

226

(4)

7.3.4 A consequence of strong convergence

230

(1)

7.4 Algebraic convergence and qc deformation space

230

(10)

7.4.1 Projective structures and representations

230

(4)

7.4.2 Global geometry of quasiconformal deformation spaces

234

(6)

Appendix

240

(4)

References

244

(7)

Index

251

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Hyperbolic Manifolds and Kleinian Groups

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Hyperbolic Manifolds and Kleinian Groups

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