Outer circles : an introduction to hyperbolic 3-manifolds / A. Marden.

Format:

Book

Author/Creator:

Marden, Albert.

Contributor:

Anne and Joseph Trachtman Memorial Book Fund.

Language:

English

Subjects (All):

Three-manifolds (Topology).

Physical Description:

xvii, 427 pages : illustrations (partly color) ; 26 cm

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2007.

Summary:

We live in a three-dimensional space, our universe; what are the possible kinds of geometry it can have? The most common are euclidean, spherical, and hyperbolic. The first two are familiar, but in the last 30 years it has been discovered that the vast majority of spaces have only hyperbolic geometry. Outer circles describes the basic mathematics needed for building and exploring hyperbolic spaces and makes clear the grand design of the subject and its central role in modern geometry and topology.

The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study.

Contents:

1 Hyperbolic space and its isometries 1

1.1 Mobius transformations 1

1.2 Hyperbolic geometry 6

1.3 The circle or sphere at infinity 11

1.4 Gaussian curvature 15

1.5 Further properties of Mobius transformations 18

1.6 Exercises and explorations 23

2 Discrete groups 49

2.1 Convergence of Mobius transformations 49

2.2 Discreteness 51

2.3 Elementary discrete groups 55

2.4 Kleinian groups 58

2.5 Quotient manifolds and orbifolds 62

2.5.1 Two fundamental algebraic theorems 68

2.7 Fuchsian and Schottky groups 74

2.8 Riemannian metrics and quasiconformal mappings 78

2.9 Exercises and explorations 83

3 Properties of hyperbolic manifolds 105

3.1 The Ahlfors Finiteness Theorem 105

3.2 Tubes and horoballs 106

3.3 Universal properties 108

3.4 The thick/thin decomposition of a manifold 115

3.5 Fundamental polyhedra 116

3.6 Geometric finiteness 124

3.7 Three-manifold surgery 129

3.8 Quasifuchsian groups 134

3.9 Geodesic and measured geodesic laminations 136

3.10 The convex hull of the limit set 144

3.11 The convex core 151

3.12 The compact and relative compact core 155

3.13 Rigidity 156

3.14 Exercises and explorations 161

4 Algebraic and geometric convergence 187

4.1 Algebraic convergence 187

4.2 Geometric convergence 193

4.3 Polyhedral convergence 194

4.4 The geometric limit 197

4.5 Convergence of limit sets and regions of discontinuity 200

4.6 New parabolics 203

4.7 Acylindrical manifolds 205

4.8 Dehn surgery 207

4.9 The prototypical example 208

4.10 Manifolds of finite volume 211

4.11 The Dehn surgery theorems for finite volume manifolds 212

4.12 Exercises and explorations 218

5 Deformation spaces and the ends of manifolds 239

5.1 The representation variety 239

5.2 Homotopy equivalence 244

5.3 The quasiconformal deformation space boundary 248

5.4 The three great conjectures 250

5.5 Ends of hyperbolic manifolds 251

5.6 Tame manifolds 252

5.7 Quasifuchsian spaces 261

5.8 The quasifuchsian space boundary 265

5.9 Geometric limits at boundary points 271

5.10 Exercises and explorations 282

6 Hyperbolization 312

6.1 Hyperbolic manifolds that fiber over a circle 312

6.1.1 Automorphisms of surfaces 312

6.1.2 The Double Limit Theorem 314

6.1.3 Manifolds fibered over the circle 315

6.2 The Skinning Lemma 317

6.2.1 Hyperbolic manifolds with totally geodesic boundary 317

6.2.2 Skinning the manifold (Part II) 319

6.3 The Hyperbolization Theorem 322

6.3.1 Knots and links 325

6.4 Geometrization 327

6.5 The Orbifold Theorem 329

6.6 Exercises and Explorations 331

7 Line geometry 348

7.1 Half-rotations 348

7.2 The Lie product 349

7.3 Square roots 353

7.4 Complex distance 353

7.5 Complex distance and line geometry 355

7.6 Exercises and explorations 356

8 Right hexagons and hyperbolic trigonometry 366

8.1 Generic right hexagons 366

8.2 The sine and cosine laws for generic right hexagons 368

8.3 Degenerate right hexagons 370

8.4 Formulas for triangles, quadrilaterals, and pentagons 372

8.5 Exercises and explorations 375.

Notes:

Includes bibliographical references (pages 393-410) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

9780521839747

0521839742

OCLC:

141379079

Online:

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