Outer circles: an introduction to hyperbolic 3-manifolds

447 Pages·2007·3.673 MB·English

by  A. Marden

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OUTER CIRCLES
We live in a three-dimensional space; what sort of space is it? Can we build it from
simplegeometricobjects? Theanswerstosuchquestionshavebeenfoundinthelast
30years,andOuterCirclesdescribesthebasicmathematicsneededforthoseanswers
as well as making clear the grand design of the subjectof hyperbolic manifolds as a
whole.
The purpose of Outer Circles is to provide an account of the contemporary theory,
accessibletothosewithminimalformalbackgroundintopology,hyperbolicgeome-
try,andcomplexanalysis. Thetextexplainswhatisneeded,andprovidestheexper-
tise to use the primary tools to arrive at a thorough understanding of the big picture.
Thispictureisfurtherfilledoutbynumerousexercisesandexpositionsattheendsof
thechaptersandiscomplementedbyaprofusionofhighqualityillustrations. There
isanextensivebibliographyforfurtherstudy.
ALBERTMARDENisaProfessorofMathematicsintheSchoolofMathematicsatthe
UniversityonMinnesota.
ThediscretenesslocusintheextendedBerssliceofthehexagonalonce-punctured
torus (see Exercise 6-8). The Bers slice—the red central object—is surrounded by
other islands of discontinuity, in blue. The inward pointing cusps on the Bers slice
boundary represent geometrically finite groups and the same is presumably true for
the other components. The yellow dots are the fuchsian centers of the components.
Only a small number of islands are shown because of theoretical and computational
limitations.
ThecomputationandimageweremadebyDavidDumasofBrownUniversity;his
web sitecontainsmany beautifulrelatedimages.
OUTER CIRCLES
An Introduction to Hyperbolic 3-Manifolds
A. MARDEN
UniversityofMinnesota
CAMBRIDGEUNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521839747
© A. Marden, 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-28901-9 eBook (EBL)
ISBN-10 0-511-28901-4 eBook (EBL)
ISBN-13 978-0-521-83974-7 hardback
ISBN-10 0-521-83974-2 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
ToDorothy

Contents
List of Illustrations page xi
Preface xiii
1 Hyperbolicspaceanditsisometries 1
1.1 Möbiustransformations 1
1.2 Hyperbolic geometry 6
1.3 Thecircleorsphereatinfinity 11
1.4 Gaussiancurvature 15
1.5 FurtherpropertiesofMöbiustransformations 18
1.6 Exercisesandexplorations 23
2 Discretegroups 49
2.1 Convergence ofMöbiustransformations 49
2.2 Discreteness 51
2.3 Elementarydiscretegroups 55
2.4 Kleiniangroups 58
2.5 Quotient manifoldsand orbifolds 62
2.5.1 Twofundamentalalgebraictheorems 68
2.6 IntroductiontoRiemann surfacesandtheiruniformization 69
2.7 FuchsianandSchottkygroups 74
2.8 Riemannianmetricsandquasiconformalmappings 78
2.9 Exercisesandexplorations 83
3 Propertiesofhyperbolicmanifolds 105
3.1 TheAhlforsFinitenessTheorem 105
3.2 Tubesandhoroballs 106
3.3 Universalproperties 108
3.4 Thethick/thin decompositionofa manifold 115
3.5 Fundamentalpolyhedra 116
3.6 Geometric finiteness 124
3.7 Three-manifoldsurgery 129
3.8 Quasifuchsiangroups 134
3.9 Geodesicandmeasuredgeodesiclaminations 136
vii
viii Contents
3.10 Theconvexhullofthelimitset 144
3.11 Theconvexcore 151
3.12 Thecompactandrelativecompactcore 155
3.13 Rigidity 156
3.14 Exercisesandexplorations 161
4 Algebraicandgeometricconvergence 187
4.1 Algebraicconvergence 187
4.2 Geometricconvergence 193
4.3 Polyhedralconvergence 194
4.4 Thegeometriclimit 197
4.5 Convergenceoflimitsetsandregionsofdiscontinuity 200
4.6 Newparabolics 203
4.7 Acylindricalmanifolds 205
4.8 Dehn surgery 207
4.9 Theprototypicalexample 208
4.10 Manifoldsoffinitevolume 211
4.11 TheDehn surgerytheoremsforfinitevolumemanifolds 212
4.12 Exercisesandexplorations 218
5 Deformationspacesandtheendsofmanifolds 239
5.1 Therepresentationvariety 239
5.2 Homotopyequivalence 244
5.3 Thequasiconformaldeformationspace boundary 248
5.4 Thethreegreatconjectures 250
5.5 Endsofhyperbolicmanifolds 251
5.6 Tamemanifolds 252
5.7 Quasifuchsianspaces 261
5.8 Thequasifuchsianspace boundary 265
5.9 Geometriclimitsatboundarypoints 271
5.10 Exercisesandexplorations 282
6 Hyperbolization 312
6.1 Hyperbolicmanifoldsthatfiberoveracircle 312
6.1.1 Automorphismsofsurfaces 312
6.1.2 TheDoubleLimitTheorem 314
6.1.3 Manifoldsfiberedoverthecircle 315
6.2 TheSkinningLemma 317
6.2.1 Hyperbolicmanifoldswithtotallygeodesicboundary 317
6.2.2 Skinningthemanifold(PartII) 319
6.3 TheHyperbolizationTheorem 322
6.3.1 Knotsandlinks 325
6.4 Geometrization 327
6.5 TheOrbifoldTheorem 329
6.6 ExercisesandExplorations 331