A primer on mapping class groups

489 Pages·2011·2.509 MB·English

by  Farb B., Margalit D.

A Primer on Mapping Class Groups
PrincetonMathematicalSeries
EDITORS: PHILLIP A. GRIFFITHS, JOHN N. MATHER, AND ELIAS M.
STEIN
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49. A Primer on Mapping Class Groups by Benson Farb and Dan Mar-
galit
A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright (cid:1)c 2012byPrincetonUniversityPress
Published byPrincetonUniversityPress,41WilliamStreet,Princeton,
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ISBN978-0-691-14794-9
LibraryofCongressCataloging-in-Publication Data
Farb,Benson.
Aprimeronmappingclassgroups/BensonFarb,DanMargalit.
p.cm.–(Princeton mathematicalseries)
Includes bibliographical references andindex.
ISBN978-0-691-14794-9 (hardback)
1. Mappings (Mathematics) 2. Class groups (Mathematics) I. Margalit,
Dan,1976–II.Title.III.Series
QA360.F372011
512.7’4–dc22
2011008491
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Contents
Preface xi
Acknowledgments xiii
Overview 1
PART1. MAPPINGCLASSGROUPS 15
1. Curves,Surfaces,andHyperbolicGeometry 17
1.1 SurfacesandHyperbolicGeometry 17
1.2 SimpleClosedCurves 22
1.3 TheChangeofCoordinatesPrinciple 36
1.4 ThreeFactsaboutHomeomorphisms 41
2. MappingClassGroupBasics 44
2.1 DefinitionandFirstExamples 44
2.2 ComputationsoftheSimplestMappingClassGroups 47
2.3 TheAlexanderMethod 58
3. DehnTwists 64
3.1 DefinitionandNontriviality 64
3.2 DehnTwistsandIntersectionNumbers 69
3.3 BasicFactsaboutDehnTwists 73
3.4 TheCenteroftheMappingClassGroup 75
3.5 RelationsbetweenTwoDehnTwists 77
3.6 Cutting,Capping,andIncluding 82
4. GeneratingtheMappingClassGroup 89
4.1 TheComplexofCurves 92
4.2 TheBirmanExactSequence 96
4.3 ProofofFiniteGeneration 104
4.4 ExplicitSetsofGenerators 107
5. PresentationsandLow-dimensionalHomology 116
5.1 TheLanternRelationandH1(Mod(S);Z) 116
5.2 PresentationsfortheMappingClassGroup 124
5.3 ProofofFinitePresentability 134
viii CONTENTS
5.4 Hopf’sFormulaandH2(Mod(S);Z) 140
5.5 TheEulerClass 146
5.6 SurfaceBundlesandtheMeyerSignatureCocycle 153
6. TheSymplecticRepresentationandtheTorelliGroup 162
6.1 AlgebraicIntersectionNumberasaSymplecticForm 162
6.2 TheEuclideanAlgorithmforSimpleClosedCurves 166
6.3 MappingClassesasSymplecticAutomorphisms 168
6.4 CongruenceSubgroups,Torsion-freeSubgroups,andResidualFiniteness 176
6.5 TheTorelliGroup 181
6.6 TheJohnsonHomomorphism 190
7. Torsion 200
7.1 Finite-orderMappingClassesversusFinite-orderHomeomorphisms 200
7.2 Orbifolds,the84(g−1)Theorem,andthe4g+2Theorem 203
7.3 RealizingFiniteGroupsasIsometryGroups 213
7.4 ConjugacyClassesofFiniteSubgroups 215
7.5 GeneratingtheMappingClassGroupwithTorsion 216
8. TheDehn–Nielsen–BaerTheorem 219
8.1 StatementoftheTheorem 219
8.2 TheQuasi-isometryProof 222
8.3 TwoOtherViewpoints 236
9. BraidGroups 239
9.1 TheBraidGroup:ThreePerspectives 239
9.2 BasicAlgebraicStructureoftheBraidGroup 246
9.3 ThePureBraidGroup 248
9.4 BraidGroupsandSymmetricMappingClassGroups 253
PART2. TEICHMU¨LLERSPACEANDMODULISPACE 261
10. Teichmu¨llerSpace 263
10.1 DefinitionofTeichmu¨llerSpace 263
10.2 Teichmu¨llerSpaceoftheTorus 265
10.3 TheAlgebraicTopology 269
10.4 TwoDimensionCounts 272
10.5 TheTeichmu¨llerSpaceofaPairofPants 275
10.6 Fenchel–NielsenCoordinates 278
10.7 The9g−9Theorem 286
11. Teichmu¨llerGeometry 294
11.1 QuasiconformalMapsandanExtremalProblem 294
11.2 MeasuredFoliations 300
11.3 HolomorphicQuadraticDifferentials 308
11.4 Teichmu¨llerMapsandTeichmu¨ller’sTheorems 320
11.5 Gro¨tzsch’sProblem 325
CONTENTS ix
11.6 ProofofTeichmu¨ller’sUniquenessTheorem 327
11.7 ProofofTeichmu¨ller’sExistenceTheorem 330
11.8 TheTeichmu¨llerMetric 337
12. ModuliSpace 342
12.1 ModuliSpaceastheQuotientofTeichmu¨llerSpace 342
12.2 ModuliSpaceoftheTorus 345
12.3 ProperDiscontinuity 349
12.4 Mumford’sCompactnessCriterion 353
12.5 TheTopologyatInfinityofModuliSpace 359
12.6 ModuliSpaceasaClassifyingSpace 362
PART3. THECLASSIFICATIONANDPSEUDO-ANOSOVTHEORY 365
13. TheNielsen–ThurstonClassification 367
13.1 TheClassificationfortheTorus 367
13.2 TheThreeTypesofMappingClasses 370
13.3 StatementoftheNielsen–ThurstonClassification 376
13.4 Thurston’sGeometricClassificationofMappingTori 379
13.5 TheCollarLemma 380
13.6 ProofoftheClassificationTheorem 382
14. Pseudo-AnosovTheory 390
14.1 FiveConstructions 391
14.2 Pseudo-AnosovStretchFactors 403
14.3 PropertiesoftheStableandUnstableFoliations 408
14.4 TheOrbitsofaPseudo-AnosovHomeomorphism 414
14.5 LengthsandIntersectionNumbersunderIteration 419
15. Thurston’sProof 424
15.1 AFundamentalExample 424
15.2 ASketchoftheGeneralTheory 434
15.3 MarkovPartitions 442
15.4 OtherPointsofView 445
Bibliography 447
Index 465