Fundamentals of hyperbolic manifolds, selected expositions

348 Pages·2006·1.666 MB·English

by  Canary R., Marden A., Epstein D.

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LondonMathematicalSocietyLectureNoteSeries.328
Fundamentals of Hyperbolic
Geometry:
Selected Expositions
Editedby
RICHARD D. CANARY
UniversityofMichigan
DAVID EPSTEIN
UniversityofWarwick
ALBERT MARDEN
UniversityofMinnesota
publishedbythepresssyndicateoftheuniversityofcambridge
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©CambridgeUniversityPress2006
Thisbookisincopyright.Subjecttostatutoryexceptionand
totheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthe
writtenpermissionofCambridgeUniversityPress.
Firstpublished2006
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN0521615585paperback
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Everyefforthasbeenmadeinpreparingthisbooktoprovideaccurateandup-to-date
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standardsareconstantlychangingthroughresearchandregulation.Theauthors,editors
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attentiontoinformationprovidedbythemanufacturerofanydrugsorequipmentthatthey
plantouse.
Contents
Preface ix
Preface2005 xi
PARTI:NOTESONNOTESOFTHURSTON 1
R.D.Canary,D.B.A.Epstein,P.L.Green
ANewForeword 3
ChapterI.1.(G,X)-structures 31
I.1.1.(G,X)-structuresonamanifold 31
I.1.2.Developingmapandholonomy 32
I.1.3.Convexity 34
I.1.4.Thedevelopingmapandconvexity 37
I.1.5.Thedeformationspace 38
I.1.6.Thickenings 41
I.1.7.Varyingthestructure 44
ChapterI.2.Hyperbolicstructures 49
I.2.1.Möbiusgroups 49
I.2.2.Thethick–thindecomposition 50
I.2.3.Thenearestpointretraction 51
I.2.4.Neighbourhoodsofconvexhyperbolicmanifolds 52
I.2.5.Convexthickenings 56
ChapterI.3.Spacesofhyberbolicmanifolds 59
I.3.1.Thegeometrictopology 59
ε
I.3.2. -relationsandapproximateisometries 66
v
vi Contents
ChapterI.4.Laminations 76
I.4.1.Geodesiclaminations 76
I.4.2.Minimallaminations 80
ChapterI.5.Pleatedsurfaces 89
I.5.1.Introduction 89
I.5.2.Compactnesspropertiesofpleatedsurfaces 91
I.5.3.Realizations 105
PARTII:CONVEXHULLSINHYPERBOLICSPACE,
ATHEOREMOFSULLIVAN,ANDMEASURED
PLEATEDSURFACES 117
D.B.A.Epstein,A.Marden
ChapterII.1.Convexhulls 121
II.1.1.Introduction 121
II.1.2.Hyperbolicconvexhulls 121
II.1.3.Thenearestpointretraction 122
II.1.4.Propertiesofhyperbolicconvexhulls 125
II.1.5.Metriconconvexhullboundary 128
II.1.6.Hyperbolicconvexhulls 132
II.1.7.Limitsoflinesandplanes 134
II.1.8.Ridgelines 135
II.1.9.Theroof 136
II.1.10.Loweringtheroof 138
II.1.11.Thebendingmeasure 141
II.1.12.Theboundaryisacompletehyperbolicmanifold 144
II.1.13.Finiteapproximationstotheconvexhullboundary 148
II.1.14.Convergenceoflaminations 150
ChapterII.2.Foliationsandtheepsilondistantsurface 153
II.2.1.Introduction 153
II.2.2.Theepsilondistantsurface 153
II.2.3.Frominfinitytotheepsilonsurface 159
II.2.4.Extendingalaminationtoapairoforthogonalfoliations 166
II.2.5.Somestandardvectorfields 170
II.2.6.Lipschitzlinefieldsinthehyperbolicplane 172
II.2.7.Foliationcoordinates 176
II.2.8.Formulasinthehyperbolicplane 182
Contents vii
II.2.9.Flatequidistantsurfaces 186
II.2.10.Foliationsontheconvexhull 187
II.2.11.Threeorthogonalfields 190
II.2.12.Theequidistantsurfacefromafinitelybentconvex
hullboundary 193
II.2.13.Surfacesequidistantfromageneralconvexhullboundary 195
ρ
II.2.14.Themap 196
II.2.15.Numericalresults 208
II.2.16.Counterexample 208
ChapterII.3.Measuredpleatedsurfaces 211
II.3.1.Introduction 211
II.3.2.Finitequakebends 211
II.3.3.Norms 213
II.3.4.Productsofrotationsaboutgeodesics 214
II.3.5.Thequakebendcocycle 217
II.3.6.Thequakebendmap 219
II.3.7.Invariance 219
II.3.8.Deformations 221
II.3.9.Derivatives 222
II.3.10.Secondvariation 226
II.3.11.Varyingthelamination 228
Appendix 239
Addendum2005 255
PARTIII:EARTHQUAKESIN2-DIMENSIONAL
HYPERBOLICGEOMETRY 267
W.P.Thurston
ChapterIII.1.Earthquakesin2-dimensional
hyperbolicgeometry 269
III.1.1.Introduction 269
III.1.2.Whatarehyperbolicearthquakes? 271
III.1.3.Associatingearthquakestomapsofthecircle 276
III.1.4.Examples 281
III.1.5.Earthquakesonhyperbolicsurfaces 283
III.1.6.Themeasureandcauseofearthquakes 285
III.1.7.Quasi-symmetriesandquasi-isometries 287
viii Contents
PARTIV:LECTURESONMEASURESONLIMITSETS
OFKLEINIANGROUPS 291
S.J.Patterson
ChapterIV.1.Theproblemswithwhichweshall
beconcerned 293
ChapterIV.2.Ameasureonthelimitset 300
ChapterIV.3.Firstfruits 308
ChapterIV.4.Spectraltheory 314
ChapterIV.5.Geodesicflows 321
Preface
During the academic year 1983/84, the Science and Engineering Research
CounciloftheUnitedKingdomgavegenerousfinancialsupportfortwosym-
posia, at the Universities of Warwick and Durham, on hyperbolic geometry,
Kleiniangroupsand3-dimensionaltopology.ThesymposiumatDurhamwas
alsosponsoredbytheLondonMathematicalSociety.Iwouldliketoexpressmy
thankstoboththeSERCandtheLMSfortheirhelpandsupport.Itisapleasure
toacknowledgethehelpofmyco-organizeratDurham,PeterScott,whowas
alsoanunofficialco-organizeratWarwick.Hemadeanessentialcontribution
tothegreatsuccessofthesymposia.
Theworld’sforemostcontributorstothisveryactiveareawereallinvited,
andnearlyallofthemcame.TheactivitycentredontheUniversityofWarwick,
andclimaxedwitha2-weeklongintensivemeetingattheUniversityofDurham
during the first 2-weeks of July 1984. There was earlier a period of intense
activityduringtheEastervacationof1984,whenanumberofshortintroductory
lecturesweregiven.Thetextofthemostimportantoftheseseriesoflectures,
byS.J.Patterson,ispublishedintheseProceedings.
Thepaperspublishedherearetheresultofaninvitationtoallthoseattending
thetwoSymposiatosubmitpapers.Notallthepaperssubmittedwerethesubject
oftalksgivenduringtheSymposia–thecontentsoftheProceedingsarebasedon
theirrelevancetothesubject,andnotontheiraccuracyasdocumentsrecording
theeventsoftheSymposia.Also, anumberofimportantcontributionstothe
Symposiaarenotpublishedhere,havingbeenpreviouslypromisedelsewhere.
Oneofthefewadvantagesofbeinganeditoristhatonecanconfercertain
rights and privileges on oneself. I have taken the opportunity of accepting as
suitableforpublicationseveralratherlargepapersofwhichIwastheauthoror
co-author, andwhichhaveasubstantialelementofexposition. Thisisafield
whichisexpandingveryquickly,mainlyunderThurston’sinfluence,andmore
ix