Geometry, Topology, and Dynamics in Negative Curvature

378 Pages·2016·3.959 MB·English

by  C. S. Aravinda, F. T. Farrell, J.-F. Lafont (eds.)

LONDONMATHEMATICAL SOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
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344 AlgebraiccyclesandmotivesII, J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry, A.NEEMAN
346 Surveysincombinatorics2007, A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics, N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis, P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI, Z.CHATZIDAKIS,D.MACPHERSON,
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357 Algebraictheoryofdifferentialequations, M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory, M.R.BRIDSON,P.H.KROPHOLLER&
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366 Highlyoscillatoryproblems, B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
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368 GeometryofRiemannsurfaces, F.P.GARDINER,G.GONZA´LEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIE´
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373 Smoothness,regularityandcompleteintersection, J.MAJADAS&A.G.RODICIO
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380 Realandcomplexsingularities, M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
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382 Forcingwithrandomvariablesandproofcomplexity, J.KRAJ´ICˇEK
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384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII, R.CLUCKERS,
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385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems, B.MARCUS,K.PETERSEN&
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386 Independence-friendlylogic, A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI, C.M.CAMPBELLetal(eds)
388 GroupsStAndrews2009inBathII, C.M.CAMPBELLetal(eds)
389 Randomfieldsonthesphere, D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials, D.E.PELINOVSKY
391 Fusionsystemsinalgebraandtopology, M.ASCHBACHER,R.KESSAR&B.OLIVER
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401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory, B.DENG,J.DU&Q.FU
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405 Torsors,e´talehomotopyandapplicationstorationalpoints, A.N.SKOROBOGATOV(ed)
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410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups,
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411 Modulispaces, L.BRAMBILA-PAZ,O.GARC´IA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds)
412 Automorphismsandequivalencerelationsintopologicaldynamics, D.B.ELLIS&R.ELLIS
413 Optimaltransportation, Y.OLLIVIER,H.PAJOT&C.VILLANI(eds)
414 AutomorphicformsandGaloisrepresentationsI, F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
415 AutomorphicformsandGaloisrepresentationsII, F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
416 Reversibilityindynamicsandgrouptheory, A.G.O’FARRELL&I.SHORT
417 Recentadvancesinalgebraicgeometry, C.D.HACON,M.MUSTATAˇ&M.POPA(eds)
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419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations, J.C.MEYER&
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420 Arithmeticandgeometry, L.DIEULEFAITetal(eds)
421 O-minimalityandDiophantinegeometry, G.O.JONES&A.J.WILKIE(eds)
422 GroupsStAndrews2013, C.M.CAMPBELLetal(eds)
423 Inequalitiesforgrapheigenvalues, Z.STANIC´
424 Surveysincombinatorics2015, A.CZUMAJetal(eds)
425 Geometry,topologyanddynamicsinnegativecurvature, C.S.ARAVINDA,F.T.FARRELL&
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426 Lecturesonthetheoryofwaterwaves, T.BRIDGES,M.GROVES&D.NICHOLLS(eds)
427 RecentadvancesinHodgetheory, M.KERR&G.PEARLSTEIN(eds)
LondonMathematicalSocietyLectureNoteSeries:425
Geometry, Topology, and Dynamics
in Negative Curvature
Editedby
C. S. ARAVINDA
TIFRCentreforApplicableMathematics,Bangalore
F. T. FARRELL
TsinghuaUniversity,Beijing
J.-F. LAFONT
OhioStateUniversity,Columbus
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Contents
Listofcontributors pageix
Preface xi
1 Gapdistributionsandhomogeneousdynamics 1
JayadevS.Athreya
1 Introduction 1
2 Meta-TheoremsonGapDistributions 12
3 Fareysequences 17
4 Saddleconnections 19
5 AbstractHomogeneousDynamics 22
6 FurtherQuestions 27
References 30
2 Topologyofopennonpositivelycurvedmanifolds 32
IgorBelegradek
1 Flavorsofnegativecurvature 33
2 Manifoldsofdimensionstwoandthree 36
3 Towardsaroughclassificationofdiscreteisometrygroups 38
4 Groupsofnon-parabolicisometries 40
5 Groupswithrankoneelements:prelude 42
6 Acylindricallyhyperbolicgroupsandrankoneelements 42
7 Boundedcohomologyandrankoneelements 45
8 Monod-Shalom’sclassandrankoneelements 46
9 Groupsthatfixapointatinfinity:prelude 47
10 Groupswhosecentercontainsaparabolic 50
11 Anchoredgroupsandfixedpointsatinfinity 51
12 Homotopyobstructions(afterGromovandIzeki-Nayatani) 55
13 Homeomorphismobstructions:exploitingRfactors 56
v
vi Contents
14 Benefitsofalowercurvaturebound 58
15 Injectivityradiusgoingtozeroatinfinity 59
16 Negativelycurvedmanifoldswithuniformvolumebound 62
17 Non-asphericalendsofnonpositivelycurvedmanifolds 63
18 Riemannianhyperbolization(afterOntaneda) 66
19 Topologyofknownmanifoldsofboundednegative
curvature 68
20 Zoooffinitevolumerankonemanifolds 72
References 76
3 Cohomologieetactionsisome´triquespropressurles
espacesL 84
p
MarcBourdon
1 Introduction 84
2 Cohomologie(cid:2) 86
p
3 Actionsisome´triquespropressurlesespacesL 99
p
References 107
4 CompactClifford–KleinForms–Geometry,Topology
andDynamics 110
DavidConstantine
1 Introduction 110
2 Notation 116
3 Topology 116
4 Geometry 123
5 Dynamics 126
6 Deformationsandmodulispacesofcompactforms 131
7 Theroadahead 139
References 142
5 Asurveyonnoncompactharmonicandasymptotically
harmonicmanifolds 146
GerhardKnieper
1 Introduction 146
2 BasicsonJacobitensorsonmanifoldswithoutconjugate
points 149
3 Volumegrowthonasymptoticallyharmonicmanifolds 156
4 Densityfunctionsofharmonicmanifolds 160
5 Harmonicmanifoldswithsubexponentialvolumegrowth
areflat 163
6 Therankofanasymptoticallyharmonicmanifold 169
Contents vii
7 Characterizationofhyperbolicityforasymptotically
harmonicmanifolds 175
8 Harmonicmanifoldswithboundedasymptote 181
9 Harmonicandasymptoticallyharmonicmanifoldswithout
focalpoints 185
References 195
6 TheAtiyahconjecture 198
PeterA.Linnell
1 Introduction 198
2 NotationandTerminology 200
3 GroupvonNeumannalgebrasanddimension 201
4 FormulationoftheAtiyahconjecture 202
5 EarlyresultsontheAtiyahconjecture 206
6 ApproximationTechniques 206
7 Finiteextensions 208
8 Pro-pgroupsoffiniterank 211
9 Approximationoverarbitraryfields 216
References 218
7 Cannon-ThurstonMapsforSurfaceGroups:An
ExpositionofAmalgamationGeometryandSplitGeometry 221
MahanMj
1 Introduction 221
2 PreliminariesandAmalgamationGeometry 223
3 RelativeHyperbolicity 228
4 UniversalCoversofBuildingBlocksandElectric
Geometry 238
5 ConstructionofQuasiconvexSetsandQuasigeodesics 242
6 Cannon-ThurstonMapsforSurfacesWithoutPunctures 253
7 WeakeningtheHypothesisI:GraphQuasiconvexityand
GraphAmalgamationGeometry 255
8 WeakeningtheHypothesisII:SplitGeometry 259
References 270
8 CountingvisiblecirclesonthesphereandKleiniangroups 272
HeeOhandNimishShah
1 Introduction 272
2 Equidistributionofnormaltranslatesofahyperbolic
surface 277
3 ProofofTheorem3 282
viii Contents
4 CountingwithrespecttoEuclideancurvature 285
References 287
9 Countingarcsinnegativecurvature 289
JouniParkkonenandFre´de´ricPaulin
1 Introduction 289
2 Geometryanddynamicsinnegativecurvature 291
3 Commonperpendicularsofconvexsets 300
4 UsingEskin-McMullen’sequidistributiontheorem 309
5 Arithmeticapplications 312
6 Patterson,Bowen-Margulisandskinningmeasures 322
7 Finitevolumehyperbolicmanifolds 326
8 Themaincountingresultofcommonperpendiculars 332
9 Spectralgaps,exponentialdecayofcorrelationsanderror
terms 335
10 Gibbsmeasuresandcountingarcswithweights 337
References 340
10 Latticesinhyperbolicbuildings 345
AnneThomas
1 Buildingsandhyperbolicbuildings 346
2 Lattices 354
References 360
Contributors
JayadevS.Athreya
DepartmentofMathematics,UniversityofWashington,
PadelfordHall,Seattle,WA98915U.S.A.
E-mail:[email protected]
IgorBelegradek
SchoolofMathematics,GeorgiaInstituteofTechnology,
Atlanta,GA30332-0160U.S.A.
E-mail:[email protected]
MarcBourdon
LaboratoirePainleve´,
UMRCNRS8524,Universite´ deLille1,
59655Villeneuved’Ascq,FRANCE
E-mail:[email protected]
DavidConstantine
DepartmentofMathematicsandComputerScience,
WesleyanUniversity,Middletown,CT06459U.S.A.
E-mail:[email protected]
GerhardKnieper
FacultyofMathematics,RuhrUniversityBochum,
44780Bochum,GERMANY
Email:[email protected]
PeterA.Linnell
DepartmentofMathematics,VirginiaTech,
Blacksburg,VA24061-0123U.S.A.
E-mail:[email protected]
ix