Handbook of Teichmuller Theory, Volume III

874 Pages·2012·7.633 MB·English

by  Athanase Papadopoulos

IRMA Lectures in Mathematics and Theoretical Physics 17
Edited by Christian Kassel and Vladimir G. Turaev
Institut de Recherche Mathématique Avancée
CNRS et Université de Strasbourg
7 rue René Descartes
67084 Strasbourg Cedex
France
IRMA Lectures in Mathematics and Theoretical Physics
Edited by Christian Kassel and Vladimir G. Turaev
This series is devoted to the publication of research monographs, lecture notes, and other
material arising from programs of the Institut de Recherche Mathématique Avancée
(Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical
physics and to make them accessible to wide circles of mathematicians, physicists, and
students of these disciplines.
Previously published in this series:
1 Deformation Quantization, Gilles Halbout(Ed.)
2 Locally Compact Quantum Groups and Groupoids, Leonid Vainerman(Ed.)
3 From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi(Eds.)
4 Three courses on Partial Differential Equations, Eric Sonnendrücker(Ed.)
5 Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher(Ed.)
6 Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature
7 Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier,
Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.)
8 AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries,
Oliver Biquard(Ed.)
9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez,
C. Mitschi, C. Sabbah and R. Schäfke(Eds.)
10 Physics and Number Theory, Louise Nyssen(Ed.)
11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos(Ed.)
12 Quantum Groups, Benjamin Enriquez(Ed.)
13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos(Ed.)
14 Michel Weber, Dynamical Systems and Processes
15 Renormalization and Galois Theories, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis
(Eds.)
16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés(Ed.)
18 Strasbourg Master Class on Geometry, Athanase Papadopoulos(Ed.)
Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)
Handbook of
Teichmüller Theory
Volume III
Athanase Papadopoulos
Editor
Editor:
Athanase Papadopoulos
Institut de Recherche Mathématique Avancée
CNRS et Université de Strasbourg
7 Rue René Descartes
67084 Strasbourg Cedex
France
2010 Mathematics Subject Classification: Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60;
secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10,
30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 53A35, 53B35, 53C35, 53C50, 53C80,
53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16.
ISBN 978-3-03719-103-3
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9 8 7 6 5 4 3 2 1
Foreword
This Handbook is growing in size, reflecting the fact that Teichmüller theory has
multiplefacetsandisbeingdevelopedinseveraldirections.
Inthisnewvolume,asintheprecedingvolumes,therearechaptersthatconcernthe
fundamentaltheoryandothersthatdealwithmorespecializeddevelopments. Some
chapterstreatinmoredetailsubjectsthatwereonlybrieflyoutlinedinthepreceding
volumes, and others present general theories that were not treated there. The study
ofTeichmüllerspacescannotbedissociatedfromthatofmappingclassgroups, and
likeinthepreviousvolumes,asubstantialpartofthepresentvolumedealswiththese
groups.
Thevolumeisdividedintothefollowingfourparts:
• Themetricandtheanalytictheory,3.
• Thegrouptheory,3.
• Thealgebraictopologyofmappingclassgroupsandmodulispaces.
• Teichmüllertheoryandmathematicalphysics.
The numbers that follow the titles in the first two parts indicate that there were
partsintheprecedingvolumesthatcarrythesametitles.
This Handbook is also a place where several fields of mathematics interact. For
thepresentvolume,onecanmentionthefollowing: partialdifferentialequations,one
andseveralcomplexvariables,algebraicgeometry,algebraictopology,combinatorial
topology,3-manifolds,theoreticalphysics,andthereareseveralothers. Thisconflu-
enceofideastowardsauniquesubjectisamanifestationoftheunityandharmonyof
mathematics
InadditiontothefactofprovidingsurveysonTeichmüllertheory,severalchapters
in this volume contain expositions of theories and techniques that do not strictly
speaking belong to Teichmüller theory, but that have been used in an essential way
in the development of this theory. Such sections contribute in making this volume
and the whole set of volumes of the Handbook quite self-contained. The reader
who wants to learn the theory is thus spared some of the effort of searching into
several books and papers in order to find the material that he needs. For instance,
Chapter4containsanintroductiontoarithmeticgroupsandtheiractionsonsymmetric
spaces, with a view towards comparisons and analogies between this theory and the
theory of mapping class groups and their action on Teichmüller spaces. Chapter 5
contains an introduction to abstract simplicial complexes and their automorphisms.
Chapter 9 contains a concise survey of group homology and cohomology, and an
expositionoftheFoxcalculus,havinginmindapplicationstothetheoryoftheMagnus
representation of the mapping class group. Chapter 10 contains an exposition of
the theory of Thompson’s groups in relation with Teichmüller spaces and mapping
classgroups. ThesamechaptercontainsareviewofPenner’stheoryoftheuniversal
vi Foreword
decorated Teichmüller space and of cluster algebras. Chapter 10 and Chapter 14
contain an exposition of the dilogarithm, having in mind its use in the quantization
theoryofTeichmüllerspaceandintherepresentationtheoryofmappingclassgroups.
Chapter 11 contains a section on the intersection theory of complex varieties, as
wellasanintroductiontothetheoryofcharacteristicclassesofvectorbundles,with
applicationstotheintersectiontheoryofthemodulispaceofcurvesandofitsstable
curvecompactification. Chapter13containsanexpositionofLp-cohomology,ofthe
intersection cohomology theory for projective algebraic varieties and of the Hodge
decompositiontheoryforcompactKählermanifolds,withastressonapplicationsto
Teichmüllerandmodulispaces.
Finally,letusmentionthatseveralchaptersinthisvolumecontainopenproblems
directedtowardsfutureresearch;inparticularChapter4byJi,Chapter5byMcCarthy
andmyself,Chapter7byKorkmaz,Chapter8byHabiroandMassuyeau,Chapter9
bySakasai,Chapter10byFunar,KapoudjianandSergiescu,andChapter13byJiand
Zucker.
Uptonow,sixtydifferentauthors(someofthemwithmorethanonecontribution)
haveparticipatedtothisproject, andthereareotherauthors, workingonvolumesin
preparation. Iwouldliketothankthemallforthisfruitfulcooperationwhichweall
hopewillservegenerationsofmathematicians.
I would like to thank once more Manfred Karbe and Vladimir Turaev for their
interestandtheircare,andIreneZimmermannfortheseriousnessofherwork.
Strasbourg,April2012 AthanasePapadopoulos
Contents
Foreword...................................................................v
IntroductiontoTeichmüllertheory,oldandnew,III
byAthanasePapadopoulos...................................................1
PartA.Themetricandtheanalytictheory,3
Chapter1. QuasiconformalandBMO-quasiconformalhomeomorphisms
byJean-PierreOtal........................................................37
Chapter2. Earthquakesonthehyperbolicplane
byJunHu ................................................................ 71
Chapter3. Kerckhoff’slinesofminimainTeichmüllerspace
byCarolineSeries........................................................123
PartB.Thegrouptheory,3
Chapter4. Ataleoftwogroups: arithmeticgroupsandmappingclassgroups
byLizhenJi..............................................................157
Chapter5. Simplicialactionsofmappingclassgroups
JohnD.McCarthyandAthanasePapadopoulos..............................297
Chapter6. Onthecoarsegeometryofthecomplexofdomains
byValentinaDisarlo......................................................425
Chapter7. Minimalgeneratingsetsforthemappingclassgroupofasurface
byMustafaKorkmaz......................................................441
Chapter8. Frommappingclassgroupstomonoidsofhomologycobordisms:
asurvey
KazuoHabiroandGwénaëlMassuyeau.....................................465
viii Contents
Chapter9. AsurveyofMagnusrepresentationsformappingclassgroups
andhomologycobordismsofsurfaces
byTakuyaSakasai........................................................531
Chapter10. AsymptoticallyrigidmappingclassgroupsandThompson’s
groups
LouisFunar,ChristopheKapoudjianandVladSergiescu ..................... 595
PartC.Thealgebraictopologyofmappingclassgroupsandtheir
intersectiontheory
Chapter11. Anintroductiontomodulispacesofcurves
andtheirintersectiontheory
byDimitriZvonkine.......................................................667
Chapter12. Homologyoftheopenmodulispaceofcurves
byIbMadsen.............................................................717
Chapter13. OntheLp-cohomologyandthegeometryofmetrics
onmodulispacesofcurves
byLizhenJiandStevenZucker.............................................747
PartD.Teichmüllertheoryandmathematicalphysics
Chapter14. TheWeil–Peterssonmetricandtherenormalizedvolume
ofhyperbolic3-manifolds
byKirillKrasnovandJean-MarcSchlenker.................................779
Chapter15. DiscreteLiouvilleequationandTeichmüllertheory
byRinatM.Kashaev......................................................821
Corrigenda..............................................................853
ListofContributors ...................................................... 855
Index....................................................................857
Introduction to Teichmüller theory, old and new, III
Athanase Papadopoulos
Contents
1 PartA.Themetricandtheanalytictheory,3. . . . . . . . . . . . . . . . . 2
1.1 TheBeltramiequation . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 EarthquakesinTeichmüllerspace . . . . . . . . . . . . . . . . . . . 4
1.3 LinesofminimainTeichmüllerspace . . . . . . . . . . . . . . . . . 9
2 PartB.Thegrouptheory,3 . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Mappingclassgroupsversusarithmeticgroups . . . . . . . . . . . . 11
2.2 Simplicialactionsofmappingclassgroups . . . . . . . . . . . . . . 15
2.3 Minimalgeneratingsetsformappingclassgroups . . . . . . . . . . . 17
2.4 Mappingclassgroupsand3-manifoldtopology . . . . . . . . . . . . 18
2.5 Thompson’sgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 PartC.Thealgebraictopologyofmappingclassgroupsandmodulispaces 27
3.1 Theintersectiontheoryofmodulispace . . . . . . . . . . . . . . . . 27
3.2 ThegeneralizedMumfordconjecture. . . . . . . . . . . . . . . . . . 28
3.3 TheLp-cohomologyofmodulispace . . . . . . . . . . . . . . . . . 30
4 PartD.Teichmüllertheoryandmathematicalphysics . . . . . . . . . . . . 32
4.1 TheLiouvilleequationandnormalizedvolume . . . . . . . . . . . . 33
4.2 ThediscreteLiouvilleequationandthequantizationtheoryof
Teichmüllerspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Surveying a vast theory like Teichmüller theory is like surveying a land, and the
various chapters in this Handbook are like a collection of maps forming an atlas:
some of them give a very general overview of the field, others give a detailed view
ofsomecrowdedarea,andothersaremorefocussedoninterestingdetails. Thereare
intersections between the chapters, and these intersections are necessary. They are
alsovaluable,becausetheyarewrittenbydifferentpersons,havingdifferentideason
whatisessential,and(toreturntotheimageofageographicalatlas)usingtheirproper
colorpencilset.
The various chapters differ in length. Some of them contain proofs, when the
results presented are new, and other chapters contain only references to proofs, as it
isusualinsurveys.
Iaskedtheauthorstomaketheirtextsaccessibletoalargenumberofreaders. Of
course,thereisnoabsolutemeasureofaccessibility,andtheresponsedependsonthe
soundsenseoftheauthorandalsoonthebackgroundofthereader. Butinprinciple
2 AthanasePapadopoulos
alloftheauthorsmadeaneffortinthissense,andweallhopethattheresultisuseful
tothemathematicscommunity.
Thisintroductionservesadoublepurpose. Firstofall,itpresentsthecontentofthe
presentvolume.Atthesametime,readingthisintroductionisawayofquicklyreview-
ingsomeaspectsofTeichmüllertheory. Inthissense,theintroductioncomplements
theintroductionsIwroteforVolumesIandIIofthisHandbook.
1 PartA.The metric and the analytic theory, 3
1.1 TheBeltramiequation
Chapter 1 by Jean-Pierre Otal concerns the theory of the Beltrami equation. This is
thepartialdifferentialequation
@N(cid:2) D(cid:3)@(cid:2); (1.1)
where (cid:2)W U ! V is an orientation preserving homeomorphism between two do-
mains U of V of the complex plane and where @ and @N denote the complex partial
derivativations
(cid:2) (cid:3) (cid:2) (cid:3)
1 @(cid:2) @(cid:2) 1 @(cid:2) @(cid:2)
@(cid:2) D (cid:2)i and @N(cid:2) D Ci :
2 @x @y 2 @x @y
If (cid:2) is a solution of the Beltrami equation (1.1), then (cid:3) D @N(cid:2)=@(cid:2) is called the
complexdilatationof(cid:2).
Without entering into technicalities, let us say that the partial derivatives @(cid:2) and
@N(cid:2) of(cid:2) areallowedtobedistributionalderivativesandarerequiredtobeinL2 .U/.
loc
Thefunction(cid:3)thatdeterminestheBeltramiequationisinL1.U/,andiscalledthe
Beltramicoefficient oftheequation.
TheBeltramiequationanditssolutionconstituteanimportanttheoreticaltoolin
the analytical theory of Teichmüller spaces. For instance, the Teichmüller space of
a surface of negative Euler characteristic can be defined as some quotient space of
a space of Beltrami coefficients on the upper-half plane. As a matter of fact, this
definition is the one commonly used to endow Teichmüller space with its complex
structure.
TheclassicalgeneralresultaboutthesolutionoftheBeltramiequation(1.1)says
thatforanyBeltramicoefficient(cid:3)satisfyingk(cid:3)k <1,thereexistsaquasiconformal
1
homeomorphism(cid:2) Df(cid:2)W U !V whichsatisfiesa.e.thisequation,andthatf(cid:2) is
uniqueuptopost-compositionbyaholomorphicmap. Thereareseveralversionsand
proofsofthisexistenceanduniquenessresult. Thefirstversionissometimesattributed
to Morrey (1938), and there are versions due to Teichmüller (1943), to Lavrentieff
(1948) and to Bojarski (1955). In the final form that is used in Teichmüller theory,
theresultisattributedtoAhlforsandBers,whopublisheditintheirpaperRiemann’s