IRMA Lectures in Mathematics and Theoretical Physics 27 AH ta h n a nd ab so eo Handbook of Teichmüller Theory Pk a Volume VI po af Athanase Papadopoulos, Editor do Te pi oc uh lm o This volume is the sixth in a series dedicated to Teichmüller theory in a broad sü , l sense, including various moduli and deformation spaces, and the study of El de mapping class groups. It is divided into five parts: itr oT rh e Part A: The metric and the analytic theory. o Handbook of Part B: The group theory. ry Part C: Representation theory and generalized structures. Part D: The Grothendieck–Teichmüller theory. V Teichmüller Theory Part E: Sources. o l u m The topics surveyed include Grothendieck’s construction of the analytic e Volume VI structure of Teichmüller space, identities on the geodesic length spectrum of V I hyperbolic surfaces (including Mirzakhani’s application to the computation of Weil–Petersson volumes), moduli spaces of configurations spaces, the Teichmüller tower with the action of the Galois group on dessins d’enfants, Athanase Papadopoulos and several other actions related to surfaces. The last part consists of three papers by Teichmüller, translated into English with mathematical Editor commentaries, and a document that comprises H. Grötzsch’s comments on Teichmüller’s famous paper Extremale quasikonforme Abbildungen und quadratische Differentiale. The handbook is addressed to researchers and to graduate students. ISBN 978-3-03719-161-3 www.ems-ph.org Papadopoulos VI | IRMA 27 | FONT: Rotis Sans | Farben: Pantone 287, Pantone 116 | 170 x 240 mm | RB: 37 mm IRMA Lectures in Mathematics and Theoretical Physics 27 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. For a complete listing see our homepage at www.ems-ph.org. 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 Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) 22 Handbook of Hilbert Geometry, Athanase Papadopoulos and Marc Troyanov (Eds.) 23 Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, Lizhen Ji and Athanase Papadopoulos (Eds.) 24 Free Loop Spaces in Geometry and Topology, Janko Latschev and Alexandru Oancea (Eds.) 25 Takashi Shioya, Metric Measure Geometry. Gromov‘s Theory of Convergence and Concentration of Metrics and Measures 26 Handbook of Teichmüller Theory, Volume V, Athanase Papadopoulos (Ed.) Handbook of Teichmüller Theory Volume VI 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, 11G32, 14C05, 14H15, 14H30, 14H15, 14H60, 14H55, 14J60, 18A22, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 32G13, 32G15, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16 ISBN 978-3-03719-161-3 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2016 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 InmemoryofAlexanderGrothendieck(1928–2014) Foreword Teichmüllertheory,inabroadsense,includesthestudyofparameterspacesforgeo- metric structures on surfaces and of representations of fundamental groups of sur- facesintovariousLiegroups. Thistheoryalsoinvolvesthestudyofactionsofmap- pingclassgroupsandothergroupsonvariousspaces,includingTeichmüllerspaces, charactervarietiesofrepresentations,simplicialcomplexesbuiltfromsystemsofho- motopyclassesofsimpleclosedcurvesorarcs,spacesoflaminationsandofequiva- lence classes of foliations equipped with various kinds of structures, and there are manyothers. Techniquesfromseveralfields areused(complexanalysis,hyperbolic geometry, partial differential equations, affine differential geometry, geometric and combinatorialgrouptheory, algebraicgeometry, Kählergeometry, etc.) andthis of- tengivesseveralpointsofviewonthesameobjectstudied. ThepresentHandbookis an attempt to presentin a consistentand systematic way the various points of view, ideasandtechniquesandtherichinteractionbetweenthem. Amongthetwenty-threechaptersthatthisvolumecontains,sevenarededicatedto theideasthatAlexanderGrothendieckbroughtintoTeichmüllertheoryandthetheory ofmodulispacesofRiemannsurfaces. TothemultitudeoffieldswhichTeichmüller theory unites, Grothendieck added number theory and the actions of the absolute Galois group. The present volume is dedicated to his memory. We are especially respectfulofhiscourageousengagementagainstthemainstreamideas. Thisvolumeisdividedintofiveparts: (cid:2) PartA,Themetricandtheanalytictheory,6 (cid:2) PartB.Thegrouptheory,5 (cid:2) PartC.Representationtheoryandgeneralizedstructures,4 (cid:2) PartD.TheGrothendieck-Teichmüllertheory,2 (cid:2) PartE.Sources,3 Thenumberaftereachpartindicatesthatitisasequeltoapartcarryingthesame nameinapreviousvolumeoftheHandbook. IwouldliketothankVincentAlbergeforhishelpintheproof-reading. AthanasePapadopoulos StrasbourgandNewYork,November2015 Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii IntroductiontoTeichmüllertheory,oldandnew,VI byAthanasePapadopoulos: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Chapter1. AlexanderGrothendieck byValentinPoenaru : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 PartA.Themetricandtheanalytictheory Chapter2. OnGrothendieck’sconstructionofTeichmüllerspace byNorbertA’Campo,LizhenJi,andAthanasePapadopoulos : : : : : : : : : : : 35 Chapter3. Null-setcompactificationsofTeichmüllerspaces byVincentAlberge,HidekiMiyachi,andKen’ichiOhshika : : : : : : : : : : : : : 71 Chapter4. Mirzakhani’srecursionformulaonWeil–Peterssonvolume andapplications byYiHuang : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 95 PartB.Thegrouptheory Chapter5. Rigidityphenomenainthemappingclassgroup byJavierAramayonaandJuanSouto : : : : : : : : : : : : : : : : : : : : : : : : : : : 131 Chapter6. Harmonicvolumeanditsapplications byYuukiTadokoro : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 167 Chapter7. Torusbundlesand2-formsontheuniversalfamily ofRiemannsurfaces byRobindeJong : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 195 PartC.Representationtheoryandgeneralizedstructures Chapter8. Cubicdifferentialsinthedifferentialgeometryofsurfaces byJohnLoftinandIanMcIntosh : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 231