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 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. © 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ T E X files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Foreword This Handbook is growing in size, reflecting the fact that Teichmüller theory has multiple facets and is being developed in several directions. In this new volume, as in the preceding volumes, there are chapters that concern the fundamental theory and others that deal with more specialized developments. Some chapters treat in more detail subjects that were only briefly outlined in the preceding volumes, and others present general theories that were not treated there. The study of Teichmüller spaces cannot be dissociated from that of mapping class groups, and like in the previous volumes, a substantial part of the present volume deals with these groups. The volume is divided into the following four parts: • The metric and the analytic theory, 3. • The group theory, 3. • The algebraic topology of mapping class groups and moduli spaces. • Teichmüller theory and mathematical physics. The numbers that follow the titles in the first two parts indicate that there were parts in the preceding volumes that carry the same titles. This Handbook is also a place where several fields of mathematics interact. For the present volume, one can mention the following: partial differential equations, one and several complex variables, algebraic geometry, algebraic topology, combinatorial topology, 3-manifolds, theoretical physics, and there are several others. This conflu- ence of ideas towards a unique subject is a manifestation of the unity and harmony of mathematics In addition to the fact of providing surveys on Teichmüller theory, several chapters 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, Chapter 4 contains an introduction to arithmetic groups and their actions on symmetric 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 exposition of theFox calculus, having in mind applications to the theory of the Magnus 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 class groups. The same chapter contains a review of Penner’s theory of the universal 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 theory of Teichmüller space and in the representation theory of mapping class groups. Chapter 11 contains a section on the intersection theory of complex varieties, as well as an introduction to the theory of characteristic classes of vector bundles, with applications to the intersection theory of the moduli space of curves and of its stable curve compactification. Chapter 13 contains an exposition of L p -cohomology, of the intersection cohomology theory for projective algebraic varieties and of the Hodge decomposition theory for compact Kähler manifolds, with a stress on applications to Teichmüller and moduli spaces. Finally, let us mention that several chapters in this volume contain open problems directed towards future research; in particular Chapter 4 by Ji, Chapter 5 by McCarthy and myself, Chapter 7 by Korkmaz, Chapter 8 by Habiro and Massuyeau, Chapter 9 by Sakasai, Chapter 10 by Funar, Kapoudjian and Sergiescu, and Chapter 13 by Ji and Zucker. Up to now, sixty different authors (some of them with more than one contribution) have participated to this project, and there are other authors, working on volumes in preparation. I would like to thank them all for this fruitful cooperation which we all hope will serve generations of mathematicians. I would like to thank once more Manfred Karbe and Vladimir Turaev for their interest and their care, and Irene Zimmermann for the seriousness of her work. Strasbourg, April 2012 Athanase Papadopoulos Contents Foreword v Introduction to Teichmüller theory, old and new, III by Athanase Papadopoulos 1 Part A. The metric and the analytic theory, 3 Chapter 1. Quasiconformal and BMO-quasiconformal homeomorphisms by Jean-Pierre Otal 37 Chapter 2. Earthquakes on the hyperbolic plane by Jun Hu 71 Chapter 3. Kerckhoff’s lines of minima in Teichmüller space by Caroline Series 123 Part B. The group theory, 3 Chapter 4. A tale of two groups: arithmetic groups and mapping class groups by Lizhen Ji 157 Chapter 5. Simplicial actions of mapping class groups John D. McCarthy and Athanase Papadopoulos 297 Chapter 6. On the coarse geometry of the complex of domains by Valentina Disarlo 425 Chapter 7. Minimal generating sets for the mapping class group of a surface by Mustafa Korkmaz 441 Chapter 8. From mapping class groups to monoids of homology cobordisms: a survey Kazuo Habiro and Gwénaël Massuyeau 465 viii Contents Chapter 9. A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces by Takuya Sakasai 531 Chapter 10. Asymptotically rigid mapping class groups and Thompson’s groups Louis Funar, Christophe Kapoudjian and Vlad Sergiescu 595 Part C. The algebraic topology of mapping class groups and their intersection theory Chapter 11. An introduction to moduli spaces of curves and their intersection theory by Dimitri Zvonkine 667 Chapter 12. Homology of the open moduli space of curves by Ib Madsen 717 Chapter 13. On the L p -cohomology and the geometry of metrics on moduli spaces of curves by Lizhen Ji and Steven Zucker 747 Part D. Teichmüller theory and mathematical physics Chapter 14. The Weil–Petersson metric and the renormalized volume of hyperbolic 3-manifolds by Kirill Krasnov and Jean-Marc Schlenker 779 Chapter 15. Discrete Liouville equation and Teichmüller theory by Rinat M. Kashaev 821 Corrigenda 853 List of Contributors 855 Index 857 Introduction to Teichmüller theory, old and new, III Athanase Papadopoulos Contents 1 Part A. The metric and the analytic theory, 3 2 1.1 The Beltrami equation 2 1.2 Earthquakes in Teichmüller space 4 1.3 Lines of minima in Teichmüller space 9 2 Part B. The group theory, 3 11 2.1 Mapping class groups versus arithmetic groups 11 2.2 Simplicial actions of mapping class groups 15 2.3 Minimal generating sets for mapping class groups 17 2.4 Mapping class groups and 3-manifold topology 18 2.5 Thompson’s groups 23 3 Part C. The algebraic topology of mapping class groups and moduli spaces 27 3.1 The intersection theory of moduli space 27 3.2 The generalized Mumford conjecture 28 3.3 The L p -cohomology of moduli space 30 4 Part D. Teichmüller theory and mathematical physics 32 4.1 The Liouville equation and normalized volume 33 4.2 The discrete Liouville equation and the quantization theory of Teichmüller space 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 of some crowded area, and others are more focussed on interesting details. There are intersections between the chapters, and these intersections are necessary. They are also valuable, because they are written by different persons, having different ideas on what is essential, and (to return to the image of a geographical atlas) using their proper color pencil set. 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 is usual in surveys. I asked the authors to make their texts accessible to a large number of readers. Of course, there is no absolute measure of accessibility, and the response depends on the sound sense of the author and also on the background of the reader. But in principle 2 Athanase Papadopoulos all of the authors made an effort in this sense, and we all hope that the result is useful to the mathematics community. This introduction serves a double purpose. First of all, it presents the content of the present volume. At the same time, reading this introduction is a way of quickly review- ing some aspects of Teichmüller theory. In this sense, the introduction complements the introductions I wrote for Volumes I and II of this Handbook. 1 Part A. The metric and the analytic theory, 3 1.1 The Beltrami equation Chapter 1 by Jean-Pierre Otal concerns the theory of the Beltrami equation. This is the partial differential equation N @ D @; (1.1) where  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 @ D 1 2  @ @x  i @ @y à and N @ D 1 2  @ @x C i @ @y à : If  is a solution of the Beltrami equation (1.1), then  D N @=@ is called the complex dilatation of . Without entering into technicalities, let us say that the partial derivatives @ and N @ of  are allowed to be distributional derivatives and are required to be in L 2 loc .U /. The function  that determines the Beltrami equation is in L 1 .U /, and is called the Beltrami coefficient of the equation. The Beltrami equation and its solution constitute an important theoretical tool in 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. The classical general result about the solution of the Beltrami equation (1.1) says that for any Beltrami coefficient  satisfying kk 1 <1, there exists a quasiconformal homeomorphism  D f  W U ! V which satisfies a.e. this equation, and that f  is unique up to post-composition by a holomorphic map. There are several versions and proofs of this existence and uniqueness result. The first version is sometimes attributed 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, the result is attributed to Ahlfors and Bers, who published it in their paper Riemann’s [...]... components of the surface Introduction to Teichmüller theory, old and new, III 7 The space of quasi-symmetric maps of the circle considered as the boundary of the hyperbolic unit disk is an important tool in the theory of the universal Teichmüller space Using the correspondence between the set of quasiconformal homeomorphisms of the open unit disk and the set of quasi-symmetric homeomorphisms of the boundary... and the general theory of the Beltrami equation and its generalization that is reviewed in Chapter 1 Earthquake theory has many applications in Teichmüller theory Some of them appear in other chapters of this volume, e.g Chapter 3 by Series and Chapter 14 by Krasnov and Schlenker Before going into the details of Chapter 2, let us briefly review the evolution of earthquake theory The theory originates... lines and stretch lines, lines of minima are not associated to any metric on Teichmüller space Series made a study of lines of minima in the context of the deformation theory of Fuchsian groups She established a relation between lines of minima and bending measures for convex core boundaries of quasi-Fuchsian groups This work introduced the use of lines of minima in the study of hyperbolic 3-manifolds Series... reduction theory, Ji addresses the question of the existence of various kinds of fundamental domains (geometric, rough, measurable, etc.), and of studying finiteness and local finiteness properties of such domains in relation to questions of finite generation and of bounded generation, and other related questions on group actions Introduction to Teichmüller theory, old and new, III 15 2.2 Simplicial actions of. .. automorphism group of this subcomplex, called the 6 The one-skeleton of the complex of non-separating curves is different from the Schmutz graph of nonseparating curves Introduction to Teichmüller theory, old and new, III 17 truncated complex of domains With the exception, as usual, of a certain finite number of special surfaces, the simplicial automorphism group of the truncated complex of domains is the... most of the constructions in Chapter 8 by Habiro and Massuyeau also apply to closed surfaces Introduction to Teichmüller theory, old and new, III 19 from the mapping class group of S into the automorphism group of a free group This is an instance of the general fact that the theory of free groups is much more present in the study of mapping class groups of surfaces with boundary than in that of closed... Introduction to Teichmüller theory, old and new, III 11 line of minima, with an appropriate parametrization, is uniformly comparable (in the sense of large-scale quasi-isometry) to the Teichmüller distance between these points The proof of that result is based on previous work by Rafi It involves an analysis of which closed curves get shortened along a line of minima, and the comparison of these curves... embedding of the Teichmüller space of any surface whose universal cover is the hyperbolic disk into this universal Teichmüller space The universal Teichmüller space also appears as a basic object in the study of the Thompson groups, surveyed in Chapter 10 of this volume By lifting the earthquake deformations of hyperbolic surfaces to the universal covers, the earthquake deformation theory of any hyperbolic... that the quasiconformal distortion of a homeomorphism of the hyperbolic disk can be defined in terms of distortion of quadrilaterals in that disk Analogously, the quasi-symmetry of a homeomorphism h of the circle can be defined in terms of distortion of cross ratios of quadruples of points on that circle The parallel between these two definitions hints to another point of view on the relation between quasi-symmetry... the Bers universal Teichmüller space Here, Diff C S1 / denotes the group of orientationpreserving homeomorphisms of the circle and QS.S1 / its group of quasi-symmetric homeomorphisms, of which we talk later in this text 6 Athanase Papadopoulos of earthquake maps and distortions of homeomorphisms of the circle, as we shall see below Thurston’s 1986 proof of existence and uniqueness of left (respectively . Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook. 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.