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Foliations and the Geometry of 3–Manifolds Foliations and the Geometry of 3–Manifolds Danny Calegari California Institute of Technology CLARENDON PRESS . OXFORD 2007 iv PREFACE The pseudo-Anosov theory of taut foliations The purpose of this book is to give an exposition of the so-called “pseudo- Anosov” theory of foliations of 3-manifolds. This theory generalizes Thurston’s theory of surface automorphisms, and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Some (but by no means all) of the content of this theory can be found already in the literature, espe- cially [236], [239], [82], [95], [173], [73], [75], [72], [31], [33], [35], [40] and [37], although I hope my presentation and perspective offers something new, even to the experts. This book is not meant to be an introduction to either the theory of folia- tions in general, nor to the geometry and topology of 3-manifolds. An excellent reference for the first is [42] and [43]. Some relevant references for the second are [127], [140], [230], and [216]. Spiral of ideas One conventional school of mathematical education holds that children should be exposed to the same material year after year, but that each time they return they should be exposed to it at a “higher level”, with more nuance, and with gradually more insight and perspective. The student progresses in an ascending spiral, rising gradually but understanding what is important. This book begins with the theory of surface bundles. The first chapter is both an introduction to, and a rehearsal for the theory developed in the rest of the book. In Thurston’s theory, this is a kind of branched linear algebra: train tracks and measured foliations reduce automorphisms of surfaces to Perron-Frobenius matrices and algebraic weights. The key to this approach is that the dynamics is carried by Abelian groups and groupoids: train tracks with one dimensional leaves carry transverse measures parameterized by manifold charts, and the dy- namical system generated by a single pseudo-Anosov element can be diagonal- ized near fixed points in these co-ordinate systems. In Nielsen’s more primitive version of this theory, cruder topological tools like the Hausdorff topology and order structures on transversals are important. When we return to these ideas at a “higher dimension”, we run up against laminations without transverse mea- sures, non-Hausdorff 1-manifolds, and “recurrent” branched surfaces which carry nothing. The linear algebra does not survive (except in the best cases), but the cruder topological tools prove more resilient. v vi Unifying themes This book is not written from a single, unified, coherent perspective. The theory of taut foliations, and their relation to geometric structures on 3-manifolds is incomplete, and one should not be too eager to stuff it into a narrow framework. Under Ren´eThom’s classification system, it deserves to be denoted by a baby’s crib denoting “live mathematics”, allowing change, clarification, completing of proofs (or development of better proofs), objection, refutation. Of course, I have at- tempted to make the arguments presented in this book as complete and self- contained as space allows; but sometimes subtle issues are better treated by giving examples (or counterexamples) than by general nonsense. And yet, there are a number of themes which are significant and are re- peated again and again throughout the book. One is the importance of geome- try, especially the hyperbolic geometry of surfaces. Another is the importance of monotonicity, especially in 1 dimensional and co-dimensional dynamics. A third theme is combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated con- tinuous objects. vii Aims and scope A principal aim of this book is to expose the idea of universal circles for taut foli- ations and other dynamical objects in 3-manifolds. Many sources feed into this idea, and I have tried to collect and present some of them and to explain how they work and fit together. Some of these sources (Dehn, Moore, Poincar´e) are very old; others are very new. Their continued vitality reflects the multiplicity of contexts in which they arise. This diversity is celebrated, and there are many loose threads in the book for the reader to tease out and play with. One of the most significant omissions is that I do not give an exposition of the many important developments in the theory of genuine laminations, mainly carried out by Dave Gabai and Will Kazez, especially in the trio of papers [92], [95] and [94]. Another serious omission is that the discussion of Fenley’s recent work re- lating pseudo-Anosov flows and (asymptotic) hyperbolic geometry is cursory, and does not attempt to explain much of the content of [79] or [78]. One sort of excuse is that Fenley’s program is currently in a period of substantial excite- ment and activity, and that one expects it to look very different even by the time this book appears in print. This book can be read straight through (like a “novel”) or the reader should be able to dip into individual chapters or sections. Only Chapter 7 and 8 are really cumulative and technical, requiring the reader to have a reasonable fa- miliarity with the contents of the rest of the book. The ideal reader is me when I entered graduate school: having a little bit of familiarity with Riemann surfaces and cut-and-paste topology in dimension 2 and 3, and a generalized fear of analysis, big technical machinery, and noncon- structive arguments. Low-dimensional topology in general has a very “hands- on” flavor. There are very few technical prerequisites: one can draw pictures which accurately represent mathematical objects, and one can do experiments and calculations which are guided by physical and spatial intuitions. The ideal reader must enjoy doing these things, must be prepared to be guided by and to sharpen these intuitions, and must want to understand why a theorem is true, beyond being able to verify that some argument proves it. I think ideal readers of this kind must exist; I hope this book finds some of them, and I hope they find it useful. Danny Calegari. Pasadena, September 2006 ACKNOWLEDGEMENTS In my second year of graduate school, the MSRI held a special year on low- dimensional topology. During that period, a number of foliators met on a semi- regular basis for the “very informal foliations seminar”. This seminar was not advertised, and one day I basically wandered off the street into the middle of a 3-hour lecture by Bill Thurston. During the next few months, I learned as much as I could from Bill, and from other attendees at that seminar, which included Mark Brittenham, Alberto Candel, Joe Christy, S´ergio Fenley, Dave Gabai, Rachel Roberts and Ying-Qing Wu. It is a pleasure to acknowledge the extent to which this seminar and these people shaped and formed my mathe- matical development and my interest in the theory of foliations and 3-manifolds, and directly contributed to the tone, point of view and content of this volume. More recently, thanks to my students — Thomas Mack, Bob Pelayo, Ru- pert Venzke and Dongping Zhuang — who sat through some early versions of this material. Thanks to Don Knuth for TEX, and thanks to the Persistence of Vision Team for writing povray, which was used to produce most of the 3- dimensional figures. Thanks to Dale Rolfsen, Amie Wilkinson, Nick Makarov, Dave Witte-Morris, Nathan Dunfield, Joel Kramer and Toby Colding for com- ments on drafts of specific chapters. Special thanks to Benson Farb for clear-headed and penetrating comments and criticisms, and for moral support. Special thanks to Bill Thurston for being so generous with time and ideas, and for creating so much of the substance of this book. Finally, super-special thanks to Tereez, Lisa and Anna, for all their love. viii CONTENTS 1 Surfacebundles 1 1.1 Surfaces and mapping class groups 1 1.2 Geometric structures on manifolds 7 1.3 Automorphisms of tori 9 1.4 PSL(2, Z) and Euclidean structures on tori 10 1.5 Geometric structures on mapping tori 11 1.6 Hyperbolic geometry 12 1.7 Geodesic laminations 17 1.8 Train tracks 32 1.9 Singular foliations 37 1.10 Quadratic holomorphic differentials 41 1.11 Pseudo-Anosov automorphisms of surfaces 44 1.12 Geometric structures on general mapping tori 45 1.13 Peano curves 46 1.14 Laminations and pinching 48 2 The topology of S1 50 2.1 Laminations of S1 50 2.2 Monotone maps 54 2.3 Pullback of monotone maps 57 2.4 Pushforward of laminations 58 2.5 Left-invariant orders 59 2.6 Circular orders 63 2.7 Homological characterization of circular groups 68 2.8 Bounded cohomology and Milnor–Wood 74 2.9 Commutators and uniformly perfect groups 80 2.10 Rotation number and Ghys’ theorem 84 2.11 Homological characterization of laminations 87 2.12 Laminar groups 88 2.13 Groups with simple dynamics 90 2.14 Convergence groups 93 2.15 Examples 96 2.16 Analytic quality of groups acting on I and S1 105 3 Minimalsurfaces 113 3.1 Connections, curvature 113 3.2 Meancurvature 116 3.3 Minimal surfaces in R3 118 3.4 The second fundamental form 121 ix x CONTENTS 3.5 Minimal surfaces and harmonic maps 123 3.6 Stableandleastareasurfaces 124 3.7 Existence theorems 130 3.8 Compactness theorems 132 3.9 Monotonicity and barrier surfaces 134 4 Taut foliations 137 4.1 Definition of foliations 137 4.2 Foliated bundles and holonomy 140 4.3 Basic constructions and examples 144 4.4 Volume-preserving flows and dead-ends 155 4.5 Calibrations 158 4.6 Novikov’s theorem 161 4.7 Palmeira’s theorem 167 4.8 Branching and distortion
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