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Calegari D. Foliations and the Geometry of 3-Manifolds (OUP OXFORD MATHEMATICAL MONOGRAPHS Series Editors J.M. BALL W.T. GOWERS N.J. HITCHIN L. NIRENBERG R. PENROSE A. WILES OXFORD MATHEMATICAL MONOGRAPHS For a full list of titles please visit http://www.oup.co.uk/academic/science/maths/series/omm/ Donaldson and Kronheimer: The geometry of four-manifolds, paperback Woodhouse: Geometric quantization, second edition, paperback Hirschfeld: Projective geometries over finite fields, second edition Evans and Kawahigashi: Quantum symmetries of operator algebras Klingen: Arithmetical similarities: Prime decomposition and finite group theory Matsuzaki and Taniguchi: Hyperbolic manifolds and Kleinian groups Macdonald: Symmetric functions and Hall polynomials, second edition, paperback Catto, Le Bris and Lions: Mathematical theory of theormodynamic limits: Thomas-Fermi type models McDuff and Salamon: Introduction to symplectic topology, paperback Holschneider: Wavelets: An analysis tool, paperback Goldman: Complex hyperbolic geometry Colbourn and Rosa: Triple systems Kozlov,Maz’ya and Movchan: Asymptotic analysis of fields in multi-structures Maugin: Nonlinear waves in elastic crystals Dassios and Kleinman: Low frequency scattering Ambrosio, Fusco and Pallara: Functions of bounded variation and free discontinuity problems Slavyanov and Lay: Special functions: A unified theory based on singularities Joyce: Compact manifolds with special holonomy Carbone and Semmes: A graphic apology for symmetry and implicitness Boos: Classical and modern methods in summability Higson and Roe: Analytic K-homology Semmes: Some novel types of fractal geometry Iwaniec and Martin: Geometric function theory and nonlinear analysis Johnson and Lapidus: The Feynman integral and Feynman’s operational calculus, paperback Lyons and Qian: System control and rough paths Ranicki: Algebraic and geometric surgery Ehrenpreis: The radon transform Lennox and Robinson: The theory of infinite soluble groups Ivanov: The Fourth Janko Group Huybrechts: Fourier-Mukai transforms in algebraic geometry Hida: Hilbert modular forms and Iwasawa theory Boffi and Buchsbaum: Threading homology through algebra Vazquez: The Porous Medium Equation Benzoni-Gavage and Serre: Multi-dimensional hyperbolic partial differential equations Calegari: Foliations and the geometry of 3-manifolds Foliations and the Geometry of 3-Manifolds Danny Calegari California Institute of Technology 1 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Danny Calegari, 2007 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 978–0–19–857008–0 13579108642 For Tereez This page intentionally left blank 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, especially [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 foliations 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 dynamical system generated by a single pseudo-Anosov element can be diago- nalized near fixed points in these co-ordinate systems. In Nielsen’s more primi- tive 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 trans- verse measures, 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. 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 vii viii PREFACE incomplete, and one should not be too eager to stuff it into a narrow frame- work. Under René Thom’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 repeated again and again throughout the book. One is the importance of geometry, espe- cially the hyperbolic geometry of surfaces. Another is the importance of mono- tonicity, especially in 1 dimensional and co-dimensional dynamics.Athird theme is combinatorial approximation, using finite combinatorical objects such as train- tracks, branched surfaces and hierarchies to carry more complicated continuous objects. Aims and scope A principal aim of this book is to expose the idea of universal circles for taut foliations 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, Poincaré) 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 relating pseudo-Anosov flows and (asymptotic) hyperbolic geometry is cursory, PREFACE ix 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 excitement 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 familiarity 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.
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