топология / Farb, Margalit, A primer on mapping class groups

A primer on mapping class groups Version 4.08

A primer on mapping class groups

Version 4.08

Benson Farb and Dan Margalit

PRINCETON UNIVERSITY PRESS

PRINCETON AND OXFORD

Contents

Preface

In the Winter quarter of 2001, the first author gave a graduate course on mapping class groups at the University of Chicago. His (then) graduate student, the second author, took detailed notes, and the project grew into this text.

Our goal here is to explain as many important theorems, examples, and techniques as possible, as quickly and directly as possible, while at the same time giving (nearly) full details and keeping the text (nearly) self-contained. This book contains some simplifications of known approaches and proofs, the exposition of some results that are not readily available, and some new material as well. We have tried to incorporate many of the “gr eatest hits” of the subject, as well as its small quirks.

These notes should be viewed as a first pass through the theory ; in particular there are important topics not covered here. Most notable among these is the theory of measured laminations and the Thurston compactification of Teichm ¨uller space; these are discussed briefly in Chapte r 15 but are not given a full treatment.

There are a number of other references that cover various of the topics we cover here (and more). We would especially like to mention the books by Abikoff [1], Birman [20], Casson–Bleiler [40], Fathi–Laud enbach–Po´enaru [54], and Hubbard, as well as the survey papers by Harer [72] and Ivanov [88]. The works of Bers on Teichm ¨uller's theorems and on the Nielsen– Thurston classification theorem are particularly influenti al on this book [11, 12].

The first author learned much of what he knows about these topi cs from his advisor Bill Thurston, his teacher Curt McMullen, and his collaborators Lee Mosher and Howard Masur. This book in particular owes a debt to notes he took from a course given by McMullen at Berkeley in 1991.

Benson Farb and Dan Margalit

Chicago and Boston, May 2009

Acknowledgments

We would like to thank Mohammed Abouzaid, Yael Algom-Kfir, Je ssica Banks, Mark Bell, Joan Birman, Jeff Brock, Xuanting Cai, Pallavi Dani, Florian Deloup, John Franks, Siddhartha Gadgil, Bill Goldman, Michael Handel, John Harer, Peter Horn, Nikolai Ivanov, Jamie Jorgensen, Krishna Kaipa, Keiko Kawamuro, Richard Kent, Sarah Kitchen, Mustafa Korkmaz, Feng Luo, Joseph Maher, Kathryn Mann, Howard Masur, Jon McCammond, Curt McMullen, Ben McReynolds, Guido Mislin, Lee Mosher, Ziggy Nitecki, Jeremy Pecharich, Simon Rose, Keefe San Agustin, Travis Schedler, Paul Seidel, Ignat Soroko, Steven Spallone, Harold Sultan, Macky Suzuki, Genevieve Walsh, Richard Webb, Alexander Wickens, Bert Wiest, Jenny Wilson, Rebecca Winarski, Stefan Witzel, Kevin Wortman, Alex Wright, and Bruno Zimmermann for comments, questions, and insights.

We would like to thank Robert Bell, Tara Brendle, Jeff Carlson, Meredith Casey, Tom Church, Spencer Dowdall, Moon Duchin, David Dumas, Jeff Frazier, Daniel Groves, Asaf Hadari, Thomas Koberda, Justin Malestein, Johanna Mangahas, Erika Meucci, Catherine Pfaff, Kasra Rafi , and Saul Schleimer for thorough comments on large portions of the book.

We are especially indebted to Mladen Bestvina, Ken Bromberg, Allen Hatcher, Chris Leininger, and Andy Putman for extensive discussions on the material

in this book.

The first author would like to thank the second author for ever ything he has done to bring this project to its current form. His hard work, thorough understanding of the material, and ability to simplify proofs and explain concepts clearly, has been invaluable. The first author would also lik e to thank the second author for all he has taught him about mapping class groups.

The second author would like to thank the first author for intr oducing him to and teaching him about the subject of mapping class groups, and for the opportunity to work on this project together. The first autho r's enthusiasm, vision, and mathematical wisdom are both a joy and an inspiration. The second author would also like to thank the University of Utah for its support during the writing of this book.