A Primer on Mapping Class Groups

by Benson Farb, Dan Margalit

Overview

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.



A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Product Details

ISBN-13:	9780691147949
Publisher:	Princeton University Press
Publication date:	10/16/2011
Series:	Princeton Mathematical Series , #49
Pages:	488
Product dimensions:	6.30(w) x 9.30(h) x 1.90(d)

About the Author

Benson Farb is professor of mathematics at the University of Chicago. He is the editor of Problems on Mapping Class Groups and Related Topics and the coauthor of Noncommutative Algebra. Dan Margalit is assistant professor of mathematics at Georgia Institute of Technology.

Table of Contents

• Frontmatter, pg. i
• Contents, pg. vii
• Preface, pg. xi
• Acknowledgments, pg. xiii
• Overview, pg. 1
• Chapter One. Curves, Surfaces, and Hyperbolic Geometry, pg. 17
• Chapter Two. Mapping Class Group Basics, pg. 44
• Chapter Three. Dehn Twists, pg. 64
• Chapter Four. Generating The Mapping Class Group, pg. 89
• Chapter Five. Presentations And Low-Dimensional Homology, pg. 116
• Chapter Six. The Symplectic Representation and the Torelli Group, pg. 162
• Chapter Seven. Torsion, pg. 200
• Chapter Eight. The Dehn–Nielsen–Baer Theorem, pg. 219
• Chapter Nine. Braid Groups, pg. 239
• Chapter Ten. Teichmüller Space, pg. 263
• Chapter Eleven. Teichmüller Geometry, pg. 294
• Chapter Twelve. Moduli Space, pg. 342
• Chapter Thirteen. The Nielsen–Thurston Classification, pg. 367
• Chapter Fourteen. Pseudo-Anosov Theory, pg. 390
• Chapter Fifteen. Thurston’S Proof, pg. 424
• Bibliography, pg. 447
• Index, pg. 465

Editorial Reviews

"Winner of the Steele Prize for Mathematical Exposition, American Mathematical Society"

"It is clear that a lot of care has been taken in the production of this book, something that indicates the authors' love for the subject. This book should now become the standard text for the subject."—-Stephen P Humphries, Mathematical Reviews

"[T]his is a very pleasant and appealing book and it is an excellent reference for any reader willing to learn about this fascinating part of mathematics."—-Raquel Díaz, Álvaro Martínez, European Mathematical Society

From the Publisher

It is clear that a lot of care has been taken in the production of this book, something that indicates the authors' love for the subject. This book should now become the standard text for the subject.

Mathematical Reviews - Stephen P Humphries

It is clear that a lot of care has been taken in the production of this book, something that indicates the authors' love for the subject. This book should now become the standard text for the subject.

Mathematical Reviews - Stephen P. Humphries

It is clear that a lot of care has been taken in the production of this book, something that indicates the authors' love for the subject. This book should now become the standard text for the subject.
— Stephen P Humphries

Mathematical Reviews