A Primer on Mapping Class Groups

by Benson Farb, Dan Margalit

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.

Genres: Mathematics

488 pages, Hardcover

First published January 1, 2011

Reviews

[Steve Dalton · Rating 5 out of 5]

This is a great book. It is well-written, engaging, and covers not only the core material, but the tangential topics as well. In fact, every time I was reading something and thought "I'm gonna have to look up blank", where blank was group actions, or branched coverings, etc., the next section of the book would be about exactly that topic. It is so well-organized that every single section seems to lead naturally into the next, and the clarity of the arguments and proofs is first-rate.

I enjoyed the detailed introduction quite a bit, because it paints a convincing picture of where you're going and how all these things tie together. Much better than the usual "why'd I write this book, and here's who I thank". Part 1 was exceptional, providing a great foundation for the study of mapping class groups.

Parts 2 and 3 were not as great. I found some of the details on Teichmuller geometry to be a bit sketchy, and the section on train tracks is very uninspiring and poorly developed: there are much better intros to train tracks out there. That's not to say everything is a loss in these parts: the section on quadratic holomorphic differentials is really enlightening, especially for someone who has had no contact with them before (like me).

Despite some minor shortcomings, this is a must-read book for anyone working in this or neighboring fields. It has become the standard reference for most things MCG-related, and I now fully understand (and appreciate) why.

[Chris · Rating 5 out of 5]

Everyone who knows their shit is interested in this book, and for good reason.