**Proceedings of Symposia in Pure Mathematics**

Volume: 74;
2006;
371 pp;
Hardcover

MSC: Primary 58; 20; 30; 14; 57;

Print ISBN: 978-0-8218-3838-9

Product Code: PSPUM/74

List Price: $97.00

AMS Member Price: $77.60

MAA Member Price: $87.30

**Electronic ISBN: 978-0-8218-9380-7
Product Code: PSPUM/74.E**

List Price: $97.00

AMS Member Price: $77.60

MAA Member Price: $87.30

# Problems on Mapping Class Groups and Related Topics

Share this page *Edited by *
*Benson Farb*

The appearance of mapping class groups in mathematics is
ubiquitous. The book presents 23 papers containing problems about
mapping class groups, the moduli space of Riemann surfaces,
Teichmüller geometry, and related areas. Each paper focusses
completely on open problems and directions. The problems range in
scope from specific computations, to broad programs. The goal is to
have a rich source of problems which have been formulated explicitly
and accessibly.

The book is divided into four parts. Part I contains problems on
the combinatorial and (co)homological group-theoretic aspects of
mapping class groups, and the way in which these relate to problems in
geometry and topology. Part II concentrates on connections with
classification problems in 3-manifold theory, the theory of symplectic
4-manifolds, and algebraic geometry. A wide variety of problems, from
understanding billiard trajectories to the classification of Kleinian
groups, can be reduced to differential and synthetic geometry problems
about moduli space. Such problems and connections are discussed in
Part III. Mapping class groups are related, both concretely and
philosophically, to a number of other groups, such as braid groups,
lattices in semisimple Lie groups, and automorphism groups of free
groups. Part IV concentrates on problems surrounding these
relationships.

This book should be of interest to anyone studying geometry,
topology, algebraic geometry or infinite groups. It is meant to
provide inspiration for everyone from graduate students to senior
researchers.

#### Readership

Graduate students and research mathematicians interested in mapping class groups and applications.

# Table of Contents

## Problems on Mapping Class Groups and Related Topics

- Contents vii8 free
- Preface ix10 free
- I. Cohomological, Combinatorial and Algebraic Structure 112 free
- Four Questions about Mapping Class Groups 314
- Some Problems on Mapping Class Groups and Moduli Space 1122
- Finiteness and Torelli Spaces 5768
- Fifteen Problems about the Mapping Class Groups 7182
- Problems on Homomorphisms of Mapping Class Groups 8192
- The Mapping Class Group and Homotopy Theory 91102
- Probing Mapping Class Groups Using Arcs 97108
- Relations in the Mapping Class Group 115126

- II. Connections with 3-manifolds, Symplectic Geometry and Algebraic Geometry 121132
- III. Geometric and Dynamical Aspects 187198
- Mapping Class Group Dynamics on Surface Group Representations 189200
- Geometric Properties of the Mapping Class Group 215226
- Problems on Billiards, Flat Surfaces and Translation Surfaces 233244
- Problems in the Geometry of Surface Group Extensions 245256
- Surface Subgroups of Mapping Class Groups 257268
- Weil-Petersson Perspectives 269280

- IV. Braid Groups, Out(F[sub(n)]) and other Related Groups 283294
- Braid Groups and Iwahori-Heck Algebras 285296
- Automorphism Groups of Free Groups, Surface Groups and Free Abelian Groups 301312
- Problems: Braid Groups, Homotopy, Cohomology and Representations 317328
- Cohomological Structure of the Mapping Class Group and Beyond 329340
- From Braid Groups to Mapping Class Groups 355366