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, Teichmuller geometry, and related areas. Each paper focuses 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 semi simple 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.
I. Cohomological, combinatorial and algebraic structure: Four questions about mapping class groups by M. Bestvina Some problems on mapping class groups and moduli space by B. Farb Finiteness and Torelli spaces by R. Hain Fifteen problems about the mapping class groups by N. V. Ivanov Problems on homomorphisms of mapping class groups by M. Korkmaz The mapping class group and homotopy theory by I. Madsen Probing mapping class groups using arcs by R. C. Penner Relations in the mapping class group by B. Wajnryb II. Connections with 3-manifolds, symplectic geometry and algebraic geometry: Mapping class group factorizations and symplectic 4-manifolds: Some open problems by D. Auroux The topology of 3-manifolds, Heegaard distances and the mapping class group of a 2-manifold by J. S. Birman Lefschetz pencils and mapping class groups by S. K. Donaldson Open problems in Grothendieck-Teichmuller theory by P. Lochak and L. Schneps III. Geometry and dynamical aspects: Mapping class group dynamics on surface group representations by W. M. Goldman Geometric properties of the mapping class group by U. Hamenstadt Problems on billiards, flat surfaces and translation surfaces by P. Hubert, H. Masur, T. Schmidt, and A. Zorich Problems in the geometry of surface group extensions by L. Mosher Surface subgroups of mapping class groups by A. W. Reid Weil-Petersson perspectives by S. A. Wolpert IV. Braid groups, Out$(F_n)$ and other related groups: Braid groups and Iwahori-Hecke algebras by S. Bigelow Automorphism groups of free groups, surface groups and free abelian groups by M. R. Bridson and K. Vogtmann Problems: Braid groups, homotopy, cohomology, and representations by F. R. Cohen Cohomological structure of the mapping class group and beyond by S. Morita From braid groups to mapping class groups by L. Paris.
