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Book DetailsProceedings of Symposia in Pure MathematicsVolume: 74; 2006; 371 ppMSC: Primary 58; 20; 30; 14; 57;
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 grouptheoretic 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 3manifold theory, the theory of symplectic 4manifolds, 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.ReadershipGraduate students and research mathematicians interested in mapping class groups and applications.

Table of Contents

I. Cohomological, combinatorial and algebraic structures

Mladen Bestvina  Four questions about mapping class groups [ MR 2264129 ]

Benson Farb  Some problems on mapping class groups and moduli space [ MR 2264130 ]

Richard Hain  Finiteness and Torelli spaces [ MR 2264131 ]

Nikolai V. Ivanov  Fifteen problems about the mapping class groups [ MR 2264532 ]

Mustafa Korkmaz  Problems on homomorphisms of mapping class groups [ MR 2264533 ]

Ib Madsen  The mapping class group and homotopy theory [ MR 2264534 ]

R. C. Penner  Probing mapping class groups using arcs [ MR 2264535 ]

Bronislaw Wajnryb  Relations in the mapping class group [ MR 2264536 ]

II. Connections with 3manifolds, symplectic geometry and algebraic geometry

Denis Auroux  Mapping class group factorizations and symplectic 4manifolds: some open problems [ MR 2264537 ]

Joan S. Birman  The topology of 3manifolds, Heegaard distance and the mapping class group of a 2manifold [ MR 2264538 ]

S. K. Donaldson  Lefschetz pencils and mapping class groups [ MR 2264539 ]

Pierre Lochak and Leila Schneps  Open problems in GrothendieckTeichmüller theory [ MR 2264540 ]

III. Geometric and dynamical aspects

William M. Goldman  Mapping class group dynamics on surface group representations [ MR 2264541 ]

Ursula Hamenstädt  Geometric properties of the mapping class group [ MR 2264542 ]

Pascal Hubert, Howard Masur, Thomas Schmidt and Anton Zorich  Problems on billiards, flat surfaces and translation surfaces [ MR 2264543 ]

Lee Mosher  Problems in the geometry of surface group extensions [ MR 2264544 ]

Alan W. Reid  Surface subgroups of mapping class groups [ MR 2264545 ]

Scott A. Wolpert  WeilPetersson perspectives [ MR 2264546 ]

IV. Braid groups, $\mathrm {Out}(F_n)$ and other related groups

Stephen Bigelow  Braid groups and IwahoriHecke algebras [ MR 2264547 ]

Martin R. Bridson and Karen Vogtmann  Automorphism groups of free groups, surface groups and free abelian groups [ MR 2264548 ]

F. R. Cohen  Problems: Braid groups, homotopy, cohomology, and representations [ MR 2264549 ]

Shigeyuki Morita  Cohomological structure of the mapping class group and beyond [ MR 2264550 ]

Luis Paris  From braid groups to mapping class groups [ MR 2264551 ]


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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 grouptheoretic 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 3manifold theory, the theory of symplectic 4manifolds, 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.
Graduate students and research mathematicians interested in mapping class groups and applications.

I. Cohomological, combinatorial and algebraic structures

Mladen Bestvina  Four questions about mapping class groups [ MR 2264129 ]

Benson Farb  Some problems on mapping class groups and moduli space [ MR 2264130 ]

Richard Hain  Finiteness and Torelli spaces [ MR 2264131 ]

Nikolai V. Ivanov  Fifteen problems about the mapping class groups [ MR 2264532 ]

Mustafa Korkmaz  Problems on homomorphisms of mapping class groups [ MR 2264533 ]

Ib Madsen  The mapping class group and homotopy theory [ MR 2264534 ]

R. C. Penner  Probing mapping class groups using arcs [ MR 2264535 ]

Bronislaw Wajnryb  Relations in the mapping class group [ MR 2264536 ]

II. Connections with 3manifolds, symplectic geometry and algebraic geometry

Denis Auroux  Mapping class group factorizations and symplectic 4manifolds: some open problems [ MR 2264537 ]

Joan S. Birman  The topology of 3manifolds, Heegaard distance and the mapping class group of a 2manifold [ MR 2264538 ]

S. K. Donaldson  Lefschetz pencils and mapping class groups [ MR 2264539 ]

Pierre Lochak and Leila Schneps  Open problems in GrothendieckTeichmüller theory [ MR 2264540 ]

III. Geometric and dynamical aspects

William M. Goldman  Mapping class group dynamics on surface group representations [ MR 2264541 ]

Ursula Hamenstädt  Geometric properties of the mapping class group [ MR 2264542 ]

Pascal Hubert, Howard Masur, Thomas Schmidt and Anton Zorich  Problems on billiards, flat surfaces and translation surfaces [ MR 2264543 ]

Lee Mosher  Problems in the geometry of surface group extensions [ MR 2264544 ]

Alan W. Reid  Surface subgroups of mapping class groups [ MR 2264545 ]

Scott A. Wolpert  WeilPetersson perspectives [ MR 2264546 ]

IV. Braid groups, $\mathrm {Out}(F_n)$ and other related groups

Stephen Bigelow  Braid groups and IwahoriHecke algebras [ MR 2264547 ]

Martin R. Bridson and Karen Vogtmann  Automorphism groups of free groups, surface groups and free abelian groups [ MR 2264548 ]

F. R. Cohen  Problems: Braid groups, homotopy, cohomology, and representations [ MR 2264549 ]

Shigeyuki Morita  Cohomological structure of the mapping class group and beyond [ MR 2264550 ]

Luis Paris  From braid groups to mapping class groups [ MR 2264551 ]