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Problems on Mapping Class Groups and Related Topics
 
Edited by: Benson Farb University of Chicago, Chicago, IL
Front Cover for Problems on Mapping Class Groups and Related Topics
Available Formats:
Hardcover ISBN: 978-0-8218-3838-9
Product Code: PSPUM/74
List Price: $103.00
MAA Member Price: $92.70
AMS Member Price: $82.40
Electronic ISBN: 978-0-8218-9380-7
Product Code: PSPUM/74.E
List Price: $97.00
MAA Member Price: $87.30
AMS Member Price: $77.60
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $154.50
MAA Member Price: $139.05
AMS Member Price: $123.60
Front Cover for Problems on Mapping Class Groups and Related Topics
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Problems on Mapping Class Groups and Related Topics
Edited by: Benson Farb University of Chicago, Chicago, IL
Available Formats:
Hardcover ISBN:  978-0-8218-3838-9
Product Code:  PSPUM/74
List Price: $103.00
MAA Member Price: $92.70
AMS Member Price: $82.40
Electronic ISBN:  978-0-8218-9380-7
Product Code:  PSPUM/74.E
List Price: $97.00
MAA Member Price: $87.30
AMS Member Price: $77.60
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $154.50
MAA Member Price: $139.05
AMS Member Price: $123.60
  • Book Details
     
     
    Proceedings of Symposia in Pure Mathematics
    Volume: 742006; 371 pp
    MSC: 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 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
     
     
    • 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 3-manifolds, symplectic geometry and algebraic geometry
    • Denis Auroux - Mapping class group factorizations and symplectic 4-manifolds: some open problems [ MR 2264537 ]
    • Joan S. Birman - The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold [ MR 2264538 ]
    • S. K. Donaldson - Lefschetz pencils and mapping class groups [ MR 2264539 ]
    • Pierre Lochak and Leila Schneps - Open problems in Grothendieck-Teichmüller 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 - Weil-Petersson perspectives [ MR 2264546 ]
    • IV. Braid groups, $\mathrm {Out}(F_n)$ and other related groups
    • Stephen Bigelow - Braid groups and Iwahori-Hecke 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 ]
  • Additional Material
     
     
  • Request Review Copy
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Volume: 742006; 371 pp
MSC: 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 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.

  • 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 3-manifolds, symplectic geometry and algebraic geometry
  • Denis Auroux - Mapping class group factorizations and symplectic 4-manifolds: some open problems [ MR 2264537 ]
  • Joan S. Birman - The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold [ MR 2264538 ]
  • S. K. Donaldson - Lefschetz pencils and mapping class groups [ MR 2264539 ]
  • Pierre Lochak and Leila Schneps - Open problems in Grothendieck-Teichmüller 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 - Weil-Petersson perspectives [ MR 2264546 ]
  • IV. Braid groups, $\mathrm {Out}(F_n)$ and other related groups
  • Stephen Bigelow - Braid groups and Iwahori-Hecke 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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