Handbook of Group Actions（群作用手册）（第 I 卷）

Part Ⅰ Geometries and General Group Actions
  Geometry of Singular Space Shing-Tung Yau
    1  The development of modern geometry that influenced our concept of space
    2  Geometry of singular spaces
    3  Geometry for Einstein equation and special holonomy group
    4  The Laplacian and the construction of generalized Riemannian geometry in terms of operators
    5  Differential topology of the operator geometry
    6  Inner product on tangent spaces and Hodge theory
    7  Gauge groups, convergence of operator manifolds and Yang-Mills theory
    8  Generalized manifolds with special holonomy groups
    9  Maps, subspaces and sigma models
    10  Noncompact manifolds
    11  Discrete spaces
    12  Conclusion
    13  Appendix
    References
  A Summary of Topics Related to Group Actions
  Lizhen Ji
    1  Introduction
    2  Different types of groups
    3  Different types of group actions
    4  How do group actions arise
    5  Spaces which support group actions
    6  Compact transformation groups
    7  Noncompact transformation groups
    8  Quotient spaces of discrete group actions
    9  Quotient spaces of Lie groups and algebraic group actions
    I0  Understanding groups via actions
    11  How to make use of symmetry
    12  Understanding and classifying nonlinear actions of groups
    13  Applications of finite group actions in combinatorics
    14  Applications in logic
    15  Groups and group actions in algebra
    16  Applications in analysis
    17  Applications in probability
    18  Applications in number theory
    19  Applications in algebraic geometry
    20  Applications in differential geometry
    21  Applications in topology
    22  Group actions and symmetry in physics
    23  Group actions and symmetry in chemistry
    24  Symmetry in biology and the medical sciences
    25  Group actions and symmetry in material science and engineering
    26  Symmetry in arts and architecture
    27  Group actions and symmetry in music
    28  Symmetries in chaos and fractals
    29  Acknowledgements and references
    References
Part Ⅱ Mapping Class Groups and Teichmiiller Spaces
  Actions of Mapping Class Groups
  Athanase Papadopoulos
    1  Introduction
    2  Rigidity and actions of mapping class groups
    3  Actions on foliations and laminations
    4  Some perspectives
    References
  The Mapping Class Group Action on the Horofunction Compactification of Teichmiiller Space
  Weixu Su
    1  Introduction
    2  Background
    3  Thurston's compactification of Teichmiiller space
    4  Compactification of Teichmfiller space by extremal length
    5  Analogies between the Thurston metric and the Teichmiiller metric
    6  Detour cost and Busemann points
    7  The extended mapping class group as an isometry group
    8  On the classification of mapping class actions on Thurston's metric
    9  Some questions
    References
  Schottky Space and Teichmiiller Disks
  Frank Herrlich
    1  Introduction
    2  Schottky coverings
    3  Schottky space
    4  Schottky and Teichmfiller space
    5  Schottky space as a moduli space
    6  Teichmiiller disks
    7  Veech groups
    8  Horizontal cut systems
    9  Teichmiiller disks in Schottky space
    References
  Topological Characterization of the Asymptotically Trivial Mapping Class Group
  Ege Fujikawa
    1  Introduction
    2  Preliminaries
    3  Discontinuity of the Teichmfiller modular group action
    4  The intermediate Teichmiiller space
    5  Dynamics of the Teichmiiller modular group
    6  A fixed point theorem for the asymptotic Teichmiiller modular group
    7  Periodicity of asymptotically Teichmiiller modular transformation
    References
  The Universal Teichmiiller Space and Diffeomorphisms of the Circle with HSlder Continuous Derivatives
  Katsuhiko Matsuzaki
    1  Introduction
    2  Quasisymmetric automorphisms of the circle
    3  The universal Teichmiiller space
    4  Quasisymmetric functions on the real line
    5  Symmetric automorphisms and functions
    6  The small subspace
    7  Diffeomorphisms of the circle with HSlder continuous derivatives
    8  The Teichmiiller space of circle diffeomorphisms
    References
  On the Johnson Homomorphisms of the Mapping Class Groups of Surfaces
  Takao Satoh
    1  Introduction
    2  Notation and conventions
    3  Mapping class groups of surfaces
    4  Johnson homomorphisms of Aut Fn
    5  Johnson homomorphisms of A4g,1
    6  Some other applications of the Johnson homomorphisms
    Acknowledgements
    References
Part Ⅲ Hyperbolic Manifolds and Locally Symmetric Spaces
  The Geometry and Arithmetic of Kleinian Groups
  Gaven J. Martin
    1  Introduction
    2  The volumes of hyperbolic orbifolds
    3  The Margulis constant for Kleinian groups
    4  The general theory
    5  Basic concepts
    6  Two-generator groups
    7  Polynomial trace identities and inequalities
    8  Arithmetic hyperbolic geometry
    9  Spaces of discrete groups, p, q E {3, 4, 5}
    10  (p, q, r)-Kleinian groups
    References
  Weakly Commensurable Groups, with Applications to Differential Geometry
  Gopal Prasad and Andrei S. Rapinchuk
    1  Introduction
    2  Weakly commensurable Zariski-dense subgroups
    3  Results on weak commensurability of S-arithmetic groups
    4  Absolutely almost simple algebraic groups having the same maximal tori
    5  A finiteness result
    6  Back to geometry
    Acknowledgements
    References
Part Ⅳ: Knot Groups
  Representations of Knot Groups into SL(2, C) and Twisted Alexander Polynomials
  Takayuki Morifuji
    1  Introduction
    2  Alexander polynomials
    3  Representations of knot groups into SL(2, C)
    4  Deformations of representations of knot groups
    5  Twisted Alexander polynomials
    6  Twisted Alexander polynomials of hyperbolic knots
    Acknowledgements
    References
  Meridional and Non-meridional Epimorphisms between Knot Groups
  Masaaki Suzuki
    1  Introduction
    2  Some relations on the set of knots
    3  Twisted Alexander polynomial and epimorphism
    4  Meridional epimorphisms
    5  Non-meridional epimorphisms
    6  The relation≥on the set of prime knots
    7  Simon's conjecture and other problems
    Acknowledgements
    References