群和群作用是数学研究的重要对象，拥有强大的力量并且富于美感，这可以通过它广泛出现在诸多不同的科学领域体现出来。
此多卷本手册由相关领域专家撰写的一系列综述文章组成，首次系统地展现了群作用及其应用，内容囊括经典主题的讨论、近来的热点专业问题的论述，有些文章还涉及相关的历史。《群作用手册（第1卷）》填补了数学著作中的一项空白，适合于从初学者到相关领域专家的各个层次读者阅读。
- 目录
- 前辅文
- Part I: Geometries and General Group Actions
- Geometry of Singular Space
- Shing-Tung Yau
- 1 The development of modern geometry that influenced ourconcept of space
- 2 Geometry of singular spaces
- 3 Geometry for Einstein equation and special holonomy group
- 4 The Laplacian and the construction of generalized Riemanniangeometry 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-Millstheory
- 8 Generalized manifolds with special holonomy groups
- 9 Maps, subspaces and sigmamodels
- 10 Noncompactmanifolds
- 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
- 10 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 Teichm¨uller Spaces Actions of Mapping Class Groups
- Athanase Papadopoulos
- 1 Introduction
- 2 Rigidity and actions ofmapping class groups
- 3 Actions on foliations and laminations
- 4 Some perspectives
- References
- The Mapping Class Group Action on the Horofunction Compactification of Teichm¨uller Space
- Weixu Su
- 1 Introduction
- 2 Background
- 3 Thurston’s compactification of Teichm¨uller space
- 4 Compactification of Teichm¨uller space by extremal length
- 5 Analogies between the Thurston metric and the Teichm¨uller 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 Teichm¨uller Disks
- Frank Herrlich
- 1 Introduction
- 2 Schottky coverings
- 3 Schottky space
- 4 Schottky and Teichm¨uller space
- 5 Schottky space as amoduli space
- 6 Teichm¨uller disks
- 7 Veech groups
- 8 Horizontal cut systems
- 9 Teichm¨uller disks in Schottky space
- References
- Topological Characterization of the Asymptotically Trivial Mapping Class Group
- Ege Fujikawa
- 1 Introduction
- 2 Preliminaries
- 3 Discontinuity of the Teichm¨uller modular group action
- 4 The intermediate Teichm¨uller space
- 5 Dynamics of the Teichm¨uller modular group
- 6 A fixed point theorem for the asymptotic Teichm¨uller modular group
- 7 Periodicity of asymptotically Teichm¨uller modular transformation
- References
- The Universal Teichm¨uller Space and Diffeomorphisms of the Circle with H¨older Continuous Derivatives
- Katsuhiko Matsuzaki
- 1 Introduction
- 2 Quasisymmetric automorphisms of the circle
- 3 The universal Teichm¨uller space
- 4 Quasisymmetric functions on the real line
- 5 Symmetric automorphisms and functions
- 6 The small subspace
- 7 Diffeomorphisms of the circle with H¨older continuous derivatives
- 8 The Teichm¨uller space of circle diffeomorphisms
- References
- On the Johnson Homomorphisms of the Mapping Class Groups of urfaces
- Takao Satoh
- 1 Introduction
- 2 Notation and conventions
- 3 Mapping class groups of surfaces
- 4 Johnson homomorphisms of Aut Fn
- 5 Johnson homomorphisms of Mg,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 JMartin
- 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 ∈ {3, 4, 5}
- 10 (p, q, r)-Kleinian groups
- References
- Weakly Commensurable Groups, with Applications to Differential Geometry
- Gopal Prasad and Andrei SRapinchuk
- 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 Therelation≥ on the set of prime knots
- 7 Simon’s conjecture and other problems
- Acknowledgements
- References
