群作用手册

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

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。