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 SL2, C and Twisted Alexander Polynomials
Takayuki Morifuji
1 Introduction
2 Alexander polynomials
3 Representations of knot groups into SL2, 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
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