Part Ⅰ Geometries and General Group Actions  Geometry of Singular Space Shing-Tung Yau    1  The development of modern geometry that influenced our conce 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 ReferencesPart 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 ReferencesPart 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 ReferencesPart 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