Handbook of Group Actions

Part Ⅰ Geometries and General Group ActionsGeometry of Singular Space Shing-Tung Yau1  The development of modern geometry that influenced our concept of space2  Geometry of singular spaces3  Geometry for Einstein equation and special holonomy group4  The Laplacian and the construction of generalized Riemannian geometry in terms of operators5  Differential topology of the operator geometry6  Inner product on tangent spaces and Hodge theory7  Gauge groups, convergence of operator manifolds and Yang-Mills theory8  Generalized manifolds with special holonomy groups9  Maps, subspaces and sigma models10  Noncompact manifolds11  Discrete spaces12  Conclusion13  AppendixReferencesA Summary of Topics Related to Group ActionsLizhen Ji1  Introduction2  Different types of groups3  Different types of group actions4  How do group actions arise5  Spaces which support group actions6  Compact transformation groups7  Noncompact transformation groups8  Quotient spaces of discrete group actions9  Quotient spaces of Lie groups and algebraic group actionsI0  Understanding groups via actions11  How to make use of symmetry12  Understanding and classifying nonlinear actions of groups13  Applications of finite group actions in combinatorics14  Applications in logic15  Groups and group actions in algebra16  Applications in analysis17  Applications in probability18  Applications in number theory19  Applications in algebraic geometry20  Applications in differential geometry21  Applications in topology22  Group actions and symmetry in physics23  Group actions and symmetry in chemistry24  Symmetry in biology and the medical sciences25  Group actions and symmetry in material science and engineering26  Symmetry in arts and architecture27  Group actions and symmetry in music28  Symmetries in chaos and fractals29  Acknowledgements and referencesReferencesPart Ⅱ Mapping Class Groups and Teichmiiller SpacesActions of Mapping Class GroupsAthanase Papadopoulos1  Introduction2  Rigidity and actions of mapping class groups3  Actions on foliations and laminations4  Some perspectivesReferencesThe Mapping Class Group Action on the Horofunction Compactification of Teichmiiller SpaceWeixu Su1  Introduction2  Background3  Thurston's compactification of Teichmiiller space4  Compactification of Teichmfiller space by extremal length5  Analogies between the Thurston metric and the Teichmiiller metric6  Detour cost and Busemann points7  The extended mapping class group as an isometry group8  On the classification of mapping class actions on Thurston's metric9  Some questionsReferencesSchottky Space and Teichmiiller DisksFrank Herrlich1  Introduction2  Schottky coverings3  Schottky space4  Schottky and Teichmfiller space5  Schottky space as a moduli space6  Teichmiiller disks7  Veech groups8  Horizontal cut systems9  Teichmiiller disks in Schottky spaceReferencesTopological Characterization of the Asymptotically Trivial Mapping Class GroupEge Fujikawa1  Introduction2  Preliminaries3  Discontinuity of the Teichmfiller modular group action4  The intermediate Teichmiiller space5  Dynamics of the Teichmiiller modular group6  A fixed point theorem for the asymptotic Teichmiiller modular group7  Periodicity of asymptotically Teichmiiller modular transformationReferencesThe Universal Teichmiiller Space and Diffeomorphisms of the Circle with HSlder Continuous DerivativesKatsuhiko Matsuzaki1  Introduction2  Quasisymmetric automorphisms of the circle3  The universal Teichmiiller space4  Quasisymmetric functions on the real line5  Symmetric automorphisms and functions6  The small subspace7  Diffeomorphisms of the circle with HSlder continuous derivatives8  The Teichmiiller space of circle diffeomorphismsReferencesOn the Johnson Homomorphisms of the Mapping Class Groups of SurfacesTakao Satoh1  Introduction2  Notation and conventions3  Mapping class groups of surfaces4  Johnson homomorphisms of Aut Fn5  Johnson homomorphisms of A4g,16  Some other applications of the Johnson homomorphismsAcknowledgementsReferencesPart Ⅲ Hyperbolic Manifolds and Locally Symmetric SpacesThe Geometry and Arithmetic of Kleinian GroupsGaven J. Martin1  Introduction2  The volumes of hyperbolic orbifolds3  The Margulis constant for Kleinian groups4  The general theory5  Basic concepts6  Two-generator groups7  Polynomial trace identities and inequalities8  Arithmetic hyperbolic geometry9  Spaces of discrete groups, p, q E {3, 4, 5}10  (p, q, r)-Kleinian groupsReferencesWeakly Commensurable Groups, with Applications to Differential GeometryGopal Prasad and Andrei S. Rapinchuk1  Introduction2  Weakly commensurable Zariski-dense subgroups3  Results on weak commensurability of S-arithmetic groups4  Absolutely almost simple algebraic groups having the same maximal tori5  A finiteness result6  Back to geometryAcknowledgementsReferencesPart Ⅳ: Knot GroupsRepresentations of Knot Groups into SL(2, C) and Twisted Alexander PolynomialsTakayuki Morifuji1  Introduction2  Alexander polynomials3  Representations of knot groups into SL(2, C)4  Deformations of representations of knot groups5  Twisted Alexander polynomials6  Twisted Alexander polynomials of hyperbolic knotsAcknowledgementsReferencesMeridional and Non-meridional Epimorphisms between Knot GroupsMasaaki Suzuki1  Introduction2  Some relations on the set of knots3  Twisted Alexander polynomial and epimorphism4  Meridional epimorphisms5  Non-meridional epimorphisms6  The relation≥on the set of prime knots7  Simon's conjecture and other problemsAcknowledgementsReferences