離散群幾何

內(nèi)容概要

　　This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds：see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known （but virtually unobtainable） FencheI-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo-metrical ideas to be found in that manuscript, as well as some more recent material.

書籍目錄

CHAPTER 1Preliminary Material1.1.Notation1.2.Inequalities1.3.Algebra1.4.Topology1.5.Topological Groups1.6.AnalysisCHAPTER 2Matrices2.1.Non-singular Matrices2.2.The Metric Structure2.3.Discrete Groups2.4.Quaternions2.5.Unitary MatricesCHAPTER 3M6bius Transformations on Rn3.1.The M6bius Group on Rn3.2.Properties of M6bius Transformations3.3.The Poincar6 Extension3.4.Self-mappings of the Unit Ball3.5.The General Form of a M6bius Transformation3.6.Distortion Theorems3.7.The Topological Group Structure3.8.NotesCHAPTER 4Complex M6bius Transformations4.1.Representations by Quaternions4.2.Representation by Matrices4.3.Fixed Points and Conjugacy Classes4.4.Cross Ratios4.5.The Topology on,M4.6.NotesCHAPTER 5Discontinuous Groups5.1.The Elementary Groups5.2, Groups with an Invariant Disc5.3.Discontinuous Groups5.4.Jrgensen's Inequality5.5.NotesCHAPTER 6Riemann Surfaces6.1.Riemann Surfaces6.2.Quotient Spaces6.3.Stable SetsCHAPTER 7Hyperbolic GeometryFundamental Concepts7.1.The Hyperbolic Plane7.2.The Hyperbolic Metric7.3.The Geodesics7.4.The Isometries7.5.Convex Sets7.6.AnglesHyperbolic Trigonometry7.7.Triangles7.8.Notation7.9.The Angle of Parallelism7.10.Triangles with a Vertex at Infinity7.11.Right-angled Triangles7.12.The Sine and Cosine Rules7,13.The Area of a Triangle7.14.The Inscribed CirclePolygons7.15.The Area of a Polygon7.16.Convex Polygons7,17.Quadrilaterals7.18.Pentagons7.19.HexagonsThe Geometry of Geodesics7.20.The Distance of a Point from a Line7.21.The Perpendicular Bisector of a Segment7.22.The Common Orthogonal of Disjoint Geodesics7.23.The Distance Between Disjoint Geodesics7,24.The Angle Between Intersecting Geodesics7.25.The Bisector of Two Geodesics7.26.TransversalsPencils of Geodesics7.27.The General Theory of Pencils7.28.Parabolic Pencils7.29.Elliptic Pencils7.30.Hyperbolic PencilsThe Geometry of lsometries7.31.The Classification of Isometries7.32.Parabolic Isometrics7.33.Elliptic Isometries7.34.Hyperbolic Isometries7.35.The Displacement Function7.36.Isometric Circles7.37.Canonical Regions7.38.The Geometry of Products of Isometries7.39.The Geometry of Commutators7.40.NotesCHAPTER 8Fuchsian Groups8.1.Fuchsian Groups8.2.Purely Hyperbolic Groups8.3.Groups Without Elliptic Elements8.4.Criteria for Discreteness8.5.The Nielsen Region8.6.NotesCHAPTER 9Fundamental Domains9.1.Fundamental Domains9.2.Locally Finite Fundamental Domains9.3.Convex Fundamental Polygons9.4.The Dirichlet Polygon9.5.Generalized Dirichlet Polygons9.6.Fundamental Domains for Coset Decompositions9.7.Side-Pairing Transformations9.8.Poincare's Theorem9.9.NotesCHAPTER 10Finitely Generated Groups10.1.Finite Sided Fundamental Polygons10.2.Points of Approximation10.3.Conjugacy Classes10.4.The Signature of a Fuchsian Group10.5.The Number of Sides of a Fundamental Polygon10.6.Triangle Groups10.7.NotesCHAPTER 11Universal Constraints on Fuchsian Groupsi1.1.Uniformity of Discreteness11.2.Universal Inequalities for Cycles of Vertices11.3.Hecke Groups11.4.Trace Inequalities11.5.Three Elliptic Elements of Order Two11.6.Universal Bounds on the Displacement Function11.7.Canonical Regions and Quotient Surfaces11.8.NotesReferencesIndex

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