離散群幾何

《離散群幾何》是2011年7月1日世界圖書出版公司出版的圖書，作者是（英國）比爾登（AlanF.Beardon）。

基本介紹

• 中文名：離散群幾何
• 別名：the geometry of discrete groups
• 作者：(英國)比爾登（AlanF.Beardon）
• 類別：高等數學
• 出版社：世界圖書出版公司
• 出版時間：2011年7月1日
• 頁數：337 頁
• 定價：26.00
• 開本：24 開
• 裝幀：平裝
• ISBN：9787510037559 
• 正文語種：簡體中文

內容簡介

《離散群幾何(英文)》主要內容簡介：ThistextisintendedtoserveasanintroductiontothegeometryoftheactionofdiscretegroupsofMobiustransformations.Thesubjectmatterhasnowbeenstudiedwithchangingpointsofemphasisforoverahundredyears,themostrecentdevelopmentsbeingconnectedwiththetheoryof3-manifolds:see,forexample,thepapersofPoincare[77]andThurston[101].About1940,thenowwell-known（butvirtuallyunobtainable）FencheI-Nielsenmanuscriptappeared.Sadly,themanuscriptneverappearedinprint,andthismoremodesttextattemptstodisplayatleastsomeofthebeautifulgeo-metricalideastobefoundinthatmanuscript,aswellassomemorerecentmaterial.

目錄

CHAPTER 1

Preliminary Material

1.1.Notation

1.2.Inequalities

1.3.Algebra

1.4.Topology

1.5.Topological Groups

1.6.Analysis

CHAPTER 2

Matrices

2.1.Non-singular Matrices

2.2.The Metric Structure

2.3.Discrete Groups

2.4.Quaternions

2.5.Unitary Matrices

CHAPTER 3

M6bius Transformations on Rn

3.1.The M6bius Group on Rn

3.2.Properties of M6bius Transformations

3.3.The Poincar6 Extension

3.4.Self-mappings of the Unit Ball

3.5.The General Form of a M6bius Transformation

3.6.Distortion Theorems

3.7.The Topological Group Structure

3.8.Notes

CHAPTER 4

Complex M6bius Transformations

4.1.Representations by Quaternions

4.2.Representation by Matrices

4.3.Fixed Points and Conjugacy Classes

4.4.Cross Ratios

4.5.The Topology on,M

4.6.Notes

CHAPTER 5

Discontinuous Groups

5.1.The Elementary Groups

5.2, Groups with an Invariant Disc

5.3.Discontinuous Groups

5.4.Jrgensen's Inequality

5.5.Notes

CHAPTER 6

Riemann Surfaces

6.1.Riemann Surfaces

6.2.Quotient Spaces

6.3.Stable Sets

CHAPTER 7

Hyperbolic Geometry

Fundamental Concepts

7.1.The Hyperbolic Plane

7.2.The Hyperbolic Metric

7.3.The Geodesics

7.4.The Isometries

7.5.Convex Sets

7.6.Angles

Hyperbolic Trigonometry

7.7.Triangles

7.8.Notation

7.9.The Angle of Parallelism

7.10.Triangles with a Vertex at Infinity

7.11.Right-angled Triangles

7.12.The Sine and Cosine Rules

7,13.The Area of a Triangle

7.14.The Inscribed Circle

Polygons

7.15.The Area of a Polygon

7.16.Convex Polygons

7,17.Quadrilaterals

7.18.Pentagons

7.19.Hexagons

The Geometry of Geodesics

7.20.The Distance of a Point from a Line

7.21.The Perpendicular Bisector of a Segment

7.22.The Common Orthogonal of Disjoint Geodesics

7.23.The Distance Between Disjoint Geodesics

7,24.The Angle Between Intersecting Geodesics

7.25.The Bisector of Two Geodesics

7.26.Transversals

Pencils of Geodesics

7.27.The General Theory of Pencils

7.28.Parabolic Pencils

7.29.Elliptic Pencils

7.30.Hyperbolic Pencils

The Geometry of lsometries

7.31.The Classification of Isometries

7.32.Parabolic Isometrics

7.33.Elliptic Isometries

7.34.Hyperbolic Isometries

7.35.The Displacement Function

7.36.Isometric Circles

7.37.Canonical Regions

7.38.The Geometry of Products of Isometries

7.39.The Geometry of Commutators

7.40.Notes

CHAPTER 8

Fuchsian Groups

8.1.Fuchsian Groups

8.2.Purely Hyperbolic Groups

8.3.Groups Without Elliptic Elements

8.4.Criteria for Discreteness

8.5.The Nielsen Region

8.6.Notes

CHAPTER 9

Fundamental Domains

9.1.Fundamental Domains

9.2.Locally Finite Fundamental Domains

9.3.Convex Fundamental Polygons

9.4.The Dirichlet Polygon

9.5.Generalized Dirichlet Polygons

9.6.Fundamental Domains for Coset Decompositions

9.7.Side-Pairing Transformations

9.8.Poincare's Theorem

9.9.Notes

CHAPTER 10

Finitely Generated Groups

10.1.Finite Sided Fundamental Polygons

10.2.Points of Approximation

10.3.Conjugacy Classes

10.4.The Signature of a Fuchsian Group

10.5.The Number of Sides of a Fundamental Polygon

10.6.Triangle Groups

10.7.Notes

CHAPTER 11

Universal Constraints on Fuchsian Groups

i1.1.Uniformity of Discreteness

11.2.Universal Inequalities for Cycles of Vertices

11.3.Hecke Groups

11.4.Trace Inequalities

11.5.Three Elliptic Elements of Order Two

11.6.Universal Bounds on the Displacement Function

11.7.Canonical Regions and Quotient Surfaces

11.8.Notes

References

Index