这本《离散群几何》由Alan F.Beardon著，主要内容：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) Fenchel-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.

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 Poincare 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. Jergensen'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 Isometries

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 Isometrics

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. Poincar6'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

 11.1. 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