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
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.
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