Preface
1. Hyperbolic geometry
1.1. The hyperbolic metric
1.2. Geodesics
1.3. Isometrics
1.4. Hyperbolic area and the Gauss-Bonnet formula
1.5. Hyperbolic trigonometry
1.6. Comparison between hyperbolic, spherical and Euclidean trigonometry
Exercises for Chapter 1
2. Fuchsian groups
2.1. The group PSL(2,R)
2.2. Discrete and properly discontinuous groups
2.3. Algebraic properties of Fuchsian groups
2.4. Elementary groups
Exercises for Chapter 2
3. Fundamental regions
3.1. Definition of a fundamental region
3.2. The Dirichlet region
3.3. Isometric circles and the Ford fundamental region
3.4. The limit set of [ ]
3.5. Structure of a Dirichlet region
3.6. Connection with Riemann surfaces and homogeneous spaces
Exercises for Chapter 3
4. Geometry of Fuchsian groups
4.1. Geometrically finite Fuchsian groups
4.2. Cocompact Fuchsian groups
4.3. The signature of a Fuchsian group
4.4. Fuchsian groups generated by reflections
4.5. Fuchsian groups of the first kind
4.6. Finitely generated Fuchsian groups
Exercises for Chapter 4
5. Arithmetic Fuchsian groups
5.1. Definitions of arithmetic Fuchsian groups
5.2. Fuchsian groups derived from quaternion algebras
5.3. Criteria for arithmeticity
5.4. Compactness of [ ] for Fuchsian groups derived from division quaternion algebras
5.5. The modular group and its subgroups
5.6. Examples
Exercises for Chapter 5
Hints for Selected Exercises
Bibliography
Index