Description: 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

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