Contents Preface ix Preface to the AMS Edition xi Introduction 1 Chapter 0. Harmonic Analysis on the Euclidean Plane 3 Chapter 1. Harmonic Analysis on the Hyperbolic Plane 7 1.1. The upper half-plane 7 1.2. H as a homogeneous space 12 1.3. The geodesic polar coordinates 15 1.4. Group decompositions 16 1.5. The classification of motions 17 1.6. The Laplace operator 19 1.7. Eigenfunctions of Δ 19 1.8. The invariant integral operators 26 1.9. The Green function on H 32 Chapter 2. Fuchsian Groups 37 2.1. Definitions 37 2.2. Fundamental domains 38 2.3. Basic examples 41 2.4. The double coset decomposition 45 v
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