Introduction to Spectral Theory on Hyperbolic Surfaces

Contents 1. Hyperbolic geometry 1 2. Fuchsian groups and hyperbolic surfaces 4 3. Spectrum and resolvent 11 4. Spectral theory: finite-area case 16 5. Spectral theory: infinite-area case 20 6. Selberg trace formula 23 7. Arithmetic surfaces 33 References 44 1. Hyperbolic geometry In complex analysis, we learn that the upper half-plane H = {Im z > 0} has a large group of conformal automorphisms, consisting of Möbius transformations of the form (1.1) γ : z → az + b cz + d , where a, b, c, d ∈ R and ad − bc > 0. The Schwarz Lemma implies that all au-tomorphisms of H are of this type. Since the transformation is unchanged by a simultaneous rescaling of a, b, c, d, the conformal automorphism group of H is identified with the matrix group