Riemann surfaces and their automorphism groups GAC2010 workshop lecture notes Harish-Chandra Research Institute Allahabad, India Gareth A. Jones School of Mathematics University of Southampton Southampton SO17 1BJ, U.K.
[email protected] Riemann surfaces provide plenty of interesting groups and problems asociated with them: these include finite groups, since most compact Riemann surfaces have finite au- tomorphism groups, and also finitely presented infinite groups, arising from the uniformi- sation of Riemann surfaces by Fuchsian groups. In many cases, these problems can be approached by using computational techniques. Background reading. For an elementary introduction to Riemann surfaces and their automorphisms,: G. A. Jones and D. Singerman, Complex Functions, Cambridge University Press. At a rather more advanced level, and very comprehensive: H. M. Farkas and I. Kra, Riemann Surfaces, Springer. For background on the combinatorial group theory aspects of Fuchsian and related groups: H. Zieschang, E. Vogt and H-D. Coldewey, Surfaces and Planar Discontinuous Groups, Springer LNM 835, 1980. For useful techniques for studying automorphism groups, though some of the results have now been overtaken: R. D. M. Accola, Topics in the Theory of Riemann Surfaces, Springer LNM 1595, 1994. For applications of character theory to Riemann surface automorphism groups: T. Breuer, Characters and Automorphism Groups of Compact Riemann Surfaces, LMS Lecture Notes 280, 2000. 1 1 Riemann surfaces 1.1 Definitions Informally, a Riemann surface X is a connected Hausdorff space which looks locally like a subset of the complex plane. More precisely we require that every point p ∈ X has an open neighbourhood U with a homeomorphism φ : U → V between U and an open subset V ⊆ C.