Riemann Surfaces as Orbit Spaces of Fuchsian Groups

Moore, M. J.

Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 612-616

Résumé

A Fuchsian group is a discrete subgroup of the hyperbolic group, L.F. (2, R), of linear fractional transformations each such transformation mapping the complex upper half plane D into itself. If Γ is a Fuchsian group, the orbit space D/Γ has an analytic structure such that the projection map p: D → D/Γ, given by p(z) = Γz, is holomorphic and D/Γ is then a Riemann surface.If N is a normal subgroup of a Fuchsian group Γ, then N is a Fuchsian group and S = D/N is a Riemann surface. The factor group, G = Γ/N, acts as a group of automorphisms (biholomorphic self-transformations) of S for, if γ ∊ Γ and z ∊ D, then γN ∊ G, Nz ∊ S, and (γN) (Nz) = Nγz. This is easily seen to be independent of the choice of γ in its N-coset and the choice of z in its N-orbit.Conversely, if S is a compact Riemann surface, of genus at least two, then S can be identified with D/K, where K is a Fuchsian group acting without fixed points in D.

Détail

DOI : 10.4153/CJM-1972-055-9

Bibliographie

[1] 1. Ahlfors, L. V. and Sario, L., Riemann surfaces (Princeton University Press, Princeton, N.J., 1960). Google Scholar

[2] 2. Armstrong, M. A., On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965), 639–646. Google Scholar

[3] 3. Lehner, J., Discontinuous groups and automorphic functions, Math. Surveys (Amer. Math. Soc, Providence, R.I., 1964). Google Scholar

[4] 4. Macbeath, A. M., Discontinuous groups and birational transformations, Proceedings of Summer School in Geometry and Topology (Dundee, 1961). Google Scholar