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Riemann Surfaces as Orbit Spaces of Fuchsian Groups

Published online by Cambridge University Press:  20 November 2018

M. J. Moore
M. J. Moore*
Affiliation:
Carleton University, Ottawa, Ontario
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 612 - 616
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Ahlfors, L. V. and Sario, L., Riemann surfaces (Princeton University Press, Princeton, N.J., 1960).Google Scholar
2. Armstrong, M. A., On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965), 639–646.Google Scholar
3. Lehner, J., Discontinuous groups and automorphic functions, Math. Surveys (Amer. Math. Soc, Providence, R.I., 1964).Google Scholar
4. Macbeath, A. M., Discontinuous groups and birational transformations, Proceedings of Summer School in Geometry and Topology (Dundee, 1961).Google Scholar