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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

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