Why do we stop at Fuchsian groups (I.e. discrete subgroups of automorphisms of the hyperbolic plane) when we study things like quotients and what not?
Is there a maximalist or universality property behind that distinction?
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Depends on what kind of whatnot you want to study. It's a fact that any hyperbolic Riemann surface (that is, any connected complex manifold of dimension one except the Riemann sphere, complex tori, or the cylinder) can be written as the quotient of the upper half-plane by a Fuchsian group.
So the study of Riemann surfaces is basically equivalent to the study of Fuchsian groups.
Albert
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Could you tell me where I can find the proof and more details of the fact you mentioned? – Aolong Li Mar 13 '17 at 16:52
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@AolongLi any book on Riemann surfaces. for example, the beginning of Hubbard's book "Teichmüller theory and applications to geometry, topology, and dynamics" (volume 1) – Albert Apr 20 '17 at 19:59