Every finitely generated group is the automorphism group of a Riemann surface (compact for finite groups), proved via maximal non-arithmetic triangle groups and Margulis' commensurability theorem.
Greenberg, Maximal Fuchsian groups, Bull
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A short proof of Greenberg's Theorem
Every finitely generated group is the automorphism group of a Riemann surface (compact for finite groups), proved via maximal non-arithmetic triangle groups and Margulis' commensurability theorem.