Let ∑g denote a closed orientable surface of genus g ≥ 2. Let G be a nontrivial finite group. If G can be embedded in the group of orientation-preserving self- homeomorphisms of ∑g, then we say that G acts on ∑g. In this case, ∑g and be realized as a Riemann surface and G as a subgroup of its automorphism group. For each fixed g, there can be only finitely many finite groups G that act on ∑g, since by a famous result of Hurwitz [11] the order of G is bounded above by 84(g-1).
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