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In this case, $(S,\\beta)$ is called an origami pair and ${\\rm Aut}(S,\\beta)$ is the group of conformal automorphisms $\\phi$ of $S$ such that $\\beta=\\beta \\circ \\phi$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of ${\\rm Aut}(S,\\beta)$ for a suitable origami pair $(S,\\beta)$. 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Hidalgo","submitted_at":"2019-07-24T20:01:31Z","abstract_excerpt":"A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $\\beta:S \\to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,\\beta)$ is called an origami pair and ${\\rm Aut}(S,\\beta)$ is the group of conformal automorphisms $\\phi$ of $S$ such that $\\beta=\\beta \\circ \\phi$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of ${\\rm Aut}(S,\\beta)$ for a suitable origami pair $(S,\\beta)$. 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