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 $(S,\beta)$. It is also known that $G$ can be realized as a group of conformal automorphisms of a Riemann surface $X$ of genus $g \geq 2$ and with quotient orbifold $X/G$ also of genus $\gamma \geq 2$. Given a conformal action of $G$ on a surface $X$ as before, we prove that there is an origami pair $(S,\beta)$, where $S$ has genus $g$ and $G \cong {\rm Aut}(S,\beta)$ such that the actions of ${\rm Aut}(S,\beta)$ on $S$ and that of $G$ on $X$ are topologically equivalent.

中文翻译：

---

折纸曲线的自同构群

一个封闭的黎曼曲面 $S$（至少一个属）被称为折纸曲线，如果它承认一个非常量全纯映射 $\beta:S\to E$ 最多具有一个分支值，其中 $E$ 是一个第一类黎曼曲面。在这种情况下，$(S,\beta)$ 称为折纸对，${\rm Aut}(S,\beta)$ 是 $S$ 的共形自同构 $\phi$ 的群，使得 $\beta =\beta\circ\phi$。令 $G$ 是一个有限群。众所周知，对于合适的折纸对 $(S,\beta)$，$G$ 可以实现为 ${\rm Aut}(S,\beta)$ 的子群。众所周知，$G$ 可以被实现为属 $g \geq 2$ 的黎曼面 $X$ 的一组共形自同构，并且具有同样属于 $\gamma \geq 2 的商轨道 $X/G$ $. 给定 $G$ 在表面 $X$ 上的保形作用，我们证明存在折纸对 $(S,\beta)$，