Automorphism groups of origami curves

Abstract

A closed Riemann surface S (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $\beta :S \rightarrow E$ 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 $\mathrm{Aut}(S,\beta )$ 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 $\mathrm{Aut}(S,\beta )$ 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 $g \ge 2$ and with quotient orbifold X/G of genus $\gamma \ge 1$. 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 $G \cong \mathrm{Aut}(S,\beta )$ such that the actions of $\mathrm{Aut}(S,\beta )$ on S and that of G on X are topologically equivalent.