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.