Automorphism groups of origami curves

摘要

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 -> 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 Aut(S, beta) is the group of conformal automorphisms phi of S such that beta = beta circle phi f. Let G be a finite group. It is a known fact that G can be realized as a subgroup of 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 >= 2 and with quotient orbifold X/G of genus gamma >= 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 congruent to Aut(S, beta) such that the actions of Aut(S, beta) on S and that of G on X are topologically equivalent.