Computing equivalence classes of finite group actions on orientable surfaces

Abstract

This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface Sg of genus g≥2. With each such action of a group G on Sg one can associate the fundamental group Γ=π(O) of the quotient orbifold O=Sg/G, isomorphic to a Fuchsian group determined completely by orbifold's signature. The Riemann existence theorem reduces the problem of the existence of an action of G on Sg to a purely group-theoretical problem of deciding whether there is an smooth epimorphism mapping the Fuchsian group Γ onto the group G. Using computer algebra systems such as Magma or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus g≥2 becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to topological equivalence. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set of finite group actions on Sg with the corresponding signature. In this paper we derive several results on the topological equivalence of finite group actions on Riemann surfaces. As an application, we derive complete lists of finite group actions of genus g≤8 distinguished up to the topological equivalence.