Extending Harvey's Surface Kernel Maps

TL;DR

This paper extends Harvey's classification of surface kernel maps from cyclic groups to arbitrary finite groups, enabling explicit computation of fundamental group generators and group actions on surfaces.

Contribution

It generalizes Harvey's uniqueness result to finite groups beyond cyclic ones, facilitating the analysis of automorphism groups of Riemann surfaces.

Findings

Extended Harvey's result to finite groups

Provided explicit generators for fundamental groups

Demonstrated group action on homology for S_3

Abstract

Let $S$ be a compact Riemann surface and $G$ a group of conformal automorphisms of $S$ with $S_0 = S/G$. $S$ is a finite regular branched cover of $S_0$. If $U$ denotes the unit disc, let $\Gamma$ and $\Gamma_0$ be the Fuchsian groups with $S = U/{\Gamma}$ and $S_0 = U/{\Gamma_0}$. There is a group homomorphism of $\Gamma_0$ onto $G$ with kernel $\Gamma$ and this is termed a surface kernel map. Two surface kernel maps are equivalent if they differ by an automorphism of $\Gamma_0$. In his 1971 paper Harvey showed that when $G$ is a cyclic group, there is a unique simplest representative for this equivalence class. His result has played an important role in establishing subsequent results about conformal automorphism groups of surfaces. We extend his result to some surface kernel maps onto arbitrary finite groups. These can be used along with the Schreier-Reidemeister Theory to find a set of generators for $\Gamma$ and the action of $G$ as an outer automorphism group on the fundamental group of $S$ putting the action on the fundamental group and the induced action on homology into a relatively simple format. As an example we compute generators for the fundamental group and a homology basis together with the action of $G$ when $G$ is ${\mathcal{S}_3$, the symmetric group on three letters. The action of $G$ shows that the homology basis found is not an adapted homology basis.