Subvarieties of moduli space determined by finite groups acting on surfaces

Abstract:

Suppose the finite group $G$ acts as orientation preserving homeomorphisms of the oriented surface $S$ of genus $g$. This determines an irreducible subvariety $\mathcal {M}_g^{[G]}$ of the moduli space ${\mathcal {M}_g}$ of Riemann surfaces of genus $g$ consisting of all surfaces with a group ${G_1}$ of holomorphic homeomorphisms of the same topological type as $G$. This family is determined by an equivalence class of epimorphisms $\psi$ from a Fuchsian group $\Gamma$ to $G$ whose kernel is isomorphic to the fundamental group of $S$. To examine the singularity of ${\mathcal {M}_g}$ along this family one needs to know the full automorphism group of a generic element of $\mathcal {M}_g^{[G]}$. In $\S 2$ we show how to compute this from $\psi$. Let $\mathcal {M}_g^G$ denote the locus of all Riemann surfaces of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. In $\S 3$ we show that the irreducible components of this subvariety do not necessarily correspond to the families above, that is, the components cannot be put into a one-to-one correspondence with the topological actions of $G$. In $\S 4$ we examine the actions of $G$ on the spaces of holomorphic $k$-differentials and on the first homology. We show that when $G$ is not cyclic, the characters of these actions do not necessarily determine the topological type of the action of $G$ on $S$.