Subvarieties of Moduli Space Determined by Finite Groups Acting on Surfaces

Suppose the finite group G acts as orientation preserving homeomorphisms of the oriented surface S of genus g . This determines an irreducible subvariety Jtg of the moduli space J?g of Riemann surfaces of genus g consisting of all surfaces with a group Gx of holomorphic homeomorphisms of the same topological type as G. This family is determined by an equivalence class of epimorphisms <// from a Fuchsian group r to G whose kernel is isomorphic to the fundamental group of S . To examine the singularity of Jfg along this family one needs to know the full automorphism group of a generic element of Jig ' . In §2 we show how to compute this from y/ . Let JtP denote the locus of all Riemann surfaces of genus g whose automorphism group contains a subgroup isomorphic to G . In §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 §4 we examine the actions of G on the spaces of holomorphic ^-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 oî G on S .