GENERATING-FUNCTIONS FOR FINITE-GROUP ACTIONS ON SURFACES

For a fixed finite group G, the numbers N-g of equivalence classes of orientation-preserving actions of G on closed orientable surfaces Sigm a(g) of genus g can be encoded by a generating function Sigma N(g)z(g) . When equivalence is determined by the isomorphism class of the quoti ent orbifold Sigma(g)/G, we show that the generating function is ratio nal. When equivalence is topological conjugacy, me examine the cases w here G is abelian and show that the generating function is again ratio nal in the cases where G is cyclic.