One dimensional equisymmetric strata in moduli space with genus 1 quotient surfaces

The complex orbifold structure of the moduli space of Riemann surfaces of genus g (g≥2) produces a stratification into complex subvarieties named equisymmetric strata. Each equisymmetric stratum is formed by the surfaces where the group of automorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two structurally different types. Type 1 equisymmetric strata correspond to Riemann surfaces where the group of automorphisms produces a quotient surface of genus zero, while those of Type 2 appear when such a quotient is a surface of genus one. Type 1 equisymmetric strata have been extensively studied by the authors of the present work in a previous recent paper, we now focus on Type 2 strata. We first establish the existence of such strata and their frequency of occurrence in moduli spaces. As a main result we obtain a complete description of Type 2 strata as coverings of the sphere branched over three points (Belyi curves) and where certain isolated points (punctures) have to be eliminated. Finally, we study in detail the doubly infinite family of Type 2 strata whose automorphism groups have order the product of two primes.