On Riemann surfaces of genus g with 4g automorphisms

We determine, for all genus g≥2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g≠ 3,6,12,15 or 30, these surfaces form a real Riemann surface Fg in the moduli space Mg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real forms of real Riemann surfaces in Fg. The set of real Riemann surfaces in Fg consists of three intervals its closure in the Deligne-Mumford compactification of Mg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in Fg.