Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.