Automorphism groups of pseudo-real Riemann surfaces of low genus

A pseudo-real Riemann surface admits anticonformal automorphisms but no anticonformal involution. We obtain the classification of actions and groups of automorphisms of pseudo-real Riemann surfaces of genera 2, 3 and 4. For instance the automorphism group of a pseudo-real Riemann surface of genus 4 is either C 4 or C 8 or the Fröbenius group of order 20, and in the case of C 4 there are exactly two possible topological actions. Let M K PR,g be the set of surfaces in the moduli space M K g corresponding to pseudo-real Riemann surfaces. We obtain the equisymmetric stratification of M K PR,g for genera g = 2, 3, 4, and as a consequence we have that M K PR,g is connected for g = 2, 3 but M K PR,4 has three connected components.