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 _PR,g ^K be the set of surfaces in the moduli space M _g ^K corresponding to pseudo-real Riemann surfaces. We obtain the equisymmetric stratification of M _PR,g ^K for genera g = 2, 3, 4, and as a consequence we have that M _PR,g ^K is connected for g = 2, 3 but M _PR,4 ^K has three connected components.