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 Frobenius 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-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.