Automorphism groups of pseudo-real Riemann surfaces of low genus

Abstract

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 C4 or C8 or the Frobenius group of order 20, and in the case of C4 there are exactly two possible topological actions. Let MPR,gK be the set of surfaces in the moduli space MgK corresponding to pseudo-real Riemann surfaces. We obtain the equisymmetric stratification of MPR,gK for genera g = 2, 3, 4, and as a consequence we have that MPR,gK is connected for g = 2, 3 but MPR,4K has three connected components.