On compact Riemann surfaces with dihedral groups of automorphisms

Abstract

We study compact Riemann surfaces of genus $g\geq2$ having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group $D_N$. The question of extendability of the action of $D_N$ is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.