Lifting involutions to ramified covers of Riemann surfaces

Abstract.

Let X be a Riemann surface which admits a group H of automorphisms and assume that X is a branched cover of its orbit space under H.We derive necessary and sufficient conditions which determine when an involution of the orbit space lifts to an automorphism of X. We extend the well known result that if the orbit space is hyperelliptic and H is abelian and fixed point free, then the hyperelliptic involution lifts to an automorphism of X; we prove that if no Weierstrass points of the orbit space are ramified in X, and if the ramification indices of all non-Weierstrass points are relatively prime in pairs, then the hyperelliptic involution lifts for any abelian group H. It is possible to use our methods to obtain conditions for lifting automorphisms of arbitrary order and we give the conditions for lifting automorphisms of order three.