A necessary and sufficient condition for lifting the hyperelliptic involution

Abstract

Let X denote a Riemann surface which possesses a fixed point free group of automorphisms with a hyperelliptic orbit space. A criterion is proved which determines whether the hyperelliptic involution lifts to an automorphism of X. Necessary and sufficient conditions are stated which determine when a lift of the hyperelliptic involution is fixed point free. A complete determination is made of the abelian groups which may arise as automorphism groups of surfaces which possess a fixed point free lift. Hyperelliptic Riemann surfaces are natural objects of interest and have been studied quite extensively. They have simple defining equations and, since they admit an involution, they constitute a family of Riemann surfaces whose members admit a nontrivial automorphism. Covering surfaces of hyperelliptic surfaces have also been closely examined. The question of when a hyperelliptic surface can have a hyperelliptic cover was investigated in [2], [5], [8], and [10]. It was shown in [6] that if a Riemann surface X admits an abelian, fixed point free automorphism group H, then the hyperelliptic involution lifts to X. In addition, in [6] it was stated that if H is cyclic of prime order, then the lift of the hyperelliptic involution is never fixed point free. Further results concerning when the hyperelliptic involution lifts to a covering surface are contained in [1]. Let X be a compact Riemann surface which admits a fixed point free automorphism group H < Aut(X) with a hyperelliptic orbit space. In this paper we give necessary and sufficient conditions which determine when the hyperelliptic involution lifts to X. In addition, we give necessary and sufficient conditions which determine when such a lift is fixed point free. A complete determination is made of the abelian groups which may arise as automorphism groups of surfaces which possess a fixed point free lift. These results are combined with results in [12] to yield specific examples of automorphism groups yielding a hyperelliptic orbit space.