Sufficient conditions for a group of automorphisms of a Riemann surface to be its full automorphism group

Abstract

There are few examples in the literature of Riemann surfaces whose defining algebraic equations and full automorphism groups are completely determined. Although explicit examples of Riemann surfaces which admit automorphisms may be constructed by the use of symmetries in the defining equations of the surface, determining whether the admitted automorphisms constitute the full automorphism group is usually intractable. In this paper, it is proved that for many groups a simple lifting criterion determines whether the admitted automorphisms form the full automorphism group. The criterion is employed to give numerous examples of Riemann surfaces whose defining equations and full automorphism groups are determined.