Asymmetric and pseudo-symmetric hyperelliptic surfaces

Abstract

In this paper we examine hyperelliptic Riemann surfaces which possess an anticonformal automorphism but are not symmetric. We determine that all such surfaces must have a full automorphism group which is either cyclic, or an abelian extension of a cyclic group by ℤ2. We give defining equations for all of these hyperelliptic surfaces and show how they can be constructed by using NEC groups. As a special case, we determine all hyperelliptic surfaces which are pseudo-symmetric but not symmetric.