A characterization of q-hyperelliptic compact planar Klein surfaces

Abstract

A compact Klein surface X is called q-hyperelliptic if there is an involution $\phi$ of X such that the quotient surface $X/<\phi >$ has algebraic genus q. If X is represented as D/$\Gamma$ where D is the unit disc and $\Gamma$ a non-Euclidean crystallographic group, then $<\phi >\cong \Gamma\sb 1/\Gamma$. Necessary and sufficient conditions on $\Gamma\sb 1$ are determined so that X is both planar (i.e. a two-sphere with holes) and q-hyperelliptic. One of the consequences of this is that the subspace of the Teichmüller space of those compact planar surfaces of fixed algebraic genus p which are q-hyperelliptic is a submanifold of dimension 2p-q-1 in the cases where $p>4q+1$. The methods involved are standard using the structure of NEC groups.