A characterization of q-hyperelliptic compact planar Klein surfaces
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 \(<\phi >\cong \Gamma_ 1/\Gamma\). Necessary and sufficient conditions on \(\Gamma_ 1\) 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.
scientific article; zbMATH DE number 4152810
scientific article