Geometrical characterization of \(p\)-hyperelliptic planar Klein surfaces.
Let us consider \(q\)-hyperelliptic Klein surfaces \(X=H^2/\Gamma\), where \(\Gamma\) is a surface NEC-group with signature \(\sigma(\Gamma)=(0,+,[-1],\{(-1)^k\})\), called planar surfaces. These surfaces are characterized by the existence of an order two automorphism \(\phi\), called \(q\)-hyperelliptic involution, such that the quotient \(X/\langle\phi\rangle\) has algebraic genus \(q\). The main purpose of this work is the geometrical study of \(q\)-hyperelliptic involutions. For this aim, first the author constructs fundamental regions of the NEC-group \(\Gamma\) that are right-angled hyperbolic polygons. After that the author obtains canonical regions from Wilkie regions. By considering the existence of a symmetric canonical right-angled region so the \(q\)-hyperelliptic character of planar surfaces is characterized. Besides these the author considers the Teichmüller spaces of \(q\)-hyperelliptic planar Klein surfaces. Then the dimension of these spaces is calculated and it is described geometrically by means of congruence classes of polygons.
scientific article; zbMATH DE number 1995506