Computational Methods and Function Theory 2 (2002), No. 1, 267--279
Copyright Heldermann Verlag 2002
Beatriz Estrada
bestra@mat.uned.es, Departamento de Matemáticas Fundamentales, Facultad de Ciencias, U.N.E.D., Paseo Senda del Rey, 9, 28040 Madrid, Spain.
A compact Klein surface $X$ is a compact surface with a dianalytic structure. Such a surface can be seen as the quotient of the hyperbolic plane $\mathbb{H}^{2}$ under the action of a non-Euclidean crystallographic group (NEC group)~$\Gamma$. The $q$-hyperelliptic Klein 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 $q$. In this work, the geometry of the $q$-hyperelliptic involution is studied for planar surfaces. It is made by constructing fundamental regions that are right-angled hyperbolic polygons. These polygons are also interesting in the study of the Teichm\"{u}ller space of planar $q$-hyperelliptic Klein surfaces.
Keywords: Non-euclidean crystallograhic groups, Klein surfaces, fundamental regions.
MSC 2000: 20H10, 30F50, 30F60.
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