A Theoretical Algorithm to get a Schottky Uniformization from a Fuchsian one

Abstract

Riemann surfaces appear in many different areas of mathematics and physics, as in algebraic geometry, the theory of moduli spaces, topological field theories, cosmology, quantum chaos and integrable systems. A closed Riemann surface may be described in many different forms; for instance, as algebraic curves and by means of different topological classes of uniformizations. The highest uniformization corresponds to Fuchsian groups and the lowest ones to Schottky groups. In this note we discuss a theoretical algorithm which relates a Schottky group from a given Fuchsian group both uniformizing the same closed Riemann surface.