Geometric Schottky groups and non-compact hyperbolic surfaces with infinite genus

Abstract

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface Sm,s with s ends and exactly m of them with infinite genus, such that m,s∈N and 1<m≤s, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group Γm,s such that the quotient space H/Γm,s is homeomorphic to Sm,s and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.