Geometric Schotkky groups and non compact hyperbolic surface 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 $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\le s$, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group ${\Gamma }_{m,s}$ such that the quotient space $\mathbb{H}/{\Gamma }_{m,s}$ is homeomorphic to $S_{m,s}$ 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.