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arXiv · 1805.04553

Geometric Schotkky groups and non compact hyperbolic surface with infinite genus

John A. Arredondo·Camilo Ramírez Maluendas
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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 $1<m\leq s$, 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.

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John A. Arredondo, Camilo Ramírez Maluendas. 2018-05-11. Geometric Schotkky groups and non compact hyperbolic surface with infinite genus. https://arxiv.org/abs/1805.04553

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