Geometric Schotkky groups and non compact hyperbolic surface with infinite genus

TL;DR

This paper constructs explicit Fuchsian groups with infinitely many generators to model non-compact hyperbolic surfaces of infinite genus and describes their geometric structure using hyperbolic polygons and Möbius transformations.

Contribution

It provides a detailed construction of geometric Schottky groups for non-compact surfaces with infinite genus and describes their generators explicitly.

Findings

Explicit generators for Fuchsian groups modeling infinite genus surfaces

Construction of hyperbolic polygons with infinitely many sides

Description of side-pairing Möbius transformations

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.