On Infinitely generated Fuchsian groups of some infinite genus surfaces

TL;DR

This paper constructs explicit infinitely generated Fuchsian groups for certain infinite genus surfaces, providing new examples of hyperbolic structures on these complex surfaces.

Contribution

It explicitly constructs infinitely generated Fuchsian groups for specific infinite genus surfaces, expanding the understanding of hyperbolic structures on such surfaces.

Findings

Explicit Fuchsian groups for infinite genus surfaces

Hyperbolic surfaces homeomorphic to the Infinite Loch Ness monster, Cantor tree, and Blooming Cantor tree

New examples of hyperbolic structures on complex surfaces

Abstract

In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $\Gamma<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}/\Gamma$ is a hyperbolic surface homeomorphic to $S$.