On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree

Abstract

In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

Keywords

• infinite loch ness monster
• cantor tree
• blooming cantor tree
• geometric schottky groups
• noncompact surfaces
• 20h10
• 57n05
• 57n16