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

John A Arredondo,Camilo Ramírez Maluendas

doi:10.1515/coma-2020-0004

John A Arredondo, Camilo Ramírez Maluendas

Open Access

https://doi.org/10.1515/coma-2020-0004

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Abstract

AbstractIn 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.

Highlights

A classical problem during the 19th century, in which several authors had been focused e.g., Klein, Schwarz, and Poincaré among others, known as the uniformization problem, [1], said that: being S a Riemann surface, nd all domains S ⊂ Cand holomorphic functions t : S → S such that at each point p ∈ S, t is a local uniformizing variable at p
It will means that the triplet (S, S, t) is a covering space with base space S, total space S, and holomorphic projection t
Of all non-compact Riemann surfaces we focus on three of them: the In nite Loch Ness monster, the Cantor tree and the Blooming Cantor tree

Summary

Introduction

A classical problem during the 19th century, in which several authors had been focused e.g., Klein, Schwarz, and Poincaré among others, known as the uniformization problem, [1], said that: being S a Riemann surface, nd all domains S ⊂ Cand holomorphic functions t : S → S such that at each point p ∈ S, t is a local uniformizing variable at p. F ∈Γ (ii) the intersection of the interior sets f (D) ∩ D = ∅, for each f ∈ Γ \ {Id}; (iii) the boundary of R (in the closure of H, i.e., in H ∪ R ∪ {∞} consists of limits points of Γ, and a nite or countable collection of curves inside H (with the possible exception of its end points), called its sides;. The quotient space S = (H − K)/Γ is well-de ned and via π : (H − K) → S the projection map z → [z] It comes with a hyperbolic structure, it means, S is a Riemann surface (see e.g., [14], [20, Theorem 18.2]). If R is a locally nite fundamental domain for the Fuchsian group Γ, the quotient space H/Γ is homeomorphic to R/Γ (see [3, Theorem 9.2.4 ]).

5: Objects

10: Half circles