Parabolicity of zero-twist tight flute surfaces and uniformization of the Loch Ness monster

TL;DR

This paper explores the geometric properties of zero-twist flute surfaces, linking their parabolicity to the divergence of a series, and constructs an uncountable family of hyperbolic surfaces homeomorphic to the Loch Ness Monster.

Contribution

It establishes a connection between the divergence of series and the parabolic type of zero-twist flute surfaces, and introduces a new family of hyperbolic Loch Ness Monster surfaces.

Findings

Zero-twist flute surfaces are parabolic iff the series sum diverges.

Fuchsian groups of the first kind correspond to divergent series.

Constructs an uncountable family of Loch Ness Monster surfaces.

Abstract

We study the zero-twist flute surface and we associate to each one of them a sequence of positive real numbers $\mathbf{x}=(x_{n})_{n\in\mathbb{N}_{0}}$, with a torsion-free Fuchsian group $\Gamma_{\mathbf{x}}$ such that the convex core of $\mathbb{H}^2/\Gamma_{\mathbf{x}}$ is isometric to a zero-twist tight flute surface $S_{\mathbf{x}}$. Moreover, we prove that the Fuchsian group $\Gamma_{\mathbf{x}}$ is of the first kind if and only if the series $\sum x_{n}$ diverges. As consequence of the recent work of Basmajian, Hakobian and {\v{S}}ari{\'c}, we obtain that the zero-twist flute surface $S_{\mathbf{x}}$ is of parabolic type if and only $\sum x_{n}$ diverges. In addition, we present an uncountable family of hyperbolic surfaces homeomorphic to the Loch Ness Monster. More precisely, we associate to each sequence $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}$, where $y_{n}=(a_{n},b_{n},c_{n},d_{n},e_{n})\in \mathbb{R}^5$ and $a_n\leq b_n \leq c_n \leq d_n \leq e_n \leq a_{n+1}$, a Fuchsian group $G_{\mathbf{y}}$ such that $\mathbb{H}^2/G_{\mathbf{y}}$ is homeomorphic to the Loch Ness Monster.