Parabolicity of Zero-Twist Tight Flute Surfaces and Uniformization of the Loch Ness Monster

Abstract

In this paper we study the zero-twist flute surface and we associate to each one of them a sequence of positive real numbers. Conversely, we associate to each sequence of positive real numbers $\mathbf{x}=(x_{n})_{n\in\mathbb{N}_{0}}$ 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 first kind if and only if the series $\sum x_{n}$ diverges. As consequence of the recent work of Basmajian, Hakobian and {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.