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

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 {Š}ari{ć}, 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 $a_n\le b_n\le c_n\le d_n\le e_n\le a_{n+1}$, a Fuchsian group $G_{\mathbf{y}}$ such that ${\mathbb{H}}^2/G_{\mathbf{y}}$ is homeomorphic to the Loch Ness Monster.