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

We study the zero-twist flute surfaces and we associate to each one of them a sequence of positive real numbers , with a torsion-free Fuchsian group such that the convex core of is isometric to a zero-twist tight flute surface . Moreover, we prove that the Fuchsian group is of the first kind if and only if the series diverges. As consequence of the recent work of Basmajian, Hakobian and Šarić, we obtain that the zero-twist flute surface is of parabolic type if and only diverges. In addition, we associate to each sequence , where and , a Fuchsian group such that is homeomorphic to the Loch Ness Monster.