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

This is the authors' abstract. We don't add key points for this paper.

We study the zero-twist flute surfaces and we associate to each one of them a sequence of positive real numbers x=(xₙ)_n₀ , with a torsion-free Fuchsian group Γₓ such that the convex core of H²/Γₓ is isometric to a zero-twist tight flute surface Sₓ . Moreover, we prove that the Fuchsian group Γₓ is of the first kind if and only if the series ∑ xₙ diverges. As consequence of the recent work of Basmajian, Hakobian and Šarić, we obtain that the zero-twist flute surface Sₓ is of parabolic type if and only ∑ xₙ diverges. In addition, we associate to each sequence y=(yₙ)ₙ , where yₙ=(aₙ,bₙ,cₙ,dₙ,eₙ)⁵ and aₙ<bₙ<cₙ<dₙ<eₙ<aₙ₊₁ , a Fuchsian group Gy such that H²/Gy is homeomorphic to the Loch Ness Monster.