零扭曲紧槽表面的抛物性和尼斯湖水怪的均匀化
Parabolicity of zero-twist tight flute surfaces and uniformization of the Loch Ness monster
摘要：
我们研究了零扭曲的长笛表面，我们关联到他们中的每一个一个序列的正实数$\mathbf{x}=(x_{n})_{n\in\mathbb{N}_{0}}$，与扭转免费Fuchsian群$Γ_{\mathbf{x}}$，使凸核心的$\mathbb{H}^2/Γ_{\mathbf{x}}$是等距的零扭曲紧长笛表面$S_{\mathbf{x}}$。证明了Fuchsian群$Γ_{\mathbf{x}}$是第一类群当且仅当级数$\sum x_{n}$发散.作为Basmajian，Hakobian和{Kazari}{ć}最近工作的结果，我们得到了零捻槽纹曲面$S_{\mathbf{x}}$是抛物型的当且仅当$\sum x_{n}$发散.此外，我们提出了一个不可数家庭的双曲曲面同胚的尼斯湖水怪。更准确地说，我们将每个序列$\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}$（其中$y_{n}=(a_{n},b_{n},c_{n},d_{n},e_{n})\in \mathbb{R}^5$和$a_n\leq b_n \leq c_n \leq d_n \leq e_n \leq a_{n+1}$）与Fuchsian群$G_{\mathbf{y}}$相关联，使得$\mathbb{H}^2/G_{\mathbf{y}}$同胚于尼斯湖水怪。
英文摘要：
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 $Γ_{\mathbf{x}}$ such that the convex core of $\mathbb{H}^2/Γ_{\mathbf{x}}$ is isometric to a zero-twist tight flute surface $S_{\mathbf{x}}$. Moreover, we prove that the Fuchsian group $Γ_{\mathbf{x}}$ 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 $a_n\leq b_n \leq c_n \leq d_n \leq e_n \leq a_{n+1}$, a Fuchsian group $G_{\mathbf{y}}$ such that $\mathbb{H}^2/G_{\mathbf{y}}$ is homeomorphic to the Loch Ness Monster.
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