The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with s ends and exactly m of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\le s$, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group ${\mathrm{\Gamma }}_{m,s}$ such that the quotient space $\mathbb{H}/{\mathrm{\Gamma }}_{m,s}$ is homeomorphic to $S_{m,s}$ and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

中文翻译：

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具有无限属的几何肖特基群和非紧致双曲曲面

非紧致黎曼曲面的拓扑类型由其端部空间和无限大的端部决定。本文针对非紧致黎曼曲面${\mathrm{小号}}_{米，s}$带有s末端，并且恰好有m个具有无限的属，这样$米，s\in \mathbb{ñ}$ 和 $\mathrm{1个}<米\le s$，我们给出了Fuchsian（几何肖特基）群的无限生成子集的精确描述 ${\mathrm{\Gamma }}_{米，s}$ 这样商空间 $\mathbb{H}/{\mathrm{\Gamma }}_{米，s}$ 是同胚的 ${\mathrm{小号}}_{米，s}$并具有无限的面积。对于此构造，我们展示了一个具有无限数量边的双曲多边形，并给出了Mobius变换的集合，这些变换成对地标识了边。