The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves.

中文翻译：

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Dessins D'enfants 和尼斯湖水怪上的一些全纯结构

紧黎曼曲面上的婴儿设计理论是紧致可定向曲面上的二分映射，是用于研究紧黎曼曲面与有理数域的绝对伽罗瓦群之间的分支覆盖的组合对象。在本文中，我们展示了该理论如何自然地扩展到非紧致可定向表面，特别是，我们观察到尼斯湖水怪（LNM；具有正好一个末端的无限属的表面）允许无限多的规则dessins d'婴儿（手性或自反）。此外，我们研究了 LNM 上的不同全纯结构，这些全纯结构来自紧黎曼曲面的同调覆盖，以及无限超椭圆曲线和无限超椭圆曲线。