Dessins D’enfants and Some Holomorphic Structures on the Loch Ness Monster

Abstract

Abstract 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.