Dessins d'enfants and some holomorphic structures on the Loch Ness Monster

TL;DR

This paper extends the theory of dessins d'enfants to non-compact surfaces, specifically demonstrating that the Loch Ness monster surface admits infinitely many regular dessins d'enfants and exploring related holomorphic structures.

Contribution

It introduces a natural extension of dessins d'enfants to non-compact surfaces and shows the Loch Ness monster admits infinitely many regular dessins d'enfants, also studying associated holomorphic structures.

Findings

Loch Ness monster admits infinitely many regular dessins d'enfants

Extended dessins d'enfants theory to non-compact surfaces

Analyzed holomorphic structures from homology covers and infinite curves

Abstract

The classical theory of dessin d'enfants, 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 (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 Loch Ness monster, which come from homology covers of compact Riemann surfaces, infinite hyperelliptic and infinite superelliptic curves.