A new $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant of dessins d'enfants

Abstract

We study the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the category of Belyi functions (finite, \'{e}tale covers of $\mathbb{P}^1_{\overline{\mathbb{Q}}}\setminus \{0,1,\infty\}$). We describe a new combinatorial $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant for whose monodromy cycle types above $0$ and $\infty$ are the same. We use a version of our invariant to prove that $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of Belyi functions whose monodromy cycle types above 0 and $\infty$ are the same; the proof of this result involves a version of Belyi's Theorem for odd degree morphisms. Using our invariant, we obtain that for all $k < 2^{\sqrt{\frac{2}{3}}}$ and all positive integers $N$, there is an $n \le N$ such that the set of degree $n$ Belyi functions of a particular rational Nielsen class must split into at least $\Omega\left(k^{\sqrt{N}}\right)$ Galois orbits.