A new Gal(ℚ¯/ℚ)-invariant of dessins d'enfants

Abstract

We study the action of Gal ( Q ¯ / Q ) on the category of Belyĭ functions (finite étale covers of P Q ¯ 1 ∖ { 0 , 1 , ∞ } ). We describe a new combinatorial Gal ( Q ¯ / Q ) -invariant for Belyĭ functions whose monodromy cycle types above 0 and ∞ are the same. We use a version of our invariant to prove that Gal ( Q ¯ / Q ) acts faithfully on the set of Belyĭ functions whose monodromy cycle types above 0 and ∞ are the same; the proof of this result involves a version of Belyĭ's Theorem for meromorphic functions of odd degree. Using our invariant, we obtain that for all k < 2 2 3 and all positive integers N, there exists a positive integer n ⩽ N such that the set of degree n Belyĭ functions of a particular rational Nielsen class must split into at least Ω ( k N ) Galois orbits.