Weighted Trees with Primitive Edge Rotation Groups

Abstract

Let R, S ∈ ℂ[x] be two coprime polynomials of the same degree with prescribed multiplicities of their roots. A classical problem of number theory actively studied during the last half-century is, What could be the minimum degree of the difference T = R − S? The theory of dessins d’enfants implies that such a minimum is attained if and only if the rational function f = R/T is a Belyi function for a bicolored plane map all of whose faces except the outer one are of degree 1. Such maps are called weighted trees, since they can be conveniently represented by plane trees whose edges are endowed with positive integral weights.

It is well known that the absolute Galois group (the automorphism group of the field $\overline{\mathbb{Q}}$ of algebraic numbers) acts on dessins. An important invariant of this action is the edge rotation group, which is also the monodromy group of a ramified covering corresponding to the Belyi function. In this paper, we classify all weighted trees with primitive edge rotation groups. There are, up to the color exchange, 184 such trees, which are subdivided into (at least) 85 Galois orbits and generate 34 primitive groups (the highest degree is 32). This result may also be considered as a contribution to the classification of covering of genus 0 with primitive monodromy groups in the framework of the Guralnick–Thompson conjecture.