A Unique Chief Series in the arboreal Galois Group of Belyi Maps

TL;DR

This paper characterizes the normal subgroups forming a unique chief series in the arboreal Galois groups of Belyi maps and explores discriminants of polynomial iterates to identify special properties of postcritically finite polynomials.

Contribution

It provides a complete description of the normal subgroups in these Galois groups and links discriminant properties to the structure of PCF polynomials.

Findings

Normal subgroups form a unique chief series

Discriminant calculations predict perfect squares in fields

Identifies a new PCF cubic polynomial with the same Galois group

Abstract

We give a complete description of the normal subgroups of arboreal Galois groups of Belyi maps. The normal groups form a unique chief series. We also carefully compute the discriminate of the iterate of a polynomial minus an algebraic number, which allows us to predict when a such discriminate is a perfect square in the base field or intermediate field for a postcritically finite polynomials (PCF). As a consequence we are able to find another PCF cubic polynomial that has the same arboreal Galois group as the one of Belyi maps.