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

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
