Arboreal Galois groups of postcritically finite quadratic polynomials: The periodic case

TL;DR

This paper explicitly constructs the arboreal Galois group for certain quadratic polynomials with periodic critical points over fields of characteristic not 2, focusing on the case where the critical point is periodic.

Contribution

It provides an explicit construction of the arboreal Galois group for postcritically finite quadratic polynomials with periodic critical points.

Findings

Explicit construction of arboreal Galois groups

Applicable to fields of characteristic not 2

Focus on periodic critical points

Abstract

We provide an explicit construction of the arboreal Galois group for the postcritically finite polynomial $f(z) = z^2 +c$, where $c$ belongs to some arbitrary field of characteristic not equal to $2$. In this first of two papers, we consider the case that the critical point is periodic.