Arboreal Galois groups for quadratic rational functions with colliding critical points

TL;DR

This paper investigates the structure of arboreal Galois groups associated with quadratic rational functions with colliding critical points, providing new descriptions, proofs, and criteria for maximality of these groups.

Contribution

It offers a new description of Pink's subgroup, a new proof of Pink's theorem, and necessary and sufficient conditions for the arboreal Galois group to be maximal.

Findings

New description of Pink's subgroup $M_{ ext{ell}}$

A new proof of Pink's theorem

Criteria for the Galois group to be the full subgroup

Abstract

Let $K$ be a field, and let $f\in K(z)$ be rational function. The preimages of a point $x_0\in P^1(K)$ under iterates of $f$ have a natural tree structure. As a result, the Galois group of the resulting field extension of $K$ naturally embeds into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup $M_{\ell}$ that this so-called arboreal Galois group $G_{\infty}$ must lie in if $f$ is quadratic and its two critical points collide at the $\ell$-th iteration. After presenting a new description of $M_{\ell}$ and a new proof of Pink's theorem, we state and prove necessary and sufficient conditions for $G_{\infty}$ to be the full group $M_{\ell}$.