Large arboreal Galois representations

TL;DR

This paper investigates conditions under which the Galois group acts maximally on preimage trees generated by polynomial iterates, providing new examples and partial confirmations of existing conjectures, especially over the rationals.

Contribution

It constructs examples of polynomials of every even degree with maximal Galois action on preimage trees over 0, advancing the understanding of arboreal Galois representations.

Findings

Existence of polynomials of every even degree with full Galois action over 0.

Partial verification of Odoni's conjecture for specific cases.

Analysis of Galois groups as monodromy groups of ramified covers.

Abstract

Given a field $K$, a polynomial $f \in K[x]$, and a suitable element $t \in K$, the set of preimages of $t$ under the iterates $f^{\circ n}$ carries a natural structure of a $d$-ary tree. We study conditions under which the absolute Galois group of $K$ acts on the tree by the full group of automorphisms. When $K=\mathbb{Q}$ we exhibit examples of polynomials of every even degree with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of $K=F(t)$ and $f \in F[x]$ in which the corresponding Galois groups are the monodromy groups of the ramified covers $f^{\circ n}: \mathbb{P}^1_F \to \mathbb{P}^1_F$.