Large arboreal Galois representations

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 $f^{\circ n}:{\mathbb{P}}_F^1\to {\mathbb{P}}_F^1$.