Large arboreal Galois representations.

Given a field K , a polynomial f ∈ K [ x ] of degree d , and a suitable element t ∈ K , the set of preimages of t under the iterates f ∘ 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 d ≥ 20 is even and K = Q we exhibit examples of polynomials 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 ∈ F [ x ] in which the corresponding Galois groups are the monodromy groups of the ramified covers f ∘ n : P F 1 → P F 1. [ABSTRACT FROM AUTHOR]