On arboreal Galois representations of rational functions

The action of the absolute Galois group Gal(K-sep/K) of a global field K on a tree T(phi, alpha) of iterated preimages of alpha is an element of P-1(K) under phi is an element of K(x) with deg(pi) >=> 2 induces a homomorphism rho : Gal(K-sep/K) -> Aut(T(phi, alpha)), which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes [5,6] about the size of the group G(phi, alpha) := im rho = lim(<- n) Gal(K(phi(-n)(alpha))/K). Specifically, we consider two cases for the pair (phi, alpha): (1) phi is such that the sequence {a(n)} defined by a(0) = alpha and a(n) = phi(a(n-1)) is periodic, and (2) phi commutes with a nontrivial Mobius transformation that fixes alpha. In the first case, we resolve a question posed by Jones [5] about the size of G(phi, alpha), and taking K = Q, we describe the Galois groups of iterates of polynomials' phi is an element of Z[x] that have the form phi(x) = x(2) + kx or phi(x) = x(2) - (k + 1)x + k. When K = Q and phi is an element of Z[x], arboreal Galois representations are a useful tool for studying the arithmetic dynamics of phi. In the case of phi(x) = x(2) + kx for k is an element of Z, we employ a result of Jones [4] regarding the size of the group G(psi, 0), where psi(x) = x(2) - kx + k, to obtain a zero-density result for primes dividing terms of the sequence {a(n)} defined by a(0) is an element of Z and a(n) = phi(a(n-1)). In the second case, we resolve a conjecture of Jones [5] about the size of a certain subgroup C(phi, alpha) subset of Aut(T(phi, alpha)) that contains G(phi, alpha), and we present progress toward the proof of a conjecture of Jones and Manes [6] concerning the size of G(phi, alpha) as a subgroup of C(phi, alpha). (C) 2015 Elsevier Inc. All rights reserved.