On arboreal Galois representations of rational functions

The action of the absolute Galois group Gal(Ksep/K)Gal(Ksep/K) of a global field K   on a tree T(ϕ,α)T(ϕ,α) of iterated preimages of α∈P1(K)α∈P1(K) under ϕ∈K(x)ϕ∈K(x) with deg⁡(ϕ)≥2deg⁡(ϕ)≥2 induces a homomorphism ρ:Gal(Ksep/K)→Aut(T(ϕ,α))ρ:Gal(Ksep/K)→Aut(T(ϕ,α)), which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes [5] and [6] about the size of the group G(ϕ,α):=imρ=lim←n⁡Gal(K(ϕ−n(α))/K). Specifically, we consider two cases for the pair (ϕ,α)(ϕ,α): (1) ϕ   is such that the sequence {an}{an} defined by a0=αa0=α and an=ϕ(an−1)an=ϕ(an−1) is periodic, and (2) ϕ commutes with a nontrivial Möbius transformation that fixes α.In the first case, we resolve a question posed by Jones [5] about the size of G(ϕ,α)G(ϕ,α), and taking K=QK=Q, we describe the Galois groups of iterates of polynomials ϕ∈Z[x]ϕ∈Z[x] that have the form ϕ(x)=x2+kxϕ(x)=x2+kx or ϕ(x)=x2−(k+1)x+kϕ(x)=x2−(k+1)x+k. When K=QK=Q and ϕ∈Z[x]ϕ∈Z[x], arboreal Galois representations are a useful tool for studying the arithmetic dynamics of ϕ  . In the case of ϕ(x)=x2+kxϕ(x)=x2+kx for k∈Zk∈Z, we employ a result of Jones [4] regarding the size of the group G(ψ,0)G(ψ,0), where ψ(x)=x2−kx+kψ(x)=x2−kx+k, to obtain a zero-density result for primes dividing terms of the sequence {an}{an} defined by a0∈Za0∈Z and an=ϕ(an−1)an=ϕ(an−1).In the second case, we resolve a conjecture of Jones [5] about the size of a certain subgroup C(ϕ,α)⊂Aut(T(ϕ,α))C(ϕ,α)⊂Aut(T(ϕ,α)) that contains G(ϕ,α)G(ϕ,α), and we present progress toward the proof of a conjecture of Jones and Manes [6] concerning the size of G(ϕ,α)G(ϕ,α) as a subgroup of C(ϕ,α)C(ϕ,α).