Finite index theorems for iterated Galois groups of unicritical polynomials

Let K be the function field of a smooth irreducible curve defined over Q¯. Let f ∈ K[x] be of the form f(x) = xq + c, where q = pr, r ≥ 1, is a power of the prime number p, and let β ∈ K̄. For all n ∈ ℕ ∪ {∞}, the Galois groups Gn(β) = Gal(K(f-n(β))/K(β)) embed into [Cq]n, the n-fold wreath product of the cyclic group Cq. We show that if f is not isotrivial, then [[Cq]∞ : G∞(β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if f1(x) = xq + c1 and f2(x) = xq + c2 are two such distinct polynomials, then the fields ∪∞n=1 K(f-n1 (β)) and ∪∞n=1 K(f-n2 (β)) are disjoint over a finite extension of K.