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 ƒ ∈ K[x] be of the form ƒ(x) = xq + c, where q = pr, r ≥ 1, is a power of the prime number p, and let β ∈ K. For all n ∈ N ∪∞, the Galois groups Gn(β) = Gal(K(ƒ−n(β))/K(β)) embed into [Cq]n, the n-fold wreath product of the cyclic group Cq. We show that if ƒ is not isotrivial, then [[Cq]∞ : G∞ (β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if ƒ1(x) = xq + c1 and ƒ2(x) = xq + c2 are two such distinct polynomials, then the fields ∪n=1∞ K(ƒ1−n(β)) and ∪n=1∞ K(ƒ2−n(β)) are disjoint over a finite extension of K. [ABSTRACT FROM AUTHOR]