Finite index theorems for iterated Galois groups of unicritical polynomials

Let $K$ be the function field of a smooth, irreducible curve defined over $\bar{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$isapoweroftheprimenumber$p$,andlet$\beta\in \overline{K}$\mathrm{.}Forall$n\in$\mathbb{N}$\cup{\infty}$,theGaloisgroups$G_n(\beta)=\mathop{\rm{Gal}}(K(f^{-n}(\beta))/K(\beta))$embedinto$[C_q]^n$, the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is not isotrivial, then $[[C_q{]}^{\mathrm{\infty }}:G_{\mathrm{\infty }}(\beta )]<\mathrm{\infty }$ unless $\beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$ and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields ${\bigcup }_{n=1}^{\mathrm{\infty }}K(f_1^{-n}(\beta ))$ and $\bigcup_{n=1}^\infty K(f_2^{-n}(\beta))$aredisjointoverafiniteextensionof$K$.