Profinite Iterated Monodromy Groups of Unicritical Polynomials

Abstract

Let $f(x)=a{x}^d+b\in K[x]$ be a unicritical polynomial with degree $d\ge 2$ which is coprime to $\mathrm{char}K$. We provide an explicit presentation for the profinite iterated monodromy group of $f$, analyze the structure of this group, and use this analysis to determine the constant field extension in $K(f^{-\infty }(t))/K(t)$.