Overgroups of the arboreal representation of PCF polynomial

Abstract

Consider a number field $K$ and a rational function $f$ of degree greater than 1 over $K$. By taking preimages of $\hat{I}\pm \in K$ under successive iterates of $f$, an infinite $d$-ary tree $T_{\infty }$ rooted at $\hat{I}\pm$ can be constructed. An edge is assigned between two preimages $x$ and $y$ if $f(x)=y$. The absolute Galois group of $K$, acting on $T_{\infty }$ through tree automorphisms, generates a subgroup ${\text{Gal}}_f^{\infty }(\hat{I}\pm )$ in the group of all automorphisms of $T_{\infty }$, $\text{Aut}(T_{\infty })$. We have discovered a new class of natural overgroups in which the image of the Galois representation attached to a PCF polynomial must reside. Moreover, we have found that the image of the Galois representation of a new PCF polynomial is isomorphic to one of these overgroups. We also investigate the structure of these overgroups for specific maps, such as normalized dynamical Belyi polynomials, and show that the normal subgroups of these overgroups form a unique chief series. This allows us to bound the number of generators through group-theoretic analysis.