The arboreal finite index problem

Let $K$ be a global field, $f \in K(x)$, and $b \in K$. Let $K_n$ be the splitting field of $f^n(x)-b$, where $f^n$ denotes composition. The projective limit of the groups $Gal(K_n/K)$ embeds into the automorphism group of an infinite rooted tree. A major problem in arithmetic dynamics is to study when the index is finite. The motivation is to find a dynamical analogue of Serre’s celebrated open image theorem. I solve this problem for cubic polynomials over function fields by proving a list of necessary and sufficient conditions for finite index. For number fields, the proof is conditional on abc and a form of Vojta’s conjecture.