The Minkowski dimension of the image of an arboreal Galois representation

Let $f:{\mathbb{P}}^1\to {\mathbb{P}}^1$ be a rational map of degree $d\ge 2$ defined over a number field $K$ and let $\alpha \in {\mathbb{P}}^1(K)$. We consider the lower and upper Minkowski dimensions of the arboreal Galois group $G_{f,\alpha }$ associated to the pair $(f,\alpha )$, which is naturally a subgroup of the automorphism group of the infinite $d$-ary rooted tree whose vertices are indexed by the backward orbit $f^{-\mathrm{\infty }}(\alpha )$. We state conjectures on the existence of Minkowski dimension, as well as proposed characterizations of cases in which it takes its minimal and maximal values. We establish basic cases in which the upper Minkowski dimension of $G_{f,\alpha }$ is not maximal, and we establish basic cases in which it is minimal. We show that abelian automorphism groups always have vanishing Minkowski dimension, and as a consequence, that one of our conjectures implies a conjecture of Andrews-Petsche on pairs $(f,\alpha )$ with abelian arboreal Galois group.