The Minkowski dimension of the image of an arboreal Galois representation
Chifan Leung, Clayton Petsche
TL;DR
The paper develops a framework to measure the size of arboreal Galois representations G_{f,α} via Minkowski (Hausdorff) dimensions on automorphism groups of infinite d-ary rooted trees. It establishes fundamental dimensional bounds, proves 0-dimensionality for abelian images and several small-image families (power, Chebyshev, Lattès maps), and analyzes how dimensions behave under base extension, conjugacy, and iterate-sharing. It also proves non-large image results for post-critical and periodic base points, and shows that genus-1 cases yield always zero dimension, linking to torsion-based arguments. Collectively, these results support conjectures characterizing when arboreal Galois representations have maximal, minimal, or intermediate dimensions, and they connect to broader questions such as the Andrews-Petsche conjecture on abelian arboreal Galois groups.
Abstract
Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree $d\geq2$ defined over a number field $K$ and let $α\in\mathbb{P}^1(K)$. We consider the lower and upper Minkowski dimensions of the arboreal Galois group $G_{f,α}$ associated to the pair $(f,α)$, which is naturally a subgroup of the automorphism group of the infinite $d$-ary rooted tree whose vertices are indexed by the backward orbit $f^{-\infty}(α)$. 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 $G_{f,α}$ 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 $(f,α)$ with abelian arboreal Galois group.