The Minkowski dimension of the image of an arboreal Galois representation

Abstract

We consider the Minkowski dimension of the arboreal Galois group $G_{f,\alpha}$ associated to a rational map $f:\mathbb{P}^1\to\mathbb{P}^1$ and a base point $\alpha\in\mathbb{P}^1(K)$. This is a subgroup of the automorphism group of the infinite $d$-ary rooted tree whose vertices are indexed by the backward orbit $f^{-\infty}(\alpha)$. We show that the Minkowski dimension exists for the profinite iterated monodromy groups $G_f^\mathrm{arith}$ and $G_f^\mathrm{geom}$, and that these two groups have the same dimension. We prove a dichotomy theorem stating that $G_f^\mathrm{arith}$ and $G_f^\mathrm{geom}$ are either the full tree automorphism group or else have non-maximal dimension. We identify several cases of interest in which dimension non-maximality $\overline{\dim}(G_{f,\alpha})<1$ holds, including the cases of postcritical base point, the case of periodic base point, the case in which $f$ is a nontrivial iterate, and the postcritically finite case. We identify several cases of interest in which dimension minimality $\dim(G_{f,\alpha})=0$ holds, including the power, Chebyshev, Latt\`es, and abelian cases. We formulate a conjecture on dimension minimality for quadratic polynomials, which if true would imply the $d=2$ case of a conjecture of Andrews-Petsche on abelian arboreal Galois groups.