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, Lattès, 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.