Fractal dimensions and profinite groups

Abstract

Let $T$ be a finitely branching rooted tree such that any node has at least two successors. The path space $[T]$ is an ultrametric space: for distinct paths $f,g$ let $d(f,g)= 1/|T_n|$, where $T_n$ denotes the $n$-th level of the tree, and $n$ is largest such that $f(n)= g(n)$. Let $S$ be a subtree of $T$ without leaves that is level-wise uniformly branching, in the sense that the number of successors of a node only depends on its level. We~show that the Hausdorff and lower box dimensions coincide for~$[S]$, and the packing and upper box dimensions also coincide. We give geometric proofs, as well as proofs based on the point-to-set principles. We use the first result to reprove a theorem of Barnea and Shalev on the Hausdorff dimension of closed subgroups of a profinite group $G$, referring only on the geometric structure of the closed subgroup in the canonical path space given by an inverse system for $G$. We obtain an analogous theorem for the packing dimension.