Fractal dimensions and profinite groups

TL;DR

This paper explores the fractal dimensions of path spaces in rooted trees and applies these findings to determine the Hausdorff and packing dimensions of closed subgroups in profinite groups, offering geometric insights and new proofs.

Contribution

It establishes the equivalence of Hausdorff and lower box dimensions, as well as packing and upper box dimensions, for certain ultrametric spaces derived from trees, and applies these results to profinite groups.

Findings

Hausdorff and lower box dimensions coincide for [S]

Packing and upper box dimensions coincide for [S]

Reproves a theorem on the Hausdorff dimension of closed subgroups in 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.