Restricted Hausdorff spectra of $q$-adic automorphisms

TL;DR

This paper determines the Hausdorff spectra of $q$-adic automorphisms and subclasses of self-similar groups, introduces a new formula for Hausdorff dimension, and provides examples of pro-$p$ groups with zero Hausdorff dimension, challenging existing conjectures.

Contribution

It offers a complete characterization of the Hausdorff spectra for $q$-adic automorphisms and subclasses, along with a new explicit formula for Hausdorff dimension and novel examples of pro-$p$ groups.

Findings

Complete Hausdorff spectrum of $q$-adic automorphisms determined.

New explicit formula for Hausdorff dimension of self-similar groups.

Examples of pro-$p$ groups with zero Hausdorff dimension provided.

Abstract

Firstly, we completely determine the self-similar Hausdorff spectrum of the group of $q$-adic automorphisms where $q$ is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products. Secondly, we provide for every prime $p$ the first examples of just infinite branch pro-$p$ groups with zero Hausdorff dimension in $\Gamma_p$, giving strong evidence against a well-known conjecture of Boston.