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An algorithm to compute the Hausdorff dimension of regular branch groups

Jorge Fariña-Asategui

TL;DR

The paper solves the problem of computing the Hausdorff dimension of closures of regular branch groups by deriving an explicit algorithm that expresses $\mathrm{hdim}_{W_H}(\overline{G})$ in terms of a branch structure, namely $\mathrm{hdim}_{W_H}(\overline{G})=\frac{1}{\log|H|}\bigl(\log|G_1|-S_G(1/m)\bigr)$. It formalizes the relevant objects (level stabilizers, sections, self-similarity, and regular-branchness) and introduces the sequence $s_n(G)$ and its generating function $S_G(x)$ to capture growth via $\log|G_n|=\alpha m^n+\beta$ for large $n$, enabling a computational approach. The method is implemented in GAP and demonstrated on GGS-groups acting on the 4-adic tree, yielding explicit dimension values depending on defining vectors, and providing a practical criterion to discard non-viable branch structures. Overall, the work furnishes a concrete, computable framework for assessing fractal dimensions of a broad class of profinite groups arising from self-similar actions on rooted trees, with direct applications to GGS-groups and related generalized regular branch groups.

Abstract

An explicit algorithm is given for the computation of the Hausdorff dimension of the closure of a regular branch group in terms of an arbitrary branch structure. We implement this algorithm in GAP and apply it to a family of GGS-groups acting on the 4-adic tree.

An algorithm to compute the Hausdorff dimension of regular branch groups

TL;DR

The paper solves the problem of computing the Hausdorff dimension of closures of regular branch groups by deriving an explicit algorithm that expresses in terms of a branch structure, namely . It formalizes the relevant objects (level stabilizers, sections, self-similarity, and regular-branchness) and introduces the sequence and its generating function to capture growth via for large , enabling a computational approach. The method is implemented in GAP and demonstrated on GGS-groups acting on the 4-adic tree, yielding explicit dimension values depending on defining vectors, and providing a practical criterion to discard non-viable branch structures. Overall, the work furnishes a concrete, computable framework for assessing fractal dimensions of a broad class of profinite groups arising from self-similar actions on rooted trees, with direct applications to GGS-groups and related generalized regular branch groups.

Abstract

An explicit algorithm is given for the computation of the Hausdorff dimension of the closure of a regular branch group in terms of an arbitrary branch structure. We implement this algorithm in GAP and apply it to a family of GGS-groups acting on the 4-adic tree.

Paper Structure

This paper contains 8 sections, 5 theorems, 20 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groups acting on rooted trees
  4. Hausdorff dimension
  5. Proof of the main result
  6. An application: GGS-groups acting on the 4-adic tree
  7. Hausdorff dimension
  8. Discarding branch structures

Key Result

Theorem 1

Let $G\le W_H\le \mathrm{Aut}~T_m$ be a regular branch group branching over a finite-index subgroup $K\le G$. Then the closure $\overline{G}$ of $G$ in $W_H$ is regular branch over its $\Omega(G:K)$th level stabilizer. Therefore the Hausdorff dimension of $\overline{G}$ in $W_H$ is Furthermore, the logarithmic indices of the quotients $G_n$ are given by for all $n\ge \Omega(G:K)$, where

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2.1: see JorgeSpectra
  • Lemma 3.1
  • proof
  • proof : Proof of \ref{['Theorem: algorithm Hausdorff dimension']}
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof