Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hausdorff dimension of the generalized Brunner-Sidki-Vieira Groups

Jorge Fariña-Asategui, Mikel E. Garciarena

Abstract

We compute the Hausdorff dimension of the closure of the generalized Brunner-Sidki-Vieira group acting on the $m$-adic tree for $m\ge 2$, providing the first examples of self-similar topologically finitely generated closed subgroups of transcendental Hausdorff dimension in the group of $m$-adic automorphisms.

The Hausdorff dimension of the generalized Brunner-Sidki-Vieira Groups

Abstract

We compute the Hausdorff dimension of the closure of the generalized Brunner-Sidki-Vieira group acting on the -adic tree for , providing the first examples of self-similar topologically finitely generated closed subgroups of transcendental Hausdorff dimension in the group of -adic automorphisms.
Paper Structure (7 sections, 11 theorems, 86 equations, 1 figure)

This paper contains 7 sections, 11 theorems, 86 equations, 1 figure.

Table of Contents

  1. introduction
  2. Preliminaries and the generalized Brunner-Sidki-Vieira group
  3. The full automorphism group
  4. Subgroups of $\mathrm{Aut}~\mathcal{T}$
  5. Hausdorff dimension
  6. The generalized Brunner-Sidki-Vieira group $H$
  7. Proof of the main result

Key Result

Theorem A

Let $H$ be the generalized Brunner-Sidki-Vieira group acting on the $m$-adic tree. Then the Hausdorff dimension of its closure in $\Gamma_m$ is where the parameter $\tau$ is defined as

Figures (1)

  • Figure 1: The generators of the generalized Brunner-Sidki-Vieira group acting on the 4-adic tree.

Theorems & Definitions (18)

  • Theorem A
  • Theorem 2.1: see Jorge
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4: see BSV2
  • Proposition 2.5: see BSV2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 8 more