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On a question of Abért and Virág

Jorge Fariña-Asategui

TL;DR

The paper investigates how Abért and Virág's dimension-theoretic results extend from 1-dimensional to higher-dimensional subgroups of $W_p$ and more generally $W_H$. It provides a negative answer in general by constructing counterexamples $G_K$ with positive Hausdorff dimension but lacking weakly branch structure, while establishing a positive theory for self-similar positive-dimensional subgroups: these are weakly branch, have full Hausdorff spectrum, and force nontrivial normal subgroups to have maximal dimension. A key technical advance is a generalized perfection result for Hausdorff dimension in $W_H$, using a streamlined bound on abelianizations instead of previous sublinear generators bounds. The work thereby delineates when positive-dimensional actions can occur in self-similar groups, links dimension to branch structure, and supplies both spectral and centrality consequences that sharpen our understanding of Hausdorff dimensions in iterated wreath products.

Abstract

Abért and Virág proved in 2005 that the Hausdorff dimension of a non-trivial normal subgroup of a level-transitive 1-dimensional subgroup of the group of $p$-adic automorphisms $W_p$ is always 1. They further asked whether the same holds replacing 1-dimensional with positive dimensional. On the one hand, we provide a negative answer in general by giving counterexamples where the non-trivial normal subgroups are not all 1-dimensional. Furthermore, these counterexamples are pro-$p$ subgroups of $W_p$ with positive Hausdorff dimension in $W_p$ but with non-trivial center, and thus not weakly branch. On the other hand, we restrict ourselves to the class of self-similar groups and answer the question of Abért and Virág in the positive in this case. Along the way, we generalize a result of Abért and Virág on the closed subgroups of $W_p$ being perfect in the sense of Hausdorff dimension to closed subgroups of any iterated wreath product $W_H$ and show that self-similar positive-dimensional subgroups of $W_H$ do not satisfy any group law.

On a question of Abért and Virág

TL;DR

The paper investigates how Abért and Virág's dimension-theoretic results extend from 1-dimensional to higher-dimensional subgroups of and more generally . It provides a negative answer in general by constructing counterexamples with positive Hausdorff dimension but lacking weakly branch structure, while establishing a positive theory for self-similar positive-dimensional subgroups: these are weakly branch, have full Hausdorff spectrum, and force nontrivial normal subgroups to have maximal dimension. A key technical advance is a generalized perfection result for Hausdorff dimension in , using a streamlined bound on abelianizations instead of previous sublinear generators bounds. The work thereby delineates when positive-dimensional actions can occur in self-similar groups, links dimension to branch structure, and supplies both spectral and centrality consequences that sharpen our understanding of Hausdorff dimensions in iterated wreath products.

Abstract

Abért and Virág proved in 2005 that the Hausdorff dimension of a non-trivial normal subgroup of a level-transitive 1-dimensional subgroup of the group of -adic automorphisms is always 1. They further asked whether the same holds replacing 1-dimensional with positive dimensional. On the one hand, we provide a negative answer in general by giving counterexamples where the non-trivial normal subgroups are not all 1-dimensional. Furthermore, these counterexamples are pro- subgroups of with positive Hausdorff dimension in but with non-trivial center, and thus not weakly branch. On the other hand, we restrict ourselves to the class of self-similar groups and answer the question of Abért and Virág in the positive in this case. Along the way, we generalize a result of Abért and Virág on the closed subgroups of being perfect in the sense of Hausdorff dimension to closed subgroups of any iterated wreath product and show that self-similar positive-dimensional subgroups of do not satisfy any group law.

Paper Structure

This paper contains 11 sections, 16 theorems, 42 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Groups acting on rooted trees
  4. Hausdorff dimension
  5. A counterexample in the general setting
  6. Proof of \ref{['Theorem: counterexample']}
  7. Further properties of the groups $G_K$
  8. Perfectness of the Hausdorff dimension
  9. The self-similar case
  10. Hausdorff dimension of rigid stabilizers
  11. Proof of the main results

Key Result

Theorem 1.1

Let $G\le_c W_p$ be a level-transitive closed subgroup with Hausdorff dimension 1 in $W_p$. Then any non-trivial normal closed subgroup $N\trianglelefteq_c G$ has Hausdorff dimension 1 in $G$.

Theorems & Definitions (24)

  • Theorem 1.1: see AbertVirag
  • Theorem A
  • Theorem 1.2: see AbertVirag
  • Theorem B
  • Theorem C
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Lemma 2.1: see FiniteSpectrum
  • Theorem 2.2: see RestrictedSpectra
  • ...and 14 more