On a question of Ab\'ert and Vir\'ag

TL;DR

This paper investigates the Hausdorff dimension of normal subgroups within p-adic automorphism groups, providing counterexamples in general but positive results within self-similar groups, and generalizing previous theorems.

Contribution

It offers counterexamples to a conjecture about positive-dimensional normal subgroups and establishes positive results for self-similar groups, extending existing theorems.

Findings

Counterexamples of non-1-dimensional normal subgroups with positive Hausdorff dimension

Self-similar groups of positive dimension do not satisfy any group law

Generalization of Hausdorff dimension results to iterated wreath products

Abstract

Ab\'ert and Vir\'ag 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 Ab\'ert and Vir\'ag in the positive in this case. Along the way, we generalize a result of Ab\'ert and Vir\'ag 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.