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

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.