On a question of Abért and Virág

Abért and Virág proved in 2005 [J. Amer. Math. Soc. 18 (2005), 157–192] that the Hausdorff dimension of a non-trivial normal subgroup of a level-transitive 1-dimensional subgroup of the group of p p -adic automorphisms W p 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 p subgroups of W p W_p with positive Hausdorff dimension in W p 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 W_p being perfect in the sense of Hausdorff dimension to closed subgroups of any iterated wreath product W H W_H and show that self-similar positive-dimensional subgroups of W H W_H do not satisfy any group law.