Spinal groups: semidirect product decompositions and Hausdorff dimension

Abstract

Abstract. Let p be a prime, and let G denote a Sylow pro-p subgroup of the group of automorphisms of the p-adic rooted tree. By using probabilistic methods, Abért and Virág [2] have shown that the Hausdor¤ dimension of a finitely generated closed subgroup of G can be any number in the interval ½0; 1. In the case p ¼ 2, Siegenthaler has provided examples of subgroups of G of transcendental Hausdor¤ dimension which arise as closures of spinal groups. In this paper, we show that the situation is completely di¤erent for p > 2, since the Hausdor¤ dimension of the closure of a spinal group is always a rational number in that case. (Here, spinal groups are constructed over spines with one vertex at every level, as in the case p ¼ 2.) Furthermore, we determine the set S of all rational numbers that appear as Hausdor¤ dimensions of spinal groups. A key ingredient in our approach to this problem is provided by a general procedure for decomposing spinal groups as a semidirect product, which allows us to reduce to the case of 2-generator spinal groups.

GGS-groups: Order of congruence quotients and Hausdorff dimension

“…The determination of the Hausdorff dimension of closed subgroups of Γ has received special attention in the last few years (see [2,9,16,17]). The most natural choice is to calculate the Hausdorff dimension with respect to the metric induced by the filtration of Γ given by the level stabilizers Stab Γ (n).…”

On the finitely generated Hausdorff spectrum of spinal groups

“…We choose an infinite path P = (p n ) n≥0 , starting at the root. Following the definition in [7], if we consider, for every n ≥ 1, and immediate descendant s n of p n−1 not lying in P , we say that the sequence S = (s n ) n≥1 is a spine of the tree T . An element g ∈ G is a spinal automorphism if the support of g is contained in S.…”

“…In [12] Siegenthaler explicitly computed the Hausdorff dimension of level-transitive spinal groups. Fernandez-Alcober and Zugadi-Reizabal [7] give an explicit set of values for the dimension of certain spinal groups. In his thesis [8] B. Klopsch has shown that branch groups have full subgroup Hausdorff spectrum [0, 1].…”

Hausdorff dimension of the second Grigorchuk group

“…This was first applied by Abercrombie [1], Barnea and Shalev [3] in the more general setting of profinite groups. Key results concerning the Hausdorff dimension of Grigorchuk-type groups were established in [2,8,9,11,21,22]. It is proved in [14] that the closure of the first Grigorchuk group has Hausdorff dimension 5/8, however the computation of the Hausdorff dimension of the second Grigorchuk group does not appear to be recorded anywhere in the literature.…”