Journal of Group Theory
Editor-in-Chief: Parker, Christopher W. / Wilson, John S.
Managing Editor: Khukhro, Evgenii I. / Kramer, Linus
6 Issues per year
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Spinal groups: semidirect product decompositions and Hausdorff dimension
Let p be a prime, and let Γ denote a Sylow pro-p subgroup of the group of automorphisms of the p-adic rooted tree. By using probabilistic methods, Abért and Virág [J. Amer. Math. Soc. 18: 157–192, 2005] have shown that the Hausdorff dimension of a finitely generated closed subgroup of Γ can be any number in the interval [0, 1]. In the case p = 2, Siegenthaler has provided examples of subgroups of Γ of transcendental Hausdorff dimension which arise as closures of spinal groups. In this paper, we show that the situation is completely different for p > 2, since the Hausdorff 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 = 2.) Furthermore, we determine the set S of all rational numbers that appear as Hausdorff 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.