If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients Gn = G/ StabG(n) for every n. If G is defined by the vector e = (e1, . . . , ep−1) ∈ F p−1 p , the determination of the order of Gn is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.
Let G be a finitely generated pro-p group, equipped with the ppower series P : G i = G p i , i ∈ N 0 . The associated metric and Hausdorff dimension function hdim P G :, the Hausdorff spectrum of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, hspec P (G) consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of G.Conversely, it is a long-standing open question whether |hspec P (G)| < ∞ implies that G is p-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that G is soluble.Furthermore, we explore the problem and related questions also for other filtration series, such as the lower p-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for p > 2, that every countably based pro-p group G with an open subgroup mapping onto Z p ⊕ Z p admits a filtration series S such that hspec S (G) contains an infinite real interval.
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.
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