Hausdorff dimensions in p-adic analytic groups

Abstract

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.

A pro-p group with infinite normal Hausdorff spectra

“…The concept of Hausdorff dimension has led to interesting applications in the context of profinite groups; see [4] and the references given therein. Let G be a countably based infinite profinite group and consider a filtration series S of G, that is, a descending chain G = G 0 ⊇ G 1 ⊇ .…”

“…Throughout we will be concerned with pro-p groups, where p denotes an odd prime; in Appendix A we indicate how our results extend to p = 2. We recall that even for well structured groups, such as p-adic analytic pro-p groups G, the Hausdorff dimension function and the Hausdorff spectrum of G are known to be sensitive to the choice of S; compare [4]. However, for a finitely generated pro-p group G there are natural choices for S, such as the p-power series P, the Frattini series F, the lower p-series L and the modular dimension subgroup series D; see Section 2.…”

“…We continue to use the notation set up in Section 3 and establish that ξ = hdim P G (Z) = 1 /3 and η = hdim P G (H) = 1, with respect to the p-power series P. In view of Corollary 4.2 this proves Theorem 1.1 for the p-power series. Indeed, hdim P G (H) = 1 is already a consequence of [4,Prop. 4.2].…”

“…First suppose that S is one of the filtration series P, D, F on G. By Remark 6.6, the group H has strong Hausdorff dimension 1 in G with respect to S. As every finitely generated subgroup of H is finite, it follows from [4,Thm. 5.4] that hspec S (G) = [0, 1].…”

“…Choose m, n ∈ N such that 1 ≤ m < p n /2 and In a similar, but much more straightforward way, we see that ZK has strong Hausdorff dimension hdim L G (ZK) = 2 5 + 2 5 m p n + 1 5 = 1 5 3 + 2m /p n . An application of [4,Thm. 5.4]…”