Hausdorff dimensions in p-adic analytic groups

Abstract

Let G be a finitely generated pro-p group, equipped with the p-power series $P:{G_i} = {G^{{P^i}}}$, i ∈ ℕ₀. The associated metric and Hausdorff dimension function $hdim_G^P:\{{X|X\subseteq{G}\}}\rightarrow[0,1]$ give rise to

$$hspe{c^P}(G) = \{ h\dim _G^P(H)|H \leqslant G\} \subseteq [0,1],$$

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.