Hausdorff dimensions in p-adic analytic groups

Abstract

Let G be a finitely generated pro-p group, equipped with the p-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum 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 the finiteness of the Hausdorff spectrum 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 odd primes p, that every countably based pro-p group G with an open subgroup mapping onto 2 copies of the p-adic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.