Hausdorff dimensions in p-adic analytic groups

Klopsch, Benjamin and Thillaisundaram, Anitha and Zugadi-Reizabal, Amaia (2019) Hausdorff dimensions in p-adic analytic groups. Israel Journal of Mathematics . ISSN 0012 -2172

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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.

Keywords:pro-p groups, Hausdorff dimension, filtration series, p-adic analytic groups, soluble groups
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:31157
Deposited On:07 Mar 2018 09:33

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