Klopsch, Benjamin, Thillaisundaram, Anitha and ZugadiReizabal, Amaia
(2019)
Hausdorff dimensions in padic analytic groups.
Israel Journal of Mathematics, 231
(1).
pp. 123.
ISSN 00212172
Full content URL: https://doi.org/10.1007/s118560191852z
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Abstract
Let G be a finitely generated prop group, equipped with the ppower 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 padic 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 longstanding open question whether the finiteness of the Hausdorff spectrum implies that G is padic 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 pseries, 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 prop group G with an open subgroup mapping onto 2 copies of the padic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.
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