Hausdorff dimensions in p-adic analytic groups
(2019) In Israel Journal of Mathematics 231.- Abstract
- Let G be a finitely generated pro-p group, equipped with the p-power series P:Gi=GPi, i ∈ ℕ0. The associated metric and Hausdorff dimension function hdimGP:{X|X⊆G}→[0,1] give rise to hspecP(G)={hdimGP(H)|H≤G}⊆[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, hspecP(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 |hspecP(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... (More)
- Let G be a finitely generated pro-p group, equipped with the p-power series P:Gi=GPi, i ∈ ℕ0. The associated metric and Hausdorff dimension function hdimGP:{X|X⊆G}→[0,1] give rise to hspecP(G)={hdimGP(H)|H≤G}⊆[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, hspecP(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 |hspecP(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 ℤp ⨁ ℤp admits a filtration series S such that hspecS(G) contains an infinite real interval. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/c530e305-2b29-484b-b92f-eba23accc7a9
- author
- Klopsch, Benjamin ; Thillaisundaram, Anitha LU and Zugadi-Reizabal, Amaia
- publishing date
- 2019
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Israel Journal of Mathematics
- volume
- 231
- publisher
- Hebrew University Magnes Press
- external identifiers
-
- scopus:85067055363
- ISSN
- 0021-2172
- DOI
- 10.1007/s11856-019-1852-z
- language
- English
- LU publication?
- no
- id
- c530e305-2b29-484b-b92f-eba23accc7a9
- date added to LUP
- 2024-06-07 14:27:30
- date last changed
- 2024-08-07 10:34:50
@article{c530e305-2b29-484b-b92f-eba23accc7a9, abstract = {{Let G be a finitely generated pro-p group, equipped with the p-power series P:Gi=GPi, i ∈ ℕ0. The associated metric and Hausdorff dimension function hdimGP:{X|X⊆G}→[0,1] give rise to hspecP(G)={hdimGP(H)|H≤G}⊆[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, hspecP(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 |hspecP(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 ℤp ⨁ ℤp admits a filtration series S such that hspecS(G) contains an infinite real interval.}}, author = {{Klopsch, Benjamin and Thillaisundaram, Anitha and Zugadi-Reizabal, Amaia}}, issn = {{0021-2172}}, language = {{eng}}, publisher = {{Hebrew University Magnes Press}}, series = {{Israel Journal of Mathematics}}, title = {{Hausdorff dimensions in p-adic analytic groups}}, url = {{http://dx.doi.org/10.1007/s11856-019-1852-z}}, doi = {{10.1007/s11856-019-1852-z}}, volume = {{231}}, year = {{2019}}, }