Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Hausdorff dimensions in p-adic analytic groups

Klopsch, Benjamin ; Thillaisundaram, Anitha LU and Zugadi-Reizabal, Amaia (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:
author
; and
publishing date
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)|&lt;∞ 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 &gt; 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}},
}