Abstract
Let G be a finitely generated pro-p group, equipped with the p-power series \(P:{G_i} = {G^{{P^i}}}\), i ∈ ℕ0. The associated metric and Hausdorff dimension function \(hdim_G^P:\{{X|X\subseteq{G}\}}\rightarrow[0,1]\) give rise to
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.
Similar content being viewed by others
References
A. G. Abercrombie, Subgroups and subrings of profinite rings, Mathematical Proceedings of the Cambridge Philosophical Society 116 (1994), 209–222.
M. Abért and B. Virág, Dimension and randomness in groups acting on rooted trees, Journal of the American Mathematical Society 18 (2005), 157–192.
Y. Barnea and B. Klopsch, Index-subgroups of the Nottingham group, Advances inMathematics 180 (2003), 187–221.
Y. Barnea and A. Shalev, Hausdorff dimension, pro-p groups, and Kac–Moody algebras, Transactions of the American Mathematical Society 349 (1997), 5073–5091.
J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p Groups, Cambridge Studies in Advanced Mathematics, Vol. 61, Cambridge University Press, Cambridge, 1999.
M. Ershov, New just-infinite pro-p groups of finite width of the Nottingham group, Journal of Algebra 275 (2004), 419–449.
M. Ershov, On the commensurator of the Nottingham group, Transactions of the American Mathematical Society 362 (2010), 6663–6678.
K. J. Falconer, Fractal Geometry, John Wiley & Sons, Chicester, 1990.
G. A. Fernández-Alcober, E. Giannelli and J. González-Sánchez, Hausdorff dimension in R-analytic profinite groups, Journal of Group Theory 20 (2017), 579–587.
G. A. Fernández-Alcober and A. Zugadi-Reizabal, GGS-groups: order of congruence quotients and Hausdorff dimension, Transactions of the American Mathematical Society 366 (2014), 1993–2017.
Y. Glasner, Strong approximation in random towers of graphs, Combinatorica 34 (2014), 139–172.
A. Jaikin-Zapirain and B. Klopsch, Analytic groups over general pro-p domains, Journal of the London Mathematical Society 76 (2007), 365–383.
B. Klopsch, Substitution Groups, Subgroup Growth and Other Topics, PhD. Thesis, University of Oxford, 1999.
M. Lazard, Groupes analytiques p-adiques, Institut des Hautes Études Sientifiques. Publications Mathématiques 26 (1965), 389–603.
J.-P. Serre, Local fields, Graduate Texts in Mathematics, Vol. 67, Springer-Verlag, New York–Berlin, 1979.
E. Zelmanov, On periodic compact groups, Israel Journal of Mathematics 77 (1992), 83–95.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author acknowledges support from the Alexander von Humboldt Foundation and thanks Heinrich-Heine-Universität Düsseldorf for its hospitality.
The third author was supported by the Spanish Government, grant MTM2011-28229-C02-02, partly FEDER funds, and by the Basque Government, grant IT-460-10.
Rights and permissions
About this article
Cite this article
Klopsch, B., Thillaisundaram, A. & Zugadi-Reizabal, A. Hausdorff dimensions in p-adic analytic groups. Isr. J. Math. 231, 1–23 (2019). https://doi.org/10.1007/s11856-019-1852-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1852-z