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Hausdorff dimensions in p-adic analytic groups

  • Benjamin Klopsch
  • Anitha ThillaisundaramEmail author
  • Amaia Zugadi-Reizabal
Article
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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
$$hspe{c^P}(G) = \{ h\dim _G^P(H)|H \leqslant G\} \subseteq [0,1],$$
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.

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References

  1. [1]
    A. G. Abercrombie, Subgroups and subrings of profinite rings, Mathematical Proceedings of the Cambridge Philosophical Society 116 (1994), 209–222.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Abért and B. Virág, Dimension and randomness in groups acting on rooted trees, Journal of the American Mathematical Society 18 (2005), 157–192.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Y. Barnea and B. Klopsch, Index-subgroups of the Nottingham group, Advances inMathematics 180 (2003), 187–221.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Y. Barnea and A. Shalev, Hausdorff dimension, pro-p groups, and Kac–Moody algebras, Transactions of the American Mathematical Society 349 (1997), 5073–5091.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    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.Google Scholar
  6. [6]
    M. Ershov, New just-infinite pro-p groups of finite width of the Nottingham group, Journal of Algebra 275 (2004), 419–449.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Ershov, On the commensurator of the Nottingham group, Transactions of the American Mathematical Society 362 (2010), 6663–6678.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    K. J. Falconer, Fractal Geometry, John Wiley & Sons, Chicester, 1990.zbMATHGoogle Scholar
  9. [9]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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.Google Scholar
  11. [11]
    Y. Glasner, Strong approximation in random towers of graphs, Combinatorica 34 (2014), 139–172.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Jaikin-Zapirain and B. Klopsch, Analytic groups over general pro-p domains, Journal of the London Mathematical Society 76 (2007), 365–383.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Klopsch, Substitution Groups, Subgroup Growth and Other Topics, PhD. Thesis, University of Oxford, 1999.zbMATHGoogle Scholar
  14. [14]
    M. Lazard, Groupes analytiques p-adiques, Institut des Hautes Études Sientifiques. Publications Mathématiques 26 (1965), 389–603.MathSciNetzbMATHGoogle Scholar
  15. [15]
    J.-P. Serre, Local fields, Graduate Texts in Mathematics, Vol. 67, Springer-Verlag, New York–Berlin, 1979.Google Scholar
  16. [16]
    E. Zelmanov, On periodic compact groups, Israel Journal of Mathematics 77 (1992), 83–95.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Benjamin Klopsch
    • 1
  • Anitha Thillaisundaram
    • 2
    Email author
  • Amaia Zugadi-Reizabal
    • 3
  1. 1.Mathematisches InstitutHeinrich-Heine-UniversitätDüsseldorfGermany
  2. 2.School of Mathematics and PhysicsUniversity of LincolnLincolnEngland
  3. 3.Department of MathematicsUniversity of the Basque Country UPV/EHUBilbaoSpain

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