Skip to main content
SpringerLink
Log in

Trou dimensionnel dans les groupes de Lie compacts semisimples via les séries de Fourier

  • Published:
Journal d'Analyse Mathématique Aims and scope

Résumé

Soit G un groupe de Lie réel compact semisimple de dimension d. Nous démontrons ici qu'il existe un α0 < d tel que toute partie borélienne de G de dimension de Hausdorff strictement supérieure à α0 engendre G comme groupe abstrait. Une propriété analogue de «trou dimensionnel» est aussi démontrée pour les groupes spéciaux linéaires SL r+1(ℤ p ) sur les entiers p-adiques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Références

  1. A. G. Abercrombie, Subgroups and subrings of profinite rings, Math. Proc. Cambridge Philos. Soc. 116 (1994), 209–222.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Babai, N. Nikolov, and L. Pyber, Product growth and mixing in finite groups, Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2008, pp. 248–257.

    Google Scholar 

  3. N. Bourbaki, Groupes et Algèbres de Lie, Masson, 1982.

  4. J. Bourgain, On the Erdos-Volkmann and Katz-Tao ring conjectures, Geom. Funct. Anal. 13 (2003), 334–365.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bourgain, Mordell's exponential sum revisited, J. Amer. Math. Soc. 18 (2005), 477–499.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bourgain, The discretized sum-product and projection theorems, J. Anal. Math. 112 (2010), 193–236.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Bourgain and A. Gamburd, Expansion and random walks in SLd (ℤ/pnℤ), Eur. Math. Soc. 10 (JEMS)(2008), 987–1011.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Bourgain and A. Gamburd, On the spectral gap for finitely generated subgroups of SU(2), Invent. Math. 171 (2008), 83–121.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL 2({ie331-1}p), Ann. of Math. (2) 167 (2008), 625–642.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Breuillard, Lecture notes on approximate groups, Preliminary Notes for a Course on Approximate Groups given at IHP, February–March 2011, available at www.math.u-psud.fr/breuilla/ClermontLectures.pdf.

  11. G. A. Edgar and C. Miller. Borel subrings of the reals, Proc. Amer. Math. Soc. 131 (2003), 1121–1129.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Erdos and K. J. Volkmann, Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math 221 (1966), 203–208.

    MathSciNet  Google Scholar 

  13. K. J. Falconer, Rings of fractional dimension Mathematika, 31 (1984), 25–27.

    Article  MathSciNet  MATH  Google Scholar 

  14. K. J. Falconer, Geometry of Fractals, Wiley, 2003.

  15. W. T. Gowers, Quasirandom groups, Combin. Probab. Comput. 17, (2008), 363–387.

    MathSciNet  MATH  Google Scholar 

  16. A. Knapp, Lie Groups Beyond an Introduction, Second Edition. Birkhäuser, 2002.

  17. J. F. Le Gall Intégration, probabilités et processus aléatoires, Notes d'un cours fait à l'E.N.S., 2006, available at www.math.u-psud.fr/breuilla/ClermontLectures.pdf.

  18. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.

  19. P. Sarnak and X. Xue, Bounds for multiplicities of automorphic representations, Duke Math. J. 64 (1991), 207–227.

    Article  MathSciNet  MATH  Google Scholar 

  20. N. de Saxcé, Subgroups of fractional dimension in nilpotent or solvable Lie groups, Mathematika 59 (2013), 497–511.

    Article  MATH  Google Scholar 

  21. A. Weil, L'Intégration dans les Groupes Topologiques et ses Applications, Hermann, 1940.

Download references

Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem, 91904, Israel

    Nicolas De Saxcé

Authors
  1. Nicolas De Saxcé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas De Saxcé.

Rights and permissions

Reprints and permissions

About this article

Cite this article

De Saxcé, N. Trou dimensionnel dans les groupes de Lie compacts semisimples via les séries de Fourier. JAMA 120, 311–331 (2013). https://doi.org/10.1007/s11854-013-0022-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-013-0022-1

Search

Navigation