Skip to main content
Log in

A product theorem in simple Lie groups

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove a discretized Product Theorem for general simple Lie groups, in the spirit of Bourgain’s Discretized Sum-Product Theorem.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Y. Benoist and N. de Saxcé. Spectral gap in compact simple Lie groups. preprint arXiv:1405.1808, 2014.

  2. J. Bourgain. On the Erdős–Volkmann and Katz–Tao ring conjectures. GAFA, 13, 2003.

  3. Bourgain J.: The discretized sum-product and projection theorems. Journal d’Analyse Mathématique. 112, 193–236 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Bourgain and A. Gamburd. On the spectral gap for finitely generated subgroups of SU(2). Inventiones Mathematicae, 171:83–121, 2008.

  5. Bourgain J., Gamburd A.: A spectral gap theorem in SU(d). Journal of the European Mathematical Society, (5)14:1455–1511, 2012.

  6. J. Bourgain and A. Yehudayoff. Expansion in \({SL(2,\mathbb{R})}\) and monotone expansion. GAFA, 23:1–41, 2013.

  7. Jean Bourgain, Alex Furman, Elon Lindenstrauss, and Shahar Mozes. Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus. J. Amer. Math. Soc., (1)24:231–280, 2011.

  8. Breuillard E., Green B.J., Tao T.: Approximate subgroups of linear groups. GAFA. 21, 774–819 (2011)

    MATH  MathSciNet  Google Scholar 

  9. A. Eskin, S. Mozes, and H. Oh. On uniform exponential growth for linear groups. Invent. Math., (1)160:1–30, 2005.

  10. H. A. Helfgott. Growth and generation in \({SL(2,\mathbb{Z}/p\mathbb{Z})}\). Annals of Mathematics, 167:601–623, 2008.

  11. N.H. Katz and T. Tao Some connections between Falconer’s distance set conjecture and sets of Furstenburg type. New York Mathematical Journal, 7:149–187, 2001.

  12. S. Łojasiewicz. Ensembles semi-analytiques. Notes from a course given in Orsay, 2006. available at https://perso.univ-rennes1.fr/michel.coste.

  13. L. Pyber and E. Szabó. Growth in finite groups of Lie type of bounded rank. preprint arXiv:1005.1858, 2010.

  14. N. de Saxcé. Borelian subgroups of simple Lie groups. preprint arXiv:1408.1579, 2014.

  15. T.C. Tao. Product set estimates for non-commutative groups. Combinatorica, 28:547–594, 2008.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nicolas de Saxcé.

Additional information

The author is supported by ERC AdG Grant 267259.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Saxcé, N. A product theorem in simple Lie groups. Geom. Funct. Anal. 25, 915–941 (2015). https://doi.org/10.1007/s00039-015-0326-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-015-0326-7

Keywords

Navigation