Skip to main content
SpringerLink
Log in

A spectral gap theorem in simple Lie groups

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.

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

Similar content being viewed by others

Notes

  1. Otherwise, one should use polynomials \(P_{I_0,g}(v)\) defining the subvariety \(\{v \,|\, g\cdot v\pm v=0\}\).

References

  1. Aoun, R.: Transience of algebraic varieties in linear groups and application to generic Zariski density. Annales de l’Institut Fourier 63(5), 2049–2080 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benoist, Y.: Sous-groupes discrets des groupes de Lie. Notes from the 1997 European Summer School in Group Theory, Luminy, July 7–18 (1997)

  3. Benoist, Y., Quint, J.-F.: Random walks on reductive groups (2013). http://www.math.u-bordeaux1.fr/~jquint/

  4. Bourbaki, N.: Groupes et algèbres de Lie. Masson (1982)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgain, J., Gamburd, A.: A spectral gap theorem in \(SU(d)\). J. Eur. Math. Soc. 14(5), 1455–1511 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Breuillard, E.: Approximate groups and superstrong approximation. In: Proceedings of the conference Groups St Andrews 2013 (2014)

  9. Kesten, H.: Symmetric random walks on groups. Trans. Am. Math. Soc. 92(2), 336–354 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  10. Knapp, A.: Lie Groups Beyond an Introduction, 2nd edn. Birkhäuser, Basel (2002)

    MATH  Google Scholar 

  11. Lindenstrauss, E., de Saxcé, N.: Hausdorff dimension and subgroups of \(SU(2)\). Israel J. Math. 209(1), 335–354 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lindenstrauss, E., Varjú, P.: Random walks in the group of Euclidean isometries and self-similar measures (2014, preprint). arXiv:1405.4426,

  13. Masser, D.W., Wüstholz, G.: Fields of large transcendence degree generated by values of elliptic functions. Invent. Math. (1983)

  14. Golsefidy, A.S., Varjú, P.: Expansion in perfect groups. Geom. Funct. Anal. 22, 1832–1891 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Saxcé, N.: Trou dimensionnel dans les groupes de Lie semisimples compacts, via les séries de Fourier. J. Anal. Math. 120, 311–331 (2013)

    Article  MathSciNet  Google Scholar 

  16. de Saxcé, N.: A product theorem in simple Lie groups. Geom. Funct. Anal. 25(3), 915–941 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the Israel Institute for Advanced Studies, where this work was done, during the 2013 Arithmetic and Dynamics semester.

Author information

Authors and Affiliations

  1. C.N.R.S.-Université Paris-Sud, 91405, Orsay, France

    Yves Benoist

  2. C.N.R.S.-Université Paris-Nord, 93430, Villetaneuse, France

    Nicolas de Saxcé

Authors
  1. Yves Benoist
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolas de Saxcé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas de Saxcé.

Additional information

N. de Saxcé was 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

Benoist, Y., de Saxcé, N. A spectral gap theorem in simple Lie groups. Invent. math. 205, 337–361 (2016). https://doi.org/10.1007/s00222-015-0636-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-015-0636-2

Search

Navigation