Skip to main content
Log in

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bézout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. A. Vavilov and S. S. Sinchuk, “Dennis-Vaserstein type decompositions,” Zap. Nauchn. Semin. POMI, 375, 48–60 (2010).

    MathSciNet  Google Scholar 

  2. N. A. Vavilov and S. S. Sinchuk, “Parabolic factorizations of split classical groups,” Zap. Nauchn. Semin. POMI, 23, No. 4, 1–30 (2011).

    MathSciNet  Google Scholar 

  3. N. A. Vavilov, A. V. Smolenskii, and B. Sury, “Unitriangular factorizations of Chevalley groups,” Zap. Nauchn. Semin. POMI, 388, 17–47 (2011).

    Google Scholar 

  4. R. Steinberg, Lectures on Chevalley Groups [Russian translation], Mir, Moscow (1975).

    Google Scholar 

  5. L. Babai, N. Nikolov, and L. Pyber, “Product growth and mixing in finite groups,” in: 19th Annual ACM SIAM Sympisium on Discrete Algorithms, ACM–SIAM (2008), pp. 248–257.

  6. J. Bourgain and A. Gamburd, “Uniform expansion bounds for Cayley graphs of SL2 (GF p),” Ann. Math., 167, 625–642 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. W. Carter, Simple Groups of Lie Type, Wilet, London (1972).

    MATH  Google Scholar 

  8. I. V. Erovenko, “Sl n (F[x]) is not boundedly generated by elementary matrices: explicit proof,” Electr. J. Linear Algebra, 11, 162–167 (2004).

    MathSciNet  MATH  Google Scholar 

  9. D. R. Grayson, “SK1 of an interesting principal ideal domain,” J. Pure Appl. Algebra, 20, 157–163 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  10. H. A. Helfgott, “Growth and generation in SL2 \( \left( {{{\mathbb{Z}} \left/ {{p\mathbb{Z}}} \right.}} \right) \),” Ann. Math., 167, 601–623 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Ischebeck, “Hauptidealringe mit nichttrivialer SK1-Gruppe,” Arch. Math., 35, 138–139 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  12. W. van der Kallen, “SL3 \( \left( {\mathbb{C}\left[ x \right]} \right) \) does not have bounded word length,” Lect. Notes Math., 966, 357–361 (1982).

    Article  Google Scholar 

  13. M. Kassabov, A. Lubotzky, and N. Nikolov, “Finite simple groups as expanders,” Proc. Nat. Acad. Sci. USA, 103, 6116–6119 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Liebeck, N. Mikolov, and A. Shalev, “A conjecture on product decompositions in simple groups,” Groups Geom. Dynamics, 4, 799–812 (2010).

    Article  MATH  Google Scholar 

  15. M. Liebeck, N. Nikolov, and A. Shalev, “Groups of Lie type as products of SL2 subgroups,” J. Algebra, 326, 201–207 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  16. D. W. Morris, “Bounded generation of SL (n, A) (after D. Carter, G. Keller, and E. Paige),” New York J. Math., 13, 383–421 (2007).

    MathSciNet  MATH  Google Scholar 

  17. N. Nikolov, “A product decomposition for the classical quasisimple groups,” J. Group Theory, 10, 43–53 (2007).

    MathSciNet  MATH  Google Scholar 

  18. N. Nikolov and L. Pyber, “Product decomposition of quasirandom groups and a Jordan type theorem,” arXiv:math/0703343 (2007).

  19. S. Sinchuk and N. Vavilov, “Parabolic factorizations of exceptional Chevalley groups,” (to appear).

  20. A. Stepanov and N. Vavilov, “On the length of commutators in Chevalley groups,” Israel Math. J., 185, 253–276 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  21. O. I. Tavgen’, “Bounded generation of normal and twisted Chevalley groups over the rings of S-integers,” Contemp. Math., 131, No. 1, 409–421 (1992).

    Article  MathSciNet  Google Scholar 

  22. L. N. Vaserstein, “Polynomial parametrization for the solution of Diophantine equations and arithmetic groups,” Ann. Math., 171, No. 2, 979–1009 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. N. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Proceedings of Conference Nonassociative Algebras and Related Topics (Hiroshima–1990), World Sci. Publ., London (1991), pp. 219–335.

  24. N. Vavilov, “A third look at weight diagrams,” Rendiconti del Rend. Sem. Mat. Univ. Padova, 204, No. 1, 201–250 (2000).

    MathSciNet  Google Scholar 

  25. N. Vavilov and E. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73–115 (1996).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to N. A. Vavilov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 394, 2011, pp. 20–32.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A., Kovach, E.I. SL2-factorizations of Chevalley groups. J Math Sci 188, 483–489 (2013). https://doi.org/10.1007/s10958-013-1145-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1145-8

Keywords

Navigation