Archiv der Mathematik

, Volume 99, Issue 5, pp 417–424 | Cite as

Diameters of Chevalley groups over local rings



Let G be a Chevalley group scheme of rank l. Let \({G_n := G(\mathbb{Z} / p^{n} \mathbb{Z})}\) be the family of finite groups for \({n \in \mathbb{N}}\) and some fixed prime number pp 0. We prove a uniform poly-logarithmic diameter bound of the Cayley graphs of G n with respect to arbitrary sets of generators. In other words, for any subset S which generates G n , any element of G n is a product of C n d elements from \({S \cup S^{-1}}\). Our proof is elementary and effective, in the sense that the constant d and the functions p 0(l) and C(l, p) are calculated explicitly. Moreover, we give an efficient algorithm for computing a short path between any two vertices in any Cayley graph of the groups G n .


Local Ring Cayley Graph Chevalley Group Chevalley Basis International Mathematics Research Notice 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abe E.: Chevalley groups over local rings. Tohoku Mathematical Journal 21, 474–494 (1969)MATHCrossRefGoogle Scholar
  2. 2.
    Babai L., Seress Á.: On the diameter of Cayley graphs of the symmetric group. Journal of Combinatorial Theory, Series A 49, 175–179 (1988)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Babai L., Seress Á.: On the diameter of permutation groups. European journal of combinatorics 13, 231–243 (1992)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    L. Babai et al., On the diameter of finite groups, Proceedings of the 31st Annual Symposium on Foundations of Computer Science (1990), 857–865.Google Scholar
  5. 5.
    Bourgain J., Gamburd A.: New results on expanders. Comptes Rendus Mathematique 342, 717–721 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    E. Breuillard, B. Green, and T. Tao, Linear approximate groups, Arxiv preprint arXiv:1001.4570 (2010).Google Scholar
  7. 7.
    R.W. Carter, Lie algebras of finite and affine type, vol. 96, Cambridge Univ Pr, 2005.Google Scholar
  8. 8.
    C.M. Dawson and M.A. Nielsen, The Solovay-Kitaev algorithm, arXiv preprint quant-ph/0505030 (2005).Google Scholar
  9. 9.
    Dinai O.: Poly-log diameter bounds for some families of finite groups. Proceedings of the American Mathematical Society 134, 3137–3142 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dinai O.: Growth in SL2 over finite fields. Journal of Group Theory 14, 273–297 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    J. Ellenberg, C. Hall, and E. Kowalski, Expander graphs, gonality and variation of Galois representations, Duke Math. Journal (to appear).Google Scholar
  12. 12.
    A. Gamburd and M. Shahshahani, Uniform diameter bounds for some families of Cayley graphs, International mathematics research notices no 71 (2004).Google Scholar
  13. 13.
    H.A. Helfgott, Growth and generation in \({SL_2(\mathbb{Z}/p \mathbb{Z})}\), arXiv preprint math/0509024 (2005).Google Scholar
  14. 14.
    H.A. Helfgott and A. Seress, On the diameter of permutation groups, arXiv preprint arXiv:1109.3550 (2011).Google Scholar
  15. 15.
    E. Kowalski, Explicit growth and expansion for SL2, preprint (2012).Google Scholar
  16. 16.
    M.A. Nielsen and I. Chuang, Quantum information and computation, Cambridge University Press, 2000.Google Scholar
  17. 17.
    Nielsen M.A., Chuang I., Grover L.K.: Quantum computation and quantum information. American Journal of Physics 70, 558 (2002)CrossRefGoogle Scholar
  18. 18.
    L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, arXiv preprint arXiv:1005.1858 (2010).Google Scholar
  19. 19.
    P.P. Varjú, Random walks in compact groups, arXiv preprint arXiv:1209.1745 (2012).Google Scholar
  20. 20.
    Weigel T.: On the rigidity of Lie lattices and just infinite powerful groups. Journal of the London Mathematical Society 62, 381–397 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsETH Zurich Ramistrasse 101ZurichSwitzerland

Personalised recommendations