Advertisement

Israel Journal of Mathematics

, Volume 76, Issue 3, pp 265–288 | Cite as

A braidlike presentation of Sp(n, p)

  • Bronislaw Wajnryb
Article

Abstract

Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupB n+1. Ifs 1, …,s n are standard generators ofB n+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s 1 p ,(s 1 s 2)6,s 1 (p+1)/2 s 2 4 s 1 (p−1)/2 s 2 −2 s 1 −1 s 2 2 and (s 1 s 2 s 3)4 A −1 s 1 −2 A, whereA=s 2 s 3 −1 s 2 (p−1)/2 s 4 s 3 2 s 4, all of them lying in the subgroupB 5. Sp(n, p) acts on a vector space and the image of the subgroupB n ofB n+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …B k+1·B k … induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem.

Keywords

Induction Hypothesis Braid Group Symplectic Group Split Epimorphism Minimal Representative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    J. Assion,Einige endliche Faktorgruppen der Zopfgruppen, Math. Z.163 (1978), 291–302.MATHCrossRefMathSciNetGoogle Scholar
  2. [B]
    J. Birman,Braids, Links and Mapping Class Groups, Ann. Math. Studies82 (1974).Google Scholar
  3. [C]
    H. S. M. Coxeter,Factor groups of the braid group, Proc. 4th Can. Math. Congress, Toronto Univ. Press, 1959, pp. 95–122.Google Scholar
  4. [M]
    B. Moishezon,Finite quotients of braid groups related to symplectic groups over ℤ/3, preprint.Google Scholar
  5. [S]
    J. G. Sunday,Presentations of the groups SL(2,m) and PSL(2,m), Can. J. Math.24 (1972), 1129–1131.MATHMathSciNetGoogle Scholar
  6. [W]
    B. Wajnryb,Symplectic groups over a field with three elements, preprint.Google Scholar

Copyright information

© Hebrew University 1991

Authors and Affiliations

  • Bronislaw Wajnryb
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations