Algebra and Logic

, Volume 39, Issue 4, pp 224–251 | Cite as

Computing commutator length in free groups

  • V. G. Bardakov
Article

Abstract

We study commutator length in free groups. (By a commutator lengthcl(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F2 (Thm. 1). Moreover, for every element z ε F′2 and for any natural m, the following estimate derives:cl(zm) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory, we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of pairs, and also for its corresponding Lie algebra PL. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F2k with free generators a1, b1, ..., ak, bk, k εN, every natural m satisfiescl(([a1, b1] ... [ak, bk])m)=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely generated group G is well known. Associated with a derived subgroup of F2 is some series depending on two variables which bears information not only on the number of elements of prescribed length but also on the number of elements of prescribed commutator length. A number of open questions are formulated.

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References

  1. 1.
    R. Z. Goldstein and E. C. Turner, “Applications of topological graph theory to group theory,”Math. Z.,165, No. 1, 1–10 (1979).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Culler, “Using surfaces to solve equations in free groups,”Topology,20, No. 2, 133–145 (1981).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Yu. Ol’shanskii, “Homomorphism diagrams of surface groups,”Sib. Mat. Zh.,30, No. 6, 150–171 (1989).MathSciNetGoogle Scholar
  4. 4.
    C. C. Edmunds and G. Rosenberger, “Powers of genus two in free groups,”Can. Math. Bull.,33, No. 3, 342–344 (1990).MATHMathSciNetGoogle Scholar
  5. 5.
    M. P. Shutzenberger, “Sur l’equationa 2+n=b 2+m c 2+p dans un group libre,”C. R. Acad. Sc. Paris, Sér. I, Math.,248, 2435–2436 (1959).Google Scholar
  6. 6.
    J. A. Comerford, L. P. Comerford, Jr., and C. C. Edmunds, “Powers as product of commutators,”Comm. Alg.,19, No. 2, 675–684 (1991).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. A. Vdovina, “Constructing of orientable Wicks forms and estimation of their number,”Comm. Alg.,23, No. 9, 3205–3222 (1995).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    A. J. Duncan and J. Howie, “The genus problem for one-relator products of locally indicable groups,”Math. Z.,208, No. 2, 225–237 (1991).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. C. Lyndon and P. E. Schupp,Combinatorial Group Theory, Springer, Berlin (1977).MATHGoogle Scholar
  10. 10.
    R. I. Grigorchuk and P. F. Kurchanov, “Some problems of group theory associated with geometry,” inAlgebra 7, Itogi Nauki Tekh., Ser. Fund. Napr., Vol 58, VINITI, Moscow (1990), pp. 191–256.Google Scholar
  11. 11.
    W. S. Massey and J. Stallings,Algebraic Topology. An Introduction [Russian translation], Mir, Moscow (1977).Google Scholar
  12. 12.
    R. Z. Goldstein and E. C. Turner, “Counting orbits of a product of permutations,”Discr. Math.,80, No. 3, 267–272 (1990).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    V. G. Bardakov, “Even permutations that are not represented as a product of two permutations of prescribed order,”Mat. Zametki,62, No. 2, 169–177 (1997).MathSciNetGoogle Scholar
  14. 14.
    M. I. Kargapolov and Yu. I. Merzlyakov,Fundamentals of Group Theory [in Russian], 4th edn., Nauka, Moscow (1996).MATHGoogle Scholar
  15. 15.
    A. I. Kostrikin,An Introduction to Algebra [in Russian], Nauka, Moscow (1977).Google Scholar
  16. 16.
    V. G. Bardakov, “Toward a theory of braid groups,”Mat. Sb.,183, No. 6, 3–42 (1992).Google Scholar
  17. 17.
    A. H. Rhemtulla, “A problem of bounded expressibility in free products,”Proc. Cambr. Phil. Soc.,64, No. 3, 573–584 (1969).CrossRefMathSciNetGoogle Scholar
  18. 18.
    V. G. Bardakov, “On a width of verbal subgroups of certain free constructions,”Algebra Logika,36, No. 5, 494–517 (1997).MATHMathSciNetGoogle Scholar
  19. 19.
    M. J. Wicks, “Commutators in free products,”J. London Math. Soc.,37, No. 4, 433–444 (1962).MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    R. Grigorchuk and P. de la Harpe, “On problems related to growth entropy and spectrum in group theory,”J. Dynamical and Control Systems,3, No. 1, 51–89 (1997).MATHCrossRefGoogle Scholar

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© Kluwer Academic/Plenum Publishers 2000

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  • V. G. Bardakov

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