# Computing commutator length in free groups

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## Abstract

We study commutator length in free groups. (By a commutator length*cl*(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 F_{2} (Thm. 1). Moreover, for every element z ε F′_{2} and for any natural m, the following estimate derives:*cl*(z^{m}) ≥ (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 P_{L}. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F_{2k} with free generators a_{1}, b_{1}, ..., a_{k}, b_{k}, k ε*N*, every natural m satisfies*cl*(([a_{1}, b_{1}] ... [a_{k}, b_{k}])^{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 F_{2} 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.

## Keywords

Arbitrary Element Orientable Surface Free Generator Zero Divisor Cyclic Shift## Preview

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## References

- 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.M. Culler, “Using surfaces to solve equations in free groups,”
*Topology*,**20**, No. 2, 133–145 (1981).MATHCrossRefMathSciNetGoogle Scholar - 3.A. Yu. Ol’shanskii, “Homomorphism diagrams of surface groups,”
*Sib. Mat. Zh.*,**30**, No. 6, 150–171 (1989).MathSciNetGoogle Scholar - 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.M. P. Shutzenberger, “Sur l’equation
*a*^{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.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.A. A. Vdovina, “Constructing of orientable Wicks forms and estimation of their number,”
*Comm. Alg.*,**23**, No. 9, 3205–3222 (1995).MATHCrossRefMathSciNetGoogle Scholar - 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.R. C. Lyndon and P. E. Schupp,
*Combinatorial Group Theory*, Springer, Berlin (1977).MATHGoogle Scholar - 10.R. I. Grigorchuk and P. F. Kurchanov, “Some problems of group theory associated with geometry,” in
*Algebra 7, Itogi Nauki Tekh., Ser. Fund. Napr.*, Vol 58, VINITI, Moscow (1990), pp. 191–256.Google Scholar - 11.W. S. Massey and J. Stallings,
*Algebraic Topology. An Introduction*[Russian translation], Mir, Moscow (1977).Google Scholar - 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.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.M. I. Kargapolov and Yu. I. Merzlyakov,
*Fundamentals of Group Theory*[in Russian], 4th edn., Nauka, Moscow (1996).MATHGoogle Scholar - 15.A. I. Kostrikin,
*An Introduction to Algebra*[in Russian], Nauka, Moscow (1977).Google Scholar - 16.
- 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.V. G. Bardakov, “On a width of verbal subgroups of certain free constructions,”
*Algebra Logika*,**36**, No. 5, 494–517 (1997).MATHMathSciNetGoogle Scholar - 19.M. J. Wicks, “Commutators in free products,”
*J. London Math. Soc.*,**37**, No. 4, 433–444 (1962).MATHCrossRefMathSciNetGoogle Scholar - 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