Commutators in metabelian groups. (Commutateurs dans les groupes métabéliens.) (French) Zbl 0769.20015

Bavard, Christophe; Meigniez, Gaël

Indag. Math., New Ser. 3, No. 2, 129-135 (1992).

Let \(G\) be a group and let \(G'\) be the commutator subgroup of \(G\). Then \(c(G)\) is defined as the least positive integer \(k\) such that every element of \(G'\) is the product of \(k\) commutators, if such an integer exists, otherwise \(c(G) = \infty\). The authors prove that if \(G_ n\) is the free metabelian group on \(n\) generators \((n \geq 2)\), then \(c(G_ 2) = 2\) and \(\left[{n\over 2}\right] \leq c(G_ n) \leq n\), where \(\left[{n\over 2}\right]\) is the integral part of \({n\over 2}\). Moreover, if \(G\) is the free metabelian group on a countable set of generators, then \(c(G) = \infty\). As a consequence \(c(G)\) is finite for every polycyclic group \(G\).

Reviewer: S.Franciosi (Napoli)

MSC:

20F12	Commutator calculus
20F05	Generators, relations, and presentations of groups
20F16	Solvable groups, supersolvable groups
20E05	Free nonabelian groups

Keywords:

commutator subgroup; product of \(k\) commutators; free metabelian group; polycyclic group

References:

[1]	Barge, J.; Ghys, E., Surfaces et cohomologie bornée, Invent. Math., 92, 509-526 (1988) · Zbl 0641.55015
[2]	Bavard, C., Longuer stable des commutateurs, L’enseignement Math., 37, 109-150 (1991) · Zbl 0810.20026
[3]	Culler, M., Using surfaces to solve equations in free groups, Topology, 20, 133-145 (1981) · Zbl 0452.20038
[4]	Dennis, R. K.; Vaserstein, L. N., On a question of M. Newman on the number of commutators, J. of Algebra, 118, 150-161 (1988) · Zbl 0649.20048
[5]	Gromov, M., Volume and bounded cohomology, Pub. Math., 56, 5-99 (1982), IHES · Zbl 0516.53046
[6]	Ivanov, N. V., Foundations of the theory of bounded cohomology, J. of Soviet Math., 37, 1090-1115 (1987) · Zbl 0612.55006
[7]	Magnus, W.; Karras, A.; Solitar, D., Combinatorial group theory (1976), Dover
[8]	Newman, M., Unimodular commutators, Proc. AMS, 101 n° 4, 605-609 (1987) · Zbl 0633.15007