If \(F\) is a non-cyclic free group of rank \(n\) (here \(n\) is a positive integer) with free generating set \(X = \{x_1 , \ldots, x_n\}\), then one can consider commutator subgroups of the form \([F,F]=F'\), \([F',F']=F''\), \([F'',F'']=F'''\) and so on and quotients of the form \(F/[F'', F]\), in order to get to the short exact sequence \[ 1 \longrightarrow F''/[F'',F] \longrightarrow F/[F'',F] \longrightarrow F/F'' \longrightarrow 1. \] It turns out that this sequence induces a central extension of \(F/F''\) which is the free metabelian group of rank \(n\). In fact \(F/[F'',F]\) is the free centre-by-metabelian group of rank \(n\).
C. K. Gupta [J. Aust. Math. Soc. 16, 294–299 (1973; Zbl 0275.20061)] proved that \(F/[F'',F]\) is torsion-free for \(n = 2\) and \(n = 3\), but for \(n \ge 4\) one has that the structure of \(F/[F'',F]\) is more complicated and in fact it contains an elementary abelian \(2\)-group of rank \[ r(n)=\frac{n!}{(n-4)! 4!} \] in its centre. In addition, Gupta provided a generating set for this torsion subgroup.
The authors of the paper under review are offering a simplified process in terms of computational group theory for the original ideas of Gupta [loc. cit.]. They offer a more general description of what happens when \(n\) is large, so one can investigate better the structure of \(F/[F'',F]\). Theorems 1.1, 1.2 and 1.3 describe the combinatorics of the words which are involved in the result of Gupta [loc. cit.], focusing on the isolators of certain large free abelian subgroups of rank \(r(n)\). Note that the theory of the isolators was introduced by P. Hall and played a fundamental role in the classification of nilpotent groups, see for instance Paragraph 2.3 of J. C. Lennox and D. J. S. Robinson [The theory of infinite soluble groups. Oxford: Clarendon Press (2004; Zbl 1059.20001)]. There are in fact many connections with the study of \(F/[F'',F]\) and the classification of nilpotent groups; for instance \(F'/[F'',F']\) turns out to be a free nilpotent group of class two and it can also be studied from the point of view of the 4-dimensional homology with coefficients in the field with two elements, as originally noted in the works of Yu. V. Kuz’min [Math. USSR, Izv. 11, 1–30 (1977; Zbl 0379.20033)] and R. Stöhr [J. Pure Appl. Algebra 46, 249–289 (1987; Zbl 0626.20018); Math. Proc. Camb. Philos. Soc. 106, No. 1, 13–28 (1989; Zbl 0686.20022)]. Homological methods are mostly involved in Section 4 of the paper under review.
Some minor corrections of the authors appeared in [ibid. 52, No. 4, (I)–(II) (2024; Zbl 07834095)].