It was first shown by C. K. Gupta that a relatively free group defined by an outer commutator word may contain non-trivial torsion. She showed [J. Aust. Math. Soc. 16, 294-299 (1973; Zbl 0275.20061)] that if \(n \geq 4\), then the free centre-by-metabelian group of rank \(n\) contains involutions. The phenomenon of torsion in relatively free groups defined by commutator identities has since received systematic attention, initially in papers of Kuz’min, and then by other authors, notably the second author of this paper. For details, see the paper. This attention has been from the homological viewpoint, and has led to interesting results of a purely homological nature. The present paper is a further contribution to this area.
Let \(F\) be a free group, let \(N \triangleleft F\), and let \(G = F/N\). Also let \(M = N/N'\), viewed as a \(\mathbb{Z} G\)-module by conjugation, be the relation module. Let \(H^{(n)}\) be \(n\)-th term of the derived series of a group \(H\), and \(\gamma_ n(H)\) be the \(n\)-th term of the lower central series of \(H\). The authors are interested in torsion in the relatively free group \(F/[N^{(n)},F]\gamma_{2^ n+1}N\). By known results, any torsion elements in this group must lie in \(N^{(n)}\gamma_{2^ n+1} N/[N^{(n)},F]\gamma_{2^ n+1}N\). Let \(T\) denote the torsion subgroup of the latter. This is the main object of interest. Homology enters the picture as follows. Let \(\wedge^ n A\) denote the \(n\)-th exterior power of a \(G\)-module \(A\), regarded as a \(G\)-module via diagonal action, and let \(\Omega^ n A = \wedge \wedge \dots \wedge A\), with \(n\) factors \(\wedge\). Then we have the following Lemma. \(N^{(n)}\gamma_{2^ n+1}N/[N^{(n)},F]\gamma_{2^ n+1}N\) is isomorphic to \(\Omega^ n M\otimes_ G\mathbb{Z}\). In particular, \(T_ n\) is isomorphic to the torsion subgroup of \(\Omega^ n M \otimes_ G \mathbb{Z}\).
Now let \(\Phi_ n\) denote the 2-torsion subgroup of \(\wedge^ n M \otimes \mathbb{Z}\). The torsion subgroup of this product has been studied in a remarkable paper of L. Kovács, Yu. Kuz’min and R. Stöhr [Mat. Sb. 182, 526-542 (1991); see Zbl 0798.20043 below]. Let \(\Phi_ n\) be its 2-torsion subgroup. Then it follows from their results that if \(G\) has no elements of order 2, then \(\Phi_ n\) is a direct sum of various homology groups of \(G\) with coefficients in \(\mathbb{Z}_ 2\), with explicitly given multiplicities. Theorem 2 states that if \(n \geq 1\), then \(T_ n\) has a direct summand isomorphic to \(\Phi_{2^ n}\), so that when \(G\) has no elements of order 2, it follows that \(T_ n\) has a certain explicit direct sum of homology groups of \(G\) with \(\mathbb{Z}_ 2\)- coefficients as a direct summand.
Theorem 1 states that the exponent of \(T_ n\) divides \(2^{n + 1}\). Theorem 2 tells us, among other things, about the actual occurrence of torsion, and tells us that in many cases, \(T_ n\) will contain involutions. In Theorem 3, \(T_ 2\) is explicitly expressed in terms of homology groups of \(G\), and it turns out that elements of order 4 can occur. But it is not known whether elements of order 8 can occur, for example. Theorem 3 also shows that \(\Phi_ 4\) is in general a proper subgroup of \(T_ 2\). Theorem 3. If \(G\) has no elements of order 2, then \(T_ n\) is isomorphic to \(H_ 8(G,\mathbb{Z}_ 2) \oplus H_ 7(G,\mathbb{Z}_ 4) \oplus H_ 6(G,\mathbb{Z}_ 2) \oplus H_ 5(G,\mathbb{Z}_ 2)\).