Dimension subgroups of free center-by-metabelian groups (Q1074707)

Let \(\Delta\) (G) be the augmentation ideal of the group ring \({\mathbb{Z}}G\). For \(n\geq 1\), the nth dimension subgroup of G is \(D_ n(G)=\{g\in G:\) \((g-1)\in \Delta (G)^ n\}\). It is easy to check that \(\gamma_ n(G)\), the nth term of the lower central series of G, is contained in \(D_ n(G)\). The validity of the reverse inequality, \(D_ n(G)\subseteq \gamma_ n(G)\), for all n, is known as the dimension subgroup problem for G. It was shown by \textit{E. Rips} [Isr. J. Math. 12, 342-346 (1972; Zbl 0267.20018)] that there is a finite 2-group G with \(D_ 4(G)\neq \gamma_ 4(G)\). However, if \(\gamma_ k(G)/\gamma_{k+1}(G)\) is torsion free for all \(k\geq 1\), then \(D_ n(G)=\gamma_ n(G)\) for all \(n\geq 1\), by a result of P. Hall and S. A. Jennings [see \textit{I. B. S. Passi}, Group rings and their augmentation ideals (Lect. Notes Math. 715), (1979; Zbl 0405.20007), Corollary 3.1]. This covers the case of a free metabelian group G, but not that of a free centre-by-metabelian group \(G=F/[F'',F]\) (F free), for which the lower central factors have elementary abelian 2-subgroups. Nevertheless, the purpose of this paper is to show that \(D_ n(G)=\gamma_ n(G)\) for all n when G is free centre-by-metabelian.