On varietal quotients defined by ideals generated by Fox derivatives (Q1192256)

Given a group \(G\), every subset of the integral group ring \(\mathbb{Z} F\) of a free group \(F\) defines, via substitution, an ideal in \(\mathbb{Z} G\). Let \({\mathcal V}\) be a variety of groups and let \(\Delta_{\mathcal V}'(G)\) be the ideal defined by the Fox derivatives of the words in the set of defining identities \(V\) of \({\mathcal V}\). Let \(\delta_{\mathcal V}(G):=G \cap(1+\Delta_{\mathcal V}'(G))\). From the fundamental formula of free differential calculus it follows that \(\delta_{\mathcal V}(G)\geq V(G)\), the verbal subgroup of \(G\) defined by the words in \(V\). The authors prove that if \({\mathcal V}\) is solvable, then \(\delta_{\mathcal V}(G)/V(G)\) is nilpotent of class at most two. In case \({\mathcal V}\) is a variety of nilpotent groups, then it is shown that \(\delta_{\mathcal V}(G)/V(G)\) is Abelian. Applied to the variety of nilpotent groups of class of most \(c\), an interesting consequence for the integral dimension subgroups of \(G\) follows, namely that \(D_ c(G)/\gamma_{c+1}(G)\) is Abelian for all \(c\geq 1\). Among other results it is also proved that \([D_ 4(G),G]=\gamma_ 5(G)\) for any group \(G\). Here \(D_ c(G):=G \cap(1+\Delta^ c(G))\), where \(\Delta(G)\) is the augmentation ideal of \(\mathbb{Z} G\) and \(\gamma_ c(G)\) denotes the \(c\)th term in the lower central series of \(G\).