Fox subgroups of free groups (Q5896371)

Following \textit{Yu. M. Gorchakov} [Algebra Logika 6, No. 3, 13--23 (1967; Zbl 0167.28702)] a subgroup \(H\) of a free group \(F\) of countable rank is called a commutator subgroup if (i) any basic commutator in \(F\backslash H\) is less than any basic commutator from \(H\); (ii) \(\gamma_{i+1}(F)H\cap \gamma_ i(F)/\gamma_{i+1}(F)\) is generated by images of basic commutators from \(H\). Here \(\gamma_ i(F)\) is the \(i\)th member of the lower central series of \(F\). Theorem: Let \(H\) be a normal commutator subgroup in \(F\) such that \(F/H\) is residually nilpotent. Denote by \(I\) the kernel of the natural homomorphism of integral group rings \({\mathbb Z}F\to {\mathbb Z}(F/H)\) and by \(\Delta\) the augmentation ideal in \({\mathbb Z}F\). Then the subgroup \(F\cap(1+\Delta^ nI)\) of \(F\) coincides with a product \(H^ n_ 1\cdot...\cdot H^ n_{n+1},\) where \(H^ n_ j\) is the product of commutators \([H\cap \gamma_{r_ 1}(F),...,H\cap \gamma_{r_ j}(F)]\) for a prescribed set of integers \(r_ 1,...,r_{n+1}\). This theorem can be used for matrix representations of relatively free polynilpotent groups [see \textit{C. K. Gupta} and \textit{N. D. Gupta}, Lect. Notes Math. 372, 318--329 (1974; Zbl 0291.20033)].