Some applications of free differential calculus in group theory (Q810145)

The author gives a generalization of the free differential calculus of Fox, and gives, without proof, some results that can be obtained by applying it. The generalization is as follows. Let F be a non-cyclic free group and \(R\triangleleft F\). Let R be freely generated by elements \(r_ i\) (i\(\in I)\), and let \(\Delta\) (R) be the augmentation ideal of \({\mathbb{Z}}R\). Then \(\Delta\) (R)\({\mathbb{Z}}F\) is freely generated as a left \({\mathbb{Z}}F\)-module by the elements \((r_ i-1)\) (i\(\in I)\), and so if \(u\in \Delta (R).{\mathbb{Z}}F\), then \(u=\sum_{i\in I}d^ R_ i(u)(r_ i- 1)\) for certain uniquely determined elements \(d^ R_ i(u)\in {\mathbb{Z}}F\). If \(R=F\), these are the usual (left) Fox derivatives of u; similarly we can obtain the right Fox derivatives. Three applications of these generalized Fox derivatives are given. Theorem 1. Let R be as above, and suppose that \(R\leq \gamma_ k(F)\), the k-th term of the lower central series of F, for some \(k\geq 2\). Let \(S_{n,m}=[\gamma_ n(R),F,...,F]\) (n\(\geq 2,m\geq 1)\) with m terms F. Then for \(m\leq 2k-2\), the abelian group \(S_{n,m}/S_{n,m+1}\) decomposes as \(D\oplus T\), where D is free abelian of infinite rank and T is a torsion group whose exponent divides 4 if \(n=2\), or n if \(n>2\). Theorem 2. Suppose that \(G=F/R\) satisfies one of the following conditions: (a) \({\mathbb{Z}}G\) has no zero divisors, (b) \(G/G'\) is not periodic. Then for \(n\geq 3\) the following are equivalent. (a) \(F/I_ R([\gamma_ n(R),F])\) is residually nilpotent, where \(I_ R(\)-) denotes the isolator in R, (b) F/R is either residually torsion-free nilpotent or discriminated by nilpotent groups of finite prime power exponent. Theorem 3. F/R is hopfian if and only if one of the groups \(F/\gamma_ n(R)\) (n\(\geq 1)\) and \(F/I_ R([\gamma_ n(R),F])\) (n\(\geq 2)\) is hopfian.