Epimorphisms between the free groups in a variety of groups
Let \(V\) be a proper variety of groups and \(F_n\) a free \(V\)-group of finite rank \(n\). Suppose \(\theta\colon F_{n+m}\to F_n\) is an epimorphism of groups. The following problem is discussed in the paper: is it the case that there exists a free basis \(x_1,\ldots,x_{n+m}\) in \(F_{n+m}\) such that \(\ker\theta\) is the normal closure of \(\{x_1,\ldots,x_m\}\) in \(F_{n+m}\)? Let \(A\) be the variety of all Abelian groups and \(V\) the product of varieties \(V=AW\) for some proper variety \(W\). Under some assumptions it is proved that the problem has a negative answer for \(n\geq 3\) and \(m=1\). In particular one can take as \(W\) the variety of all nilpotent groups of class \(c\) and \(n>c\), \(m=1\). The proof is based on the reviewer's results on non-free projective modules over group rings of soluble groups [Mat. Sb., Nov. Ser. 116(158), 232-244 (1981; Zbl 0474.16007)].
scientific article; zbMATH DE number 1384471