On torsion in free central extensions of some torsion-free groups
Let F be a free group of rank at least two, R a normal subgroup, \(\gamma_ cR\) the c-th term of the lower central series of R. The author studies torsion in groups of the form \(F/[\gamma_ cR,F]\), \(c\geq 2\). The main result says that for any prime p and any normal subgroup R such that \(G=F/R\) has no elements of order p the torsion subgroup of \(\gamma_ pR/[\gamma_ pR,F]\) is isomorphic to \(H_ 4(G,Z_ p)\), the fourth homology group of G with coefficients in \(Z_ p\). In some cases this enables to precisely compute the torsion subgroup of a group of the above kind. Another theorem is a strengthening of a result of the author [in Math. Z. 187, 259-267 (1984; Zbl 0526.20022)], and it says that for any \(c\geq 3\), \(\gamma_ cR/[\gamma_ cR,F]\) decomposes into the direct product of a free abelian group and a periodic abelian group of exponent dividing c. The main tool is the isomorphism \(\gamma_ cR/[\gamma_ cR,F]\cong L_ cM\otimes_ GZ\) where \(L_ cM\) is the c-th homogeneous component of the free Lie ring on \(M=R/R'\). This isomorphism has been established in a paper of \textit{G. Baumslag}, \textit{R. Strebel} and \textit{M. Thomson} [in J. Pure Appl. Algebra 16, 121-132 (1980; Zbl 0426.20025)].
scientific article; zbMATH DE number 4017165
scientific article