On some properties of free Abelian extensions

Yu. V. Kuz'min

Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 303-315

Résumé

Let $B$ be an arbitrary group. Present it as a factor group of a free group $F$: $B\cong F/N$, $N\vartriangleleft F$. The extension $$1\to N/N'\to F/N'\to B\to1$$ is said to be a it free Abelian extension of the group $B$ (it is free in the category of extensions of $B$ by all possible Abelian groups). The author continues his study of the integral homology groups $H_n(F/N')$. The main result is that for any $B$ the exponent of the torsion subgroup of the group $H_n(F/N')\otimes Z[1/2]$ divides $n$ (as usual, $Z[1/2]$ is the ring of $2$-rational numbers). At the end of the paper the author formulates a number of conjectures on the homology of groups of the form $F/N'$. The notion of homological identity of a group is introduced, and the problem of describing the homological identities of free solvable and free nilpotent groups is posed. Bibliography: 9 titles.