Some results on parafree groups

Abstract:

We obtain some theorems concerning parafree groups in certain varieties, which are analogs of corresponding theorems about free groups in these varieties. Our principal results are: (1) A normal subgroup $N$ of a parafree metabelian group $P$ of rank $\geqslant 2$ such that $N \cdot {\gamma _2}P$ has infinite index in $P$ is not finitely generated unless it is trivial. (2) If $x$ and $y$ are elements of a parafree group $P$ in any variety containing the variety of all metabelian groups which are independent modulo ${\gamma _2}P$, then the commutator $[x,y]$ is not a proper power.