The free multiplicative Lie algebra $L(P)$ for a finitely generated parafree group $P$

Abstract

Let $P$ be a group, $L(P)$ be the free multiplicative Lie algebra with normal subgroup ${\Gamma }_n(P)$ generated by Lie bracket ``commutators'' of weight $n$ and $\{{\gamma }_n(P)\}$ be the lower central series of $P$. We prove that ${\Gamma }_n(P)\simeq {\gamma }_n(P)$ for arbitrary $n\ge 1$ and $P$ a finitely generated parafree group such that $H_2(P,\mathbb{Z})=0=H_3(P,\mathbb{Z})$ (e.g. $P$ satisfies the Strong Parafree Conjecture), in particular Ellis's conjecture holds i.e. the above isomorphism holds for finitely generated free group $P$ but this easily implies it holds for any free group.