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 $Γ_n(P)$ generated by Lie bracket ``commutators'' of weight $n$ and ${ γ_n(P) }$ be the lower central series of $P$. We prove that $Γ_n(P) \simeq γ_n(P)$ for arbitrary $n \geq 1$ 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.