Strong Parafree Conjecture

Establish that every finitely generated parafree group has trivial second integral homology and cohomological dimension at most two.

Background

A parafree group is residually nilpotent and has the same lower-central-series quotients as a free group. The paper applies the asserted homological consequences of Baumslag’s Strong Parafree Conjecture to prove that the maps from the free multiplicative Lie algebra components $\Gamma_n(P)$ to the lower-central-series terms $\gamma_n(P)$ are isomorphisms for the relevant class of finitely generated parafree groups.

The conjecture is stronger than the vanishing assumptions used in the main theorem: the paper requires $H_2(P,\mathbb Z)=H_3(P,\mathbb Z)=0$, whereas the conjecture explicitly asserts $H_2(P,\mathbb Z)=0$ and $cd(P)\leq 2$, the latter implying the required third-homology vanishing in the settings discussed.