# Cohomologies associated to residually torsion-free nilpotent groups

This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.

A group $G$ is ${\it residually \ torsion \ free \ nilpotent}$ if for all $g \ne 1$ in $G$, there exists a normal subgroup $N$ of $G$ such that $G/N$ is a torsion-free nilpotent group and $g \notin N$.

I'm mainly interested in finitely generated groups so for the rest of this question $G$ will denote a finitely generated residually torsion-free nilpotent group. Let $E_G$ denote the Malcev Lie algebra (over $\mathbb{Q}$) of $G$. The group $G$ is ${\it 1-formal}$ if $E_G$ can be presented as a quotient of a free Lie algebra by homogeneous relations of bracket length 2, i.e., $E_G$ is quadratic. I know that the Malcev completion of $G$, $\kappa: G \to \exp E_G$, is faithful. Let $L$ denote $\text{gr}(E_G)$, the associated graded Lie algebra of $E_G$.

Are the cohomology algebras $H^*(G, \mathbb{Q})$ and $H^*(L, \mathbb{Q})$ isomorphic (as algebras)? What if, in addition, $G$ is 1-formal? Or, are there known counterexamples (in either case)?

The answer to the first question is affirmative if $G$ is a finitely generated torsion-free nilpotent group. References to proofs of this were kindly provided by Pasha Zusmanovich as an answer to: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.

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