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Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite presentability of $L(G)$ implies $G$ is finitely presented.

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  • $\begingroup$ It looks like a hopeless approach to prove finite presentability of the Torelli group? there are no general results of this kind, and in small classes where it holds, finite presentability has to be checked anyway. $\endgroup$
    – YCor
    May 6, 2014 at 8:32
  • $\begingroup$ For pro-p groups it is true: if $L(G)$ is finitely presented, then $G$ is finitely presented. But $L(G)$ is a $\mathbb{Z}_p$-Lie algebra. $\endgroup$ May 6, 2014 at 11:30
  • $\begingroup$ @Dietrich: this is really distinct since $G$ is discrete in the question, and for pro-$p$-groups $L(G)$ is defined from the $p$-series of $G$. Here $L(G)$ only reflects torsion-free nilpotent quotients of $G$. $\endgroup$
    – YCor
    May 6, 2014 at 13:44
  • $\begingroup$ Yves, does finite presentability of the Torelli group imply finite presentability of the the group? I am interested in even small classes where the result holds. Would you describe these classes. Thanks. $\endgroup$ May 6, 2014 at 16:48

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