Malcev completion

Summary

The Malcev completion of a finitely generated torsion-free nilpotent group $\Gamma$ is a simply connected nilpotent Lie group $N$ that contains $\Gamma$ as a lattice. It exists and is unique up to isomorphism, by a theorem of Malcev. It turns questions about the discrete group $\Gamma$ into questions about the Lie algebra of $N$. In this way compact nilmanifolds correspond to finitely generated torsion-free nilpotent groups.