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  "body": "\n## Summary\n\nThe Malcev completion of a finitely generated torsion-free nilpotent group $\\Gamma$ is a simply connected [[nilpotent Lie group|DK87YN458D]] $N$ that contains $\\Gamma$ as a lattice.\nIt exists and is unique up to isomorphism, by a theorem of Malcev.\nIt turns questions about the discrete group $\\Gamma$ into questions about the Lie algebra of $N$.\nIn this way compact nilmanifolds correspond to finitely generated torsion-free nilpotent groups.\n\n## Explanations\n\n### L1 human\n\nThe integer lattice $\\Z^n$ sits inside the space $\\R^n$, and $\\R^n$ is the smallest connected group in which $\\Z^n$ is spread out evenly.\nThe Malcev completion does the same for groups that are nearly commutative, the nilpotent groups.\nFor example, the $3 \\times 3$ upper triangular integer matrices with ones on the diagonal sit inside the same matrices with real entries.\nThe real group is the Malcev completion of the integer group.\n",
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