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  "body": "\n## Definition\n\nLet $\\Gamma$ be a finitely generated, torsion-free [[nilpotent group|0Q7MMQHC1P]].\nA *Malcev completion* of $\\Gamma$ is a pair $(N, \\iota)$ with three properties.\n\n1. $N$ is a connected and simply connected [[nilpotent Lie group|DK87YN458D]].\n2. $\\iota \\colon \\Gamma \\to N$ is an injective group homomorphism.\n3. $\\iota(\\Gamma)$ is a [[lattice|BTYF5Y5PFY]] in $N$: it is discrete and $N/\\iota(\\Gamma)$ is compact.\n\n## Remarks\n\nMalcev proved that a Malcev completion exists and is unique up to a unique isomorphism compatible with $\\iota$.\nMore generally, each homomorphism between two such groups $\\Gamma$ extends uniquely to a continuous homomorphism of their completions.\n\nExample: $\\Gamma = \\Z^n$ with $N = \\R^n$ and $\\iota$ the inclusion.\nThe group $\\R^n$ is abelian, hence nilpotent, and simply connected; $\\Z^n$ is discrete, and $\\R^n/\\Z^n$ is the compact torus.\nIn the same way, the integer [[Heisenberg group|ZZYCV5EX9J]] has the real Heisenberg group as Malcev completion.\n\nNon-example: $\\Gamma = \\Z$ with $N = \\R^2$ and $\\iota(k) = (k, 0)$.\nClauses 1 and 2 hold, but $\\R^2/\\iota(\\Z)$ is a cylinder, which is not compact, so clause 3 fails.\n\nThe name is also used for the rational Malcev completion of $\\Gamma$.\nIt is a torsion-free nilpotent group containing $\\Gamma$ in which each element has a unique $k$-th root for each $k \\ge 1$.\nMoreover, some positive power of each of its elements lies in $\\Gamma$.\n",
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