---
tag: V9GRK3QANS
kind: concept
name: Malcev completion
msc: ["22E25", "22E40", 20F18]
sources:
  - url: https://en.wikipedia.org/wiki/Nilmanifold
---

## Summary

The 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.
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.

## Explanations

### L1 human

The 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.
The Malcev completion does the same for groups that are nearly commutative, the nilpotent groups.
For example, the $3 \times 3$ upper triangular integer matrices with ones on the diagonal sit inside the same matrices with real entries.
The real group is the Malcev completion of the integer group.