Malcev completion

Remarks

Malcev proved that a Malcev completion exists and is unique up to a unique isomorphism compatible with $\iota$. More generally, each homomorphism between two such groups $\Gamma$ extends uniquely to a continuous homomorphism of their completions.

Example: $\Gamma = \Z^n$ with $N = \R^n$ and $\iota$ the inclusion. The group $\R^n$ is abelian, hence nilpotent, and simply connected; $\Z^n$ is discrete, and $\R^n/\Z^n$ is the compact torus. In the same way, the integer Heisenberg group has the real Heisenberg group as Malcev completion.

Non-example: $\Gamma = \Z$ with $N = \R^2$ and $\iota(k) = (k, 0)$. Clauses 1 and 2 hold, but $\R^2/\iota(\Z)$ is a cylinder, which is not compact, so clause 3 fails.

The name is also used for the rational Malcev completion of $\Gamma$. It is a torsion-free nilpotent group containing $\Gamma$ in which each element has a unique $k$-th root for each $k \ge 1$. Moreover, some positive power of each of its elements lies in $\Gamma$.