Malcev completion of a torsion-free nilpotent group
= Malcev completion of a torsion-free nilpotent group
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Every finitely generated <torsion-free group> which is a <nilpotent group> embeds as a discrete cocompact subgroup of a unique simply connected nilpotent real <Lie group>. Its dimension equals the group's <integral cohomological dimension of a group>. In dimension three the Lie algebra is either abelian or has a basis with a single nonzero bracket $[Y,T]=X$, giving the <real Heisenberg group>.