For a discrete group G there are two well known completions. The first is the Malcev
(or unipotent) completion. This is a prounipotent group [Uscr ], defined over ℚ, together
with a homomorphism ψ : G → [Uscr ] that is universal among maps from G into
prounipotent ℚ-groups. To construct [Uscr ], it suffices for us to consider the case where
G is nilpotent; the general case is handled by taking the inverse limit of the Malcev
completions of the G/ΓrG, where Γ[bull ]G
denotes the lower central series of G. If G is abelian, then
[Uscr ] = G [otimes ] ℚ. We review this construction in Section 2.