The higher lower central series

The Baer invariants Γ _n (G) of a group are central extensions of the elementsγ _n (G) of the lower central series. We show that the inclusionsγ _n +1 ⊂γ _n can be lifted to functor morphisms Γ _n+1→Γ _n and a canonical Lie algebra, analogous to Lazard’s Lie algebra, can be constructed which is explicitly computable. This is applied in various ways.