Parafree Lie algebras over Z/2 with nonzero second homology are constructed explicitly; the Z-completion of the free rank-two Lie algebra has cohomological dimension greater than two; countable parafree groups with H2 different from zero exist.
Baumslag: Groups with the same lower central sequence as a relatively free group
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Homological properties of parafree Lie algebras
Parafree Lie algebras over Z/2 with nonzero second homology are constructed explicitly; the Z-completion of the free rank-two Lie algebra has cohomological dimension greater than two; countable parafree groups with H2 different from zero exist.