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Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free nilpotent groups of the same nilpotent genus such that one has finitely generated second homology   $H_2(-,\mathbb {Z})$   and the other does not. (iv) A nontrivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first  L  2 -Betti number of a finitely generated parafree group in the same nilpotent genus as a free group of rank  r  is  r −1. It follows that the reduced  C *-algebra of the group is simple if  r ≥2, and that a version of the Freiheitssatz holds for parafree groups.

Nilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag