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Two groups are said to have the same nilpotent genus if they have the samenilpotent quotients. We answer four questions of Baumslag concerning nilpotentcompletions. (i) There exists a pair of finitely generated, residuallytorsion-free-nilpotent groups of the same nilpotent genus such that one isfinitely presented and the other is not. (ii) There exists a pair of finitelypresented, residually torsion-free-nilpotent groups of the same nilpotent genussuch that one has a solvable conjugacy problem and the other does not. (iii)There exists a pair of finitely generated, residually torsion-free-nilpotentgroups of the same nilpotent genus such that one has finitely generated secondhomology $H_2(-,\Z)$ and the other does not. (iv) A non-trivial normal subgroupof infinite index in a finitely generated parafree group cannot be finitelygenerated. In proving this last result, we establish that the first $L^2$ bettinumber of a finitely generated parafree group of rank $r$ is $r-1$. It followsthat the reduced $C^*$-algebra of the group is simple if $r\ge 2$, and that aversion of the Freiheitssatz holds for parafree groups.

Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag