and four problems of Baumslag.

**Martin R. Bridson and Alan W. Reid**

*Accepted for publication in IMRN, November 2013.*

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 $H_2(-,\Z)$ and the other does not. (iv) A non-trivial

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 of rank $r$ is $r-1$. It follows that the reduced

$C^*$-algebra of the group is simple if $r\ge 2$, and that a version of the Freiheitssatz

holds for parafree groups.

19 pages, 1 figure.