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.