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

Abstract

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 L2-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.