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

作者：Bridson Martin R^*; Reid Alan W

来源：International Mathematics Research Notices, 2015, 2015(8): 2111-2140.

DOI：10.1093/imrn/rnt353

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 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.

• 出版日期2015