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

TL;DR

This paper addresses four open questions by Baumslag on nilpotent completions of groups, demonstrating surprising differences in properties like finite presentability, solvability of the conjugacy problem, and homology among groups with the same nilpotent quotients.

Contribution

It provides counterexamples and new results showing that groups with identical nilpotent quotients can differ significantly in algebraic and computational properties.

Findings

Existence of finitely generated, residually torsion-free-nilpotent groups with same nilpotent genus but different finite presentability.

Existence of pairs with same nilpotent genus where one has solvable conjugacy problem and the other does not.

First $L^2$ betti number of a finitely generated parafree group of rank r is r-1.

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 $H_2(-,\Z)$ 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.