On Bousfield problem for the class of metabelian groups

TL;DR

This paper investigates the homological effects of localizations and completions on finitely presented metabelian groups, confirming a conjecture by Bousfield regarding the induced maps on second homology groups.

Contribution

It proves that for certain rings, the natural map from a finitely presented metabelian group to its R-completion induces an epimorphism on second homology, solving Bousfield's problem for this class.

Findings

The map induces an epimorphism on H_2(-, R) for R=Q or Z/n.

Confirms Bousfield's conjecture for metabelian groups.

Provides new insights into the homological behavior of group completions.

Abstract

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for $R=\mathbb Q$ or $R=\mathbb Z/n$ and a finitely presented metabelian group $G$, the natural map from $G$ to its $R$-completion induces an epimorphism of homology groups $H_2(-,R)$. This answers a problem of A.K. Bousfield for the class of metabelian groups.