# On Bousfield's problem for solvable groups of finite Pr\"ufer rank

## TL;DR

This paper investigates the relationship between group homology and various completions for solvable groups of finite Pr"ufer rank, showing that certain homology maps are epimorphisms and that Bousfield's localization coincides with K-completion in these cases.

## Contribution

It proves that for finitely generated solvable groups of finite Pr"ufer rank, the second homology map to R-completions is surjective, and extends results to specific finitely presented groups.

## Key findings

- H_2(G,K) to H_2(\hat G_R,K) is surjective for such groups.
- Bousfield's HK-localisation coincides with K-completion for these groups.
- H_n(G,K) to H_n(\hat G_R,K) is surjective for all n in certain semi-direct product groups.

## Abstract

For a group $G$ and $R=\mathbb Z,\mathbb Z/p,\mathbb Q$ we denote by $\hat G_R$ the $R$-completion of $G.$ We study the map $H_n(G,K)\to H_n(\hat G_R,K),$ where $(R,K)=(\mathbb Z,\mathbb Z/p),(\mathbb Z/p,\mathbb Z/p),(\mathbb Q,\mathbb Q).$ We prove that $H_2(G,K)\to H_2(\hat G_R,K)$ is an epimorphism for a finitely generated solvable group $G$ of finite Pr\"ufer rank. In particular, Bousfield's $HK$-localisation of such groups coincides with the $K$-completion for $K=\mathbb Z/p,\mathbb Q.$ Moreover, we prove that $H_n(G,K)\to H_n(\hat G_R,K)$ is an epimorphism for any $n$ if $G$ is a finitely presented group of the form $G=M\rtimes C,$ where $C$ is the infinite cyclic group and $M$ is a $C$-module.