On Bousfield's problem for solvable groups of finite Prüfer rank

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 Prüfer 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 $G=M⋊C,$ where $C$ is the infinite cyclic group and $M$ is a $C$-module.