Title: 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 $G=M\rtimes C,$ where $C$ is the infinite
cyclic group and $M$ is a $C$-module.