Subspace topologies in central extensions (Q5957523)

In 1993 \textit{P. H. Kropholler} and \textit{J. S. Wilson} constructed a countable torsion-free residually finite nilpotent group of class 2 and a finitely generated torsion-free residually finite centre-by-metabelian group, both of whose profinite completions contain an element of prime order [J. Pure Appl. Algebra 88, No. 1-3, 143-154 (1993; Zbl 0829.20044)]. In contrast, Chatzidakis has shown that the profinite completion of a torsion-free residually finite Abelian group is torsion-free [Proposition 2.1 in `Profinite groups' by \textit{J. S. Wilson}, Lond. Math. Soc. Monogr. 19, Oxford (1998; Zbl 0909.20001)]. Furthermore, \textit{W. W. Crawley-Boevey, P. H. Kropholler} and \textit{P. A. Linnell} have shown that this conclusion also holds for finitely generated torsion-free residually finite metabelian-by-finite groups [J. Pure Appl. Algebra 54, No. 2/3, 181-196 (1988; Zbl 0666.16007)]. It follows that the torsion in Kropholler and Wilson's examples lies in the centre of the group. Thus the author is motivated to study the possibilities for the topology induced on the centre by the profinite topology of a group.NEWLINENEWLINENEWLINEThe two main theorems are of the following form. Given a countable torsion-free Abelian group \(A\) with 2 countable collections \(\{A_1,A_2,\dots\}\) and \(\{B_1,B_2,\dots\}\) of subgroups of finite index, a group \(G\) of the required form is constructed with centre \(A\) such that each \(A_i\) is closed in the profinite topology on \(G\) and each \(B_j\) is not closed. Thus we have a large degree of control on the induced topology. By making suitable choices for the \(A_i\) and \(B_j\) the author can embed finite Abelian groups in various profinite completions. He concludes by proving that there exists a countable residually finite torsion-free nilpotent group \(G\) of class 2 such that every finite Abelian group embeds in the profinite completion of \(G\).