Torsion in profinite completions

Abstract

In [l] the question was raised whether the profinite completion of a torsion-free residually finite group is necessarily torsion-free. Subsequently, counter-examples were discovered by Evans [2], and more recently, Lubotsky [4] has shown that there is a finitely generated torsion-free residually finite group whose profinite completion contains a copy of every finite group. In this paper we study the situation more closely for soluble groups. The major result of the paper is the construction of finitely generated centre by metabelian counter-examples in Section 3. We begin in Section 2 by establishing some positive results showing that there are no counter-examples amongst abelian groups, finitely generated abelian by nilpotent groups or soluble minimax groups. In Section 3 we first establish counter-examples which are nilpotent of class 2: these are necessarily infinitely generated. Then we give the promised centre by metabelian examples. It is interesting to note that the dichotomy arising here between abelian by nilpotent groups and centre by metabelian groups is the one which arises in the work of Hall [3].