In recent years the authors have made detail studies of modules over a crossed product of a division ring by an Abelian group, promising us that there were group theoretic applications to come. In this current paper we get these applications, or at least some of them. There are three.
The first concerns a module \(M\) over the group ring \(RG\), where \(R\) is a commutative ring and \(G\) is a torsion-free ‘connected’ polycyclic group and gives conditions under which \(M\) is torsion-free over a sizeable section of \(RG\). Connected here means that the group can be embedded as a Zariski-connected subgroup of some \(\text{GL}(n,\mathbb Z)\).
The second, with \(M\), \(R\) and \(G\) as above, but now with \(R\) Noetherian, \(M\) finitely generated and \(G\) nilpotent, gives conditions under which a specific subgroup of \(\operatorname{Aut} G\) the authors denote by \(\text{Stab}_{\operatorname{Aut} G}M\) is arithmetic.
The third concerns a finitely generated nilpotent-nilpotent-by-finite group \(G\) with Fitting subgroup \(F\) such that no subgroup of \(G\) of finite index has a quotient isomorphic to a wreath product of a cyclic group of prime order by an infinite cyclic group. They show that the group of automorphisms induced on \(G/F\) by \(\operatorname{Aut} G\) has a subgroup of finite index acting nilpotently on \(G/F\) and that any finitely generated extension of \(G\) by a nilpotent group is again nilpotent-by-nilpotent-by-finite.