11. Separability Conditions in Free and Polycyclic Groups

This chapter and the next ones deal with abstract groups. The properties that are studied are stated in the language of the natural profinite (or, more generally, pro-\(\mathcal{C}\)) topology on an abstract group. For example subgroup separability (i.e., a finitely generated subgroup is the intersections of the subgroups of finite index that contain it) or conjugacy separability. The methods of proof use the geometric techniques developed in the previous chapters.

The second section of this chapter contains a classical theorem of Marshall Hall that says that if \(H\) is a finitely generated subgroup of a free abstract group \(\varPhi\), then \(U= H*L\), where \(U\) is a subgroup of finite index in \(\varPhi\) and \(L\) is some subgroup of \(U\). It is shown that this is in fact equivalent to saying that \(H\) is closed in the profinite topology of \(\varPhi\). A corresponding result holds for other pro-\(\mathcal{C}\) topologies, when \(\mathcal{C}\) is an extension-closed pseudovariety of finite groups. One can then deduce that the profinite topology of a finitely generated subgroup \(H\) of a free-by-finite abstract group \(R\) is precisely the topology induced from the profinite topology of \(R\). In Sect. 11.3 a more general result is proved: if \(H_{1}, \dots, H_{n}\) is a finite collection of finitely generated closed subgroups of a free abstract group \(\varPhi\) endowed with the pro-\(\mathcal{C}\) topology, then the product \(H_{1}\cdots H_{n}\) is a closed subset of \(\varPhi\). The last section records properties of abstract polycyclic-by-finite groups; these groups serve as basic building blocks for the free constructions of abstract groups studied in later chapters.