On conjugate separability of nilpotent subgroups (Q6634493)

A subgroup \(H\) of a group \(G\) is said to be malnormal in \(G\) if \(H \cap H^{x} = 1\) for \(x \in G \setminus H\). A group \(G\) is said to be conjugate separable abelian, or CSA for short, if every maximal abelian subgroup if malnormal in \(G\); equivalently, the centraliser of every non-trivial element is abelian and self-normalising. The class of CSA groups includes among others groups acting freely on \(\Lambda\)-trees and torsion-free hyperbolic groups. \par In the paper under review the author considers the following generalisation of CSA groups, which is in turn a special case of a more general framework. A group is said to be conjugate separable nilpotent of class \(k\), or \(\mathrm{CSN}_{k}\) for short, if every subgroup, which is maximal with respect to the property of having nilpotence class at most \(k\), is malnormal. Clearly \(\mathrm{CSN}_{1}\) is the same thing as CSA. \par A group is said to be commutativity transitive, or CT for short, if the binary relation of commutativity is transitive. Every CSA group is CT, but the converse does not hold. CT plays an important role in the theory of residually free groups. \N\NThe author defines a group \(G\) to be nilpotence transitive of class \(k\), or \(\mathrm{NT}_{k}\) for short, if for every pair \(N_{1}, N_{2}\) of subgroups of \(G\) of nilpotence class at most \(k\) the assumption \(N_{1} \cap N_{2} \ne 1\) implies that \(\langle N_{1}, N_{2} \rangle\) is nilpotent of class at most \(k\). \par In the paper under review the author proves that the classes \(\mathrm{CSN}_{k}\) and \(\mathrm{NT}_{k}\) are universal, and explores the connections between the two. In particular, if a certain residuality condition holds, then the two classes are shown to coincide.