Cyclic Conjugacy Separability Of Certain Hnn Extensions

Research Report No. 4/2001

Abstract

A group $G$ is called cyclic conjugacy separable if for each $x\in G$ and each cyclic subgroup $\langle y\rangle$ of $G$ such that no conjugate of $x\in G$ belongs to $\langle y\rangle$, then there exists a finite homomorphic image $\bar G$ of $G$ such that no conjugate of $\bar x\in\bar G$ belongs to $\langle\bar y\rangle$.   In this note, we give sufficient conditions for HNN extensions of cyclic conjugacy separable and subgroup separable groups to be cyclic conjugacy separable. We then apply our results to show that certain HNN extensions of finitely generated torsion-free nilpotent-by-finite groups or free-by-finite groups are cyclic conjugacy separable. Finally, one-relator groups with non-trivial centre are proved to be cyclic conjugacy separable.