The residual finiteness of certain HNN extensions of subgroup separable groups

P.C. WONG AND K.B. WONG

Research Report No. 5/2002

Abstract

A group $G$ is called cyclic subgroup separable for the cyclic subgroup $H$ if for each $x\in G\setminus H$, there exists a normal subgroup $N$ of finite index in $G$ such that $x\notin HN$. Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain HNN extensions of subgroup separable groups with associated normal subgroups are cyclic subgroup separable. We then apply our results to HNN extensions of polycyclic-by-finite groups and free-by-finite groups.