Tree products of residually finite groups

P.C. WONG  and  K.B. WONG

Research Report No. 1/2004

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 tree products and polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to tree products and polygonal products of polycyclic-by-finite groups and free-by-finite groups.