Residual finiteness of generalized free products of isomorphic groups

We give a necessary and sufficient condition for an amalgamated free product of arbitrarily many isomorphic potent and cyclic subgroup-separable groups to be potent and cyclic subgroup-separable, respectively. We then apply our results to show that certain polygonal products of free groups are potent and cyclic subgroup-separable. Further applications are made to finite tree products of polycyclic-by-finite groups and free-by-finite groups.