Separability Properties of Certain Tree Products of Groups

We introduce the concept of finite compatibility to prove certain double coset separability of tree products of central subgroup separable groups. We also prove a criterion for the conjugacy separability of generalized free products of two conjugacy separable groups amalgamating a central subgroup in one of the factors. Using this we prove that tree products of finitely many central subgroup separable and conjugacy separable groups are conjugacy separable. We apply the results to polycyclic-by-finite groups and groups of linkages of torus knots.