Tree Products of Conjugacy Separable Groups

AbstractA group G is said to be conjugacy separable if for each pair of elements x, y∈G such that x and y are not conjugate in G, there exists a finite homomorphic image Ḡ of G such that the images of x, y are not conjugate in Ḡ. In this note, we show that the tree products of finitely many conjugacy separable and subgroup separable groups amalgamating central subgroups with trivial intersections are conjugacy separable. We then apply our results to polycyclic-by-finite groups and free-by-finite groups