Hereditary conjugacy separability of free products with amalgamation

A group G is called hereditarily conjugacy separable if every finite index subgroup of G is conjugacy separable. The property is not preserved in general by the formation of free products with amalgamation. Here we find conditions (a)−(fNH) for a free product with cyclic amalgamation to preserve this property. We define a class XH to consist of all groups satisfying conditions (a)−(fNH) and prove that polycyclic-by-finite and free-by-finite groups belong to this class. We also prove that XH contains right angled Artin groups and virtual retracts of them (i.e. semidirect factors of finite index subgroups). Combining this with recent results of Wise and Agol we deduce that the fundamental group of a compact 3-manifold contains a finite index subgroup that belongs to the class XH.