Subgroup Separability of HNN-Extensions with Abelian Base Group

In the present paper we give a complete characterisation of subgroup separability of HNN-extensions with finitely generated abelian base group. We are also able to characterise subgroup separability for some families of free-by-cyclic groups and thus answer partially a question of Scott (1987, “Combinatorial Group Theory and Topology,” Annals of Mathematics Studies, Vol. 111, Princeton Univ. Press, Princeton, NJ).