Abelian subgroup separability of some one-relator groups

Following Malcev [5], we will call a subgroup M of a group G finitely separable if for any element g ∈ G, not belonging to M, there exists a homomorphism φ of group G onto some finite group X such that gφ / ∈Mφ. This is equivalent to the statement that for any element g ∈ G \M, there exists a normal subgroup N of finite index in G such that g / ∈MN . A group G is subgroup separable if each of its finitely generated subgroups is finitely separable. If every cyclic (or finitely generated abelian) subgroup of a group G is finitely separable, then group G is said to be cyclic (resp., abelian) subgroup separable. Cyclic subgroup separable groups are also called πc groups. Evidently, every πc group is residually finite (i.e., any of its one-element subsets is finitely separable). However, the converse is not true. So, cyclic subgroup separability strengthens residual finiteness. But subgroup separability is stronger; in fact, if G is a finitely presented subgroup separable group, thenG has a solvable generalized word problem (just as if G is a finitely presented residually finite group, then G has a solvable word problem). So subgroup separability is so strong that very few classes of groups satisfy it. But cyclic subgroup separability is possessed by much larger classes of groups. Hall in [1] showed that free groups are πc. In [5], Malcev proved that finitely generated torsion-free nilpotent groups are πc. Free products of πc groups are πc; further, it was established that finite extensions of πc groups are πc [6, 7]. Cyclic subgroup separability of certain generalized free products of πc groups amalgamating a cyclic subgroup was proved in [10, 8]. Some one-relator groups happen to be πc [10, 11]. In this paper, we enlarge the classes of one-relator πc groups. We will prove the following theorem.