A free product of finitely generated nilpotent groups amalgamating a cycle that is not subgroup separable

Abstract

Abstract. We exhibit a counterexample to a recent assertion concerning the subgroup separability of groups in the title. The example also serves as a simplification of work of Gitik and Rips.Denote the generalized free product of groups A and B amalgamating the common subgroup H by A * H B. Let Q be any class of groups, for example the finite groups, the finite p-groups or the nilpotent groups. A group G is said to be residually Q iff it has the following property where g ∈ G : g = 1 if and only if g = 1 in all homomorphic images G of G in Q.Likewise G is called subgroup separable (also called LERF) iff for all g ∈ G and all finitely generated subgroups H of G : g ∈ H if and only if g ∈ H in all finite images G of G.Separation properties such as these are often shared by nilpotent groups and free groups, in part because free groups are residually finitely generated torsion free nilpotent [9]. Furthermore, the proof that each free product F of free groups with cyclic amalgamation is residually finite [2] and potent [1] uses the passage from F down to generalized free products of nilpotent groups. Now in [3] it was shown that each generalized free product F as above is subgroup separable. Consequently one might well expect a similar result for generalized free products of nilpotent groups. The paper [13] attempts to prove this result in the same fashion as [3] by establishing two propositions and then appealing to Lemma 1 of [3]. Unfortunately Proposition 3.5 of [13], whilst true, seems to be insufficient.Counterexamples of the type claimed can readily be obtained by selecting suitable subgroups of the generalized free products constructed in [5]. Consider, for instance, G = A * a B where B = a, c : [a, c] = 1 and Thus G† is the free product of a finitely generated torsion free nilpotent group of class 2 with a free abelian group amalgamating a 2 , a maximal cyclic subgroup

On the finitely separability of subgroups of generalized free products

“…An example of the generalized free product with cyclic amalgamation of two subgroup separable groups, which is not a subgroup separable group, was built by E. Rips [6]. Later the more simple similar example of generalized free product with cyclic amalgamation of two finitely generated nilpotent groups was given in [7].…”

“…Let us state two another corollaries of this Theorem assertions of which are, apparently, the new. The first one is interesting to compare with the above examples of [6] and [7] and result of [9]. In the second consequence the question of subgroup separability of Baumslag -Solitar groups is considered.…”