This paper continues the study of existentially complete nilpotent groups initiated in [6]. Following [6], we let Kn denote the theory of groups nilpotent of class ≤ n and let Kn+ denote the theory of torsion-free groups nilpotent of class ≤ n. The principal results of [6] were that for n ≥ 2, neither Kn nor Kn+ has a model companion, and the classes E, F, and G of existentially complete, finitely generic and infinitely generic models of Kn are all distinct. The question of the relationships between these classes in the context of Kn was left open, however, and the proof of their distinctness for Kn+ obviously did not carry over to Kn+, because it made strong use of torsion elements.

In this paper we establish the relationships between E, F, and G for K2+. We show that all three classes are distinct. We also show that there is only one countable finitely generic model, and only one countable infinitely generic model, and that all the countable existentially complete models can be arranged in a sequence N1 ⊆ N2 ⊆ N3 ⊆ … ⊆ Nω, where Z(Nn) is the direct sum of n copies of Q. Another result is that the finite and infinite forcing companions of K2+ differ by an ∀∃∀ sentence. Finally, we show that there exist finitely generic models of K2+ in all infinite cardinalities.