Existentially complete torsion-free nilpotent groups

Extract

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