Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent p > 2

We consider the variety G2,p of nilpotent groups of class 2 of exponent p > 2 in the language L of group theory. To get the Amalgamation Property (AP) in [Bau] an additional predicate P (G) for G ∈ G2,p with G ′ ⊆ P (G) ⊆ Z(G) is introduced. Let GP2,p be the category of this groups in the extended language LP where the morphisms are embeddings. Using the class KP2,p of finite structures in GP2,p we get a Fraïssé limit D. If we build D by amalgamation then P (a) says that a will become an element of the commutator subgroup D of D in that process. In [Bau] it is shown that Th(D) is not simple. Here we point out that D has the tree property of the second kind (TP2). This is easily seen. Let L(n) be the language of group theory with n additional new constant symbols c1, . . . , cn. In L(n) we consider the class G(n) of all groups G ∈ G2,p, where G ⊆ 〈cG1 , . . . , c G n 〉 L ⊆ Z(G) and cG1 , . . . , c G n are linearly independent. 〈c G 1 , . . . c G n 〉 L denotes the subgroup generated by cG1 , . . . , c G n . G(n) is uniformly locally finite. Let K(n) be the class of finite structures in G(n). K(n) has the Hereditary Property (HP), the Joint Embedding Property (JEP) and the Amalgamation Property (AP). Hence the Fraïssé limit D(n) of the class K(n) exists. Note that D(1) is the extra special p-group considered by U. Felgner in [Fe]. In [MacSt] the corresponding bilinear alternating map is obtained as an ultraproduct of finite structures. It is a well-known example of a supersimple theory of SU-rank 1.