Embeddings and chains of free groups

We build two nonabelian CSA-groups in which maximal abelian subgroups are conjugate and divisible, as the countable unions of increasing chains of CSA-groups and by keeping the constructions as free as possible in each case. For n ≥ 1, let Fn denote the free group on n generators. We view all groups G as first-order structures 〈G, ·, , 1〉, where ·, , and 1 denote respectively the multiplication, the inverse, and the identity of the group. The following striking results are proved in a series of papers of Sela culminating in [Sel07]. Fact 1 [Sel05, Sel06a, Sel06b, Sel07] (1) For any 2 ≤ n ≤ m, the natural embedding Fn ≤ Fm is an elementary embedding. (2) For any n ≥ 2, the (common) complete theory Th (Fn) is stable. We refer to [Hod93] for model theory in general, and to [Poi87] and [Wag97] for stability theory and in particular stable groups. Let F denote the free group over countably many generators. Fact 1 has the following corollary. Corollary 2 The natural embeddings F2 ≤ · · ·Fn ≤ · · · ≤ F are all elementary. In particular each Fn is an elementary substructure of F , and Th (F ) is stable. A CSA-group is a group in which maximal abelian subgroups A are malnormal, i.e., such that A ∩ A 6= 1 implies that g is in A for any element g of the ambient group. The class of CSA-groups contains all free groups and is studied from various points of view. We refer to [JOH04, JMN08] for a model theoretic approach in combination of questions concerning particular groups [Che79, Jal01, Cor03], and to [KMRS08] for an approach more related to computational aspects in limit groups. We prove the following lemma on embeddings of torsion-free CSA-groups in which maximal abelian subgroups are cyclic.