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On the generic type of the free group. (English) Zbl 1213.03047
There has been a renewed interest in the model theory of nonabelian free groups, since Sela proved that their common first-order theory is stable. The present paper prolongates remarks of Pillay on the basic model theory of these groups, following their stability. As nonabelian free groups are connected (by early remarks of Poizat), there is a unique generic type \(p_0\) over the empty set of parameters, and for free groups \(F_n\) of finite rank realizations of \(p_0\) are precisely primitive elements. Using Whitehead automorphisms, the author proves here that \(F_n\) is covered by finitely many translates of the set of non-primitive elements. He derives that the set of primitive elements in \(F_n\) is not uniformely definable (as \(n>1\) varies), and concludes, among others, that \(p_0\) is not isolated. Then the author shows that infinite independent sets of realizations of \(p_0\) do not necessarily extend to a basis in countable free groups (contrarily to the finite ones), and in particular that elementary subgroups of the countable but not finitely generated free group \(F_{\omega}\) are not necessarily free factors. He finally proves that the fact that the weight of \(p_0\) is infinite is witnessed in \(F_{\omega}\), and that uncountable free groups are not \(\aleph_1\)-homogeneous structures.

03C60 Model-theoretic algebra
20E05 Free nonabelian groups
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