On the Generic Type of the Free Group

Abstract

We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in $F_{\omega }$. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not ${\aleph }_1$-homogeneous.

To appear in J. of Symbolic Logic

References in corpus (1)

• Diophantine Geometry over Groups VIII: Stability

Cited by in corpus (7)

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• On Superstable Expansions of Free Abelian Groups
• On groups and fields interpretable in torsion-free hyperbolic groups
• Hyperbolic towers and independent generic sets in the theory of free groups
• Maximal hyperbolic towers and weight in the theory of free groups
• Two remarks on elementary theories of groups obtained by free constructions
• On the (non) superstable part of the free group