On the generic type of the free group

Abstract

We answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed in F_ω. 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 ℵ₁-homogeneous.