On the Generic Type of the Free Group

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 ${\mathrm{\aleph }}_1$-homogeneous.