On the Generic Type of the Free Group

TL;DR

This paper investigates the properties of the generic type in free groups, showing it has infinite weight, is not isolated, and that uncountable free groups lack certain homogeneity, with implications for definability and model theory.

Contribution

It answers an open question about the infinite weight of the generic type in free groups and demonstrates non-uniform definability of primitive elements.

Findings

The infinite weight of the generic type is witnessed in F_ω.

Primitive elements are not uniformly definable in finite rank free groups.

Uncountable free groups are not $eth_1$-homogeneous.

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.