The free group has the dimensional order property

TL;DR

This paper proves that the theory of nonabelian free groups exhibits the dimensional order property, indicating complex model-theoretic behavior and the absence of a comprehensive structure theorem for certain models.

Contribution

It establishes that the theory of nonabelian free groups has the DOP, a significant property in model theory indicating high complexity.

Findings

The theory of nonabelian free groups has the DOP.

No reasonable structure theorem exists for -saturated models of this theory.

The result impacts understanding of model-theoretic complexity in group theory.

Abstract

We prove that the common theory of nonabelian free groups has the dimensional order property, or DOP, implying, for example, that there is no reasonable structure theorem for $\aleph_1$-saturated models of this theory.