Fields interpretable in the free group

TL;DR

This paper proves that no infinite field can be interpreted within the theory of nonabelian free groups and characterizes Abelian groups interpretable in this theory.

Contribution

It establishes a fundamental limitation on interpretable structures in free groups and provides a classification of Abelian groups within this framework.

Findings

No infinite field is interpretable in free groups.

Characterization of Abelian groups interpretable in free groups.

Advances understanding of the model theory of free groups.

Abstract

We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.