Elementary subgroups of the free group are free factors - a new proof

TL;DR

This paper presents a new proof that elementary subgroups of non-abelian free groups are free factors, using definability of automorphism orbits fixing large subsets.

Contribution

It introduces a novel proof technique based on automorphism orbit definability to establish the free factor property of elementary subgroups.

Findings

Elementary subgroups are free factors of free groups.

Automorphism orbit definability is key to the proof.

Provides a new perspective on subgroup structure in free groups.

Abstract

In this note we give a new proof of the fact that an elementary subgroup (in the sense of first-order theory) of a non abelian free group $\mathbb{F}$ must be a free factor. The proof is based on definability of orbits of elements of under automorphisms of $\mathbb{F}$ fixing a large enough subset of $\mathbb{F}$.