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

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 $\mathbb{F}$.