Homogeneity in the free group

Abstract

We show that any nonabelian free group $\mathbb{F}$ is strongly ${\aleph }_0$-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $Aut\left(\mathbb{F}\right)$. We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not strongly ${\aleph }_0$-homogeneous.