Homogeneity in the free group

TL;DR

This paper proves that non-abelian free groups are strongly countably homogeneous in the first-order logic sense, characterizes elements with similar properties in finitely generated groups, and discusses implications for hyperbolic surface groups.

Contribution

It establishes strong $eth_0$-homogeneity for non-abelian free groups and characterizes primitive elements in finitely generated groups.

Findings

Non-abelian free groups are strongly $eth_0$-homogeneous.

Most hyperbolic surface groups are not $eth_0$-homogeneous.

Provides a characterization of elements with primitive-like properties.

Abstract

We show that any non abelian free group $\F$ is strongly $\aleph_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $\Aut(\F)$. 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 $\aleph_0$-homogeneous.