Homogeneity and prime models in torsion-free hyperbolic groups

TL;DR

This paper proves that nonabelian free groups are homogeneous and explores their existential types, showing that certain hyperbolic groups are both existentially homogeneous and prime, providing examples of finitely generated groups with these properties.

Contribution

It establishes homogeneity of free groups and demonstrates that some torsion-free hyperbolic groups are both -homogeneous and prime, with new insights into their model-theoretic properties.

Findings

Nonabelian free groups are homogeneous.

Certain torsion-free hyperbolic groups are -homogeneous and prime.

Examples of finitely generated groups that are prime but not QFA.

Abstract

We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $\bar a$, $\bar b \in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$. We further study existential types and we show that for any tuples $\bar a, \bar b \in F^n$, if $\bar a$ and $\bar b$ have the same existential $n$-type, then either $\bar a$ has the same existential type as a power of a primitive element, or there exists an existentially closed subgroup $E(\bar a)$ (resp. $E(\bar b)$) of $F$ containing $\bar a$ (resp. $\bar b$) and an isomorphism $\sigma : E(\bar a) \to E(\bar b)$ with $\sigma(\bar a)=\bar b$. We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are $\exists$-homogeneous and prime. This gives, in particular, concrete examples of finitely generated groups which are prime and not QFA.