Homogeneity and prime models in torsion-free hyperbolic groups

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 $σ: E(\bar a) \to E(\bar b)$ with $σ(\bar a)=\bar b$. 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.

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