Homogeneity and prime models in torsion-free hyperbolic groups

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.