Abstract
We show that any nonabelian free group F of finite rank is homogeneous; that is for any tuples , having the same complete n-type, there exists an automorphism of F which sends ā to .We further study existential types and show that for any tuples , if ā and have the same existential n-type, then either ā has the same existential type as the power of a primitive element or there exists an existentially closed subgroup E(ā) (respectively ) of F containing ā (respectively ) and an isomorphism with .We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are ∃-homogeneous and prime. In particular, this gives concrete examples of finitely generated groups which are prime and not quasi axiomatizable, giving an answer to a question of A. Nies.
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