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