Submission date: 28 April 2010.

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 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.

Mathematics Subject Classification: 20F65, 20F67, 03C50

Keywords and phrases: Homogeneity, free groups, torsion-free hyperbolic groups, prime models, two-generated torsion-free hyperbolic groups.

Full text arXiv: pdf, ps.