Homogeneity and prime models in torsion-free hyperbolic groups

Ould Houcine, Abderezak

doi:10.1142/S179374421100028X

Ould Houcine, Abderezak

Confluentes Mathematici, Tome 3 (2011) no. 1, pp. 121-155

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Résumé

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 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 the power of a primitive element or there exists an existentially closed subgroup $E (\bar{a})$ (respectively $E (\bar{b})$ ) of F containing $\bar{a}$ (respectively $\bar{b}$ ) 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 ∃-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.

Publié le : 2011-03-30

DOI : 10.1142/S179374421100028X

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Ould Houcine, Abderezak. Homogeneity and prime models in torsion-free hyperbolic groups. Confluentes Mathematici, Tome 3 (2011) no. 1, pp. 121-155. doi: 10.1142/S179374421100028X

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