Homogeneity and prime models in torsion-free hyperbolic groups
Confluentes Mathematici, Volume 3 (2011) no. 1, pp. 121-155.

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 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 the power of a primitive element or there exists an existentially closed subgroup E(a¯) (respectively E(b¯)) of F containing a¯ (respectively 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. 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.

Published online:
DOI: 10.1142/S179374421100028X
Abderezak Ould Houcine 1

1
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Abderezak Ould Houcine. Homogeneity and prime models in torsion-free hyperbolic groups. Confluentes Mathematici, Volume 3 (2011) no. 1, pp. 121-155. doi : 10.1142/S179374421100028X. https://cml.centre-mersenne.org/articles/10.1142/S179374421100028X/

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