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 to .
We further study existential types and show that for any tuples , if and have the same existential n-type, then either has the same existential type as the power of a primitive element or there exists an existentially closed subgroup (respectively ) of F containing (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.
@article{CML_2011__3_1_121_0,
author = {Abderezak Ould Houcine},
title = {Homogeneity and prime models in torsion-free hyperbolic groups},
journal = {Confluentes Mathematici},
pages = {121--155},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {1},
year = {2011},
doi = {10.1142/S179374421100028X},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S179374421100028X/}
}
TY - JOUR AU - Abderezak Ould Houcine TI - Homogeneity and prime models in torsion-free hyperbolic groups JO - Confluentes Mathematici PY - 2011 SP - 121 EP - 155 VL - 3 IS - 1 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S179374421100028X/ DO - 10.1142/S179374421100028X LA - en ID - CML_2011__3_1_121_0 ER -
%0 Journal Article %A Abderezak Ould Houcine %T Homogeneity and prime models in torsion-free hyperbolic groups %J Confluentes Mathematici %D 2011 %P 121-155 %V 3 %N 1 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S179374421100028X/ %R 10.1142/S179374421100028X %G en %F CML_2011__3_1_121_0
Abderezak Ould Houcine. Homogeneity and prime models in torsion-free hyperbolic groups. Confluentes Mathematici, Tome 3 (2011) no. 1, pp. 121-155. doi : 10.1142/S179374421100028X. https://cml.centre-mersenne.org/articles/10.1142/S179374421100028X/
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