Homogeneity and prime models in torsion-free hyperbolic groups

Ould Houcine, Abderezak

doi:10.1142/S179374421100028X

Geodesic

Parcourir par

Homogeneity and prime models in torsion-free hyperbolic groups

Ould Houcine, Abderezak ¹

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

Voir la notice de l'article provenant de la source Numdam

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

Affiliations des auteurs :

Ould Houcine, Abderezak ¹

@article{CML_2011__3_1_121_0,
     author = {Ould Houcine, Abderezak},
     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 = {http://geodesic.mathdoc.fr/articles/10.1142/S179374421100028X/}
}

TY  - JOUR
AU  - Ould Houcine, Abderezak
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  - http://geodesic.mathdoc.fr/articles/10.1142/S179374421100028X/
DO  - 10.1142/S179374421100028X
LA  - en
ID  - CML_2011__3_1_121_0
ER  -

%0 Journal Article
%A Ould Houcine, Abderezak
%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 http://geodesic.mathdoc.fr/articles/10.1142/S179374421100028X/
%R 10.1142/S179374421100028X
%G en
%F CML_2011__3_1_121_0

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

Cité par Sources :