no. 1
Homogeneity and prime models in torsion-free hyperbolic groups
Confluentes Mathematici, Tome 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.

Publié le :
DOI : 10.1142/S179374421100028X
Ould Houcine, Abderezak 1

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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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. http://www.numdam.org/articles/10.1142/S179374421100028X/

[1] I. Belegradek, Geom. Dedicata 129, 119 (2007), DOI: 10.1007/s10711-007-9199-8.

[2] G. Baumslag, A. Myasnikov and V. Remeslennikov, J. Algebra 219, 16 (1999), DOI: 10.1006/jabr.1999.7881.

[3] C. Champetier and V. Guirardel, Israel J. Math. 146, 1 (2005), DOI: 10.1007/BF02773526.

[4] T. Delzant, Group Theory from a Geometrical Viewpoint (World Scientific, 1991) pp. 177–189.

[5] V. Guirardel and G. Levitt, JSJ decompositions: Definitions, existence, uniqueness. I: The JSJ deformation space, preprint .

[6] V. Guirardel and G. Levitt, JSJ decompositions: Definitions, existence, uniqueness. II. Compatibility and acylindricity, preprint .

[7] D. Groves and H. Wilton, Conjugacy classes of solutions to equations and inequations over hyperbolic groups, 2007 .

[8] W. Hodges , Model Theory , Encyclopedia of Mathematics and its Applications 42 ( Cambridge Univ. Press , 1993 ) .

[9] E. Jaligot and A. Ould Houcine, J. Algebra 280, 772 (2004).

[10] O. Kharlampovich and A. Myasnikov, Trans. Amer. Math. Soc. 350, 571 (1998), DOI: 10.1090/S0002-9947-98-01773-5.

[11] O. Kharlampovich and A. Myasnikov, J. Algebra 302, 451 (2006).

[12] I. Kapovich and R. Weidmann, Math. Z. 231, 783 (1999), DOI: 10.1007/PL00004753.

[13] R. C. Lyndon and P. E. Schupp , Combinatorial Group Theory ( Springer-Verlag , 1977 ) .

[14] D. Marker , Model Theory: An Introduction ( Springer-Verlag , 2002 ) .

[15] A. Nies, J. Algebra 263, 119 (2003), DOI: 10.1016/S0021-8693(02)00665-8.

[16] A. Nies, Internat. J. Algebra Comput. 13, 287 (2003), DOI: 10.1142/S0218196703001286.

[17] A. Nies, J. Group Theory 10, 347 (2007), DOI: 10.1515/JGT.2007.027.

[18] A. Ould Houcine, Geometric Group Theory (Birkhäuser, 2007) pp. 103–119.

[19] A. Ould Houcine, Ann. Pure Appl. Logic 154, 1 (2008).

[20] A. Ould Houcine, On finitely generated models of theories with at most countably many nonisomorphic finitely generated models, submitted .

[21] A. Ould Houcine, Commun. Algebra 38, 2825 (2010).

[22] A. Ould Houcine, J. Algebra 134, 159 (2010).

[23] A. Ould Houcine and D. Vallino, Algebraic and definable closure in free groups, in preparation .

[24] A. Yu. Ol′shanskiĭ, Izv. Akad. Nauk SSSR Ser. Mat. 44, 309 (1980).

[25] A. Yu. Ol″shanskiĭ , Geometry of Defining Relations in Groups , Mathematics and its Applications (Soviet Series) 70 ( Kluwer Academic Press , 1991 ) .

[26] F. Oger and G. Sabbagh, J. Group Theory 9, 95 (2006), DOI: 10.1515/JGT.2006.005.

[27] J. C. O’Neill and E. C. Turner, New York J. Math. 6, 107 (2000).

[28] C. Perin, Plongements élémentaires dans un groupe hyperbolique sans torsion, Thèse de doctorat, Université de Caen/Basse-Normandie, 2008 .

[29] C. Perin, Elementary embeddings in torsion-free hyperbolic groups , arXiv:0903.0945 .

[30] A. Pillay, J. Inst. Math. Jussieu 7, 375 (2008).

[31] A. Pillay, Proc. Amer. Math. Soc. 137, 3911 (2009), DOI: 10.1090/S0002-9939-09-09956-0.

[32] C. Perin and R. Sklinos, Homogeneity in the free group, preprint .

[33] E. Rips and Z. Sela, Geom. Funct. Anal. 4, 337 (1994), DOI: 10.1007/BF01896245.

[34] E. Rips and Z. Sela, Ann. Math. 146, 53 (1997), DOI: 10.2307/2951832.

[35] Z. Sela, Geom. Funct. Anal. 7, 561 (1997), DOI: 10.1007/s000390050019.

[36] Z. Sela, Israel J. Math. 150, 1 (2005).

[37] Z. Sela, Geom. Funct. Anal. 16, 537 (2006).

[38] Z. Sela, Geom. Funct. Anal. 16, 707 (2006).

[39] Z. Sela, Proc. Lond. Math. Soc. 99, 217 (2009), DOI: 10.1112/plms/pdn052.

Cité par Sources :