# Elementary embeddings in torsion-free hyperbolic groups

Annales scientifiques de l'École Normale Supérieure (2011)

- Volume: 44, Issue: 4, page 631-681
- ISSN: 0012-9593

## Access Full Article

top## Abstract

top## How to cite

topPerin, Chloé. "Elementary embeddings in torsion-free hyperbolic groups." Annales scientifiques de l'École Normale Supérieure 44.4 (2011): 631-681. <http://eudml.org/doc/272248>.

@article{Perin2011,

abstract = {We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma $, we show that the group $\Gamma $ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where $\Gamma $ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.},

author = {Perin, Chloé},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {geometric group theory; first-order logic; trees (graph theory); free groups; Tarski problem; Sela’s hyperbolic towers; elementary substructures},

language = {eng},

number = {4},

pages = {631-681},

publisher = {Société mathématique de France},

title = {Elementary embeddings in torsion-free hyperbolic groups},

url = {http://eudml.org/doc/272248},

volume = {44},

year = {2011},

}

TY - JOUR

AU - Perin, Chloé

TI - Elementary embeddings in torsion-free hyperbolic groups

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2011

PB - Société mathématique de France

VL - 44

IS - 4

SP - 631

EP - 681

AB - We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma $, we show that the group $\Gamma $ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where $\Gamma $ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.

LA - eng

KW - geometric group theory; first-order logic; trees (graph theory); free groups; Tarski problem; Sela’s hyperbolic towers; elementary substructures

UR - http://eudml.org/doc/272248

ER -

## References

top- [1] M. Bestvina, $\mathbb{R}$-trees in topology, geometry and group theory, preprint ftp://ftp.math.utah.edu/u/ma/bestvina/math/handbook.pdf, 2002. Zbl0998.57003MR1886668
- [2] M. Bestvina & M. Feighn, Outer limits, preprint http://andromeda.rutgers.edu/~feighn/papers/outer.pdf, 1994.
- [3] M. Bestvina & M. Feighn, Stable actions of groups on real trees, Invent. Math.121 (1995), 287–321. Zbl0837.20047MR1346208
- [4] B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math.180 (1998), 145–186. Zbl0911.57001MR1638764
- [5] C. C. Chang & H. J. Keisler, Model theory, third éd., Studies in Logic and the Foundations of Mathematics 73, North-Holland Publishing Co., 1990. Zbl0697.03022MR1059055
- [6] Z. Chatzidakis, Introduction to model theory, notes, http://www.logique.jussieu.fr/~zoe/papiers/MTluminy.pdf.
- [7] M. J. Dunwoody & M. E. Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math.135 (1999), 25–44. Zbl0939.20047MR1664694
- [8] M. Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol.6 (2002), 219–267. Zbl1118.20028MR1914569
- [9] K. Fujiwara & P. Papasoglu, JSJ-decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal.16 (2006), 70–125. Zbl1097.20037MR2221253
- [10] D. Gabai, The simple loop conjecture, J. Differential Geom.21 (1985), 143–149. Zbl0556.57007MR806708
- [11] V. Guirardel, Actions of finitely generated groups on $\mathbb{R}$-trees, Ann. Inst. Fourier (Grenoble) 58 (2008), 159–211. Zbl1187.20020MR2401220
- [12] V. Guirardel & G. Levitt, JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space, preprint arXiv:0911.3173. Zbl1162.20017
- [13] V. Guirardel & G. Levitt, JSJ decompositions: definitions, existence, uniqueness. II: Compatibility and acylindricity, preprint arXiv:1002.4564.
- [14] V. Guirardel & G. Levitt, Tree of cylinders and canonical splittings, preprint arXiv:0811.2383. Zbl1272.20026MR2821568
- [15] O. Kharlampovich & A. Myasnikov, Elementary theory of free non-abelian groups, J. Algebra302 (2006), 451–552. Zbl1110.03020MR2293770
- [16] J. W. Morgan & P. B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. 120 (1984), 401–476. Zbl0583.57005MR769158
- [17] F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math.94 (1988), 53–80. Zbl0673.57034MR958589
- [18] F. Paulin, The Gromov topology on $\mathbf{R}$-trees, Topology Appl.32 (1989), 197–221. Zbl0675.20033MR1007101
- [19] C. Perin, Elementary embeddings in torsion-free hyperbolic groups, Thèse de doctorat, Université de Caen Basse-Normandie, 2008. Zbl1245.20052
- [20] E. Rips & Z. Sela, Structure and rigidity in hyperbolic groups. I, Geom. Funct. Anal. 4 (1994), 337–371. Zbl0818.20042MR1274119
- [21] E. Rips & Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math.146 (1997), 53–109. Zbl0910.57002MR1469317
- [22] P. Scott, Finitely generated $3$-manifold groups are finitely presented, J. London Math. Soc.6 (1973), 437–440. Zbl0254.57003MR380763
- [23] P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc.17 (1978), 555–565. Zbl0412.57006MR494062
- [24] Z. Sela, Acylindrical accessibility for groups, Invent. Math.129 (1997), 527–565. Zbl0887.20017MR1465334
- [25] Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups. II, Geom. Funct. Anal. 7 (1997), 561–593. Zbl0884.20025MR1466338
- [26] Z. Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. I.H.É.S. 93 (2001), 31–105. Zbl1018.20034MR1863735
- [27] Z. Sela, Diophantine geometry over groups. II. Completions, closures and formal solutions, Israel J. Math. 134 (2003), 173–254. Zbl1028.20028MR1972179
- [28] Z. Sela, Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence, Israel J. Math. 143 (2004), 1–130. Zbl1088.20017MR2106978
- [29] Z. Sela, Diophantine geometry over groups. III. Rigid and solid solutions, Israel J. Math. 147 (2005), 1–73. Zbl1133.20020MR2166355
- [30] Z. Sela, Diophantine geometry over groups. ${\mathrm{V}}_{1}$. Quantifier elimination. I, Israel J. Math. 150 (2005), 1–197. Zbl1148.20022MR2249582
- [31] Z. Sela, Diophantine geometry over groups. VI. The elementary theory of a free group, Geom. Funct. Anal. 16 (2006), 707–730. Zbl1118.20035MR2238945
- [32] Z. Sela, private communication.
- [33] Z. Sela, Diophantine geometry over groups VII: The elementary theory of a hyperbolic group, to appear in Proc. of the LMS. Zbl1241.20049MR2520356
- [34] J-P. Serre, Arbres, amalgames, ${\mathrm{SL}}_{2}$, Astérisque 46 (1983). Zbl0369.20013
- [35] H. Wilton, Subgroup separability of limit groups, Thèse, Imperial College, London, 2006, http://www.math.utexas.edu/users/henry.wilton/thesis.pdf.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.