Duke Mathematical Journal

Homogeneity in the free group

Chloé Perin and Rizos Sklinos

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show that any nonabelian free group F is strongly 0-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under Aut(F). We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not strongly 0-homogeneous.

Article information

Duke Math. J., Volume 161, Number 13 (2012), 2635-2668.

First available in Project Euclid: 11 October 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E05: Free nonabelian groups
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 03C07: Basic properties of first-order languages and structures


Perin, Chloé; Sklinos, Rizos. Homogeneity in the free group. Duke Math. J. 161 (2012), no. 13, 2635--2668. doi:10.1215/00127094-1813068. https://projecteuclid.org/euclid.dmj/1349960279

Export citation


  • [BF] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), 85–101.
  • [BS] J. S. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), 217–225.
  • [Bow] B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), 145–186.
  • [CK] C. C. Chang and H. J. Kiesler, Model Theory, 3rd ed., Stud. Logic Found. Math. 73, North-Holland, Amsterdam, 1990.
  • [Cha] Z. Chatzidakis, Introduction to model theory, preprint, http://www.logique.jussieu.fr/~zoe/index.html (accessed 27 July 2012).
  • [DS] M. J. Dunwoody and M. E. Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999), 25–44.
  • [FP] K. Fujiwara and P. Papasoglu, JSJ-decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal. 16 (2006), 70–125.
  • [Gui] V. Guirardel, Actions of finitely generated groups on $\mathbb{R} $-trees, Ann. Inst. Fourier (Grenoble) 58 (2008), 159–211.
  • [GL1] V. Guirardel and G. Levitt, Tree of cylinders and canonical splittings, Geom. Topol. 15 (2011), 977–1012.
  • [GL2] Vincent Guirardel, JSJ decompositions: Definitions, existence, uniqueness, I: The JSJ deformation space, arXiv:0911.3173 [math.GR]
  • [GL3] Vincent Guirardel, JSJ decompositions: Definitions, existence, uniqueness, II: Compatibility and acylindricity, arXiv:1002.4564 [math.GR]
  • [KM] O. Kharlampovich and A. Myasnikov, Elementary theory of free nonabelian groups, J. Algebra 302 (2006), 451–552.
  • [Mar] D. Marker, Model Theory, Grad. Texts in Math. 217, Springer, New York, 2002.
  • [Mir] M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2) 168 (2008), 97–125.
  • [MS] J. W. Morgan and P. B. Shalen, Valuations, trees, and degenerations of hyperbolic structures, I, Ann. of Math. (2) 120 (1984), 401–476.
  • [NS] M. Nadel and J. Stavi, On models of the elementary theory of $(\mathbf{Z},+,1)$, J. Symbolic Logic 55 (1990), 1–20.
  • [Nie] A. Nies, Aspects of free groups, J. Algebra 263 (2003), 119–125.
  • [OH] A. Ould Houcine, Homogeneity and prime models in torsion-free hyperbolic groups, Confluentes Math. 3 (2011), 121–155.
  • [Per1] C. Perin, Elementary embeddings in torsion-free hyperbolic groups, Ph.D. dissertation, Université de Caen Basse-Normandie, Caen, France, 2008.
  • [Per2] Chloé Perin, Elementary embeddings in torsion-free hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 631–681.
  • [Per3] Chloé Perin, Erratum: Elementary embeddings in torsion-free hyperbolic groups, preprint, available at http://www-irma.u-strasbg.fr/~perin (accessed 30 July 2012).
  • [Pil1] A. Pillay, Geometric Stability Theory, Oxford Logic Guides 32, Clarendon Press, New York, 1996.
  • [Pil2] Anand Pillay, Forking in the free group, J. Inst. Math. Jussieu 7 (2008), 375–389.
  • [Pil3] Anand Pillay, On genericity and weight in the free group, Proc. Amer. Math. Soc. 137 (2009), 3911–3917.
  • [Poi1] B. Poizat, Groupes stables, avec types génériques réguliers, J. Symbolic Logic 48 (1983), 339–355.
  • [Poi2] Bruno Poizat, Stable Groups, Math. Surveys Monogr. 87, Amer. Math. Soc., Providence, 2001.
  • [RS1] E. Rips and Z. Sela, Structure and rigidity in hyperbolic groups, I, Geom. Funct. Anal. 4 (1994), 337–371.
  • [RS2] Eliyahu Rips and Zlil Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. (2) 146 (1997), 53–104.
  • [Riv] I. Rivin, Simple curves on surfaces, Geom. Dedicata 87 (2001), 345–360.
  • [Sel1] 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.
  • [Sel2] Zlil Sela, Diophantine geometry over groups, I: Makanin–Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105.
  • [Sel3] Zlil Sela, Diophantine geometry over groups, VI: The elementary theory of a free group, Geom. Funct. Anal. 16 (2006), 707–730.
  • [Sel4] Zlil Sela, Diophantine geometry over groups, VII: The elementary theory of a hyperbolic group, Proc. Lond. Math. Soc. (3) 99 (2009), 217–273.
  • [Sel5] Zlil Sela, Diophantine geometry over groups, VIII: Stability, to appear in Ann. of Math. (2), preprint, http://www.ma.huji.ac.il/~zlil/ (accessed 27 July 2012).
  • [SelP] Z. Sela, personal communication, January 2007.
  • [Ser] J.-P. Serre, Arbres, amalgames, $\mathrm{SL}_{2}$, Astérisque 46, Soc. Math. France, Paris, 1977.
  • [Skl] R. Sklinos, On the generic type of the free group, J. Symbolic Logic 76 (2011), 227–234.
  • [Wil] H. J. R. Wilton, Subgroup separability of limit groups, Ph.D. dissertation, Imperial College, London, 2006, http://www.math.utexas.edu/users/henry.wilton/thesis.pdf (accessed 27 July 2012).