Notre Dame Journal of Formal Logic

Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups

Larsen Louder, Chloé Perin, and Rizos Sklinos


We use hyperbolic towers to answer some model-theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type p0 but that there is a finitely generated model which omits p0(2). We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.

Article information

Notre Dame J. Formal Logic, Volume 54, Number 3-4 (2013), 521-539.

First available in Project Euclid: 9 August 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 03C45: Classification theory, stability and related concepts [See also 03C48]

free group hyperbolic towers stable groups generic type homogeneity


Louder, Larsen; Perin, Chloé; Sklinos, Rizos. Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups. Notre Dame J. Formal Logic 54 (2013), no. 3-4, 521--539. doi:10.1215/00294527-2143988.

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  • [1] Marker, D., Model Theory: An Introduction, vol. 217 of Graduate Texts in Mathematics, Springer, New York, 2002.
  • [2] Morgan, J. W., and P. B. Shalen, “Valuations, trees, and degenerations of hyperbolic structures, I,” Annals of Mathematics (2), vol. 120 (1984), 401–76.
  • [3] Perin, C., “Elementary embeddings in torsion-free hyperbolic groups,” Annales Scientifiques de l’École Normale Supérieure (4), vol. 44 (2011), 631–81.
  • [4] Perin, C., “Elementary embeddings in torsion-free hyperbolic groups,” preprint, arXiv:0903.0945v3 [math.GR].
  • [5] Perin, C., “Erratum: Elementary embeddings in torsion-free hyperbolic groups,” preprint, 2012.
  • [6] Perin, C., and R. Sklinos, “Homogeneity in the free group,” Duke Mathematical Journal, vol. 161 (2012), pp. 2635–58.
  • [7] Pillay, A., Geometric Stability Theory, vol. 32 of Oxford Logic Guides, Oxford University Press, New York, 1996.
  • [8] Pillay, A., An Introduction to Stability Theory, Dover Books on Mathematics, Dover, Mineola, NY, 2008.
  • [9] Pillay, A., “On genericity and weight in the free group,” Proceedings of the American Mathematical Society, vol. 137 (2009), pp. 3911–17.
  • [10] Pillay, A., private communication, June 2008.
  • [11] Sela, Z., “Diophantine geometry over groups, I: Makanin-Razborov diagrams,” Publications Mathématiques Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105.
  • [12] Sela, Z., “Diophantine geometry over groups, VI: The elementary theory of a free group,” Geometric and Functional Analysis, vol. 16 (2006), pp. 707–30.
  • [13] Sela, Z., “Diophantine geometry over groups, VII: The elementary theory of a hyperbolic group,” Proceedings of the London Mathematical Society (3), vol. 99 (2009), pp. 217–73.
  • [14] Sela, Z., private communication, May 2006.
  • [15] Sela, Z., “Diophantine geometry over groups, VIII: Stability,” to appear in Annals of Mathematics (2).
  • [16] Shelah, S., Classification Theory and The Number of Nonisomorphic Models, 2nd ed., vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1990.
  • [17] Serre, J.-P., Arbres, amalgames, $SL_{2}$, with the collaboration of Hyman Bass, vol. 46 of Astérisque, Société Mathématique de France, Paris, 1977.
  • [18] Sklinos, R., “On the generic type of the free group,” Journal of Symbolic Logic, vol. 76 (2011), pp. 227–34.