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Maximal hyperbolic towers and weight in the theory of free groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Model theory

Published online by Cambridge University Press:  18 November 2019

BENJAMIN BRÜCK
BENJAMIN BRÜCK*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany. e-mail: [email protected]
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Abstract

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 03C45: Classification theory, stability and related concepts 20E05: Free nonabelian groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 479 - 498
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Kharlampovich, O. and Myasnikov, A.. Elementary theory of free non-abelian groups. J. Algebra 302 (2006), no. 2, 451–552.CrossRefGoogle Scholar
Louder, L., Perin, C. and Sklinos, R.. Hyperbolic towers and independent generic sets in the theory of free groups. Notre Dame J. Form. Log. 54 (2013), no. 3-4, 521–539.CrossRefGoogle Scholar
Makkai, M.. A survey of basic stability theory, with particular emphasis on orthogonality and regular types. Israel J. Math. 49 (1984), no. 1-3, 181–238.CrossRefGoogle Scholar
Morgan, J. W. and Shalen, P. B.. Valuations, trees and degenerations of hyperbolic structures. I Ann. of Math. (2) 120 (1984), no. 3, 401–476.CrossRefGoogle Scholar
Perin, C.. Elementary embeddings in torsion-free hyperbolic groups. Ann. Sci. Ecole Norm. Sup. (4) 44 (2011), no. 4, 631–681.CrossRefGoogle Scholar
Pillay, A.. Geometric stability theory. Oxford Logic Guides, vol. 32. (The Clarendon Press, Oxford University Press, New York, 1996, Oxford Science Publications).Google Scholar
Pillay, A.. Forking in the free group. J. Inst. Math. Jussieu 7 (2008), no. 2, 375–389.CrossRefGoogle Scholar
Pillay, A.. On genericity and weight in the free group, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3911–3917.Google Scholar
Perin, C. and Sklinos, R.. Homogeneity in the free group. Duke Math. J. 161 (2012), no. 13, 2635–2668.CrossRefGoogle Scholar
Perin, C. and Sklinos, R.. Forking and JSJ decomposition in the free group. J. European Math. Soc. (JEMS) 18 (2016), no. 9, 1983–2017.CrossRefGoogle Scholar
Sela, Z.. Diophantine geometry over groups. VI. The elementary theory of a free group. Geom. Funct. Anal. 16 (2006), no. 3, 707–730.Google Scholar
Sela, Z.. Diophantine geometry over groups VIII: Stability. Ann. of Math. (2) 177 (2013), no. 3, 787–868.CrossRefGoogle Scholar
Serre, J.-P.. Trees. Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2003). Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.Google Scholar
Shelah, S.. Classification theory and the number of nonisomorphic models. Studies in Logic and the Foundations of Mathematics, vol. 92. (North-Holland Publishing Co., Amsterdam-New York, 1978).Google Scholar
Sklinos, R.. On the generic type of the free group. J. Symbolic Logic 76 (2011), no. 1, 227–234.CrossRefGoogle Scholar
Stallings, J. R.. Whitehead graphs on handlebodies. Geometric group theory down under (Canberra, 1996) (de Gruyter, Berlin, 1999), pp. 317–330.Google Scholar
Tent, K. and Ziegler, M.. A course in model theory. Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Whitehead, J. H. C.. On certain sets of elements in a free group. Proc. London Math. Soc. S2-41, no. 1, 48.Google Scholar