Fields definable in the free group

Dente Byron, Ayala; Sklinos, Rizos

doi:10.1090/btran/41

Fields definable in the free group
HTML articles powered by AMS MathViewer

by Ayala Dente Byron and Rizos Sklinos HTML | PDF

Trans. Amer. Math. Soc. Ser. B 6 (2019), 297-345

Abstract:

We prove that no infinite field is definable in the theory of the free group.

References

Mladen Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), no. 1, 143–161. MR 932860, DOI 10.1215/S0012-7094-88-05607-4
Mladen Bestvina and Mark Feighn, Notes on Sela’s work: limit groups and Makanin-Razborov diagrams, Geometric and cohomological methods in group theory, London Math. Soc. Lecture Note Ser., vol. 358, Cambridge Univ. Press, Cambridge, 2009, pp. 1–29. MR 2605174
Vincent Guirardel, Actions of finitely generated groups on $\Bbb R$-trees, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 159–211 (English, with English and French summaries). MR 2401220, DOI 10.5802/aif.2348
O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), no. 2, 472–516. MR 1610660, DOI 10.1006/jabr.1997.7183
Olga Kharlampovich and Alexei Myasnikov, Implicit function theorem over free groups, J. Algebra 290 (2005), no. 1, 1–203. MR 2154989, DOI 10.1016/j.jalgebra.2005.04.001
Olga Kharlampovich and Alexei Myasnikov, Elementary theory of free non-abelian groups, J. Algebra 302 (2006), no. 2, 451–552. MR 2293770, DOI 10.1016/j.jalgebra.2006.03.033
Olga Kharlampovich and Alexei Myasnikov, Definable sets in a hyperbolic group, Internat. J. Algebra Comput. 23 (2013), no. 1, 91–110. MR 3040804, DOI 10.1142/S021819671350001X
Frédéric Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1988), no. 1, 53–80 (French). MR 958589, DOI 10.1007/BF01394344
Chloé Perin, Anand Pillay, Rizos Sklinos, and Katrin Tent, On groups and fields interpretable in torsion-free hyperbolic groups, Münster J. Math. 7 (2014), no. 2, 609–621. MR 3426232, DOI 10.17879/58269757902
Chloé Perin and Rizos Sklinos, Forking and JSJ decomposition in the free group, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 9, 1983–2017. MR 3531668, DOI 10.4171/JEMS/636
Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
Anand Pillay, A note on CM-triviality and the geometry of forking, J. Symbolic Logic 65 (2000), no. 1, 474–480. MR 1782132, DOI 10.2307/2586549
Anand Pillay, Forking in the free group, J. Inst. Math. Jussieu 7 (2008), no. 2, 375–389. MR 2400726, DOI 10.1017/S1474748008000066
Zlil Sela, Diophantine geometry over groups IX: Envelopes and imaginaries.
Zlil Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105. MR 1863735, DOI 10.1007/s10240-001-8188-y
Z. Sela, Diophantine geometry over groups. II. Completions, closures and formal solutions, Israel J. Math. 134 (2003), 173–254. MR 1972179, DOI 10.1007/BF02787407
Z. Sela, Diophantine geometry over groups. III. Rigid and solid solutions, Israel J. Math. 147 (2005), 1–73. MR 2166355, DOI 10.1007/BF02785359
Z. Sela, Diophantine geometry over groups. VII. The elementary theory of a hyperbolic group, Proc. Lond. Math. Soc. (3) 99 (2009), no. 1, 217–273. MR 2520356, DOI 10.1112/plms/pdn052
Jean-Pierre Serre, Arbres, amalgames, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
Rizos Sklinos, Some model theory of the free group, Ph.D. thesis, University of Leeds, UK, 2011.
Rizos Sklinos, The free group does not have the finite cover property, Israel J. Math. 227 (2018), no. 2, 563–595. MR 3846335, DOI 10.1007/s11856-018-1748-3
Frank O. Wagner, Some remarks on one-basedness, J. Symbolic Logic 69 (2004), no. 1, 34–38. MR 2039343, DOI 10.2178/jsl/1080938823
B. I. Zil′ber, The structure of models of uncountably categorical theories, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 359–368. MR 804692