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Groups and fields with $ \operatorname{NTP}_{2}$


Authors: Artem Chernikov, Itay Kaplan and Pierre Simon
Journal: Proc. Amer. Math. Soc. 143 (2015), 395-406
MSC (2010): Primary 03C45, 03C60
DOI: https://doi.org/10.1090/S0002-9939-2014-12229-5
Published electronically: August 19, 2014
MathSciNet review: 3272764
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Abstract: $ \operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $ \operatorname {NTP}_{2}$ structures are given by ultra-products of $ p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $ \operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $ \operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.


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Additional Information

Artem Chernikov
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel
Address at time of publication: L’équipe de Logique Mathématique, IMJ-PRG, Université Paris Diderot-Paris 7, UFR de Mathématiques, case 7012, 75205 Paris Cedex 13, France
Email: art.chernikov@gmail.com

Itay Kaplan
Affiliation: Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Address at time of publication: Institute of Mathematics, Hebrew University (The Edmond J. Safra Campus), Givat Ram, Jerusalem 91904, Israel
Email: itay.kaplan@uni-muenster.de

Pierre Simon
Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel
Address at time of publication: Université Claude Bernard-Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
Email: pierre.simon@normalesup.org

DOI: https://doi.org/10.1090/S0002-9939-2014-12229-5
Received by editor(s): December 31, 2012
Received by editor(s) in revised form: February 26, 2013
Published electronically: August 19, 2014
Additional Notes: The first author was partially supported by the [European Community’s] Seventh Framework Programme [FP7/2007-2013] under grant agreement No. 238381
The second author was supported by SFB 878
Communicated by: Julia Knight
Article copyright: © Copyright 2014 American Mathematical Society