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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Definably amenable NIP groups
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by Artem Chernikov and Pierre Simon HTML | PDF
J. Amer. Math. Soc. 31 (2018), 609-641 Request permission

Abstract:

We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
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Additional Information
  • Artem Chernikov
  • Affiliation: IMJ-PRG, Université Paris Diderot, Paris 7, L’Equipe de Logique Mathématique, UFR de Mathématiques case 7012, 75205 Paris Cedex 13, France
  • MR Author ID: 974787
  • Email: chernikov@math.ucla.edu
  • Pierre Simon
  • Affiliation: Université Claude Bernard-Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
  • MR Author ID: 942320
  • Email: simon@math.univ-lyon1.fr
  • Received by editor(s): February 17, 2015
  • Received by editor(s) in revised form: November 28, 2016, and September 23, 2017
  • Published electronically: February 1, 2018
  • Additional Notes: The research leading to this paper has been partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291111 and by ValCoMo (ANR-13-BS01-0006).
    The first author was partially supported by the Fondation Sciences Mathematiques de Paris (ANR-10-LABX-0098), by the NSF (grants DMS-1600796 and DMS-1651321), and by the Sloan Foundation
    The second author was partially supported by the NSF (grant DMS-1665491) and by the Sloan Foundation.
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 31 (2018), 609-641
  • MSC (2010): Primary 03C45, 37B05, 03C60; Secondary 03C64, 22F10, 28D15
  • DOI: https://doi.org/10.1090/jams/896
  • MathSciNet review: 3787403