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Definably amenable NIP groups


Authors: Artem Chernikov and Pierre Simon
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
Published electronically: February 1, 2018
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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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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
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
Email: simon@math.univ-lyon1.fr

DOI: https://doi.org/10.1090/jams/896
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.
Article copyright: © Copyright 2018 American Mathematical Society

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