Definably amenable NIP groups
Journal of the American Mathematical Society, Tome 31 (2018) no. 3, pp. 609-641

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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.
DOI : 10.1090/jams/896

Chernikov, Artem 1 ; Simon, Pierre 2

1 IMJ-PRG, Université Paris Diderot, Paris 7, L’Equipe de Logique Mathématique, UFR de Mathématiques case 7012, 75205 Paris Cedex 13, France
2 Université Claude Bernard-Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
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Chernikov, Artem; Simon, Pierre. Definably amenable NIP groups. Journal of the American Mathematical Society, Tome 31 (2018) no. 3, pp. 609-641. doi: 10.1090/jams/896

[1] Breuillard, Emmanuel, Green, Ben, Tao, Terence The structure of approximate groups Publ. Math. Inst. Hautes Études Sci. 2012 115 221

[2] Chernikov, Artem, Kaplan, Itay Forking and dividing in 𝑁𝑇𝑃₂ theories J. Symbolic Logic 2012 1 20

[3] Chernikov, Artem, Pillay, Anand, Simon, Pierre External definability and groups in NIP theories J. Lond. Math. Soc. (2) 2014 213 240

[4] Conversano, Annalisa, Pillay, Anand Connected components of definable groups and 𝑜-minimality I Adv. Math. 2012 605 623

[5] Chernikov, Artem, Simon, Pierre Externally definable sets and dependent pairs Israel J. Math. 2013 409 425

[6] Glasner, Eli Enveloping semigroups in topological dynamics Topology Appl. 2007 2344 2363

[7] Glasner, Eli The structure of tame minimal dynamical systems Ergodic Theory Dynam. Systems 2007 1819 1837

[8] Gismatullin, Jakub, Penazzi, Davide, Pillay, Anand Some model theory of 𝑆𝐿(2,ℝ) Fund. Math. 2015 117 128

[9] Halmos, Paul R. Measure Theory 1950

[10] Hrushovski, Ehud Stable group theory and approximate subgroups J. Amer. Math. Soc. 2012 189 243

[11] Hrushovski, Ehud The Manin-Mumford conjecture and the model theory of difference fields Ann. Pure Appl. Logic 2001 43 115

[12] Hrushovski, Ehud The Mordell-Lang conjecture for function fields J. Amer. Math. Soc. 1996 667 690

[13] Hrushovski, Ehud Unidimensional theories are superstable Ann. Pure Appl. Logic 1990 117 138

[14] Hrushovski, Ehud, Pillay, Anand On NIP and invariant measures J. Eur. Math. Soc. (JEMS) 2011 1005 1061

[15] Hrushovski, Ehud, Peterzil, Ya’Acov, Pillay, Anand Groups, measures, and the NIP J. Amer. Math. Soc. 2008 563 596

[16] Hrushovski, Ehud, Pillay, Anand, Simon, Pierre Generically stable and smooth measures in NIP theories Trans. Amer. Math. Soc. 2013 2341 2366

[17] Jerison, Meyer A property of extreme points of compact convex sets Proc. Amer. Math. Soc. 1954 782 783

[18] Kechris, Alexander S. Classical descriptive set theory 1995

[19] Kerr, David, Li, Hanfeng Independence in topological and 𝐶*-dynamics Math. Ann. 2007 869 926

[20] Laskowski, Michael C. Vapnik-Chervonenkis classes of definable sets J. London Math. Soc. (2) 1992 377 384

[21] Matouå¡Ek, Jiå™Ã­ Bounded VC-dimension implies a fractional Helly theorem Discrete Comput. Geom. 2004 251 255

[22] Medvedev, Alice, Scanlon, Thomas Invariant varieties for polynomial dynamical systems Ann. of Math. (2) 2014 81 177

[23] Newelski, Ludomir Bounded orbits and measures on a group Israel J. Math. 2012 209 229

[24] Newelski, Ludomir Topological dynamics of definable group actions J. Symbolic Logic 2009 50 72

[25] Newelski, Ludomir, Petrykowski, Marcin Weak generic types and coverings of groups. I Fund. Math. 2006 201 225

[26] Phelps, Robert R. Lectures on Choquet’s theorem 2001

[27] Pillay, Anand Type-definability, compact Lie groups, and o-minimality J. Math. Log. 2004 147 162

[28] Poizat, Bruno Stable groups 2001

[29] Poizat, Bruno Groupes stables 1987

[30] Pillay, Anand, Yao, Ningyuan On minimal flows, definably amenable groups, and 𝑜-minimality Adv. Math. 2016 483 502

[31] Sela, Z. Diophantine geometry over groups VIII: Stability Ann. of Math. (2) 2013 787 868

[32] Shelah, Saharon Dependent first order theories, continued Israel J. Math. 2009 1 60

[33] Shelah, Saharon Minimal bounded index subgroup for dependent theories Proc. Amer. Math. Soc. 2008 1087 1091

[34] Shelah, Saharon Stability, the f.c.p., and superstability Ann. Math. Logic 1971 271 362

[35] Simon, Pierre A guide to NIP theories 2015

[36] Simon, Pierre Rosenthal compacta and NIP formulas Fund. Math. 2015 81 92

[37] Simon, Barry Convexity 2011

[38] Starchenko, Sergei NIP, Keisler measures and combinatorics Astérisque 2017

[39] Vapnik, V. N., ČErvonenkis, A. Ja. The uniform convergence of frequencies of the appearance of events to their probabilities Teor. Verojatnost. i Primenen. 1971 264 279

[40] Wagner, Frank O. Simple theories 2000

[41] Walters, Peter An introduction to ergodic theory 1982

[42] Zilber, Boris Uncountably categorical theories 1993

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