Definably amenable NIP groups

Chernikov, Artem; Simon, Pierre

doi:10.1090/jams/896

Definably amenable NIP groups
HTML articles powered by AMS MathViewer

by Artem Chernikov and Pierre Simon

J. Amer. Math. Soc. 31 (2018), 609-641

DOI: https://doi.org/10.1090/jams/896

Published electronically: February 1, 2018

HTML | PDF | 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.

References

Emmanuel Breuillard, Ben Green, and Terence Tao, The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221. MR 3090256, DOI 10.1007/s10240-012-0043-9
Artem Chernikov and Itay Kaplan, Forking and dividing in $\textrm {NTP}_2$ theories, J. Symbolic Logic 77 (2012), no. 1, 1–20. MR 2951626, DOI 10.2178/jsl/1327068688
Artem Chernikov, Anand Pillay, and Pierre Simon, External definability and groups in NIP theories, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 213–240. MR 3245144, DOI 10.1112/jlms/jdu019
Annalisa Conversano and Anand Pillay, Connected components of definable groups and $o$-minimality I, Adv. Math. 231 (2012), no. 2, 605–623. MR 2955185, DOI 10.1016/j.aim.2012.05.022
Artem Chernikov and Pierre Simon, Externally definable sets and dependent pairs, Israel J. Math. 194 (2013), no. 1, 409–425. MR 3047077, DOI 10.1007/s11856-012-0061-9
Eli Glasner, Enveloping semigroups in topological dynamics, Topology Appl. 154 (2007), no. 11, 2344–2363. MR 2328017, DOI 10.1016/j.topol.2007.03.009
Eli Glasner, The structure of tame minimal dynamical systems, Ergodic Theory Dynam. Systems 27 (2007), no. 6, 1819–1837. MR 2371597, DOI 10.1017/S0143385707000296
Jakub Gismatullin, Davide Penazzi, and Anand Pillay, Some model theory of $\rm {SL}(2,\Bbb R)$, Fund. Math. 229 (2015), no. 2, 117–128. MR 3315377, DOI 10.4064/fm229-2-2
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
Ehud Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25 (2012), no. 1, 189–243. MR 2833482, DOI 10.1090/S0894-0347-2011-00708-X
Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115. MR 1854232, DOI 10.1016/S0168-0072(01)00096-3
Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI 10.1090/S0894-0347-96-00202-0
Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117–138. MR 1081816, DOI 10.1016/0168-0072(90)90046-5
Ehud Hrushovski and Anand Pillay, On NIP and invariant measures, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1005–1061. MR 2800483, DOI 10.4171/JEMS/274
Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), no. 2, 563–596. MR 2373360, DOI 10.1090/S0894-0347-07-00558-9
Ehud Hrushovski, Anand Pillay, and Pierre Simon, Generically stable and smooth measures in NIP theories, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2341–2366. MR 3020101, DOI 10.1090/S0002-9947-2012-05626-1
Meyer Jerison, A property of extreme points of compact convex sets, Proc. Amer. Math. Soc. 5 (1954), 782–783. MR 65021, DOI 10.1090/S0002-9939-1954-0065021-1
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
David Kerr and Hanfeng Li, Independence in topological and $C^*$-dynamics, Math. Ann. 338 (2007), no. 4, 869–926. MR 2317754, DOI 10.1007/s00208-007-0097-z
Michael C. Laskowski, Vapnik-Chervonenkis classes of definable sets, J. London Math. Soc. (2) 45 (1992), no. 2, 377–384. MR 1171563, DOI 10.1112/jlms/s2-45.2.377
Jiří Matoušek, Bounded VC-dimension implies a fractional Helly theorem, Discrete Comput. Geom. 31 (2004), no. 2, 251–255. MR 2060639, DOI 10.1007/s00454-003-2859-z
Alice Medvedev and Thomas Scanlon, Invariant varieties for polynomial dynamical systems, Ann. of Math. (2) 179 (2014), no. 1, 81–177. MR 3126567, DOI 10.4007/annals.2014.179.1.2
Ludomir Newelski, Bounded orbits and measures on a group, Israel J. Math. 187 (2012), 209–229. MR 2891705, DOI 10.1007/s11856-011-0081-x
Ludomir Newelski, Topological dynamics of definable group actions, J. Symbolic Logic 74 (2009), no. 1, 50–72. MR 2499420, DOI 10.2178/jsl/1231082302
Ludomir Newelski and Marcin Petrykowski, Weak generic types and coverings of groups. I, Fund. Math. 191 (2006), no. 3, 201–225. MR 2278623, DOI 10.4064/fm191-3-2
Robert R. Phelps, Lectures on Choquet’s theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR 1835574, DOI 10.1007/b76887
Anand Pillay, Type-definability, compact Lie groups, and o-minimality, J. Math. Log. 4 (2004), no. 2, 147–162. MR 2114965, DOI 10.1142/S0219061304000346
Bruno Poizat, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein. MR 1827833, DOI 10.1090/surv/087
Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
Anand Pillay and Ningyuan Yao, On minimal flows, definably amenable groups, and $o$-minimality, Adv. Math. 290 (2016), 483–502. MR 3451930, DOI 10.1016/j.aim.2015.12.010
Z. Sela, Diophantine geometry over groups VIII: Stability, Ann. of Math. (2) 177 (2013), no. 3, 787–868. MR 3034289, DOI 10.4007/annals.2013.177.3.1
Saharon Shelah, Dependent first order theories, continued, Israel J. Math. 173 (2009), 1–60. MR 2570659, DOI 10.1007/s11856-009-0082-1
Saharon Shelah, Minimal bounded index subgroup for dependent theories, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1087–1091. MR 2361885, DOI 10.1090/S0002-9939-07-08654-6
Saharon Shelah, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Ann. Math. Logic 3 (1971), no. 3, 271–362. MR 317926, DOI 10.1016/0003-4843(71)90015-5
Pierre Simon, A guide to NIP theories, Lecture Notes in Logic, vol. 44, Association for Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge, 2015. MR 3560428, DOI 10.1017/CBO9781107415133
Pierre Simon, Rosenthal compacta and NIP formulas, Fund. Math. 231 (2015), no. 1, 81–92. MR 3361236, DOI 10.4064/fm231-1-5
Barry Simon, Convexity, Cambridge Tracts in Mathematics, vol. 187, Cambridge University Press, Cambridge, 2011. An analytic viewpoint. MR 2814377, DOI 10.1017/CBO9780511910135
Sergei Starchenko, NIP, Keisler measures and combinatorics, Astérisque 390 (2017), Exp. No. 1114, 303–334. Séminaire Bourbaki. Vol. 2015/2016. Exposés 1104–1119. MR 3666030
V. N. Vapnik and A. Ja. Červonenkis, The uniform convergence of frequencies of the appearance of events to their probabilities, Teor. Verojatnost. i Primenen. 16 (1971), 264–279 (Russian, with English summary). MR 0288823
Frank O. Wagner, Simple theories, Mathematics and its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000. MR 1747713, DOI 10.1007/978-94-017-3002-0
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
Boris Zilber, Uncountably categorical theories, Translations of Mathematical Monographs, vol. 117, American Mathematical Society, Providence, RI, 1993. Translated from the Russian by D. Louvish. MR 1206477, DOI 10.1090/mmono/117