Approximate subgroups
[Sous-groupes approximatifs]
Massicot, Jean-Cyrille1 ; Wagner, Frank O.2
1 tabacckludge ’Ecole normale supérieure de Rennes, Campus de Ker lann Avenue Robert Schuman, 35170 Bruz, France
2 Université Lyon 1, CNRS, Institut Camille Jordan UMR 5208 21 avenue Claude Bernard, 69622 Villeurbanne cedex, France
Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 55-63.

Étant donné un sous-groupe approximatif A définissablement moyennable d’un groupe (local) dans une structure du premier ordre, il y a un sous-groupe H type-définissable normalisé par A et contenu dans A 4 tel que tout ensemble définissable contenant H est de mesure positive.

Given a definably amenable approximate subgroup A of a (local) group in some first-order structure, there is a type-definable subgroup H normalized by A and contained in A 4 such that every definable superset of H has positive measure.

DOI : 10.5802/jep.17
Classification : 11B30, 20N99, 03C98, 20A15
Keywords: Approximate subgroup, definability, definable amenability
Mot clés : Sous-groupe approximatif, moyennable, sous-groupe type-définissable
Massicot, Jean-Cyrille 1 ; Wagner, Frank O. 2

1 tabacckludge ’Ecole normale supérieure de Rennes, Campus de Ker lann Avenue Robert Schuman, 35170 Bruz, France
2 Université Lyon 1, CNRS, Institut Camille Jordan UMR 5208 21 avenue Claude Bernard, 69622 Villeurbanne cedex, France 
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Massicot, Jean-Cyrille; Wagner, Frank O. Approximate subgroups. Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 55-63. doi : 10.5802/jep.17. http://www.numdam.org/articles/10.5802/jep.17/

[1] Breuillard, E.; Green, B.; Tao, T. The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci., Volume 116 (2012), pp. 115-221 | DOI | Numdam | MR | Zbl

[2] van den Dries, L. Approximate groups [after Hrushovski, and Breuillard, Green, Tao], Séminaire Bourbaki (2013/14) (Astérisque), Société Mathématique de France (Exp. no 1077, to appear)

[3] Eleftheriou, P. E.; Peterzil, Y. Definable quotients of locally definable groups, Selecta Math. (N.S.), Volume 18 (2012) no. 4, pp. 885-903 | DOI | MR | Zbl

[4] Gleason, A. M. Groups without small subgroups, Ann. of Math. (2), Volume 56 (1952), pp. 193-212 | MR | Zbl

[5] Goldbring, I. Hilbert’s fifth problem for local groups, Ann. of Math. (2), Volume 172 (2010) no. 2, pp. 1269-1314 | DOI | MR | Zbl

[6] Hrushovski, E. Stable group theory and approximate subgroups, J. Amer. Math. Soc., Volume 25 (2012) no. 1, pp. 189-243 | DOI | MR | Zbl

[7] Hrushovski, E.; Pillay, A. On NIP and invariant measures, J. Eur. Math. Soc. (JEMS), Volume 13 (2011) no. 4, pp. 1005-1061 | DOI | MR | Zbl

[8] Pillay, A., Private communication, 2014

[9] Sanders, T. On a nonabelian Balog-Szemerédi-type lemma, J. Aust. Math. Soc., Volume 89 (2010) no. 1, pp. 127-132 | DOI | MR | Zbl

[10] Yamabe, H. A generalization of a theorem of Gleason, Ann. of Math. (2), Volume 58 (1953), pp. 351-365 | MR | Zbl

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