Sur la théorie élémentaire des groupes libres  [ On the elementary theory of free groups ]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 922, p. 363-402

Sela has announced a complete solution of Tarski's problem, who asked around 1945 what are the finitely generated groups having the same elementary theory as a free group. We will talk about the works of Remeslennikov, Kharlampovich-Myasnikov, Sela, Champetier-Guirardel and others on the structure of the limit groups (the finitely generated groups thar are “limits”of free groups, or equivalently that have the same universal theory as a free group. We will indicate some of the tools used by Sela (including technics of Rips, Rips-Sela, Bestvina-Feighn and others on group actions on trees).

Sela a annoncé une solution complète d'un problème de Tarski, qui demanda vers 1945 quels sont les groupes de type fini qui ont la même théorie élémentaire qu'un groupe libre. Nous discuterons des travaux de Remeslennikov, Kharlampovich-Myasnikov, Sela, Champetier-Guirardel et autres sur la structure des groupes limites (les groupes de type fini qui sont “limites”de groupes libres, ou encore, qui ont la même théorie universelle qu'un groupe libre). Nous indiquerons quelques outils utilisés par Sela (dont des techniques de Rips, Rips-Sela, Bestvina-Feighn et autres sur les actions de groupes sur les arbres).

Classification:  03C60,  20E05,  20E08
Keywords: first order formula, elementary theory, Tarski's problem, free group, limit group, groups action on trees
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     author = {Paulin, Fr\'ed\'eric},
     title = {Sur la th\'eorie \'el\'ementaire des groupes libres},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     note = {talk:922},
     pages = {363-402},
     zbl = {1069.20030},
     mrnumber = {2111650},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__363_0}
}
Paulin, Frédéric. Sur la théorie élémentaire des groupes libres, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 922, pp. 363-402. http://www.numdam.org/item/SB_2002-2003__45__363_0/

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