Groupes aléatoires  [ Random groups ]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 916, p. 173-204

What are the properties of a finitely presented group “chosen at random”? The answer to this question depends on the method of sorting a group at random. One could fix the number n of generators and choose p relators at random among words of length L, and then let L go to infinity. One could also choose some finite graph, label its edges randomly by generators, and consider the group generated by these generators subject to the relations read on the cycles of the graph. In this talk, I would like to introduce the reader to some works of M. Gromov answering this kind of questions. These methods produce examples of finitely presented groups with surprising properties.

Quelles sont les propriétés d’un groupe de présentation finie “tiré au hasard” ? La réponse à cette question dépend bien entendu de la méthode choisie pour le tirage au sort. On peut par exemple fixer n générateurs et choisir p relations aléatoirement parmi les mots de longueur L, puis faire tendre L vers l’infini. On peut aussi choisir un graphe fini, étiqueter aléatoirement ses arêtes par des générateurs, et considérer le groupe engendré par ces générateurs, soumis aux relations lues sur les cycles du graphe. Dans cet exposé, je voudrais présenter des travaux de M. Gromov qui permettent de répondre à ces questions et qui mettent en évidence l’existence de groupes de présentation finie aux propriétés étonnantes.

Classification:  20F65,  20P05
Keywords: geometric group theory, hyperbolic groups, random walks, small cancellation
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     author = {Ghys, \'Etienne},
     title = {Groupes al\'eatoires},
     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:916},
     pages = {173-204},
     zbl = {1134.20306},
     mrnumber = {2111644},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__173_0}
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Ghys, Étienne. Groupes aléatoires, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 916, pp. 173-204. http://www.numdam.org/item/SB_2002-2003__45__173_0/

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