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Laurent Bartholdi; Anna Erschler
Ordering the space of finitely generated groups
(Comment ordonner l’espace des groupes de type fini)
Annales de l'institut Fourier, 65 no. 5 (2015), p. 2091-2144, doi: 10.5802/aif.2984
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Class. Math.: 20E10, 20E34, 20F65
Mots clés: Espace topologique des groupes marqués, groupes limites, variétés de groupes, croissance exponentielle non-uniforme, énoncés universels et identités

Résumé - Abstract

Nous considérons le graphe orienté dont les sommets sont les classes d’isomorphisme de groupes de type fini, avec une arête de $G$ à $H$ si, pour une partie génératrice de $H$ et une suite de parties génératrices de $G$, les boules marquées de rayon de plus en plus grand coincident dans $G$ et $H$. Nous montrons que les composantes connexes de groupes nilpotents sans torsion sont leurs variétés, et qu’il y a une arête du premier groupe de Grigorchuk vers un groupe libre.

Les flèches dans ce graphe définissent un préordre sur l’ensemble des classes d’isomorphisme de groupes de type fini. Nous montrons qu’un ordre partiel se plonge dans ce préordre si et seulement s’il est réalisable par des ensembles d’un ensemble dénombrable pour l’inclusion.

Nous montrons que tout groupe dénombrable se plonge dans un groupe de croissance exponentielle non-uniforme. En particulier, il existe des groupes de croissance exponentielle non-uniforme qui ne sont pas résiduellement de croissance subexponentielle.

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