Ordering the space of finitely generated groups

Bartholdi, Laurent; Erschler, Anna

doi:10.5802/aif.2984

Ordering the space of finitely generated groups
[Comment ordonner l’espace des groupes de type fini]

Bartholdi, Laurent ¹ ; Erschler, Anna ²

¹ Mathematisches Institut Georg-August Universität Bunsenstraße 3-5 D-37073 Göttingen (Germany)
² C.N.R.S. UMR 8553 Département de mathématiques et applications École Normale Supérieure 45 rue d’Ulm 75005 Paris (France)

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2091-2144.

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.

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from $G$ to $H$ if, for some generating set $T$ in $H$ and some sequence of generating sets $S_{i}$ in $G$ , the marked balls of radius $i$ in $(G, S_{i})$ and $(H, T)$ coincide. We show that if a connected component of this graph contains at least one torsion-free nilpotent group $G$ , then it consists of those groups which generate the same variety of groups as $G$ . We show on the other hand that the first Grigorchuk group has infinite girth, and hence belongs to the same connected component as free groups.

The arrows in the graph define a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion.

We show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space.

DOI : 10.5802/aif.2984

Classification : 20E10, 20E34, 20F65
Keywords: Topological space of marked groups, limit groups, varieties of groups, non-uniform exponential growth, universal statements and identities
Mot clés : Espace topologique des groupes marqués, groupes limites, variétés de groupes, croissance exponentielle non-uniforme, énoncés universels et identités

Affiliations des auteurs :

Bartholdi, Laurent ¹ ; Erschler, Anna ²

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     title = {Ordering the space of finitely generated groups},
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     pages = {2091--2144},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Bartholdi, Laurent; Erschler, Anna. Ordering the space of finitely generated groups. Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2091-2144. doi : 10.5802/aif.2984. https://aif.centre-mersenne.org/articles/10.5802/aif.2984/

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