Ordering the space of finitely generated groups

Bartholdi, Laurent; Erschler, Anna

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

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.

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