Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property $H_{\mathrm{FD}}$

Erschler, Anna; Zheng, Tianyi

Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1363-1402.

Nous prouvons une inégalité isopérimétrique pour les groupes. En application, nous montrons que la fonction de Følner de tout groupe de Grigorchuk à croissance intermédiaire est au moins exponentielle. En tant qu’autre application, nous obtenons des bornes inférieures sur les fonctions de Følner dans divers groupes nilpotents par cycliques. Sous une hypothèse de régularité, nous obtenons une caractérisation des fonctions de Følner de ces groupes. Comme autre application, nous évaluons le comportement asymptotique de la fonction de Følner de $Sym\left(\mathbb{Z}\right)⋊\mathbb{Z}$. Nous étudions des exemples de groupes avec la propriété de Shalom $H_{\mathrm{FD}}$ parmi les extensions d’un groupe nilpotent par un groupe cyclique. Nous montrons qu’il existe des groupes hyperboliques lacunaires avec la propriété $H_{\mathrm{FD}}$. Nous trouvons des groupes avec la propriété $H_{\mathrm{FD}}$, qui sont des produits directs de groupes hyperbolique lacunaires et ont des fonctions Følner arbitrairement grandes.

We prove an isoperimetric inequality for groups. As an application we show that any Grigrochuk group of intermediate growth has at least exponential Følner function. As another application, we obtain lower bounds on Følner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of Følner functions of these groups. As a further application, we evaluate the asymptotics of the Følner function of $Sym\left(\mathbb{Z}\right)⋊\mathbb{Z}$. We study examples of groups with Shalom’s property $H_{\mathrm{FD}}$ among nilpotent-by-cyclic groups. We show that there exist lacunary hyperbolic groups with property $H_{\mathrm{FD}}$. We find groups with property $H_{\mathrm{FD}}$, which are direct products of lacunary hyperbolic groups and have arbitrarily large Følner functions.

Classification : 20F65, 20E22, 60B15
Keywords: Følner function, isoperimetric profile, Følner sets, growth function
Mot clés : Fonction de Folner, profil isopérimétrique, ensembles de Folner, fonction de croissance

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