Densely related groups

Cornulier, Yves; Le Boudec, Adrien

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 4, pp. 619-653.

On s’intéresse à la classe des groupes densément présentés. Il s’agit des groupes de type fini (ou plus généralement des groupes localement compacts compactement engendrés) dans lesquels de nouvelles relations apparaissent à intervalles réguliers. Une relation est dite nouvelle si elle n’est pas conséquence de relations de longueur plus petites. Pour un groupe de type fini, être densément présenté est un invariant de quasi-isométrie.

On vérifie qu’un groupe densément présenté ne peut pas avoir de cône asymptotique simplement connexe. En particulier un groupe lacunaire hyperbolique n’est jamais densément présenté.

On montre que le groupe de Grigorchuk est densément présenté. On prouve également que tout groupe de type fini (non virtuellement cyclique) qui est (localement fini)-par-$\mathbb{Z}$ et qui satisfait une loi, est densément présenté. Étant donnée une classe $𝒞$ de groupes de type fini, on considère l’alternative suivante : tout groupe dans $𝒞$ est soit finiment présenté, soit densément présenté. On montre que cette alternative est satisfaite par la classe des groupes nilpotents-par-cyclique et la classe des groupes métabéliens. A contrario, cette dichotomie n’est plus vraie pour les groupes résolubles de classe $3$.

We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasi-isometry invariant among finitely generated groups.

We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related.

We prove that the Grigorchuk group is densely related. We also show that a finitely generated group that is (infinite locally finite)-by-cyclic and which satisfies a law must be densely related. Given a class $𝒞$ of finitely generated groups, we consider the following dichotomy: every group in $𝒞$ is either finitely presented or densely related. We show that this holds within the class of nilpotent-by-cyclic groups and the class of metabelian groups. In contrast, this dichotomy is no longer true for the class of $3$-step solvable groups.

Classification : 20F05, 20F65, 20F69, 20F16, 22D05

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