On dense totipotent free subgroups in full groups

Carderi, Alessandro¹ ; Gaboriau, Damien² ; Le Maître, François³

¹ Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruhe Institute of Technology, Karlsruhe, Germany
² Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, Lyon, France
³ Institut de Mathématiques de Jussieu-PRG, Université de Paris, Paris, France

Geometry & topology, Tome 27 (2023) no. 6, pp. 2297-2318.

Résumé

We study probability measure preserving (p.m.p.) nonfree actions of free groups and the associated IRSs. The perfect kernel of a countable group $\Gamma$ is the largest closed subspace of the space of subgroups of $\Gamma$ without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of $\Gamma$: those for which almost every point-stabilizer has dense conjugacy class in the perfect kernel. Equivalently, the support of the associated IRS is as large as possible, namely it is equal to the whole perfect kernel. We prove that every ergodic p.m.p. equivalence relation $\mathcal{ℛ}$ of cost $<r$ can be realized by the orbits of an action of the free group $F_r$ on $r$ generators that is totipotent and such that the image in the full group $[\mathcal{ℛ}]$ is dense. We explain why these actions have no minimal models. This also provides a continuum of pairwise orbit inequivalent invariant random subgroups of $F_r$, all of whose supports are equal to the whole space of infinite-index subgroups. We are led to introduce a property of topologically generating pairs for full groups (which we call evanescence) and establish a genericity result about their existence. We show that their existence characterizes cost $1$.

Détail

DOI : 10.2140/gt.2023.27.2297

Keywords: measurable group actions, nonfree actions, free groups, transitive actions of countable groups, IRS, space of subgroups, ergodic equivalence relations, orbit equivalence

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