On dense totipotent free subgroups in full groups

Alessandro Carderi , Damien Gaboriau (1) , François Le Maître

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated IRS's. The perfect kernel of a countable group Γ is the largest closed subspace of the space of subgroups of Γ without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of Γ: 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 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 [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 (we call evanescence) and establish a genericity result about their existence. We show that their existence characterizes cost 1.

Mots clés: Measurable group actions; non-free actions; free groups; transitive actions of countable groups; IRS; space of subgroups; ergodic equivalence relations; orbit equivalence.

Domaines: Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]

Alessandro Carderi, Damien Gaboriau, François Le Maître. On dense totipotent free subgroups in full groups. Geometry and Topology, 2023, 27 (6), pp.2297-2318. ⟨10.2140/gt.2023.27.2297⟩. ⟨hal-02932059v3⟩