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On dense totipotent free subgroups in full groups

Alessandro Carderi, Damien Gaboriau and François Le Maître

Geometry & Topology 27 (2023) 2297–2318

We study probability measure preserving (p.m.p.) nonfree actions of free groups and the associated IRSs. 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 of cost  < r can be realized by the orbits of an action of the free group Fr on r generators that is totipotent and such that the image in the full group [] is dense. We explain why these actions have no minimal models. This also provides a continuum of pairwise orbit inequivalent invariant random subgroups of Fr, 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.

measurable group actions, nonfree actions, free groups, transitive actions of countable groups, IRS, space of subgroups, ergodic equivalence relations, orbit equivalence
Mathematical Subject Classification
Primary: 37A20, 22F10
Secondary: 22F50, 37B05
Received: 16 September 2020
Revised: 5 May 2021
Accepted: 2 October 2021
Published: 25 August 2023
Proposed: Martin R Bridson
Seconded: David Fisher, Mladen Bestvina
Alessandro Carderi
Fakultät für Mathematik, Institut für Algebra und Geometrie
Karlsruhe Institute of Technology
Damien Gaboriau
Unité de Mathématiques Pures et Appliquées
École Normale Supérieure de Lyon
François Le Maître
Institut de Mathématiques de Jussieu-PRG
Université de Paris

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