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On
dense totipotent free subgroups in full groups
Alessandro Carderi, Damien Gaboriau and François Le Maître |
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Geometry & Topology 27 (2023) 2297–2318
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Abstract |
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We study probability measure preserving (p.m.p.) nonfree actions of free groups and the associated IRSs. The perfect kernel of a countable group is the largest closed subspace of the space of subgroups of without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of : 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 can be realized by the orbits of an action of the free group on 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 , 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 . |
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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Mathematical Subject Classification
Primary: 37A20, 22F10
Secondary: 22F50, 37B05
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References |
Publication
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
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Authors |
| Alessandro Carderi | |
| Fakultät für Mathematik, Institut
für Algebra und Geometrie Karlsruhe Institute of Technology Karlsruhe Germany |
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| Damien Gaboriau | |
| Unité de Mathématiques Pures et
Appliquées École Normale Supérieure de Lyon Lyon France |
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| François Le Maître | |
| Institut de Mathématiques de
Jussieu-PRG Université de Paris Paris France |
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| © 2023 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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