Abstract
We say that two free p.m.p. actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the acting groups are sofic and each has a w-normal amenable subgroup which is neither locally finite nor virtually cyclic then Shannon orbit equivalence implies that the actions have the same maximum sofic entropy. This extends a result of Austin beyond the finitely generated amenable setting and has the consequence that two Bernoulli actions of a group with the properties in question are Shannon orbit equivalent if and only if they are measure conjugate. Our arguments apply more generally to actions satisfying a sparse connectivity condition which we call property SC, and yield an entropy inequality under the assumption that one of the actions has this property.
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Notes
This bifurcation is also reflected in the two logically independent ways in which the ideas of amenability and nonamenability have been adapted to operator algebras, on the one hand through the notions of finiteness and pure infiniteness and on the other through injectivity and finite-dimensional approximation. All of this traces back to the basic Dedekindian alternative for defining what it means for a set to be finite, either as the property that every injection from the set to itself is surjective, or by the existence of a bijection between the set and \(\{ 1,\dots , n\}\) for some positive integer n.
Again this is consistent with operator algebra theory, where amenability has become synonymous with certain kinds of finite-dimensional approximation.
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Acknowledgements
The first author was partially supported by NSF grant DMS-1800633 and was affiliated with Texas A&M University while this work was done. Preliminary stages were carried out during his six-month stay in 2017-2018 at the ENS de Lyon, during which time he held ENS and CNRS visiting professorships and was supported by Labex MILYON/ANR-10-LABX-0070. He thanks Damien Gaboriau and Mikael de la Salle at the ENS for their generous hospitality. The second author was partially supported by NSF grants DMS-1600717 and DMS-1900746. We thank Robin Tucker-Drob for comments. We also thank one of the referees for pointing out that our original version of Lemma 3.9 is valid in a stronger purely Borel form and follows as such from results of [22], and that this lemma additionally yields an elementary proof of Lemma 3.30.
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Communicated by Andreas Thom.
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Kerr, D., Li, H. Entropy, Shannon orbit equivalence, and sparse connectivity. Math. Ann. 380, 1497–1562 (2021). https://doi.org/10.1007/s00208-021-02190-x
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DOI: https://doi.org/10.1007/s00208-021-02190-x