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Almost Finiteness and the Small Boundary Property

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Abstract

Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and m-comparison for some nonnegative integer m are all equivalent. The proof combines an Ornstein–Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz–Zhang and Conley–Jackson–Marks–Seward–Tucker-Drob on dynamical tilings and of Castillejos–Evington–Tikuisis–White–Winter on the Toms–Winter conjecture, this implies that crossed product \(\hbox {C}^*\)-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliott’s program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property \(\Gamma \), which confirms the Toms–Winter conjecture for such crossed products in the minimal case.

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Notes

  1. The first proof of this fact was given by Connes as an application of his celebrated result [7] that injectivity implies hyperfiniteness, whose full force is still needed to prove hyperfiniteness of the group von Neumann algebra itself.

  2. Whether or not this subequivalence is itself implemented in an approximately central way is roughly what separates \({\mathscr {Z}}\)-stability from its specialization to the nuclear setting.

  3. Nuclearity is automatic in this case since the acting group is amenable.

  4. In fact, this is more generally proved for actions satisfying the so-called topological small boundary property, which is known to be automatic for free actions on finite-dimensional spaces by Theorem 3.8 of [41].

  5. The arguments there are carried out for \(G={\mathbb Z}\) but the authors make a point that they also apply more generally to all amenable groups.

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Acknowledgements

The first author was partially supported by NSF Grant DMS-1500593. A portion of the work was 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 supported by EPSRC Grant EP/N00874X/1, the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), the European Union’s Horizon 2020 research and innovation programme under the Grant MSCA-IF-2016-746272-SCCD, and a start-up Grant of KU Leuven. Part of the work was carried out during the second author’s visit to Texas A&M in June 2017, and he thanks the first author for his hospitality. Both authors would like to thank Jianchao Wu for stimulating discussions on the subject of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    David Kerr

  2. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Box 2400, 3001, Leuven, Belgium

    Gábor Szabó

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Correspondence to David Kerr.

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Communicated by Y. Kawahigashi

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Kerr, D., Szabó, G. Almost Finiteness and the Small Boundary Property. Commun. Math. Phys. 374, 1–31 (2020). https://doi.org/10.1007/s00220-019-03519-z

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