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Mathematische Annalen

, Volume 371, Issue 1–2, pp 663–683 | Cite as

Følner tilings for actions of amenable groups

  • Clinton T. Conley
  • Steve C. Jackson
  • David Kerr
  • Andrew S. Marks
  • Brandon Seward
  • Robin D. Tucker-Drob
Article

Abstract

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G (“shapes”) with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz–Huczek–Zhang tiling theorem for countable amenable groups and strengthens the Ornstein–Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is \({\mathcal Z}\)-stable.

Notes

Acknowledgements

C.C. was partially supported by NSF Grant DMS-1500906. D.K. was partially supported by NSF Grant DMS-1500593. Part of this work was carried out while he was visiting the Erwin Schrödinger Institute (January–February 2016) and the Mittag–Leffler Institute (February–March 2016). A.M. was partially supported by NSF Grant DMS-1500974. B.S. was partially supported by ERC Grant 306494. R.T.D. was partially supported by NSF Grant DMS-1600904. Part of this work was carried out during the AIM SQuaRE: Measurable Graph Theory.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Clinton T. Conley
    • 1
  • Steve C. Jackson
    • 2
  • David Kerr
    • 3
  • Andrew S. Marks
    • 4
  • Brandon Seward
    • 5
  • Robin D. Tucker-Drob
    • 3
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of MathematicsUniversity of North TexasDentonUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA
  4. 4.UCLA Department of MathematicsLos AngelesUSA
  5. 5.Courant Institute of Mathematical SciencesNew YorkUSA

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