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


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.



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.


  1. 1.
    Bosa, J., Brown, N., Sato, Y., Tikuisis, A., White, S., Winter, W.: Covering dimension of C\(^*\)-algebras and 2-coloured classification. To appear in Mem. Am. Math. Soc.Google Scholar
  2. 2.
    Ceccherini-Silberstein, T., de la Harpe, P., Grigorchuk, R.I.: Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. (Russian) Tr. Mat. Inst. Steklova 224, 68–111 (1999); translation in Proc. Steklov Inst. Math. 224, 57–97 (1999)Google Scholar
  3. 3.
    Downarowicz, T., Huczek, D., Zhang, G.: Tilings of amenable groups. To appear in J. Reine Angew. Math.Google Scholar
  4. 4.
    Elliott, G., Gong, G., Lin, H., Niu, Z.: On the classification of simple amenable C\(^*\)-algebras with finite decomposition rank, II. arXiv:1507.03437
  5. 5.
    Elek, G., Lippner, G.: Borel oracles. An analytic approach to constant time algorithms. Proc. Am. Math. Soc. 138, 2939–2947 (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gong, G., Lin, H., Niu, Z.: Classification of finite simple amenable \({\cal{Z}}\)-stable C\(^*\)-algebras. arXiv:1501.00135
  7. 7.
    Guentner, E., Willett, R., Yu, G.: Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and C\(^*\)-algebras. Math. Ann. 367, 785–829 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hirshberg, I., Orovitz, J.: Tracially \({\cal{Z}}\)-absorbing C\(^*\)-algebras. J. Funct. Anal. 265, 765–785 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hochman, M.: Genericity in topological dynamics. Ergodic Theory Dyn. Syst. 28, 125–165 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jiang, X., Su, H.: On a simple unital projectionless C\(^*\)-algebra. Am. J. Math. 121, 359–413 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kerr, D., Li, H.: Ergodic Theory: Independence and Dichotomies. Springer, Cham (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kechris, A., Solecki, S., Todorcevic, S.: Borel chromatic numbers. Adv. Math. 141, 1–44 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lyons, R., Nazarov, F.: Perfect matchings as IID factors on non-amenable groups. Eur. J. Combin. 32, 1115–1125 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Matui, H., Sato, Y.: Strict comparison and \({\cal{Z}}\)-absorption of nuclear C\(^*\)-algebras. Acta Math. 209, 179–196 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matui, H., Sato, Y.: Decomposition rank of UHF-absorbing C\(^*\)-algebras. Duke Math. J. 163, 2687–2708 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Namioka, I.: Følner’s conditions for amenable semi-groups. Math. Scand. 15, 18–28 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ornstein, D.S., Weiss, B.: Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48, 1–141 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Phillips, N.C.: Large subalgebras. arXiv:1408.5546
  19. 19.
    Rørdam, M., Winter, W.: The Jiang–Su algebra revisited. J. Reine Angew. Math. 642, 129–155 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Szabó, G.: The Rokhlin dimension of topological \({\mathbb{Z}}^m\)-actions. Proc. Lond. Math. Soc. (3) 110, 673–694 (2015)Google Scholar
  21. 21.
    Szabó, G., Wu, J., Zacharias, J.: Rokhlin dimension for actions of residually finite groups. arXiv:1408.6096
  22. 22.
    Tikuisis, A., White, S., Winter, W.: Quasidiagonality of nuclear C\(^*\)-algebras. Ann. Math. (2) 185, 229–284 (2017)Google Scholar
  23. 23.
    Winter, W.: Nuclear dimension and \({\cal{Z}}\)-stability of pure C\(^*\)-algebras. Invent. Math. 187, 259–342 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Winter, W., Zacharias, J.: Completely positive maps of order zero. Münster J. Math. 2, 311–324 (2009)MathSciNetzbMATHGoogle Scholar

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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