Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 85–122 | Cite as

Ergodic theorems for coset spaces

  • Michael BjörklundEmail author
  • Alexander Fish


We study in this paper the validity of the Mean Ergodic Theorem along left Følner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Følner sequence in G, we provide examples where the Mean Ergodic Theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g., if G is virtually nilpotent, then the Mean Ergodic Theorem holds along any left Følner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a “sufficiently thin” subgroup, we show that the Mean Ergodic Theorem holds for the induced representation along any left Følner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Følner sequence (Fn) in L, there exists a sequence (sn) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (Fnsn) in G. Finally, we provide examples of countable (not necessarily amenable) groups G with proper, infinite-index subgroups H, so that the Pointwise Ergodic Theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset G/H.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Anatharaman-Delaroche, Approximation properties of coset spaces and their operator algebras, The Varied Landscape of Operator Theory: Conference Proceedings, Timioara, July 27, 2012, Theta, Bucharest, 2014, pp. 23–45.Google Scholar
  2. [2]
    L. Bartholdi, R. I. Grigorchuk, and Z. Sunik, Branch groups, Handbook of Algebra, vol. 3, Elsevier, North-Holland, 2003, pp. 989–1112.Google Scholar
  3. [3]
    M. Beiglböck, V. Bergelson, and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math. 223 No. 2 (2010), 416–432.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. Bekka, M, P. de la Harpe, Représentations d’un groupe faiblement équivalentes á la représentation réguliére, Bull. Soc. Math. France 122 (1994), 333–342.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s property (T), Cambridge University Press, Cambridge, 2008.CrossRefzbMATHGoogle Scholar
  6. [6]
    J. B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N. S.) 1 (1995), 411–457.zbMATHGoogle Scholar
  7. [7]
    T. Crisp, The Bost-Connes phase transition and unitary representations, J. Noncommut. Geom. (7) 1 (2013), 291–300.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. K. van Douwen, Measures invariant under actions of F2, Topology Appl. 34 (1990), 53–68.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Dudko and K. Medynets, Finite factor representations of Higman–Thompson groups, Groups Geom. Dyn. 8 (2014), 375–389.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    W. Emerson and F. Greenleaf, Group structure and the pointwise ergodic theorem for connected amenable group, Adv. Math. 14 (1974), 153–172.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Y. Glasner and N. Monod, Amenable actions, free products and a fixed point property, Bull. London Math. Soc. 39 (2007), 138–150.zbMATHGoogle Scholar
  13. [13]
    A. Gorodnik and A. Nevo, The Ergodic Theory of Lattice Subgroups, Princeton University Press, Princeton NJ, 2010.zbMATHGoogle Scholar
  14. [14]
    A. Kechris, Global Aspects of Ergodic Group Actions, Amer. Math. Soc., Providence RI, 2010.CrossRefzbMATHGoogle Scholar
  15. [15]
    A. Kechris and B. Miller, Topics in Orbit Equivalence, Springer–Verlag, Berlin, 2004.CrossRefzbMATHGoogle Scholar
  16. [16]
    A. Lubotsky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, 1994.CrossRefGoogle Scholar
  17. [17]
    G. W. Mackey, On induced representations of groups, Amer. J. Math. 73 (1951), 576–592.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N. Monod and S. Popa, On co-amenability for groups and von Neumann algebras, C. R. Acad. Sci. Canada 25 (2003), 82–87.MathSciNetzbMATHGoogle Scholar
  19. [19]
    I. Namioka, Følner’s conditions for amenable semi-groups, Math. Scand. 15 (1964), 18–28.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A. Nevo, Harmonic analysis and pointwise ergodic theorems for noncommuting transformations, J. Amer. Math. Soc. 7 (1994), 875–902.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. Peterson and A. Thom, Character rigidity for special linear groups, J. Reine Angew. Math. 716 (2016), 207–228.MathSciNetzbMATHGoogle Scholar
  22. [22]
    F. Riesz, Sur la théorie ergodique, Comm. Math. Helv. 17 (1945), 221–239.CrossRefzbMATHGoogle Scholar
  23. [23]
    W. Rudin, Fourier Analysis on Groups, John Wiley & Sons, Inc., New York, 1990.CrossRefzbMATHGoogle Scholar
  24. [24]
    K. Schmidt, Asymptotic properties of unitary representations and mixing, Proc. London Math. Soc. (3) 48 (1984), 445–460.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    E. Thoma, Über unitäre Darstellungen abzahlbarer diskreter Gruppen, Math. Ann. 153 (1964), 111–138.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. Vershik, Nonfree actions of countable groups and their characters, J. Math. Sci. (NY) 174 (2011), 1–6.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    J. von Neumann, Proof of the quasiergodic hypothesis, Proc. Nat. Acad. Sci. (USA) 18 (1932), 70–82.CrossRefGoogle Scholar
  28. [28]
    R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser Verlag, Basel, 1984.CrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsChalmers, GothenburgSweden
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

Personalised recommendations