Israel Journal of Mathematics

, Volume 149, Issue 1, pp 45–85 | Cite as

Amenability and the Liouville property



We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid action on the Poisson boundary of any invariant operator is always amenable. This approach subsumes as particular cases numerous earlier results on amenability for groups, actions, equivalence relations and foliations. For instance, we establish in a unified way topological amenability of the boundary action for isometry groups of Gromov hyperbolic spaces, Riemannian symmetric spaces and affine buildings.


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  1. [AD87]
    C. Anantharaman-Delaroche,Systèmes dynamiques non commutatifs et moyennabilité, Mathematische Annalen279 (1987), 297–315. MR 89f:46127.MATHCrossRefMathSciNetGoogle Scholar
  2. [AD02]
    C. Anantharaman-Delaroche,Amenability and exactness for dynamical systems and their C*-algebras, Transactions of the American Mathematical Society354 (2002), 4153–4178 (electronic). MR 1 926 869.MATHCrossRefMathSciNetGoogle Scholar
  3. [Ada94]
    S. Adams,Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology33 (1994), 765–783. MR 96g:58104.MATHCrossRefMathSciNetGoogle Scholar
  4. [Ada96]
    S. Adams,Reduction of cocycles with hyperbolic targets, Ergodic Theory and Dynamical Systems16 (1996), 1111–1145. MR 98i:58135.MATHMathSciNetCrossRefGoogle Scholar
  5. [ADR00]
    C. Anantharaman-Delaroche and J. Renault,Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], Vol. 36, L’Enseignement Mathématique, Geneva, 2000, With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 2001m:22005.Google Scholar
  6. [AEG94]
    S. Adams, G. A. Elliott and T. Giordano,Amenable actions of groups, Transactions of the American Mathematical Society344 (1994), 803–822. MR 94k:22010.MATHCrossRefMathSciNetGoogle Scholar
  7. [Anc90]
    A. Ancona,Théorie du potentiel sur les graphes et les variétés, École d’été de Probabilités de Saint-Flour XVIII—1988, Lecture Notes in Mathematics, Vol. 1427, Springer, Berlin, 1990, pp. 1–112. MR 92g:31012.CrossRefGoogle Scholar
  8. [AS85]
    M. T. Anderson and R. Schoen,Positive harmonic functions on complete manifolds of negative curvature, Annals of Mathematics (2)121 (1985), 429–461. MR 87a:58151.CrossRefMathSciNetGoogle Scholar
  9. [Aze70]
    R. Azencott,Espaces de Poisson des groupes localement compacts, Lecture Notes in Mathematics, Vol. 148, Springer-Verlag, Berlin, 1970. MR 58 #18748.MATHGoogle Scholar
  10. [BG02]
    P. Biane and E. Germain,Actions moyennables et fonctions harmoniques, Comptes Rendus de l’Académie des Sciences, Paris334 (2002), 355–358. MR 2003e:60173.MATHMathSciNetGoogle Scholar
  11. [Bow77]
    R. Bowen,Anosov foliations are hyperfinite, Annals of Mathematics (2)106 (1977), 549–565. MR 57 #1569.CrossRefMathSciNetGoogle Scholar
  12. [Car99]
    D. I. Cartwright,Harmonic functions on buildings of type à n, inRandom Walks and Discrete Potential Theory (Cortona, 1997), Symposia Mathematica, XXXIX, Cambridge University Press, Cambridge, 1999, pp. 104–138. MR 2002b:31010.Google Scholar
  13. [CEOO03]
    J. Chabert, S. Echterhoff and H. Oyono-Oyono,Going-down functors, the Künneth formula, and the Baum-Connes conjecture, Geometric and Functional Analysis14 (2004), 491–528. MR 2100669.MATHCrossRefMathSciNetGoogle Scholar
  14. [CFW81]
    A. Connes, J. Feldman and B. Weiss,An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems1 (1981), 431–450 (1982). MR 84h:46090.MATHMathSciNetGoogle Scholar
  15. [CHLI02]
    K. Corlette, L. Hernández Lamoneda and A. Iozzi,A vanishing theorem for the tangential de Rham cohomology of a foliation with amenable fundamental groupoid, Geometriae Dedicata103 (2004), 205–223. MR 2005b:57054.MATHCrossRefMathSciNetGoogle Scholar
  16. [CR03]
    P. Cutting and G. Robertson,Type III actions on boundaries of à n buildings, Journal of Operator Theory49 (2003), 25–44. MR 2004b:20043.MATHMathSciNetGoogle Scholar
  17. [CW89]
    A. Connes and E. J. Woods,Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks, Pacific Journal of Mathematics137 (1989), 225–243, MR 90h:46100.MATHMathSciNetGoogle Scholar
  18. [CW04]
    D. I. Cartwright and W. Woess,Isotropic random walks in a building of type à d, Mathematische Zeitschrift247 (2004), 101–135. MR 2054522.MATHCrossRefMathSciNetGoogle Scholar
  19. [Der76]
    Y. Derriennic,Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires, Annales de l’Institut Henri Poincaré Section B (N.S.)12 (1976), 111–129. MR 54 #11508.MathSciNetMATHGoogle Scholar
  20. [EG93]
    G. A. Elliott and T. Giordano,Amenable actions of discrete groups, Ergodic Theory and Dynamical Systems13 (1993), 289–318. MR 94i:22023.MATHMathSciNetGoogle Scholar
  21. [Fis87]
    A. Fisher,Convex-invariant means and a pathwise central limit theorem, Advances in Mathematics63 (1987), 213–246. MR 88g:60058.MATHCrossRefMathSciNetGoogle Scholar
  22. [FM77]
    J. Feldman and C. C. Moore,Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Transactions of the American Mathematical Society234 (1977), 289–324. MR 58 #28261a.MATHCrossRefMathSciNetGoogle Scholar
  23. [Fog69]
    S. R. Foguel,The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, No. 21, Van Nostrand Reinhold Co., New York, 1969. MR 41 #6299.MATHGoogle Scholar
  24. [Fur63]
    H. Furstenberg,A Poisson formula for semi-simple Lie groups, Annals of Mathematics (2)77 (1963), 335–386. MR 26 #3820.CrossRefMathSciNetGoogle Scholar
  25. [Fur73]
    H. Furstenberg,Boundary theory and stochastic processes on homogeneous spaces, inHarmonic Analysis on Homogeneous Spaces (Proceedings of Symposia in Pure Mathematics, Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), American Mathematical Society, Providence, R.I., 1973, pp. 193–229. MR 50 #4815.Google Scholar
  26. [Ger00]
    E. Germain,Approximate invariant means for boundary actions of hyperbolic groups, 2000, Appendix B in the book [ADR00].Google Scholar
  27. [GJT98]
    Y. Guivarc’h, L. Ji and J. C. Taylor,Compactifications of Symmetric Spaces, Progress in Mathematics, Vol. 156, Birkhäuser Boston Inc., Boston, MA, 1998. MR 2000c:31006.MATHGoogle Scholar
  28. [Gre69]
    F. P. Greenleaf,Invariant Means on Topological Groups and their Applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York, 1969. MR 40 #4776.MATHGoogle Scholar
  29. [Gro00]
    M. Gromov,Spaces and questions, Geometric and Functional Analysis (2000), Special Volume, Part I, GAFA 2000 (Tel Aviv, 1999), pp. 118–161. MR 2002e:53056.Google Scholar
  30. [Hah78]
    P. Hahn,Haar measure for measure groupoids, Transactions of the American Mathematical Society242 (1978), 1–33. MR 82a:28012.MATHCrossRefMathSciNetGoogle Scholar
  31. [Hig00]
    N. Higson,Bivariant K-theory and the Novikov conjecture, Geometric and Functional Analysis10 (2000), 563–581. MR 2001k:19009.MATHCrossRefMathSciNetGoogle Scholar
  32. [HR00]
    N. Higson and J. Roe,Amenable group actions and the Novikov conjecture, Journal für die reine und angewandte Mathematik519 (2000), 143–153. MR 2001h:57043.MATHCrossRefMathSciNetGoogle Scholar
  33. [Jac71]
    J. Jacod,Théorème de renouvellement et classification pour les chaînes semi-markoviennes, Annales de l’Institut Henri Poincaré Section B (N.S.)7 (1971), 83–129. MR MR0305496 (46 #4626).MathSciNetMATHGoogle Scholar
  34. [Jaw95]
    W. Jaworski,On the asymptotic and invariant σ-algebras of random walks on locally compact groups, Probability Theory and Related Fields101 (1995), 147–171. MR 95m:60013.MATHCrossRefMathSciNetGoogle Scholar
  35. [JKL02]
    S. Jackson, A. S. Kechris and A. Louveau,Countable Borel equivalence relations, Journal of Mathematical Logic2 (2002), 1–80. MR 2003f:03066.MATHCrossRefMathSciNetGoogle Scholar
  36. [Kai83]
    V. A. Kaimanovich,The differential entropy of the boundary of a random walk on a group, Russian Mathematical Surveys38 (1983), no. 5(233), 142–143, English translation. MR 85k:60016.CrossRefMathSciNetGoogle Scholar
  37. [Kai92]
    V. A. Kaimanovich,Measure-theoretic boundaries of Markov chains, 0–2laws and entropy, inHarmonic Analysis and Discrete Potential Theory (Frascati, 1991), Plenum, New York, 1992, pp. 145–180. MR 94h:60099.Google Scholar
  38. [Kai95]
    V. A. Kaimanovich,The Poisson boundary of covering Markov operators, Israel Journal of Mathematics89 (1995), 77–134. MR 96k:60194.MATHMathSciNetGoogle Scholar
  39. [Kai96]
    V. A. Kaimanovich,Boundaries of invariant Markov operators: the identification problem, inErgodic Theory of Z d Actions (Warwick, 1993–1994), London Mathematical Society Lecture Note Series, Vol. 228, Cambridge University Press, Cambridge, 1996, pp. 127–176. MR 97j:31008.Google Scholar
  40. [Kai97]
    V. A. Kaimanovich,Amenability, hyperfiniteness, and isoperimetric inequalities, Comptes Rendus de l’Académie des Sciences, Paris, SérieI 325 (1997), 999–1004. MR 98j:28014.MathSciNetGoogle Scholar
  41. [Kai98]
    V. A. Kaimanovich,Hausdorff dimension of the harmonic measure on trees, Ergodic Theory and Dynamical Systems18 (1998), 631–660. MR 99g:60123.MATHCrossRefMathSciNetGoogle Scholar
  42. [Kai02]
    V. A. Kaimanovich,The Poisson boundary of amenable extensions, Monatshefte für Mathematik136 (2002), 9–15. MR 2003e:60013.MATHCrossRefMathSciNetGoogle Scholar
  43. [Kai03]
    V. A. Kaimanovich,Boundary amenability of hyperbolic spaces, inDiscrete Geometric Analysis, Comtemporary Mathematics347, American Mathematical Society, Providence, RI, 2004, pp. 83–111. MR 2077032.Google Scholar
  44. [Kar67]
    F. I. Karpelevich,The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces, Transactions of the Moscow Mathematical Society1965 (1967), 51–199. MR 37 #6876.MathSciNetGoogle Scholar
  45. [Kat81]
    S. Kato,On eigenspaces of the Hecke algebra with respect to a good maximal compact subgroup of a p-adic reductive group, Mathematische Annalen257 (1981), 1–7. MR 83a:22019.MATHCrossRefMathSciNetGoogle Scholar
  46. [KF98]
    V. A. Kaimanovich and A. Fisher,A Poisson formula for harmonic projections, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques34 (1998), 209–216. MR 99d:60087.MATHCrossRefMathSciNetGoogle Scholar
  47. [KKR02]
    V. A. Kaimanovich, Y. Kifer and B.-Z. Rubshtein,Boundaries and harmonic functions for random walks with random transition probabilities, Journal of Theoretical Probability17 (2004), 605–646.MATHCrossRefMathSciNetGoogle Scholar
  48. [KL01]
    V. A. Kaimanovich and M. Lyubich,Conformal and harmonic measures on laminations associated with rational maps, Memoirs of the American Mathematical Society, no. 820, American Mathematical Society, Providence, RI, 2005.Google Scholar
  49. [KS83]
    A. Krámli and D. Szász,Random walks with internal degrees of freedom. I. Local limit theorems, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete63 (1983), 85–95. MR 85f:60098.MATHCrossRefGoogle Scholar
  50. [KV83]
    V. A. Kaimanovich and A. M. Vershik,Random walks on discrete groups: boundary and entropy, Annals of Probability11 (1983), 457–490. MR 85d:60024.MATHMathSciNetGoogle Scholar
  51. [KW02]
    V. A. Kaimanovich and W. Woess,Boundary and entropy of space homogeneous Markov chains, Annals of Probability30 (2002), 323–363. MR 2003d:60152.MATHCrossRefMathSciNetGoogle Scholar
  52. [LS84]
    T. Lyons and D. Sullivan,Function theory, random paths and covering spaces, Journal of Differential Geometry19 (1984), 299–323. MR 86b:58130.MATHMathSciNetGoogle Scholar
  53. [Mey73]
    P. A. Meyer,Limites médiales, d’après Mokobodzki, Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971–1972), Lecture Notes in Mathematics, Vol. 321, Springer, Berlin, 1973, pp. 198–204. MR 53 #8364.CrossRefGoogle Scholar
  54. [MZ98]
    A. M. Mantero and A. Zappa,Eigenfunctions of the Laplace operators for a building of type \(\tilde G_2 \), preprint, 1998.Google Scholar
  55. [MZ00]
    A. M. Mantero and A. Zappa,Eigenfunctions of the Laplace operators for a building of type \(\tilde A_2 \), The Journal of Geometric Analysis10 (2000), 339–363. MR 2001h:43010.MATHMathSciNetGoogle Scholar
  56. [MZ02]
    A. M. Mantero and A. Zappa,Eigenfunctions of the Laplace operators for buildings of type \(\tilde B_2 \), Bolletine della Unione Matematica Italiana. Serie VIII. Sezione B. Articoli di Ricerca Matematica (8)5 (2002), 163–195. MR 2003a:35137.MATHMathSciNetGoogle Scholar
  57. [MZ03]
    A. M. Mantero and A. Zappa,Remarks on harmonic functions on affine buildings, inRandom Walks and Geometry, de Gruyter, Berlin, 2004, pp. 473–485.Google Scholar
  58. [Pat88]
    A. L. T. Paterson,Amenability, Mathematical Surveys and Monographs, Vol. 29, American Mathematical Society, Provindence, RI 1988. MR 90e:43001.MATHGoogle Scholar
  59. [Pie84]
    J.-P. Pier,Amenable Locally Compact Groups, Pure and Applied Mathematics, Wiley, New York, 1984. MR 86a:43001.Google Scholar
  60. [Ren80]
    J. Renault,A groupoid approach to C*-algebras, Lecture Notes in Mathematics, Vol. 793, Springer, Berlin, 1980. MR 82h:46075.MATHGoogle Scholar
  61. [Rev84]
    D. Revuz,Markov chains, second edn., North-Holland Mathematical Library, Vol. 11, North-Holland Publishing Co., Amsterdam, 1984. MR 86a:60097.Google Scholar
  62. [Ros81]
    J. Rosenblatt,Ergodic and mixing random walks on locally compact groups, Mathematische Annalen257 (1981), 31–42. MR 83f:43002.MATHCrossRefMathSciNetGoogle Scholar
  63. [RR96]
    J. Ramagge and G. Robertson,Triangle buildings and actions of type III1/q 2, Journal of Functional Analysis140 (1996), 472–504. MR 98b:46089.MATHCrossRefMathSciNetGoogle Scholar
  64. [RS96]
    G. Robertson and T. Steger,C*-algebras arising from group actions on the boundary of a triangle building, Proceedings of the London Mathematical Society (3)72 (1996), 613–637. MR 98b:46088.MATHCrossRefMathSciNetGoogle Scholar
  65. [SCW02]
    L. Saloff-Coste and W. Woess,Transition operators on co-compact G-spaces, preprint, 2002.Google Scholar
  66. [Sol75]
    F. Solomon,Random walks in a random environment, Annals of Probability3 (1975), 1–31. MR 50 #14943.MATHGoogle Scholar
  67. [Spa87]
    R. J. Spatzier,An example of an amenable action from geometry, Ergodic Theory and Dynamical Systems7 (1987), 289–293. MR 88j:58100.MATHMathSciNetCrossRefGoogle Scholar
  68. [Sun87]
    C. Sunyach,Sur la transience et la récurrence des marches aléatoires en milieu aléatoire, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques23 (1987), 613–626. MR 89c:60080.MATHMathSciNetGoogle Scholar
  69. [SZ91]
    R. J. Spatzier and R. J. Zimmer,Fundamental groups of negatively curved manifolds and actions of semisimple groups, Topology30 (1991), 591–601. MR 92m:57047.MATHCrossRefMathSciNetGoogle Scholar
  70. [Tit86]
    J. Tits,Immeubles de type affine, inBuildings and the Geometry of Diagrams (Como, 1984), Lecture Notes in Mathematics, Vol. 1181, Springer, Berlin, 1986, pp. 159–190. MR 87h:20077.CrossRefGoogle Scholar
  71. [Val02]
    A. Valette,Introduction to the Baum-Connes Conjecture, Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel, 2002. From notes taken by Indira Chatterji, with an appendix by Guido Mislin. MR 2003f:58047.MATHGoogle Scholar
  72. [Ver78]
    A. M. Vershik,The action of PSL(2,Z)in R 1 is approximable, Uspekhi Matematicheskikh Nauk33 (1978), no. 1(199), 209–210. MR 58 #13201.MATHGoogle Scholar
  73. [Zim77]
    R. J. Zimmer,Hyperfinite factors and amenable ergodic actions, Inventiones Mathematicae41 (1977), 23–31. MR 57 #10438.MATHCrossRefMathSciNetGoogle Scholar
  74. [Zim78]
    R. J. Zimmer,Amenable ergodic group actions and an application to Poisson boundaries of random walks, Journal of Functional Analysis27 (1978), 350–372. MR 57 #12775.MATHCrossRefMathSciNetGoogle Scholar
  75. [Zim84]
    R. J. Zimmer,Ergodic Theory and Semisimple Groups, Monographs in Mathematics, Vol. 81, Birkhäuser Verlag, Basel, 1984. MR 86j:22014.MATHGoogle Scholar

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© Hebrew University 2005

Authors and Affiliations

  1. 1.CNRS UMR 6625, IRMARUniversité Rennes-1RennesFrance

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