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September  2011, 31(3): 685-707. doi: 10.3934/dcds.2011.31.685

Minimal Følner foliations are amenable

Fernando Alcalde Cuesta 1, and  Ana Rechtman 2,

1. 

Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain
2. 

Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730, United States

Received  February 2010 Revised  July 2011 Published  August 2011

For finitely generated groups, amenability and Følner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Følner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\infty$ foliations of closed manifolds that are Følner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive answer to a question of Kaimanovich. The equivalence is stated with respect to transverse invariant measures or some tangentially smooth measures. The latter include harmonic measures, and in this case the Følner condition has to be replaced by $\eta$-Følner (where the usual volume is modified by the modular form $\eta$ of the measure).
Keywords: Foliations, measurable equivalence relations, local and global means..
Mathematics Subject Classification: 37A20, 43A07, 57R3.
Citation: Fernando Alcalde Cuesta, Ana Rechtman. Minimal Følner foliations are amenable. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 685-707. doi: 10.3934/dcds.2011.31.685
References:
[1]

F. Alcalde-Cuesta and G. Hector, Intégration symplectique des variétés de Poisson régulières, (French) [Symplectic integration of regular Poisson manifolds], Israel J. Math., 90 (1995), 125-165.  Google Scholar

[2]

C. Anantharaman-Delaroche and J. Renault, "Amenable Groupoids," Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], 36, L'Enseignement Mathématique, Geneva, 2000.  Google Scholar

[3]

R. Brooks, Some Riemannian and dynamical invariants of foliations, in "Differential Geometry" (College Park, Md., 1981/1982), Progr. Math., 32, Birkhäuser, Boston, Mass., (1983), 56-72.  Google Scholar

[4]

A. Candel, The harmonic measures of Lucy Garnett, Adv. Math., 176 (2003), 187-247. doi: 10.1016/S0001-8708(02)00036-1.  Google Scholar

[5]

Y. Carrière and É. Ghys, Relations d'équivalence moyennables sur les groupes de Lie, (French) [Amenable equivalence relations on Lie groups], C. R. Acad. Sci. Paris S'er. I Math., 300 (1985), 677-680.  Google Scholar

[6]

D. M. Cass, Minimal leaves in foliations, Trans. Amer. Math. Soc., 287 (1985), 201-213. doi: 10.1090/S0002-9947-1985-0766214-2.  Google Scholar

[7]

A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems, 1 (1981), 431-450. doi: 10.1017/S014338570000136X.  Google Scholar

[8]

B. Deroin, "Laminations par Variétés Complexes," Ph.D. thesis, École Normale Supérieure de Lyon, 2003. Google Scholar

[9]

D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math., 139 (2000), 41-98. doi: 10.1007/s002229900019.  Google Scholar

[10]

L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285-311. doi: 10.1016/0022-1236(83)90015-0.  Google Scholar

[11]

É. Ghys, Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg), (French) [Construction of vector fields without periodic orbits (after Krystyna Kuperberg)], Séminaire Bourbaki, Vol. 1993/94, Astérisque, 227 (1995), 283-307.  Google Scholar

[12]

S. E. Goodman and J. F. Plante, Holonomy and averaging in foliated sets, J. Differential Geom., 14 (1979), 401-407.  Google Scholar

[13]

U. Hirsch, Some remarks on analytic foliations and analytic branched coverings, Math. Ann., 248 (1980), 139-152. doi: 10.1007/BF01421954.  Google Scholar

[14]

V. A. Kaimanovich, Brownian motion on foliations: Entropy, invariant measures, mixing, Functional Anal. Appl., 22 (1988), 326-328. doi: 10.1007/BF01077429.  Google Scholar

[15]

V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal., 1 (1992), 61-82. doi: 10.1007/BF00249786.  Google Scholar

[16]

V. A. Kaimanovich, Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 999-1004.  Google Scholar

[17]

V. A. Kaimanovich, Equivalence relations with amenable leaves need not be amenable, in "Topology, Ergodic Theory, Real Algebraic Geometry," Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, (2001), 151-166.  Google Scholar

[18]

G. Kuperberg and K. Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2), 144 (1996), 239-268. doi: 10.2307/2118592.  Google Scholar

[19]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[20]

J. Phillips, The holonomic imperative and the homotopy groupoid of a foliated manifold, Rocky Mountain J. Math., 17 (1987), 151-165. doi: 10.1216/RMJ-1987-17-1-151.  Google Scholar

[21]

J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2), 102 (1975), 327-361. doi: 10.2307/1971034.  Google Scholar

[22]

A. Rechtman, "Use and Disuse of Plugs in Foliations," Ph.D. thesis, École Normale Supérieure de Lyon, 2009. Google Scholar

[23]

G. Reeb, "Sur Certaines Propriétés Topologiques des Variétés Feuilletées," (French), Publ. Inst. Math. Univ. Strasbourg, 11, 5-89, 155-156, Actualités Sci. Ind., no. 1183, Hermann & Cie., Paris, 1952.  Google Scholar

[24]

M. Samuélidès, Tout feuilletage à croissance polynomiale est hyperfini, J. Funct. Anal., 34 (1979), 363-369. doi: 10.1016/0022-1236(79)90082-X.  Google Scholar

[25]

C. Series, Foliations of polynomial growth are hyperfinite, Israel J. Math., 34 (1979), 245-258.  Google Scholar

[26]

D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. doi: 10.1007/BF01390011.  Google Scholar

[27]

F. W. Jr. Wilson, On the minimal sets of non-singular vector fields, Ann. of Math. (2), 84 (1966), 529-536.  Google Scholar

[28]

R. J. Zimmer, Curvature of leaves in amenable foliations, Amer. J. Math., 105 (1983), 1011-1022. doi: 10.2307/2374302.  Google Scholar

show all references

References:
[1]

F. Alcalde-Cuesta and G. Hector, Intégration symplectique des variétés de Poisson régulières, (French) [Symplectic integration of regular Poisson manifolds], Israel J. Math., 90 (1995), 125-165.  Google Scholar

[2]

C. Anantharaman-Delaroche and J. Renault, "Amenable Groupoids," Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], 36, L'Enseignement Mathématique, Geneva, 2000.  Google Scholar

[3]

R. Brooks, Some Riemannian and dynamical invariants of foliations, in "Differential Geometry" (College Park, Md., 1981/1982), Progr. Math., 32, Birkhäuser, Boston, Mass., (1983), 56-72.  Google Scholar

[4]

A. Candel, The harmonic measures of Lucy Garnett, Adv. Math., 176 (2003), 187-247. doi: 10.1016/S0001-8708(02)00036-1.  Google Scholar

[5]

Y. Carrière and É. Ghys, Relations d'équivalence moyennables sur les groupes de Lie, (French) [Amenable equivalence relations on Lie groups], C. R. Acad. Sci. Paris S'er. I Math., 300 (1985), 677-680.  Google Scholar

[6]

D. M. Cass, Minimal leaves in foliations, Trans. Amer. Math. Soc., 287 (1985), 201-213. doi: 10.1090/S0002-9947-1985-0766214-2.  Google Scholar

[7]

A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems, 1 (1981), 431-450. doi: 10.1017/S014338570000136X.  Google Scholar

[8]

B. Deroin, "Laminations par Variétés Complexes," Ph.D. thesis, École Normale Supérieure de Lyon, 2003. Google Scholar

[9]

D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math., 139 (2000), 41-98. doi: 10.1007/s002229900019.  Google Scholar

[10]

L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285-311. doi: 10.1016/0022-1236(83)90015-0.  Google Scholar

[11]

É. Ghys, Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg), (French) [Construction of vector fields without periodic orbits (after Krystyna Kuperberg)], Séminaire Bourbaki, Vol. 1993/94, Astérisque, 227 (1995), 283-307.  Google Scholar

[12]

S. E. Goodman and J. F. Plante, Holonomy and averaging in foliated sets, J. Differential Geom., 14 (1979), 401-407.  Google Scholar

[13]

U. Hirsch, Some remarks on analytic foliations and analytic branched coverings, Math. Ann., 248 (1980), 139-152. doi: 10.1007/BF01421954.  Google Scholar

[14]

V. A. Kaimanovich, Brownian motion on foliations: Entropy, invariant measures, mixing, Functional Anal. Appl., 22 (1988), 326-328. doi: 10.1007/BF01077429.  Google Scholar

[15]

V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal., 1 (1992), 61-82. doi: 10.1007/BF00249786.  Google Scholar

[16]

V. A. Kaimanovich, Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 999-1004.  Google Scholar

[17]

V. A. Kaimanovich, Equivalence relations with amenable leaves need not be amenable, in "Topology, Ergodic Theory, Real Algebraic Geometry," Amer. Math. Soc. Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, (2001), 151-166.  Google Scholar

[18]

G. Kuperberg and K. Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2), 144 (1996), 239-268. doi: 10.2307/2118592.  Google Scholar

[19]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[20]

J. Phillips, The holonomic imperative and the homotopy groupoid of a foliated manifold, Rocky Mountain J. Math., 17 (1987), 151-165. doi: 10.1216/RMJ-1987-17-1-151.  Google Scholar

[21]

J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2), 102 (1975), 327-361. doi: 10.2307/1971034.  Google Scholar

[22]

A. Rechtman, "Use and Disuse of Plugs in Foliations," Ph.D. thesis, École Normale Supérieure de Lyon, 2009. Google Scholar

[23]

G. Reeb, "Sur Certaines Propriétés Topologiques des Variétés Feuilletées," (French), Publ. Inst. Math. Univ. Strasbourg, 11, 5-89, 155-156, Actualités Sci. Ind., no. 1183, Hermann & Cie., Paris, 1952.  Google Scholar

[24]

M. Samuélidès, Tout feuilletage à croissance polynomiale est hyperfini, J. Funct. Anal., 34 (1979), 363-369. doi: 10.1016/0022-1236(79)90082-X.  Google Scholar

[25]

C. Series, Foliations of polynomial growth are hyperfinite, Israel J. Math., 34 (1979), 245-258.  Google Scholar

[26]

D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. doi: 10.1007/BF01390011.  Google Scholar

[27]

F. W. Jr. Wilson, On the minimal sets of non-singular vector fields, Ann. of Math. (2), 84 (1966), 529-536.  Google Scholar

[28]

R. J. Zimmer, Curvature of leaves in amenable foliations, Amer. J. Math., 105 (1983), 1011-1022. doi: 10.2307/2374302.  Google Scholar

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