Minimal Følner foliations are amenable

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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

Abstract
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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 and 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.
[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.
[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.
[4]	A. Candel, The harmonic measures of Lucy Garnett, Adv. Math., 176 (2003), 187-247. doi: 10.1016/S0001-8708(02)00036-1.
[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.
[6]	D. M. Cass, Minimal leaves in foliations, Trans. Amer. Math. Soc., 287 (1985), 201-213. doi: 10.1090/S0002-9947-1985-0766214-2.
[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.
[8]	B. Deroin, "Laminations par Variétés Complexes," Ph.D. thesis, École Normale Supérieure de Lyon, 2003.
[9]	D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math., 139 (2000), 41-98. doi: 10.1007/s002229900019.
[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.
[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.
[12]	S. E. Goodman and J. F. Plante, Holonomy and averaging in foliated sets, J. Differential Geom., 14 (1979), 401-407.
[13]	U. Hirsch, Some remarks on analytic foliations and analytic branched coverings, Math. Ann., 248 (1980), 139-152. doi: 10.1007/BF01421954.
[14]	V. A. Kaimanovich, Brownian motion on foliations: Entropy, invariant measures, mixing, Functional Anal. Appl., 22 (1988), 326-328. doi: 10.1007/BF01077429.
[15]	V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal., 1 (1992), 61-82. doi: 10.1007/BF00249786.
[16]	V. A. Kaimanovich, Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 999-1004.
[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.
[18]	G. Kuperberg and K. Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2), 144 (1996), 239-268. doi: 10.2307/2118592.
[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.
[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.
[21]	J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2), 102 (1975), 327-361. doi: 10.2307/1971034.
[22]	A. Rechtman, "Use and Disuse of Plugs in Foliations," Ph.D. thesis, École Normale Supérieure de Lyon, 2009.
[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.
[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.
[25]	C. Series, Foliations of polynomial growth are hyperfinite, Israel J. Math., 34 (1979), 245-258.
[26]	D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. doi: 10.1007/BF01390011.
[27]	F. W. Jr. Wilson, On the minimal sets of non-singular vector fields, Ann. of Math. (2), 84 (1966), 529-536.
[28]	R. J. Zimmer, Curvature of leaves in amenable foliations, Amer. J. Math., 105 (1983), 1011-1022. doi: 10.2307/2374302.

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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.
[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.
[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.
[4]	A. Candel, The harmonic measures of Lucy Garnett, Adv. Math., 176 (2003), 187-247. doi: 10.1016/S0001-8708(02)00036-1.
[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.
[6]	D. M. Cass, Minimal leaves in foliations, Trans. Amer. Math. Soc., 287 (1985), 201-213. doi: 10.1090/S0002-9947-1985-0766214-2.
[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.
[8]	B. Deroin, "Laminations par Variétés Complexes," Ph.D. thesis, École Normale Supérieure de Lyon, 2003.
[9]	D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math., 139 (2000), 41-98. doi: 10.1007/s002229900019.
[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.
[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.
[12]	S. E. Goodman and J. F. Plante, Holonomy and averaging in foliated sets, J. Differential Geom., 14 (1979), 401-407.
[13]	U. Hirsch, Some remarks on analytic foliations and analytic branched coverings, Math. Ann., 248 (1980), 139-152. doi: 10.1007/BF01421954.
[14]	V. A. Kaimanovich, Brownian motion on foliations: Entropy, invariant measures, mixing, Functional Anal. Appl., 22 (1988), 326-328. doi: 10.1007/BF01077429.
[15]	V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal., 1 (1992), 61-82. doi: 10.1007/BF00249786.
[16]	V. A. Kaimanovich, Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 999-1004.
[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.
[18]	G. Kuperberg and K. Kuperberg, Generalized counterexamples to the Seifert conjecture, Ann. of Math. (2), 144 (1996), 239-268. doi: 10.2307/2118592.
[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.
[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.
[21]	J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2), 102 (1975), 327-361. doi: 10.2307/1971034.
[22]	A. Rechtman, "Use and Disuse of Plugs in Foliations," Ph.D. thesis, École Normale Supérieure de Lyon, 2009.
[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.
[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.
[25]	C. Series, Foliations of polynomial growth are hyperfinite, Israel J. Math., 34 (1979), 245-258.
[26]	D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255. doi: 10.1007/BF01390011.
[27]	F. W. Jr. Wilson, On the minimal sets of non-singular vector fields, Ann. of Math. (2), 84 (1966), 529-536.
[28]	R. J. Zimmer, Curvature of leaves in amenable foliations, Amer. J. Math., 105 (1983), 1011-1022. doi: 10.2307/2374302.

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