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Minimal Følner foliations are amenable
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 |
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. |
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. |
[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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