Invariant percolation and measured theory of nonamenable groups [after Gaboriau-Lyons, Ioana, Epstein]

Houdayer, Cyril

Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042. Avec table par noms d'auteurs de 1948/49 à 2009/10., Astérisque, no. 348 (2012), Exposé no. 1039, 36 p.

MR Zbl | 1 citation dans Numdam

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     title = {Invariant percolation and measured theory of nonamenable groups [after {Gaboriau-Lyons,} {Ioana,} {Epstein]}},
     booktitle = {S\'eminaire Bourbaki Volume 2010/2011 Expos\'es 1027-1042. Avec table par noms d'auteurs de 1948/49 \`a 2009/10.},
     series = {Ast\'erisque},
     note = {talk:1039},
     pages = {339--374},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {348},
     year = {2012},
     mrnumber = {3051202},
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Houdayer, Cyril. Invariant percolation and measured theory of nonamenable groups [after Gaboriau-Lyons, Ioana, Epstein], dans Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042. Avec table par noms d'auteurs de 1948/49 à 2009/10., Astérisque, no. 348 (2012), Exposé no. 1039, 36 p. http://www.numdam.org/item/AST_2012__348__339_0/

Bibliographie
Cité par

[1] S. I. Adyan - "Random walks on free periodic groups", Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), p. 1139-1149, 1343. | MR | Zbl

[2] C. Anantharaman-Delaroche - "Amenable correspondences and approximation properties for von Neumann algebras", Pacific J. Math. 171 (1995), p. 309-341. | DOI | MR | Zbl

[3] I. Benjamini, R. Lyons, Y. Peres & O. Schramm - "Group-invariant percolation on graphs", Geom. Fund. Anal. 9 (1999), p. 29-66. | DOI | MR | Zbl

[4] I. Benjamini & O. Schramm - "Percolation beyond $𝐙^{d}$ , many questions and a few answers", Electron. Comm. Probab. 1 (1996), p. 71-82. | DOI | EuDML | MR | Zbl

[5] S. I. Bezuglyĭ & V. Y. Golodets - "Hyperfinite and ${II}_{1}$ actions for nonamenable groups", J. Funct. Anal. 40 (1981), p. 30-44. | DOI | MR | Zbl

[6] N. P. Brown & N. Ozawa - C*-algebras and finite-dimensional approximations, Graduate Studies in Math., vol. 88, Amer. Math. Soc., 2008. | MR | Zbl

[7] M. Burger - "Kazhdan constants for $SL (3, 𝐙)$ ", J. reine angew. Math. 413 (1991), p. 36-67. | EuDML | MR | Zbl

[8] I. Chifan & A. Ioana - "Ergodic subequivalence relations induced by a Bernoulli action", Geom. Funct. Anal 20 (2010), p. 53-67. | DOI | MR | Zbl

[9] A. Connes - "Classification of injective factors. Cases ${II}_{1}, {II}_{\infty}, {II}_{λ}, λ \neq 1$ ", Ann. of Math. 104 (1976), p. 73-115. | DOI | MR | Zbl

[10] A. Connes, "A factor of type ${II}_{1}$ with countable fundamental group", J. Operator Theory 4 (1980), p. 151-153. | MR | Zbl

[11] A. Connes, J. Feldman & B. Weiss - "An amenable equivalence relation is generated by a single transformation", Ergodic Theory Dynam. Systems 1 (1981), p. 431-450. | DOI | MR | Zbl

[12] A. Connes & B. Weiss - "Property T and asymptotically invariant sequences", Israel J. Math. 37 (1980), p. 209-210. | DOI | MR | Zbl

[13] H. A. Dye - "On groups of measure preserving transformation. I", Amer. J. Math. 81 (1959), p. 119-159. | DOI | MR | Zbl

[14] H. A. Dye, "On groups of measure preserving transformations. II", Amer. J. Math. 85 (1963), p. 551-576. | DOI | MR | Zbl

[15] I. Epstein - "Orbit inequivalent actions of non-amenable groups", preprint arXiv:0707.4215.

[16] J. Feldman & C. C. Moore - "Ergodic equivalence relations, cohomology, and von Neumann algebras. I", Trans. Amer. Math. Soc. 234 (1977), p. 289-324. | DOI | MR | Zbl

[17] J. Feldman & C. C. Moore, "Ergodic equivalence relations, cohomology, and von Neumann algebras. II", Trans. Amer. Math. Soc. 234 (1977), p. 325-359. | DOI | MR | Zbl

[18] D. Gaboriau - "Coût des relations d'équivalence et des groupes", Invent. Math. 139 (2000), p. 41-98. | DOI | MR | Zbl

[19] D. Gaboriau, "Examples of groups that are measure equivalent to the free group", Ergodic Theory Dynam. Systems 25 (2005), p. 1809-1827. | DOI | MR | Zbl

[20] D. Gaboriau, "Invariant percolation and harmonic Dirichlet functions", Geom. Funct. Anal. 15 (2005), p. 1004-1051. | DOI | MR | Zbl

[21] D. Gaboriau, "Orbit equivalence and measured group theory", in Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, 2010, p. 1501-1527. | MR | Zbl

[22] D. Gaboriau & R. Lyons - "A measurable-group-theoretic solution to von Neumann's problem", Invent. Math. 177 (2009), p. 533-540. | DOI | MR | Zbl

[23] D. Gaboriau & S. Popa - "An uncountable family of nonorbit equivalent actions of $𝔽_{n}$ ", J. Amer. Math. Soc. 18 (2005), p. 547-559. | DOI | MR | Zbl

[24] S. L. Gefter & V. Y. Golodets - "Fundamental groups for ergodic actions and actions with unit fundamental groups", Publ. Res. Inst. Math. Sci. 24 (1988), p. 821-847. | DOI | MR | Zbl

[25] O. Häggström & Y. Peres - "Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously", Probab. Theory Related Fields 113 (1999), p. 273-285. | DOI | MR | Zbl

[26] G. Hjorth - "A converse to Dye's theorem", Trans. Amer. Math. Soc. 357 (2005), p. 3083-3103. | DOI | MR | Zbl

[27] G. Hjorth, "A lemma for cost attained", Ann. Pure Appl. Logic 143 (2006), p. 87-102. | DOI | MR | Zbl

[28] A. Ioana - "A relative version of Connes' $χ (M)$ invariant and existence of orbit inequivalent actions", Ergodic Theory Dynam. Systems 27 (2007), p. 1199-1213. | DOI | MR | Zbl

[29] A. Ioana, "Non-orbit equivalent actions of $𝔽_{n}$ ", Ann. Sci. Éc. Norm. Supér. 42 (2009), p. 675-696. | DOI | EuDML | Numdam | MR | Zbl

[30] A. Ioana, "Orbit inequivalent actions for groups containing a copy of $𝔽_{2}$ ", Invent. Math. 185 (2011), p. 55-73. | DOI | MR | Zbl

[31] A. Ioana, A. S. Kechris & T. Tsankov - "Subequivalence relations and positive-definite functions", Groups Geom. Dyn. 3 (2009), p. 579-625. | DOI | MR | Zbl

[32] A. Ioana, S. Popa & S. Vaes - "A class of superrigid group von Neumann algebras", preprint arXiv:1007.1412. | DOI | MR | Zbl

[33] D. Kazhdan - "Connection of the dual space of a group with the structure of its subgroups", Fund. Anal. Appl. 1 (1967), p. 63-65. | DOI | MR | Zbl

[34] A. S. Kechris - Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, Amer. Math. Soc., 2010. | DOI | MR | Zbl

[35] A. S. Kechris & B. D. Miller - Topics in orbit equivalence, Lecture Notes in Math., vol. 1852, Springer, 2004. | MR | Zbl

[36] H. Kesten - "Full Banach mean values on countable groups", Math. Scand. 7 (1959), p. 146-156. | DOI | EuDML | MR | Zbl

[37] Y. Kida - "Orbit equivalence rigidity for ergodic actions of the mapping class group", Geom. Dedicata 131 (2008), p. 99-109. | DOI | MR | Zbl

[38] G. Levitt - "On the cost of generating an equivalence relation", Ergodic Theory Dynam. Systems 15 (1995), p. 1173-1181. | DOI | MR | Zbl

[39] R. Lyons - "Phase transitions on nonamenable graphs. Probabilistic techniques in equilibrium and nonequilibrium statistical physics", J. Math. Phys. 41 (2000), p. 1099-1126. | DOI | Zbl

[40] R. Lyons & Y. Peres - "Probability on trees and networks", in preparation http://mypage.iu.edu/˜rdlyons/. | DOI | Zbl

[41] R. Lyons, Y. Peres & O. Schramm - "Minimal spanning forests", Ann. Probab. 34 (2006), p. 1665-1692. | DOI | Zbl

[42] R. Lyons & O. Schramm - "Indistinguishability of percolation clusters", Ann. Probab. 27 (1999), p. 1809-1836. | DOI | Zbl

[43] G. A. Margulis - "Finitely-additive invariant measures on Euclidean spaces", Ergodic Theory Dynam. Systems 2 (1982), p. 383-396. | DOI | Zbl

[44] B. Mohar - "Isoperimetric inequalities, growth, and the spectrum of graphs", Linear Algebra Appl. 103 (1988), p. 119-131. | DOI | Zbl

[45] N. Monod & Y. Shalom - "Orbit equivalence rigidity and bounded cohomology", Ann. of Math. 164 (2006), p. 825-878. | DOI | Zbl

[46] F. J. Murray & J. Von Neumann - "On rings of operators. IV", Ann. of Math. 44 (1943), p. 716-808. | DOI | Zbl

[47] F. J. Murray & J. Von Neumann - "On rings of operators", Ann. of Math. 37 (1936), p. 116-229. | DOI | JFM | Zbl

[48] J. Von Neumann - "Zur allgemeinen Theorie des Maßes", Fundam. Math. 13 (1929), p. 73-116. | DOI | EuDML | JFM

[49] C. M. Newman & L. S. Schulman - "Infinite clusters in percolation models", J. Statist. Phys. 26 (1981), p. 613-628. | DOI | Zbl

[50] A. J. Ol'Šanskiĭ- "On the question of the existence of an invariant mean on a group", Uspekhi Mat. Nauk 35 (1980), p. 199-200. | Zbl

[51] A. Y. Ol'Shanskii & M. V. Sapir - "Non-amenable finitely presented torsion-by-cyclic groups", Publ. Math. IHÉS 96 (2002), p. 43-169. | DOI | EuDML | Numdam | Zbl

[52] D. S. Ornstein & B. Weiss - "Ergodic theory of amenable group actions. I. The Rohlin lemma", Bull. Amer. Math. Soc. (N.S.) 2 (1980), p. 161-164. | DOI | Zbl

[53] N. Ozawa - "A Kurosh-type theorem for type ${II}_{1}$ factors", Int. Math. Res. Not. 2006 (2006), Art. ID 97560, 21. | Zbl

[54] I. Pak & T. Smirnova-Nagnibeda - "On non-uniqueness of percolation on nonamenable Cayley graphs", C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), p. 495-500. | DOI | Zbl

[55] S. Popa - "On a class of type ${II}_{1}$ factors with Betti numbers invariants", Ann. of Math. 163 (2006), p. 809-899. | DOI | Zbl

[56] S. Popa, "Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions", J. Inst. Math. Jussieu 5 (2006), p. 309-332. | DOI | Zbl

[57] S. Popa, "Strong rigidity of ${II}_{1}$ factors arising from malleable actions of $w$ -rigid groups. I", Invent. Math. 165 (2006), p. 369-408. | DOI | Zbl

[58] S. Popa, "Deformation and rigidity for group actions and von Neumann algebras", in International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, p. 445-477. | Zbl

[59] S. Popa, "On Ozawa's property for free group factors", Int. Math. Res. Not. 2007 (2007), Art. ID rnm036, 10. | Zbl

[60] S. Popa, "On the superrigidity of malleable actions with spectral gap", J. Amer. Math. Soc. 21 (2008), p. 981-1000. | DOI | Zbl

[61] S. Popa & S. Vaes - "Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups", Adv. Math. 217 (2008), p. 833-872. | DOI | Zbl

[62] K. Schmidt - "Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions", Ergodic Theory Dynam. Systems 1 (1981), p. 223-236. | DOI | Zbl

[63] I. M. Singer - "Automorphisms of finite factors", Amer. J. Math. 77 (1955), p. 117-133. | DOI | Zbl

[64] R. Szwarc - "An analytic series of irreducible representations of the free group", Ann. Inst. Fourier (Grenoble) 38 (1988), p. 87-110. | DOI | EuDML | Numdam | Zbl

[65] Á. Timár - "Ends in free minimal spanning forests", Ann. Probab. 34 (2006), p. 865-869. | DOI | Zbl

[66] S. Vaes - "Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)", Séminaire Bourbaki, vol. 2005/2006, exp. n° 961, Astérisque 311 (2007), p. 237-294. | EuDML | Numdam | Zbl

[67] S. Vaes, "Rigidity for von Neumann algebras and their invariants", in Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, 2010, p. 1624-1650. | Zbl

[68] R. J. Zimmer - Ergodic theory and semisimple groups, Monographs in Math., vol. 81, Birkhäuser, 1984. | Zbl