Damien Gaboriau showed recently that the Betti numbers of measured foliations with contractile leaves are invariants of the associated equivalence relation. Sorin Popa used this result, together with rigidity properties of type II factors whose fundamental group is trivial.
Damien Gaboriau a montré récemment que les nombres de Betti des feuilletages mesurés à feuilles contractiles sont des invariants de la relation d’équivalence associée. Sorin Popa a utilisé ce résultat joint à des propriétés de rigidité des facteurs de type II pour en déduire l’existence de facteurs de type II dont le groupe fondamental est trivial.
Mot clés : nombres de Betti $L^2$, feuilletage, facteur de type II${}_1$, groupe fondamental d’un facteur de type II${}_1$
Keywords: $L^2$ Betti numbers, foliation, type II${}_1$ factor, fundamental group of a type II${}_1$ factor
@incollection{SB_2002-2003__45__321_0, author = {Connes, Alain}, title = {Nombres de {Betti} $L^2$ et facteurs de type ${\rm II}_1$}, booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923}, series = {Ast\'erisque}, note = {talk:920}, pages = {321--333}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {294}, year = {2004}, zbl = {1133.46032}, language = {fr}, url = {http://archive.numdam.org/item/SB_2002-2003__45__321_0/} }
TY - CHAP AU - Connes, Alain TI - Nombres de Betti $L^2$ et facteurs de type ${\rm II}_1$ BT - Séminaire Bourbaki : volume 2002/2003, exposés 909-923 AU - Collectif T3 - Astérisque N1 - talk:920 PY - 2004 SP - 321 EP - 333 IS - 294 PB - Association des amis de Nicolas Bourbaki, Société mathématique de France PP - Paris UR - http://archive.numdam.org/item/SB_2002-2003__45__321_0/ LA - fr ID - SB_2002-2003__45__321_0 ER -
%0 Book Section %A Connes, Alain %T Nombres de Betti $L^2$ et facteurs de type ${\rm II}_1$ %B Séminaire Bourbaki : volume 2002/2003, exposés 909-923 %A Collectif %S Astérisque %Z talk:920 %D 2004 %P 321-333 %N 294 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://archive.numdam.org/item/SB_2002-2003__45__321_0/ %G fr %F SB_2002-2003__45__321_0
Connes, Alain. Nombres de Betti $L^2$ et facteurs de type ${\rm II}_1$, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 920, pp. 321-333. http://archive.numdam.org/item/SB_2002-2003__45__321_0/
[1] “Elliptic operators, discrete groups and von Neumann algebras”, in Colloque “Analyse et Topologie” en l'honneur d'Henri Cartan, Astérisque, vol. 32, Société Mathématique de France, 1976, p. 43-72. | Numdam | MR | Zbl
-[2] “On the method for constructing irreducible finite index subfactors of Popa”, Pacific J. Math. 161 (1993), p. 201-231. | MR | Zbl
-[3] “The -cohomology of negatively curved Riemannian symmetric spaces”, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), p. 95-105. | MR | Zbl
-[4] “Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups”, Amer. J. Math. 107 (1984), p. 455-500. | MR | Zbl
& -[5] “-cohomology and group cohomology”, Topology 25 (1986), p. 189-215. | MR | Zbl
& -[6] Groups with the Haagerup property (Gromov's a-T-menability), Progress in Math., Birkhäuser, 2001. | MR | Zbl
, , , & -[7] “Group factors of the Haagerup type”, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), p. 174-177. | MR | Zbl
-[8] “Classification of injective factors”, Ann. of Math. 104 (1976), p. 73-115. | MR | Zbl
-[9] -, “Sur la théorie non-commutative de l'intégration”, in Algèbres d'opérateurs, Séminaire Les Plans-sur-Bex, 1978, Lect. Notes in Math., vol. 725, Springer, Berlin, 1979, p. 19-143. | Zbl
[10] -, “A type II factor with countable fundamental group”, J. Operator Theory 4 (1980), p. 151-153. | MR | Zbl
[11] -, “Correspondences”, Notes manuscrites, 1980.
[12] -, “Classification des facteurs”, Proc. Symp. Pure Math., vol. 38, American Mathematical Society, 1982, p. 43-109. | MR
[13] -, Noncommutative geometry, Academic Press, 1994. | MR
[14] “An amenable equivalence relation is generated by a single transformation”, Ergodic Theory Dynam. Systems 4 (1982), p. 431-450. | MR | Zbl
, & -[15] “A II factor with two non-conjugate Cartan subalgebras”, Bull. Amer. Math. Soc. (N.S.) 6 (1982), p. 211-212. | MR | Zbl
& -[16] -, “Property T for von Neumann algebras”, Bull. London Math. Soc. 17 (1985), p. 57-62. | MR | Zbl
[17] “Completely bounded multipliers and the Fourier algebra of a simple Lie group of real rank one”, Invent. Math. 96 (1989), p. 507-549. | MR | Zbl
& -[18] “Sur les relations entre l'espace dual d'un groupe et la structure de ses sous-groupes fermés”, in Sém. Bourbaki (1967/1968), collection hors série de la S.M.F., vol. 10, Société Mathématique de France, 1995, exp. no 343, p. 507-528. | Numdam | MR | Zbl
& -[19] “Sous-anneaux abéliens maximaux dans les facteurs de type fini”, Ann. of Math. 59 (1954), p. 279-286. | MR | Zbl
-[20] “On groups of measure preserving transformations I, II”, Amer. J. Math. 81 (1959), p. 119-159, & 85 (1963), p. 551-576. | MR | Zbl
-[21] “Ergodic equivalence relations, cohomology, and von Neumann algebras I, II”, Trans. Amer. Math. Soc. 234 (1977), p. 289-324, 325-359. | MR | Zbl
& -[22] “Coût des relations d'équivalence et des groupes”, Invent. Math. 139 (2000), p. 41-98. | MR | Zbl
-[23] -, “Invariants de relations d’équivalence et de groupes”, Publ. Math. Inst. Hautes Études Sci. (2002). | Numdam | Zbl
[24] “Prime factors”, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), p. 12762-12763. | MR | Zbl
-[25] “T-property and nonisomorphic factors of type II and III”, J. Funct. Anal. 70 (1987), p. 80-89. | MR | Zbl
& -[26] “An example of non-nuclear C-algebra which has the metric approximation property”, Invent. Math. 50 (1979), p. 279-293. | MR | Zbl
-[27] La propriété T de Kazhdan pour les groupes localement compacts, Astérisque, vol. 175, Société Mathématique de France, 1989. | Numdam | Zbl
& -[28] “A lemma for cost attained”, UCLA preprint, 2002. | MR | Zbl
-[29] “Problems on von Neumann algebras”, Baton Rouge Conference 1867, unpublished. | Zbl
-[30] “Connection of the dual space of a group with the structure of its closed subgroups”, Functional Anal. Appl. 1 (1967), p. 63-65. | MR | Zbl
-[31] “On the cost of generating an equivalence relation ”, Ergodic Theory Dynam. Systems 15 (1995), p. 1173-1181. | MR | Zbl
-[32] “Dimension theory of arbitrary modules over finite von Neumann algebras and -Betti numbers II. Applications to Grothendieck groups, -Euler characteristics and Burnside groups”, J. reine angew. Math. 496 (1998), p. 213-236. | MR | Zbl
-[33] “Discrete groups of motion of manifolds of non-positive curvature”, Am. Math. Soc. Translations. 109 (1977), p. 33-45. | Zbl
-[34] -, “Finitely-additive invariant measures on Euclidian spaces”, Ergodic Theory Dynam. Systems 2 (1982), p. 383-396. | MR
[35] “Orbit equivalence rigidity and bounded cohomology”, preprint. | MR | Zbl
& -[36] “Ergodic theory of amenable group actions”, Bull. Amer. Math. Soc. (N.S.) 2 (1980), p. 161-164. | MR | Zbl
& -[37] “Solid von Neumann algebras”, math/0302082. | Zbl
-[38] -, “There is no separable universal II-factor”, math/0210411.
[39] “Some prime factorisation results for II factors”, math/0302240. | Zbl
& -[40] “On a class of type II factors with Betti numbers invariants”. | Zbl
-[41] -, “Strong rigidity of II factors coming from malleable actions of weakly rigid groups”, math/0305306. | Zbl
[42] -, “Correspondences”, INCREST preprint, unpublished, 1986.
[43] -, “Free independent sequences in type II factors and related problems”, in Recent advances in operator algebras (Orléans, 1992), Astérisque, vol. 232, Société Mathématique de France, 1995, p. 187-202. | Numdam | MR | Zbl
[44] “Cartan subalgebras and bimodule decomposition of II factors”, Math. Scand. 92 (2003), p. 93-102. | MR | Zbl
& -[45] C-algebras and W-algebras, Springer-Verlag, Berlin-Heidelberg-New York, 1971. | MR | Zbl
-[46] “Sur le produit tensoriel relatif d'espaces de Hilbert”, J. Operator Theory 9 (1983), p. 237-252. | MR | Zbl
-[47] Arbres, amalgames, SL(2), Astérisque, vol. 46, Société Mathématique de France, 1977. | Numdam | MR | Zbl
-[48] “The analogues of entropy and of Fisher's information theory in free probability II”, Invent. Math. 118 (1994), p. 411-440. | MR | Zbl
-[49] -, “The absence of Cartan subalgebras”, Geom. Funct. Anal. 6 (1996), p. 172-199. | MR
[50] Ergodic theory and semisimple groups, Birkhäuser-Verlag, Boston, 1984. | MR | Zbl
-