Nombres de Betti $L^2$ et facteurs de type ${\mathrm{II}}_1$

Connes, Alain

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 920, pp. 321-333.

Damien Gaboriau a montré récemment que les nombres de Betti $L^2$ 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${}_1$ pour en déduire l’existence de facteurs de type II${}_1$ dont le groupe fondamental est trivial.

Damien Gaboriau showed recently that the $L^2$ 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${}_1$ factors whose fundamental group is trivial.

Classification : 46L35, 57R30
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

[1] M. Atiyah - “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] F. Boca - “On the method for constructing irreducible finite index subfactors of Popa”, Pacific J. Math. 161 (1993), p. 201-231. | MR | Zbl

[3] A. Borel - “The $L^2$-cohomology of negatively curved Riemannian symmetric spaces”, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), p. 95-105. | MR | Zbl

[4] J. De Cannière & U. Haagerup - “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] J. Cheeger & M. Gromov - “$L^2$-cohomology and group cohomology”, Topology 25 (1986), p. 189-215. | MR | Zbl

[6] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg & A. Valette - Groups with the Haagerup property (Gromov's a-T-menability), Progress in Math., Birkhäuser, 2001. | MR | Zbl

[7] M. Choda - “Group factors of the Haagerup type”, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), p. 174-177. | MR | Zbl

[8] A. Connes - “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${}_1$ 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] A. Connes, J. Feldman & B. Weiss - “An amenable equivalence relation is generated by a single transformation”, Ergodic Theory Dynam. Systems 4 (1982), p. 431-450. | MR | Zbl

[15] A. Connes & V. F. R. Jones - “A II${}_1$ 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] M. Cowling & U. Haagerup - “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] C. Delaroche & A. Kirillov - “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] J. Dixmier - “Sous-anneaux abéliens maximaux dans les facteurs de type fini”, Ann. of Math. 59 (1954), p. 279-286. | MR | Zbl

[20] H. Dye - “On groups of measure preserving transformations I, II”, Amer. J. Math. 81 (1959), p. 119-159, & 85 (1963), p. 551-576. | MR | Zbl

[21] J. Feldman & C. C. Moore - “Ergodic equivalence relations, cohomology, and von Neumann algebras I, II”, Trans. Amer. Math. Soc. 234 (1977), p. 289-324, 325-359. | MR | Zbl

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

[23] -, “Invariants ${\ell }^2$ de relations d’équivalence et de groupes”, Publ. Math. Inst. Hautes Études Sci. (2002). | Numdam | Zbl

[24] L. Ge - “Prime factors”, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), p. 12762-12763. | MR | Zbl

[25] V. Y. Golodets & N. I. Nesonov - “T-property and nonisomorphic factors of type II and III”, J. Funct. Anal. 70 (1987), p. 80-89. | MR | Zbl

[26] U. Haagerup - “An example of non-nuclear C${}^{*}$-algebra which has the metric approximation property”, Invent. Math. 50 (1979), p. 279-293. | MR | Zbl

[27] P. De La Harpe & A. Valette - La propriété T de Kazhdan pour les groupes localement compacts, Astérisque, vol. 175, Société Mathématique de France, 1989. | Numdam | Zbl

[28] G. Hjorth - “A lemma for cost attained”, UCLA preprint, 2002. | MR | Zbl

[29] R. V. Kadison - “Problems on von Neumann algebras”, Baton Rouge Conference 1867, unpublished. | Zbl

[30] D. Kazhdan - “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] G. Levitt - “On the cost of generating an equivalence relation ”, Ergodic Theory Dynam. Systems 15 (1995), p. 1173-1181. | MR | Zbl

[32] W. Luck - “Dimension theory of arbitrary modules over finite von Neumann algebras and $L^2$-Betti numbers II. Applications to Grothendieck groups, $L^2$-Euler characteristics and Burnside groups”, J. reine angew. Math. 496 (1998), p. 213-236. | MR | Zbl

[33] G. Margulis - “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] N. Monod & Y. Shalom - “Orbit equivalence rigidity and bounded cohomology”, preprint. | MR | Zbl

[36] D. Ornstein & B. Weiss - “Ergodic theory of amenable group actions”, Bull. Amer. Math. Soc. (N.S.) 2 (1980), p. 161-164. | MR | Zbl

[37] N. Ozawa - “Solid von Neumann algebras”, math/0302082. | Zbl

[38] -, “There is no separable universal II${}_1$-factor”, math/0210411.

[39] N. Ozawa & S. Popa - “Some prime factorisation results for II${}_1$ factors”, math/0302240. | Zbl

[40] S. Popa - “On a class of type II${}_1$ factors with Betti numbers invariants”. | Zbl

[41] -, “Strong rigidity of II${}_1$ 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${}_1$ 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] S. Popa & D. Shlyakhtenko - “Cartan subalgebras and bimodule decomposition of II${}_1$ factors”, Math. Scand. 92 (2003), p. 93-102. | MR | Zbl

[45] S. Sakai - C${}^{*}$-algebras and W${}^{*}$-algebras, Springer-Verlag, Berlin-Heidelberg-New York, 1971. | MR | Zbl

[46] J.-L. Sauvageot - “Sur le produit tensoriel relatif d'espaces de Hilbert”, J. Operator Theory 9 (1983), p. 237-252. | MR | Zbl

[47] J.-P. Serre - Arbres, amalgames, SL(2), Astérisque, vol. 46, Société Mathématique de France, 1977. | Numdam | MR | Zbl

[48] D. Voiculescu - “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] R. Zimmer - Ergodic theory and semisimple groups, Birkhäuser-Verlag, Boston, 1984. | MR | Zbl