Invariants ${\ell }^2$ de relations d’équivalence et de groupes

Gaboriau, Damien

Publications Mathématiques de l'IHÉS, Tome 95 (2002), pp. 93-150.

[Ada90] S. Adams, Trees and amenable equivalence relations, Ergodic Theory Dynamical Systems, 10(1) (1990), 1-14. | MR | Zbl

[AS90] S. Adams and R. Spatzier, Kazhdan groups, cocycles and trees, Amer. J. Math., 112(2) (1990), 271-287. | MR | Zbl

[Ati76] M. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan (Orsay, 1974), pp. 43-72. Astérisque, No. 32-33. Paris: Soc. Math. France, 1976. | Numdam | Zbl

[BCH94] P. Baum, A. Connes and N. Higson, Classifying space for proper actions and K-theory of group C-algebras, In C-algebras: 1943-1993 (San Antonio, TX, 1993), pp. 240-291. Providence, RI: Amer. Math. Soc., 1994. | MR | Zbl

[BV97] M. Bekka and A. Valette, Group cohomology, harmonic functions and the first $L^2$-Betti number, Potential Anal., 6(4) (1997), 313-326. | MR | Zbl

[Bor85] A. Borel, The $L^2$-cohomology of negatively curved Riemannian symmetric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 95-105. | MR | Zbl

[Bro82] K. Brown, Cohomology of groups, New York: Springer, 1982. | MR | Zbl

[CFW81] A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems, 1(4), 431-450 (1982), 1981. | MR | Zbl

[CG86] J. Cheeger and M. Gromov, $L^2$-cohomology and group cohomology, Topology, 25(2) (1986), 189-215. | MR | Zbl

[Co79] A. Connes, Sur la théorie non commutative de l'intégration, In Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), pp. 19-143. Lect. Notes 725. Berlin: Springer, 1979. | Zbl

[Co82] A. Connes, A survey of foliations and operator algebras, In Operator algebras and applications, Part I (Kingston, Ont., 1980), pp. 521-628. Providence, RI: Amer. Math. Soc., 1982. | MR | Zbl

[CZ89] M. Cowling and R. Zimmer, Actions of lattices in sp(1,n), Ergodic Theory Dynamical Systems, 9(2) (1989), 221-237. | MR | Zbl

[Dix69] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Gauthier-Villars Editeur, Paris, 1969. Deuxième édition, revue et augmentée, Cahiers Scientifiques, Fasc. XXV. | Zbl

[Dye59] H. Dye, On groups of measure preserving transformations, I, Amer. J. Math., 81 (1959), 119-159. | MR | Zbl

[Dye63] H. Dye, On groups of measure preserving transformations, II, Amer. J. Math., 85 (1963), 551-576. | MR | Zbl

[Eck00] B. Eckmann, Introduction to l2-methods in topology: reduced l2-homology, harmonic chains, l2-Betti numbers, Israel J. Math., 117 (2000), 183-219. Notes prepared by Guido Mislin. | MR | Zbl

[Ele97] G. Elek, Betti numbers and Euler's formula for combinatorial foliations, Manuscripta Math., 92(2) (1997), 239-247. | Zbl

[FM77a] J. Feldman and C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, Trans. Amer. Math. Soc., 234(2) (1977), 289-324. | MR | Zbl

[FM77b] J. Feldman and C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, II, Trans. Amer. Math. Soc., 234(2) (1977), 325-359. | MR | Zbl

[FSZ89] J. Feldman, C. Sutherland and R. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynamical Systems, 9(2) (1989), 239-269. | MR | Zbl

[Fur99a] A. Furman, Gromov's measure equivalence and rigidity of higher rank lattices, Ann. of Math. (2), 150(3) (1999), 1059-1081. | Zbl

[Fur99b] A. Furman, Orbit equivalence rigidity, Ann. of Math. (2), 150(3) (1999), 1083-1108. | MR | Zbl

[Gab00a] D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math., 139(1) (2000), 41-98. | Zbl

[Gab00b] D. Gaboriau, Sur la (co-)homologie $L^2$ des actions préservant une mesure. C. R. Acad. Sci. Paris Sér. I Math., 330(5) (2000), 365-370. | MR | Zbl

[Gro93] M. Gromov, Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), pp. 1-295. Cambridge: Cambridge Univ. Press, 1993. | MR | Zbl

[Har00] P. De La Harpe, Topics in geometric group theory, Chicago, IL: University of Chicago Press, 2000. | MR | Zbl

[Hae71] A. Haefliger, Homotopy and integrability, In Manifolds-Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133-163. Berlin: Springer, 1971. | MR | Zbl

[Hjo02] G. Hjorth, A lemma for cost attained, Manuscrit, 2002. | Zbl

[HL91] J. Heitsch and C. Lazarov, Homotopy invariance of foliation Betti numbers, Invent. Math., 104(2) (1991), 321-347. | MR | Zbl

[ Jol01] P. JOLISSAINT, Approximation properties for measure equivalent groups, Manuscrit, 2001.

[Kec01] A. KECHRIS, Lectures on costs of equivalence relations and groups, Prépublication Caltech, 2001.

[Kri70] W. Krieger, On constructing non-isomorphic hyperfinite factors of type III, J. Functional Analysis, 6 (1970), 97-109. | MR | Zbl

[Lev95] G. Levitt, On the cost of generating an equivalence relation, Ergodic Theory Dynam. Systems, 15(6) (1995), 1173-1181. | MR | Zbl

[Luc94] W. Lück, $L^2$-torsion and 3-manifolds, In Low-dimensional topology (Knoxville, TN, 1992), pp. 75-107. Cambridge, MA: Internat. Press, 1994. | MR | Zbl

[Luc98] W. Lück, $L^2$ -invariants of regular coverings of compact manifolds and CW-complexes, à paraître in “Handbook of Geometry”, Elsevier, 1998. | Zbl

[Luc98a] W. Lück, Dimension theory of arbitrary modules over finite von Neumann algebras and $L^2$-Betti numbers, I, Foundations, J. Reine Angew. Math., 495 (1998), 135-162. | MR | Zbl

[Luc98b] W. Lück, 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), 213-236. | MR | Zbl

[Moo82] C. Moore, Ergodic theory and von Neumann algebras, In Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 179-226. Providence, RI: Amer. Math. Soc., 1982. | MR | Zbl

[MS02] N. MONOD and Y. SHALOM, en préparation. Communication personnelle.

[MvN36] F. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2), 37 (1936), 116-229. | MR | Zbl

[MvN43] F. Murray and J. Von Neumann, On rings of operators, IV, Ann. of Math. (2), 44 (1943), 716-808. | MR | Zbl

[OW80] D. Ornstein and B. Weiss, Ergodic theory of amenable group actions, I, The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2(1) (1980), 161-164. | MR | Zbl

[Pau99] F. Paulin, Propriétés asymptotiques des relations d'équivalences mesurées discrètes, Markov Process. Related Fields, 5(2) (1999), 163-200. | Zbl

[Sch87] K. Schmidt, Some solved and unsolved problems concerning orbit equivalence of countable group actions, Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986), pp. 171-184, Leipzig: Teubner, 1987. | MR | Zbl

[Shl99a] D. Shlyakhtenko, Free Fisher information with respect to a completely positive map and cost of equivalence relations, MSRI preprint 1999-030, 1999. | MR | Zbl

[Shl99b] D. Shlyakhtenko, Microstates free entropy and cost of equivalence relations, preprint, http://xxx.lanl.gov/abs/math.OA/9912224. | MR

[Sun87] V. Sunder, An invitation to von Neumann algebras, New York: Springer, 1987. | MR | Zbl

[Zim78] R. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis, 27(3) (1978), 350-372. | MR | Zbl

[Zim84] R. Zimmer, Ergodic theory and semisimple groups, Basel: Birkhäuser Verlag, 1984. | MR | Zbl