On the superrigidity of malleable actions with spectral gap

Popa, Sorin ¹

¹ Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-155505

Journal of the American Mathematical Society, Tome 21 (2008) no. 4, pp. 981-1000

Résumé

We prove that if a countable group $\Gamma$ contains infinite commuting subgroups $H, H’\subset \Gamma$ with $H$ non-amenable and $H’$ “weakly normal” in $\Gamma$, then any measure preserving $\Gamma$-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli $\Gamma$-action) is cocycle superrigid. If in addition $H’$ can be taken non-virtually abelian and $\Gamma \curvearrowright X$ is an arbitrary free ergodic action while $\Lambda \curvearrowright Y=\mathbb {T}^{\Lambda }$ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II$_{1}$ factors $L^{\infty }X \rtimes \Gamma \simeq L^{\infty }Y \rtimes \Lambda$ comes from a conjugacy of the actions.

Détail

DOI : 10.1090/S0894-0347-07-00578-4

Bibliographie

[1] Bader, Uri, Shalom, Yehuda Factor and normal subgroup theorems for lattices in products of groups Invent. Math. 2006 415 454

[2] Cherix, Pierre-Alain, Cowling, Michael, Jolissaint, Paul, Julg, Pierre, Valette, Alain Groups with the Haagerup property 2001

[3] Christensen, Erik Subalgebras of a finite algebra Math. Ann. 1979 17 29

[4] Connes, A. Classification of injective factors. Cases 𝐼𝐼₁, 𝐼𝐼_{∞}, 𝐼𝐼𝐼_{𝜆}, 𝜆̸ Ann. of Math. (2) 1976 73 115

[5] Connes, Alain Sur la classification des facteurs de type 𝐼𝐼 C. R. Acad. Sci. Paris Sér. A-B 1975

[6] Connes, A. A factor of type 𝐼𝐼₁ with countable fundamental group J. Operator Theory 1980 151 153

[7] Connes, A. Classification des facteurs 1982 43 109

[8] Connes, A., Feldman, J., Weiss, B. An amenable equivalence relation is generated by a single transformation Ergodic Theory Dynam. Systems 1981

[9] Connes, A., Jones, V. A 𝐼𝐼₁ factor with two nonconjugate Cartan subalgebras Bull. Amer. Math. Soc. (N.S.) 1982 211 212

[10] Connes, A., Jones, V. Property 𝑇 for von Neumann algebras Bull. London Math. Soc. 1985 57 62

[11] Schmidt, Klaus Asymptotically invariant sequences and an action of 𝑆𝐿(2,𝑍) on the 2-sphere Israel J. Math. 1980 193 208

[12] Dye, H. A. On groups of measure preserving transformations. II Amer. J. Math. 1963 551 576

[13] Feldman, Jacob, Moore, Calvin C. Ergodic equivalence relations, cohomology, and von Neumann algebras. I Trans. Amer. Math. Soc. 1977 289 324

[14] Furman, Alex Orbit equivalence rigidity Ann. of Math. (2) 1999 1083 1108

[15] Furstenberg, Harry Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions J. Analyse Math. 1977 204 256

[16] Gaboriau, Damien Coût des relations d’équivalence et des groupes Invent. Math. 2000 41 98

[17] Gaboriau, Damien Invariants 𝑙² de relations d’équivalence et de groupes Publ. Math. Inst. Hautes Études Sci. 2002 93 150

[18] Gaboriau, Damien, Popa, Sorin An uncountable family of nonorbit equivalent actions of 𝔽_{𝕟} J. Amer. Math. Soc. 2005 547 559

[19] Haagerup, Uffe An example of a nonnuclear 𝐶*-algebra, which has the metric approximation property Invent. Math. 1978/79 279 293

[20] Hjorth, Greg A converse to Dye’s theorem Trans. Amer. Math. Soc. 2005 3083 3103

[21] Hjorth, Greg, Kechris, Alexander S. Rigidity theorems for actions of product groups and countable Borel equivalence relations Mem. Amer. Math. Soc. 2005

[22] Každan, D. A. On the connection of the dual space of a group with the structure of its closed subgroups Funkcional. Anal. i Priložen. 1967 71 74

[23] Mcduff, Dusa Central sequences and the hyperfinite factor Proc. London Math. Soc. (3) 1970 443 461

[24] Monod, Nicolas Superrigidity for irreducible lattices and geometric splitting J. Amer. Math. Soc. 2006 781 814

[25] Monod, Nicolas, Shalom, Yehuda Orbit equivalence rigidity and bounded cohomology Ann. of Math. (2) 2006 825 878

[26] Murray, F. J., Von Neumann, J. On rings of operators Ann. of Math. (2) 1936 116 229

[27] Murray, F. J., Von Neumann, J. On rings of operators. IV Ann. of Math. (2) 1943 716 808

[28] Ornstein, Donald S., Weiss, Benjamin Ergodic theory of amenable group actions. I. The Rohlin lemma Bull. Amer. Math. Soc. (N.S.) 1980 161 164

[29] Ozawa, Narutaka Solid von Neumann algebras Acta Math. 2004 111 117

[30] Ozawa, Narutaka A Kurosh-type theorem for type 𝐼𝐼₁ factors Int. Math. Res. Not. 2006

[31] Ozawa, Narutaka, Popa, Sorin Some prime factorization results for type 𝐼𝐼₁ factors Invent. Math. 2004 223 234

[32] Popa, Sorin Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions J. Inst. Math. Jussieu 2006 309 332

[33] Popa, Sorin Some rigidity results for non-commutative Bernoulli shifts J. Funct. Anal. 2006 273 328

[34] Popa, Sorin Strong rigidity of 𝐼𝐼₁ factors arising from malleable actions of 𝑤-rigid groups. I Invent. Math. 2006 369 408

[35] Popa, Sorin Strong rigidity of 𝐼𝐼₁ factors arising from malleable actions of 𝑤-rigid groups. II Invent. Math. 2006 409 451

[36] Popa, Sorin On a class of type 𝐼𝐼₁ factors with Betti numbers invariants Ann. of Math. (2) 2006 809 899

[37] Popa, Sorin, Sasyk, Roman On the cohomology of Bernoulli actions Ergodic Theory Dynam. Systems 2007 241 251

[38] Powers, Robert T., Størmer, Erling Free states of the canonical anticommutation relations Comm. Math. Phys. 1970 1 33

[39] Schmidt, Klaus Asymptotically invariant sequences and an action of 𝑆𝐿(2,𝑍) on the 2-sphere Israel J. Math. 1980 193 208

[40] Shalom, Yehuda Measurable group theory 2005 391 423

[41] Singer, I. M. Automorphisms of finite factors Amer. J. Math. 1955 117 133

[42] Zimmer, Robert J. Strong rigidity for ergodic actions of semisimple Lie groups Ann. of Math. (2) 1980 511 529

[43] Zimmer, Robert J. Ergodic theory and semisimple groups 1984

[44] Zimmer, Robert J. Extensions of ergodic group actions Illinois J. Math. 1976 373 409