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On the superrigidity of malleable actions with spectral gap
Author: Sorin Popa
Journal: J. Amer. Math. Soc. 21 (2008), 981-1000
MSC (2000): Primary 46L35; Secondary 37A20, 22D25, 28D15
Published electronically: September 26, 2007
Abstract: 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.
Uri Bader and Yehuda Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math. 163 (2006), no. 2, 415–454. MR 2207022, DOI https://doi.org/10.1007/s00222-005-0469-5
Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette, Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov's a-T-menability. MR 1852148
Erik Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979), no. 1, 17–29. MR 543091, DOI https://doi.org/10.1007/BF01420203
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI https://doi.org/10.2307/1971057
Alain Connes, Sur la classification des facteurs de type ${\rm II}$, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 1, Aii, A13–A15 (French, with English summary). MR 377534
A. Connes, A factor of type ${\rm II}_{1}$ with countable fundamental group, J. Operator Theory 4 (1980), no. 1, 151–153. MR 587372
A. Connes, Classification des facteurs, Operator algebras and applications, Part 2 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 43–109 (French). MR 679497
A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI https://doi.org/10.1017/s014338570000136x
A. Connes and V. Jones, A ${\rm II}_{1}$ factor with two nonconjugate Cartan subalgebras, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 211–212. MR 640947, DOI https://doi.org/10.1090/S0273-0979-1982-14981-3
A. Connes and V. Jones, Property $T$ for von Neumann algebras, Bull. London Math. Soc. 17 (1985), no. 1, 57–62. MR 766450, DOI https://doi.org/10.1112/blms/17.1.57
Klaus Schmidt, Asymptotically invariant sequences and an action of ${\rm SL}(2,\,{\bf Z})$ on the $2$-sphere, Israel J. Math. 37 (1980), no. 3, 193–208. MR 599454, DOI https://doi.org/10.1007/BF02760961
H. A. Dye, On groups of measure preserving transformations. II, Amer. J. Math. 85 (1963), 551–576. MR 158048, DOI https://doi.org/10.2307/2373108
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI https://doi.org/10.1090/S0002-9947-1977-0578656-4
Alex Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999), no. 3, 1083–1108. MR 1740985, DOI https://doi.org/10.2307/121063
[Fu2]Fu2 A. Furman: On Popa's Cocycle Superrigidity Theorem, math.DS/0608364, preprint 2006.
Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 498471, DOI https://doi.org/10.1007/BF02813304
Damien Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math. 139 (2000), no. 1, 41–98 (French, with English summary). MR 1728876, DOI https://doi.org/10.1007/s002229900019
Damien Gaboriau, Invariants $l^2$ de relations d'équivalence et de groupes, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 93–150 (French). MR 1953191, DOI https://doi.org/10.1007/s102400200002
Damien Gaboriau and Sorin Popa, An uncountable family of nonorbit equivalent actions of $\Bbb F_n$, J. Amer. Math. Soc. 18 (2005), no. 3, 547–559. MR 2138136, DOI https://doi.org/10.1090/S0894-0347-05-00480-7
Uffe Haagerup, An example of a nonnuclear $C^{\ast } $-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI https://doi.org/10.1007/BF01410082
Greg Hjorth, A converse to Dye's theorem, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3083–3103. MR 2135736, DOI https://doi.org/10.1090/S0002-9947-04-03672-4
Greg Hjorth and Alexander S. Kechris, Rigidity theorems for actions of product groups and countable Borel equivalence relations, Mem. Amer. Math. Soc. 177 (2005), no. 833, viii+109. MR 2155451, DOI https://doi.org/10.1090/memo/0833
[I1]I1 A. Ioana: A relative version of Connes $\chi (M)$ invariant, Erg. Theory and Dyn. Systems 27 (2007), 1199–1213. [I2]I2 A. Ioana: Rigidity results for wreath product II$_{1}$ factors, math.OA/0606574. [I3]I3 A. Ioana: Existence of uncountable families of orbit inequivalent actions for groups containing $\mathbb {F}_{2}$, preprint 2007. [IPeP]IPeP A. Ioana, J. Peterson, S. Popa: Amalgamated free products of w-rigid factors and calculation of their symmetry groups, math.OA/0505589, to appear in Acta Math.
D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390
Dusa McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443–461. MR 281018, DOI https://doi.org/10.1112/plms/s3-21.3.443
Nicolas Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc. 19 (2006), no. 4, 781–814. MR 2219304, DOI https://doi.org/10.1090/S0894-0347-06-00525-X
Nicolas Monod and Yehuda Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006), no. 3, 825–878. MR 2259246, DOI https://doi.org/10.4007/annals.2006.164.825
F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI https://doi.org/10.2307/1968693
F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716–808. MR 9096, DOI https://doi.org/10.2307/1969107
Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753, DOI https://doi.org/10.1090/S0273-0979-1980-14702-3
Narutaka Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111–117. MR 2079600, DOI https://doi.org/10.1007/BF02441087
Narutaka Ozawa, A Kurosh-type theorem for type $\rm II_1$ factors, Int. Math. Res. Not. , posted on (2006), Art. ID 97560, 21. MR 2211141, DOI https://doi.org/10.1155/IMRN/2006/97560
Narutaka Ozawa and Sorin Popa, Some prime factorization results for type ${\rm II}_1$ factors, Invent. Math. 156 (2004), no. 2, 223–234. MR 2052608, DOI https://doi.org/10.1007/s00222-003-0338-z
[Pe]Pe J. Peterson, $L^{2}$-rigidity in von Neumann algebras, math.OA/0605033. [P1]P1 S. Popa: Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups, math.GR/0512646, to appear in Invent. Math. DOI: 10.1007/s00 222-007-0063-0.
Sorin Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu 5 (2006), no. 2, 309–332. MR 2225044, DOI https://doi.org/10.1017/S1474748006000016
Sorin Popa, Some rigidity results for non-commutative Bernoulli shifts, J. Funct. Anal. 230 (2006), no. 2, 273–328. MR 2186215, DOI https://doi.org/10.1016/j.jfa.2005.06.017
Sorin Popa, Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. I, Invent. Math. 165 (2006), no. 2, 369–408. MR 2231961, DOI https://doi.org/10.1007/s00222-006-0501-4
Sorin Popa, Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. II, Invent. Math. 165 (2006), no. 2, 409–451. MR 2231962, DOI https://doi.org/10.1007/s00222-006-0502-3
[P6]P6 S. Popa: Deformation and rigidity in the study of II$_{1}$ factors, Mini-Course at College de France, Nov. 2004. [P7]P7 S. Popa: On Ozawa's property for free group factors, math.OA/0607561, Intern. Math. Res. Notices, Vol. 2007, rnm036, 10 pages, DOI: 10.1093/imrn/rnm036, June 22, 2007.
Sorin Popa, On a class of type ${\rm II}_1$ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), no. 3, 809–899. MR 2215135, DOI https://doi.org/10.4007/annals.2006.163.809
[P9]P9 S. Popa: Deformation and rigidity for group actions and von Neumann algebras, in "Proceedings of the International Congress of Mathematicians� (Madrid 2006), Volume I, EMS Publishing House, Zurich 2006/2007, pp. 445–479.
Sorin Popa and Roman Sasyk, On the cohomology of Bernoulli actions, Ergodic Theory Dynam. Systems 27 (2007), no. 1, 241–251. MR 2297095, DOI https://doi.org/10.1017/S0143385706000502
[PV]PV S. Popa, S. Vaes: Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, math.OA/0605456, to appear in Advances in Math.
Robert T. Powers and Erling Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 1–33. MR 269230
Yehuda Shalom, Measurable group theory, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 391–423. MR 2185757
I. M. Singer, Automorphisms of finite factors, Amer. J. Math. 77 (1955), 117–133. MR 66567, DOI https://doi.org/10.2307/2372424
[V]V S. Vaes: Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) Séminaire Bourbaki, exposéé 961. Astérisque (to appear).
Robert J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2) 112 (1980), no. 3, 511–529. MR 595205, DOI https://doi.org/10.2307/1971090
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417
Robert J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), no. 3, 373–409. MR 409770
[BSh]BSh U. Bader, Y. Shalom: Factor and normal subgroup theorems for lattices in products of groups, Invent. Math. 163 (2006), 415–454. [CCJJV]CCJJV P. Cherix, M. Cowling, P. Jolissaint, P. Julg, A. Valette: "Groups with Haagerup property�, Birkh$\ddot {\text \textrm {a}}$user Verlag, Basel, Berlin, Boston, 2000. [Ch]Ch E. Christensen: Subalgebras of a finite algebra, Math. Ann. 243 (1979), 17-29. [C1]C1 A. Connes: Classification of injective factors, Ann. of Math. 104 (1976), 73-115. [C2]C2 A. Connes: Sur la classification des facteurs de type II, C. R. Acad. Sci. Paris 281 (1975), 13-15. [C3]C3 A. Connes: A type II$_{1}$ factor with countable fundamental group, J. Operator Theory 4 (1980), 151-153. [C4]C4 A. Connes: Classification des facteurs, Proc. Symp. Pure Math. 38, Amer. Math. Soc. 1982, 43-109. [CFW]CFW A. Connes, J. Feldman, B. Weiss: An amenable equivalence relation is generated by a single transformation, Erg. Theory Dyn. Sys. 1 (1981), 431-450. [CJ1]CJ1 A. Connes, V.F.R. Jones: A II$_{1}$ factor with two non-conjugate Cartan subalgebras, Bull. Amer. Math. Soc. 6 (1982), 211-212. [CJ2]CJ2 A. Connes, V.F.R. Jones: Property (T) for von Neumann algebras, Bull. London Math. Soc. 17 (1985), 57-62. [CW]CW A. Connes, B. Weiss: Property (T) and asymptotically invariant sequences, Israel J. Math. 37 (1980), 209-210. [Dy]Dy H. Dye: On groups of measure preserving transformations, II, Amer. J. Math. 85 (1963), 551–576. [FM]FM J. Feldman, C. C. Moore: Ergodic equivalence relations, cohomology and von Neumann algebras, I, II, Trans. AMS 234 (1977), 289–359. ; [Fu1]Fu1 A. Furman: Orbit equivalence rigidity, Ann. Math. 150 (1999), 1083-1108. [Fu2]Fu2 A. Furman: On Popa's Cocycle Superrigidity Theorem, math.DS/0608364, preprint 2006. [F]F H. Furstenberg: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, J. d'Analyse Math. 31 (1977) 204-256. [G1]G1 D. Gaboriau: Cout des rélations d'équivalence et des groupes, Invent. Math. 139 (2000), 41–98. [G2]G2 D. Gaboriau: Invariants $L^2$ de rélations d'équivalence et de groupes, Publ. Math. IHES, 95 (2002), 93–150. [GP]GP D. Gaboriau, S. Popa: An Uncountable Family of Non Orbit Equivalent Actions of $\mathbb {F}_{n}$, Journal of AMS 18 (2005), 547-559. [Ha]Ha U. Haagerup: An example of a non-nuclear $C^{*}$-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293. [H]H G. Hjorth: A converse to Dye's theorem, Trans AMS 357 (2004), 3083-3103. [HK]HK G. Hjorth, A. Kechris: "Rigidity theorems for actions of product groups and countable Borel equivalence relations�, Memoirs of the Amer. Math. Soc. 177, No. 833, 2005. [I1]I1 A. Ioana: A relative version of Connes $\chi (M)$ invariant, Erg. Theory and Dyn. Systems 27 (2007), 1199–1213. [I2]I2 A. Ioana: Rigidity results for wreath product II$_{1}$ factors, math.OA/0606574. [I3]I3 A. Ioana: Existence of uncountable families of orbit inequivalent actions for groups containing $\mathbb {F}_{2}$, preprint 2007. [IPeP]IPeP A. Ioana, J. Peterson, S. Popa: Amalgamated free products of w-rigid factors and calculation of their symmetry groups, math.OA/0505589, to appear in Acta Math. [K]K D. Kazhdan: Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. and its Appl. 1 (1967), 63-65. [Mc]Mc D. McDuff: Central sequences and the hyperfinite factor, Proc. London Math. Soc. 21 (1970), 443-461. [Mo]Mo N. Monod: Superrigidity for irreducible lattices and geometric splitting, Journal Amer. Math. Soc. 19 (2006) 781-814. [MoSh]MoSh N. Monod, Y. Shalom: Orbit equivalence rigidity and bounded cohomology, Annals of Math. 164 (2006). [MvN1]MvN1 F. Murray, J. von Neumann: On rings of operators, Ann. Math. 37 (1936), 116-229. [MvN2]MvN2 F. Murray, J. von Neumann: Rings of operators IV, Ann. Math. 44 (1943), 716-808. [OW]OW D. Ornstein, B. Weiss: Ergodic theory of amenable group actions I. The Rohlin Lemma, Bull. A.M.S. (1) 2 (1980), 161-164. [O1]O1 N. Ozawa: Solid von Neumann algebras, Acta Math. 192 (2004), 111-117. [O2]O2 N. Ozawa: A Kurosh type theorem for type II$_{1}$ factors, Int. Math. Res. Notices, 2006, math.OA/0401121. [OP]OP N. Ozawa, S. Popa: Some prime factorization results for type II$_{1}$ factors, Invent Math. 156 (2004), 223-234. [Pe]Pe J. Peterson, $L^{2}$-rigidity in von Neumann algebras, math.OA/0605033. [P1]P1 S. Popa: Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups, math.GR/0512646, to appear in Invent. Math. DOI: 10.1007/s00 222-007-0063-0. [P2]P2 S. Popa: Some computations of $1$-cohomology groups and construction of non orbit equivalent actions, Journal of the Inst. of Math. Jussieu 5 (2006), 309-332 (math.OA/0407199). [P3]P3 S. Popa: Some rigidity results for non-commutative Bernoulli shifts, J. Fnal. Analysis 230 (2006), 273-328. [P4]P4 S. Popa: Strong Rigidity of II$_{1}$ Factors Arising from Malleable Actions of $w$-Rigid Groups I, Invent. Math. 165 (2006), 369-408 (math.OA/0305306). [P5]P5 S. Popa: Strong Rigidity of II$_{1}$ Factors Arising from Malleable Actions of $w$-Rigid Groups II, Invent. Math. 165 (2006), 409-452 (math.OA/0407137). [P6]P6 S. Popa: Deformation and rigidity in the study of II$_{1}$ factors, Mini-Course at College de France, Nov. 2004. [P7]P7 S. Popa: On Ozawa's property for free group factors, math.OA/0607561, Intern. Math. Res. Notices, Vol. 2007, rnm036, 10 pages, DOI: 10.1093/imrn/rnm036, June 22, 2007. [P8]P8 S. Popa: On a class of type II$_{1}$ factors with Betti numbers invariants, Ann. Math. 163 (2006), 809-889 (math.OA/0209310). [P9]P9 S. Popa: Deformation and rigidity for group actions and von Neumann algebras, in "Proceedings of the International Congress of Mathematicians� (Madrid 2006), Volume I, EMS Publishing House, Zurich 2006/2007, pp. 445–479. [PS]PS S. Popa, R. Sasyk: On the cohomology of Bernoulli actions, Erg. Theory Dyn. Sys. 26 (2006), 1-11 (math.OA/0310211). [PV]PV S. Popa, S. Vaes: Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups, math.OA/0605456, to appear in Advances in Math. [PoSt]PoSt R. Powers, E. Størmer: Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 1-33. [Sc]Sc K. Schmidt: Asymptotically invariant sequences and an action of $SL(2, \mathbb {Z})$ on the $2$-sphere, Israel. J. Math. 37 (1980), 193-208. [Sh]Sh Y. Shalom: Measurable group theory, In "European Congress of Mathematics� (Stockholm 2004), European Math Soc, Zurich 2005, 391-424. [Si]Si I. M. Singer: Automorphisms of finite factors, Amer. J. Math. 77 (1955), 117–133. [V]V S. Vaes: Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) Séminaire Bourbaki, exposéé 961. Astérisque (to appear). [Z1]Z1 R. Zimmer: Strong rigidity for ergodic actions of seimisimple Lie groups, Ann. of Math. 112 (1980), 511-529. [Z2]Z2 R. Zimmer: "Ergodic Theory and Semisimple Groups�, Birkhauser, Boston, 1984. [Z3]Z3 R. Zimmer: Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409.
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Sorin Popa
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-155505
Email: popa@math.ucla.edu
Keywords: von Neumann algebras, II$_{1}$ factors, Bernoulli actions, spectral gap, orbit equivalence, cocycles
Received by editor(s): October 24, 2006
Additional Notes: Research was supported in part by NSF Grant 0601082.