On the structural theory of II 1 factors of negatively curved groups IONUT CHIFAN (joint with THOMAS SINCLAIR) Vanderbilt University Workshop “ II 1 factors: rigidity, symmetry and classification” IHP, Paris, May, 2011 Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Summary ◮ Cocycles, quasi-cocycles, arrays associated with group representations π : Γ → O ( H ) and the “small cancellation” property ◮ Structural results for von Neumann algebras arising from such groups { L Γ, L ∞ ( X ) ⋊ Γ } ; structure of normalizers for certain subalgebras, uniqueness of Cartan subalgebra and applications to W ∗ -superrigidity; some structural results for the orbit equivalence class of such groups ◮ Brief outline of our approach Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Cocycles Let π : Γ → O ( H ) be an orthogonal representation. Definition A cocycle is a map c : Γ → H satisfying the cocycle identity c ( γ 1 γ 2 ) = π ( γ 1 ) c ( γ 2 ) + c ( γ 1 ) , for all γ 1 , γ 2 ∈ Γ. A cocycle is proper if the map γ → � c ( γ ) � is proper, i.e., for all K > 0, { γ ∈ Γ : � c ( γ ) � ≤ K } < ∞ . � Examples: large classes of amalgamated free products, HNN extensions, etc Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Quasi-cocycles Definition Let ( π, H ) be a representation of Γ. A quasi-cocycle is a map q : Γ → H satisfying the cocycle identity up to bounded error, i.e., there exists D ≥ 0 such that � q ( γ 1 γ 2 ) − π γ 1 ( q ( γ 2 )) − q ( γ 1 ) � ≤ D , for all γ 1 , γ 2 ∈ Γ . � Examples: Gromov hyperbolic groups admit proper quasi-cocycles into left regular representation (Mineyev, Monod, Shalom ’04) Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Arrays Definition Let ( π, H ) be a representation of Γ. An array is a map q : Γ → H satisfying the following properties : 1. π γ q ( γ − 1 ) = − q ( γ ) (anti-symmetry); 2. sup λ ∈ Γ � π γ q ( λ ) − q ( γλ ) � < ∞ for all γ ∈ Γ (bounded equivariance); 3. γ → � q ( γ ) � is proper. proper cocycle ⇓ proper quasi-cocycle ⇓ array Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Examples ◮ If Γ is a lattice in Sp ( n , 1), n ≥ 2 then Γ has no proper (even unbounded) cocycle into any representation. (Delorme, Guichardet) ◮ Z 2 ⋊ SL 2 ( Z ) has no proper quasi-cocycle into any representation (Burger - Monod), but has an array into a representation weakly contained in the left regular representation. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Arrays “prevent” commutation: A group Γ that admits an array into the left regular representation does not have non-amenable subgroups Λ < Γ with infinite centralizer C Γ (Λ). Proof. Λ is non-amenable � there exist K > 0 and F ⊂ Λ finite s.t. for all ξ ∈ ℓ 2 (Γ) we have � ξ � ≤ K � s ∈F � π s ( ξ ) − ξ � . For all λ ∈ C Γ (Λ): � � q ( λ ) � ≤ � π s ( q ( λ ) − q ( λ ) � K 1 s ∈F � � q ( s λ ) − q ( λ ) � + K 2 ≤ K 1 1 |F| s ∈F � � q ( λ s ) − q ( λ ) � + K 2 = K 1 1 |F| s ∈F � � − π λ s q ( s − 1 λ − 1 ) + π λ ( q ( λ − 1 )) � + K 2 = K 1 1 |F| s ∈F � − π s − 1 q ( λ − 1 ) − ( q ( s − 1 λ − 1 )) � + K 2 1 |F| ≤ 2 K 2 � 1 |F| = K 1 s ∈F Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Solidity and strong solidity Definition (Ozawa) A II 1 factor M is called solid if for any diffuse A ⊂ M the relative commutant A ′ ∩ M is amenable. � Any solid, non-amenable factor M is prime, i.e., M ≇ M 1 ¯ ⊗ M 2 , for any M 1 , M 2 diffuse factors. Definition (Ozawa-Popa) A II 1 factor M is called strongly solid if for any amenable subalgebra A ⊂ M its normalizing algebra N M ( A ) ′′ is amenable. � Any strongly solid, non-amenable factor M does not have Cartan subalgebra. In particular, it cannot be decomposed as group measure space construction M ∼ = L ∞ ( X ) ⋊ Γ. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
History ◮ Popa (’81) - L F S is prime and has no Cartan subalgebras for any S uncountable ◮ Voiculescu (’96) - L F n , n ≥ 2 have no Cartan subalgebras ◮ Ge (’98) - L F n , n ≥ 2 is prime ◮ Ozawa (’03) - L Γ solid, Γ Gromov hyperbolic ◮ Popa (’06) M solid, M admits“free malleable” deformation ◮ Peterson (’06) L Γ solid, b : Γ → ℓ 2 Γ, proper cocycle ◮ Ozawa - Popa (’07) L F n , n ≥ 2 strongly solid ◮ Ozawa - Popa (’08) L Γ strongly solid, Γ a lattice in SO (2 , 1), SO (3 , 1), or SU (1 , 1) ◮ Ozawa (’08) L ( Z 2 ⋊ SL 2 ( Z )) is solid ◮ Sinclair (’10) L Γ strongly solid, Γ a lattice in SO ( n , 1) or SU ( n , 1) Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
History ◮ Popa (’81) - L F S is prime and has no Cartan subalgebras for any S uncountable ◮ Voiculescu (’96) - L F n , n ≥ 2 have no Cartan subalgebras ◮ Ge (’98) - L F n , n ≥ 2 is prime ◮ Ozawa (’03) - L Γ solid, Γ Gromov hyperbolic ◮ Popa (’06) - M solid, M admits“free malleable” deformation ◮ Peterson (’06) - L Γ solid, b : Γ → ℓ 2 Γ, proper cocycle ◮ Ozawa - Popa (’07) - L F n , n ≥ 2 strongly solid ◮ Ozawa - Popa (’08) - L Γ strongly solid, Γ a lattice in SO (2 , 1), SO (3 , 1), or SU (1 , 1) ◮ Ozawa (’08) - L ( Z 2 ⋊ SL 2 ( Z )) is solid ◮ Sinclair (’10) - L Γ strongly solid, Γ a lattice in SO ( n , 1) or SU ( n , 1) Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Main results: solidity and strong solidity Theorem (C - Sinclair, ’11) Let Γ be an icc, exact group that admits an array into the left regular representation. Then L Γ is solid. � It recovers some of the earlier results of Ozawa ’03 and Peterson ’06. Theorem (C - Sinclair, ’11) Let Γ be an icc, weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. Then L Γ is strongly solid. � Examples: all hyperbolic groups (by De Canniere-Haagerup, Cowling-Haagerup, Ozawa) Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Main results: unique prime decomposition Theorem (C - Sinclair ’11) Let { Γ i } n i =1 be exact, icc groups that admit an array into the left ⊗ L Γ n ∼ regular representation. If L Γ 1 ¯ ⊗ L Γ 2 ¯ ⊗ · · · ¯ = N 1 ¯ ⊗ N 2 ¯ ⊗ · · · ¯ ⊗ N m then n=m and there exist t 1 · · · t n = 1 such that after a permutation of indices ( L Γ i ) t i ∼ = N i for all 1 ≤ i ≤ n. � This result was proven by Ozawa - Popa ’03 for Γ i hyperbolic or a lattice is a rank one, connected, simple, Lie group, and by Peterson ’06, for Γ i admitting a proper 1-cocycle into the left regular representation. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Theorem (C - Sinclair - Udrea ’11) Let { Γ i } n i =1 be an icc, weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. If A ⊂ L Γ 1 ¯ ⊗ L Γ 2 ¯ ⊗ · · · ¯ ⊗ L Γ n = M is an amenable subalgebra such that A ′ ∩ M is amenable (e.g. A is either a MASA or an irreducible subfactor of M) then N M ( A ) ′′ is amenable. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Main results: unique Cartan subalgebra Theorem (Ozawa - Popa ’07) For any F n � X free, ergodic, p.m.p. weakly compact action L ∞ ( X ) is the unique Cartan subalgebra of L ∞ ( X ) ⋊ F n , up to unitary conjugation. Theorem (C - Sinclair ’11) Let Γ be an weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. For any Γ � X p.m.p. weakly compact action L ∞ ( X ) is the unique Cartan subalgebra of L ∞ ( X ) ⋊ Γ , up to unitary conjugation. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Main results: unique Cartan subalgebra, cont. Theorem (C - Sinclair - Udrea ’11) Let Γ i be an icc, weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. For any Γ 1 × Γ 2 × · · · × Γ n � X p.m.p. weakly compact action L ∞ ( X ) is the unique Cartan subalgebra of L ∞ ( X ) ⋊ (Γ 1 × Γ 2 × · · · × Γ n ) , up to unitary conjugation. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
Main results: W ∗ -superrigidity results Theorem (Ioana, ’08) Let Γ be a property (T) group and Γ � X be a profinite, free, ergodic, p.m.p. action. If Λ � Y is any p.m.p. action that is orbit equivalent to Γ � X then the two actions are virtually conjugate. Theorem (C - Sinclair ’11) Let Γ be an icc, property (T), hyperbolic group (e.g. Γ a lattice in Sp ( n , 1) with n ≥ 2 ). Then any p.m.p. compact action Γ � X is virtually W ∗ -superrigid. � Applying the previous theorems we obtain the same result for actions by products of such groups. Ionut Chifan(Vanderbilt University) II 1 factors of negatively curved groups
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