W -superrigidity for Bernoulli actions and wreath product group von - PDF document

W ∗ -superrigidity for Bernoulli actions and wreath product group von Neumann algebras Lecture notes VNG2011 – Institut Henri Poincar´ e – Paris, May 2011 by Stefaan Vaes (1)(2) These lecture notes are written in a very informal style. They give an account of the author’s lectures at the Institut Henri Poincar´ e in Paris, May 2011 during the program Von Neumann algebras and ergodic theory of group actions at the Centre ´ Emile Borel. It is not excluded that certain statements are not entirely accurate. Please send your ques- tions and comments to stefaan.vaes@wis.kuleuven.be. Lecture 1 – May 10, 2011 1 Wreath product groups and Bernoulli actions All results in these lectures deal with the II 1 factors given by wreath product groups and/or Bernoulli actions. Before stating the main results, we introduce the following notations. If H, Γ are countable groups, we denote by H ≀ Γ := H (Γ) ⋊ Γ the wreath product group. Here • H (Γ) is the direct sum of copies of H index by Γ. So, H (Γ) = { ξ : Γ → H | ξ g = e for all but finitely many g ∈ Γ } . • Then Γ � H (Γ) by automorphisms, shifting the indices: ( g · ξ ) h = ξ g − 1 h . More generally, given Γ � I , an action of a countable group Γ on a countable set I , put H ≀ I Γ := H ( I ) ⋊ Γ. This is called a generalized wreath product group. Recall the Bernoulli action Γ � ( X 0 , µ 0 ) Γ . We call ( X 0 , µ 0 ) the base probability space. We similarly consider the generalized Bernoulli actions Γ � ( X 0 , µ 0 ) I given Γ � I . (1) Partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and K.U.Leuven BOF research grant OT/08/032. (2) Department of Mathematics; K.U.Leuven; Celestijnenlaan 200B; B–3001 Leuven (Belgium). E-mail: stefaan.vaes@wis.kuleuven.be Lecture 1 – page 1

2 Main results The whole lecture series centers around the proof (or outline of the proof) of the following results. Theorem I (Ioana-Popa-V [IPV10]) . For ‘quite some’ Γ � I the groups G := Z n Z ≀ I Γ , n square-free , are W ∗ -superrigid: whenever L G ∼ = LΛ , we must have G ∼ = Λ . In case where n = 2 , 3, any isomorphism θ : L G → LΛ must be group-like: there must exist a group isomorphism δ : G → Λ, a character ω : G → T and a unitary V ∈ LΛ such that θ ( u g ) = ω ( g ) V v δ ( g ) V ∗ for all g ∈ Γ . In this expression we denote by ( u g ) g ∈ Γ the canonical unitaries that generate L G and we denote by ( v s ) s ∈ Λ the unitaries that generate LΛ. A large family of Γ � I for which Theorem I holds arises as follows. Let S be an infinite amenable group and Γ 0 a non-amenable group. Put Γ = Γ 0 ≀ S . Put I = Γ /S and observe that one identifies I = Γ ( S ) 0 . Then, Z is a W ∗ -superrigid group for every square-free integer n . n Z ≀ I Γ We prove in parallel the following striking and strong result by Ioana. Theorem II (Ioana [Io10]) . The Bernoulli action Γ � ( X, µ ) = ( X 0 , µ 0 ) Γ of any icc property (T) group is W ∗ -superrigid: if L ∞ ( X ) ⋊ Γ ∼ = L ∞ ( Y ) ⋊ Λ for any other free action Λ � ( Y, η ) , then Γ � X must be conjugate with Λ � Y . For an overview of earlier (and later) W ∗ -superrigidity theorems, we refer to Ioana’s lectures and to [PV09]. Both conceptually and technically Theorems I and II rely heavily on the proof of the following result by Popa. Historically this was the first theorem where conjugacy of group actions was deduced from the mere isomorphism of group measure space II 1 factors. Theorem III (Popa [Po03, Po04]) . Let Γ � ( X, µ ) be any free ergodic pmp action of any icc property (T) group. Let Λ � ( Y, η ) = ( Y 0 , η 0 ) Λ be the Bernoulli action of an arbitrary icc group. If L ∞ ( X ) ⋊ Γ ∼ = L ∞ ( Y ) ⋊ Λ , then Γ � X must be conjugate with Λ � Y . Lecture 1 – page 2

3 Wreath product groups versus Bernoulli actions If H is an abelian group, L H = L ∞ � H , where the unitaries ( u g ) g ∈ H generating L H correspond to the functions ( ω �→ ω ( g )) g ∈ H generating L ∞ � H . Then also L( H ( I ) ) = L ∞ ( � H I ), so that finally L( H ≀ I Γ) = L ∞ ( � H I ) ⋊ Γ . We work throughout with wreath product group von Neumann algebras rather than Bernoulli crossed products. Formally this means that we only prove results for Bernoulli actions where the base space has a uniform probability measure. But all results are of course true as well for arbitrary base spaces. 4 Why does Thm I look less elegant than Thm II Compared to the very neat Theorem II by Ioana, the formulation of Theorem I looks a bit messy. One might hope to prove a W ∗ -superrigidity theorem for plain wreath product groups H ≀ Γ of, say, property (T) groups. But things turn out to be more subtle. Theorem IV. Let Γ be an icc property (T) group and H a non-trivial abelian group. Assume that L( H ≀ Γ) ∼ = LΛ for some countable group Λ . Then, Λ ∼ = Σ ⋊ Γ for some abelian group Σ on which Γ acts by group automorphisms in such H Γ are conjugate. a way that the pmp actions Γ � � Σ and Γ � � The crucial point that we prove right away, is that there are typically a lot of non-isomorphic H Γ and hence groups Σ ⋊ Γ such that Γ � � Σ is conjugate to the Bernoulli action Γ � � L(Σ ⋊ Γ) ∼ = L( H ≀ Γ). Theorem (Ioana-Popa-V [IPV10]) . If Γ is a torsion free group and H a non-trivial finite abelian group, there exists a torsion free group Λ such that L( H ≀ Γ) ∼ = LΛ . In particular, Λ �∼ = H ≀ Γ . Proof. Step 1: assume Γ = Z . α We construct a torsion free abelian group Σ 0 and an action Z � Σ 0 by group automorphisms such that Z � � Σ 0 is conjugate with the Bernoulli action Z � { 1 , . . . , | H |} Z . α Take Σ 0 = Z [ | H | − 1 ] and define Z � Σ 0 generated by the automorphism of multiplication by | H | . We have L H = ℓ ∞ ( { 1 , . . . , | H |} ) ⊂ L ∞ T = L Z ⊂ LΣ 0 , where we view ℓ ∞ ( { 1 , . . . , | H |} ) ⊂ L ∞ T as the functions that are constant on the intervals that arise by cutting the circle in | H | equal pieces. Lecture 1 – page 3

Exercise. The subalgebras α k (L H ) ⊂ LΣ 0 , k ∈ Z , are independent and generate LΣ 0 . Hence Z � � Σ 0 is the correct Bernoulli action. Step 2: co-induction given Z ≤ Γ. We first recall the general co-induction construction given a pmp action Γ 0 � ( Y, η ) and a larger group Γ 0 ≤ Γ. Choose a section ρ : Γ / Γ 0 → Γ with ρ ( e Γ 0 ) = e . Put ω : Γ × Γ / Γ 0 → Γ 0 : ω ( g, t ) = ρ ( gt ) − 1 gρ ( t ) . Then ω is a 1-cocycle in the sense that ω ( gh, t ) = ω ( g, ht ) ω ( h, t ) . The pmp action Γ � Z = Y Γ / Γ 0 given by ( g − 1 · ξ ) t = ω ( g, t ) − 1 · ξ gt is called the co-induced action. Up to conjugacy it does not depend on the choice of section ρ : Γ / Γ 0 → Γ. Exercises • The action Γ � Z is pmp. • If Γ 0 � ( Y, η ) is a Bernoulli action, then the co-induced action is also Bernoulli with the same base space. • If Γ 0 � ( Y, η ) was the dual of an action Γ 0 � Σ 0 by automorphisms of a countable abelian group Σ 0 , then also the co-induced action is the dual of an action of Γ by automorphisms of Σ (Γ / Γ 0 ) . 0 The previous list of exercises also ends the proof: choose an embedding Z ֒ → Γ and co-induce the Z � Σ 0 given by step 1. Remark. In concrete cases one can vary the embedding Z ֒ → Γ and prove that the resulting groups Σ ⋊ Γ are mutually non-isomorphic. In particular, it is proven in [IPV10] that for all n ≥ 2 there are infinitely many non-isomorphic groups Λ satisfying L( Z 2 Z ≀ PSL( n, Z )) ∼ = LΛ . The following set of exercises provides an abstract characterization of co-induced actions. Exercises • The map π : Z → Y : ξ �→ ξ e Γ 0 is a Γ 0 -equivariant pmp factor map. • The maps ( ξ �→ π ( g − 1 · ξ )) g ∈ Γ / Γ 0 are independent and generate the σ -algebra on Z . • The previous two properties provide an abstract characterization of the co-induced action up to conjugacy. Lecture 1 – page 4