RIGIDITY OF GROUP ACTIONS II. Orbit Equivalence in Ergodic Theory - PowerPoint PPT Presentation

RIGIDITY OF GROUP ACTIONS II. Orbit Equivalence in Ergodic Theory Alex Furman (University of Illinois at Chicago) March 1, 2007

Ergodic Theory of II 1 Group Actions II 1 Systems: ◮ Γ – discrete countable group ◮ ( X , B , µ ) – std prob space ∼ = ([0 , 1] , Borel , Lebesgue ) ◮ Γ � ( X , µ ) – ergodic m.p. ( γ ∗ µ = µ, ∀ γ ∈ Γ)

Ergodic Theory of II 1 Group Actions II 1 Systems: ◮ Γ – discrete countable group ◮ ( X , B , µ ) – std prob space ∼ = ([0 , 1] , Borel , Lebesgue ) ◮ Γ � ( X , µ ) – ergodic m.p. ( γ ∗ µ = µ, ∀ γ ∈ Γ) T Quotient Maps: Γ � ( X , µ ) − → Γ � ( Y , ν ) ◮ T : X → Y with T ∗ µ = ν and T ( γ. x ) = γ. T ( x ) a.e. ( γ ∈ Γ)

Ergodic Theory of II 1 Group Actions II 1 Systems: ◮ Γ – discrete countable group ◮ ( X , B , µ ) – std prob space ∼ = ([0 , 1] , Borel , Lebesgue ) ◮ Γ � ( X , µ ) – ergodic m.p. ( γ ∗ µ = µ, ∀ γ ∈ Γ) T Quotient Maps: Γ � ( X , µ ) − → Γ � ( Y , ν ) ◮ T : X → Y with T ∗ µ = ν and T ( γ. x ) = γ. T ( x ) a.e. ( γ ∈ Γ) Standing Convention: ◮ Everything is measurable and considered modulo null sets

Orbit Equivalence Orbit Equivalence: Γ � ( X , µ ) is OE to Λ � ( Y , ν ) if T ( X , µ ) → ( Y , ν ) meas space iso , T (Γ . x ) = Λ . T ( x )

Orbit Equivalence Orbit Equivalence: Γ � ( X , µ ) is OE to Λ � ( Y , ν ) if T ( X , µ ) → ( Y , ν ) meas space iso , T (Γ . x ) = Λ . T ( x ) equivalently, an iso T : R Γ , X ∼ = R Λ , Y of orbit relations . R Γ , X = { ( x , x ′ ) ∈ X × X | Γ . x = Γ . x ′ }

Orbit Equivalence Orbit Equivalence: Γ � ( X , µ ) is OE to Λ � ( Y , ν ) if T ( X , µ ) → ( Y , ν ) meas space iso , T (Γ . x ) = Λ . T ( x ) equivalently, an iso T : R Γ , X ∼ = R Λ , Y of orbit relations . R Γ , X = { ( x , x ′ ) ∈ X × X | Γ . x = Γ . x ′ } Equivalence Relations (Feldman-Moore, 1977) - Axiomatization; types: I n , II 1 , II ∞ , III λ , 0 ≤ λ ≤ 1. - Axiomatization of L ∞ ( X ) ֒ → vN ( R ) ... - Every R = R Γ , X for some countable Γ � ( X , µ )

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE.

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE. Theorem (Zimmer, 1981) Let G 1 , G 2 be simple Lie groups, rk ( G 1 ) ≥ 2 , Γ i < G i lattices, and Γ i � ( X i , µ i ) erg free II 1 actions. Then R Γ 1 , X 1 ∼ = R Γ 2 , X 2 = ⇒ G 1 ≃ G 2 , and iso of induced actions .

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE. Theorem (Zimmer, 1981) Let G 1 , G 2 be simple Lie groups, rk ( G 1 ) ≥ 2 , Γ i < G i lattices, and Γ i � ( X i , µ i ) erg free II 1 actions. Then R Γ 1 , X 1 ∼ = R Γ 2 , X 2 = ⇒ G 1 ≃ G 2 , and iso of induced actions . T Proof: induced actions G i � Y i = ( G i × Γ i X i ) are OE: Y 1 − → Y 2

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE. Theorem (Zimmer, 1981) Let G 1 , G 2 be simple Lie groups, rk ( G 1 ) ≥ 2 , Γ i < G i lattices, and Γ i � ( X i , µ i ) erg free II 1 actions. Then R Γ 1 , X 1 ∼ = R Γ 2 , X 2 = ⇒ G 1 ≃ G 2 , and iso of induced actions . T Proof: induced actions G i � Y i = ( G i × Γ i X i ) are OE: Y 1 − → Y 2 Cocycle α : G 1 × Y 1 → G 2 from T ( g 1 . y 1 ) = α ( g 1 , y 1 ) . T ( y 1 )

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE. Theorem (Zimmer, 1981) Let G 1 , G 2 be simple Lie groups, rk ( G 1 ) ≥ 2 , Γ i < G i lattices, and Γ i � ( X i , µ i ) erg free II 1 actions. Then R Γ 1 , X 1 ∼ = R Γ 2 , X 2 = ⇒ G 1 ≃ G 2 , and iso of induced actions . T Proof: induced actions G i � Y i = ( G i × Γ i X i ) are OE: Y 1 − → Y 2 Cocycle α : G 1 × Y 1 → G 2 from T ( g 1 . y 1 ) = α ( g 1 , y 1 ) . T ( y 1 ) Cocycle Superrigidity: α ∼ ρ : G 1 → G 2 (isomorphism) = G

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE. Theorem (Zimmer, 1981) Let G 1 , G 2 be simple Lie groups, rk ( G 1 ) ≥ 2 , Γ i < G i lattices, and Γ i � ( X i , µ i ) erg free II 1 actions. Then R Γ 1 , X 1 ∼ = R Γ 2 , X 2 = ⇒ G 1 ≃ G 2 , and iso of induced actions . T Proof: induced actions G i � Y i = ( G i × Γ i X i ) are OE: Y 1 − → Y 2 Cocycle α : G 1 × Y 1 → G 2 from T ( g 1 . y 1 ) = α ( g 1 , y 1 ) . T ( y 1 ) Cocycle Superrigidity: α ∼ ρ : G 1 → G 2 (isomorphism) = G Conjugation map gives G � Y 1 ∼ = G � Y 2

First Impressions Theorem (Ornstein-Weiss, 1980) All ergodic II 1 actions of all amenable groups are OE. Theorem (Zimmer, 1981) Let G 1 , G 2 be simple Lie groups, rk ( G 1 ) ≥ 2 , Γ i < G i lattices, and Γ i � ( X i , µ i ) erg free II 1 actions. Then R Γ 1 , X 1 ∼ = R Γ 2 , X 2 = ⇒ G 1 ≃ G 2 , and iso of induced actions . T Proof: induced actions G i � Y i = ( G i × Γ i X i ) are OE: Y 1 − → Y 2 Cocycle α : G 1 × Y 1 → G 2 from T ( g 1 . y 1 ) = α ( g 1 , y 1 ) . T ( y 1 ) Cocycle Superrigidity: α ∼ ρ : G 1 → G 2 (isomorphism) = G Conjugation map gives G � Y 1 ∼ = G � Y 2 Example SL n ( Z ) � T n are pairwise non-OE for n ≥ 3.

Basic Questions ◮ Axiomatization and study of abstract eq. rel. (Feldman-Moore 1977)

Basic Questions ◮ Axiomatization and study of abstract eq. rel. (Feldman-Moore 1977) ◮ Orbit structure of specific actions (Descriptive Set Theory)

Basic Questions ◮ Axiomatization and study of abstract eq. rel. (Feldman-Moore 1977) ◮ Orbit structure of specific actions (Descriptive Set Theory) ◮ R Γ , X properties of Γ and Γ � ( X , µ ) �

Basic Questions ◮ Axiomatization and study of abstract eq. rel. (Feldman-Moore 1977) ◮ Orbit structure of specific actions (Descriptive Set Theory) ◮ R Γ , X properties of Γ and Γ � ( X , µ ) � ◮ Given R how to get R = R Γ , X ?

Basic Questions ◮ Axiomatization and study of abstract eq. rel. (Feldman-Moore 1977) ◮ Orbit structure of specific actions (Descriptive Set Theory) ◮ R Γ , X properties of Γ and Γ � ( X , µ ) � ◮ Given R how to get R = R Γ , X ? ◮ Given Γ how many R Γ , X ?

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L )

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L ) ◮ F ( R ) = { µ ( A ) µ ( B ) : R| A × A ∼ = R| B × B A , B ⊂ X }

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L ) ◮ F ( R ) = { µ ( A ) µ ( B ) : R| A × A ∼ = R| B × B A , B ⊂ X } ◮ Out ( R ) = Aut ( R ) / Inn ( R ) ◮ amenability, (T), Haagerup property, . . .

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L ) ◮ F ( R ) = { µ ( A ) µ ( B ) : R| A × A ∼ = R| B × B A , B ⊂ X } ◮ Out ( R ) = Aut ( R ) / Inn ( R ) ◮ amenability, (T), Haagerup property, . . . ◮ Treeability and anti-treeability (Adams)

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L ) ◮ F ( R ) = { µ ( A ) µ ( B ) : R| A × A ∼ = R| B × B A , B ⊂ X } ◮ Out ( R ) = Aut ( R ) / Inn ( R ) ◮ amenability, (T), Haagerup property, . . . ◮ Treeability and anti-treeability (Adams) ◮ cost( R ) (Levitt, Gaboriau) ◮ β (2) n ( R ) (Gaboriau)

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L ) ◮ F ( R ) = { µ ( A ) µ ( B ) : R| A × A ∼ = R| B × B A , B ⊂ X } ◮ Out ( R ) = Aut ( R ) / Inn ( R ) ◮ amenability, (T), Haagerup property, . . . ◮ Treeability and anti-treeability (Adams) ◮ cost( R ) (Levitt, Gaboriau) ◮ β (2) n ( R ) (Gaboriau) ◮ “Ergodic dimension” . . . (Gaboriau)

II 1 Equivalence Relations Invariants of a II 1 relation R on ( X , µ ): ◮ vN ( R ) and L ∞ ( X ) ֒ → vN ( R ) ◮ Cohomologies H n ( R , T 1 ), H 1 ( R , L ) ◮ F ( R ) = { µ ( A ) µ ( B ) : R| A × A ∼ = R| B × B A , B ⊂ X } ◮ Out ( R ) = Aut ( R ) / Inn ( R ) ◮ amenability, (T), Haagerup property, . . . ◮ Treeability and anti-treeability (Adams) ◮ cost( R ) (Levitt, Gaboriau) ◮ β (2) n ( R ) (Gaboriau) ◮ “Ergodic dimension” . . . (Gaboriau) Weal/Stable iso R 1 ≃ R 2 if R 1 | A × A ∼ = R 2 | B × B compression index: ind( R 1 : R 2 ) := µ ( B ) /µ ( A ).