RIGIDITY OF GROUP ACTIONS. II. Orbit Equivalence in Ergodic Theory

Transcription

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

2 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)

3 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) Quotient Maps: Γ (X, µ) T Γ (Y, ν) T : X Y with T µ = ν and T (γ.x) = γ.t (x) a.e. (γ Γ)

4 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) Quotient Maps: Γ (X, µ) T 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

5 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)

6 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 = RΛ,Y of orbit relations. R Γ,X = {(x, x ) X X Γ.x = Γ.x }

7 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 = RΛ,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)

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

9 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 Γ1,X 1 = RΓ2,X 2 = G 1 G 2, and iso of induced actions.

10 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 Γ1,X 1 = RΓ2,X 2 = G 1 G 2, and iso of induced actions. Proof: induced actions G i Y i = (G i Γi X i ) are OE: Y 1 Y2 T

11 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 Γ1,X 1 = RΓ2,X 2 = G 1 G 2, and iso of induced actions. Proof: induced actions G i Y i = (G i Γi X i ) are OE: Y 1 Y2 Cocycle α : G 1 Y 1 G 2 from T (g 1.y 1 ) = α(g 1, y 1 ).T (y 1 ) T

12 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 Γ1,X 1 = RΓ2,X 2 = G 1 G 2, and iso of induced actions. Proof: induced actions G i Y i = (G i Γi X i ) are OE: Y 1 Y2 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 T

13 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 Γ1,X 1 = RΓ2,X 2 = G 1 G 2, and iso of induced actions. Proof: induced actions G i Y i = (G i Γi X i ) are OE: Y 1 Y2 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 Y2 T

14 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 Γ1,X 1 = RΓ2,X 2 = G 1 G 2, and iso of induced actions. Proof: induced actions G i Y i = (G i Γi X i ) are OE: Y 1 Y2 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 Y2 Example SL n (Z) T n are pairwise non-oe for n 3. T

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

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

17 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)

18 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?

19 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?

20 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)

21 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 }

22 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,...

23 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)

24 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)

25 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)

26 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 = R2 B B compression index: ind(r 1 : R 2 ) := µ(b)/µ(a).

27 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 = R2 B B compression index: ind(r 1 : R 2 ) := µ(b)/µ(a). (Weak) Morphisms R 1 R 2 in/sur/bi-jective morphisms

28 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X.

29 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg,

30 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, X G/Γ

31 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, X G/Γ Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y

32 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, X G/Γ Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y

33 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, X G/Γ Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y Remarks virtual iso for actions means iso = modulo finite groups and fin index

34 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, X G/Γ Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y Remarks virtual iso for actions means iso = modulo finite groups and fin index = ind(r Γ,X : R Λ,Y ) Q

35 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y Remarks virtual iso for actions means iso = modulo finite groups and fin index = ind(r Γ,X : R Λ,Y ) Q

36 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y Remarks virtual iso for actions means iso = modulo finite groups and fin index = ind(r Γ,X : R Λ,Y ) Q π : X G/Γ π Γ π X π OE to Γ X

37 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y or π : X G/Γ π so that Λ Γ π and Λ Y Γ π X π. Remarks virtual iso for actions means iso = modulo finite groups and fin index = ind(r Γ,X : R Λ,Y ) Q π : X G/Γ π Γ π X π OE to Γ X

38 Rigidity of Orbit Structures of Higher Rank Lattices Theorem (Zimmer 1981) Let Γ < G simple rk(g) 2. Then any R X,Γ remembers Lie(G) and G G Γ X. Theorem (F. 1999) Let Γ < G simple rk(g) 2, Γ (X, µ) erg, Let Λ be any group, Λ (Y, ν) free erg, R Γ,X R Λ,Y Then Γ Λ and Γ X Λ Y or π : X G/Γ π so that Λ Γ π and Λ Y Γ π X π. Remarks virtual iso for actions means iso = modulo finite groups and fin index = ind(r Γ,X : R Λ,Y ) Q {vol(g/γ)/vol(g/γ )} π : X G/Γ π Γ π X π OE to Γ X

39 A question of Feldman and Moore Theorem (Feldman-Moore 1977) For any countable relation R there exists an action of a countable group Γ with R = R Γ,X

40 A question of Feldman and Moore Theorem (Feldman-Moore 1977) For any countable relation R there exists an action of a countable group Γ with R = R Γ,X Question: (Feldman-Moore) Can one always find an essentially free action of some group?

41 A question of Feldman and Moore Theorem (Feldman-Moore 1977) For any countable relation R there exists an action of a countable group Γ with R = R Γ,X Question: (Feldman-Moore) Can one always find an essentially free action of some group? Theorem (F. 1999) There exist II 1 (and II ) relations that cannot be generated by a free action of any group.

42 A question of Feldman and Moore Theorem (Feldman-Moore 1977) For any countable relation R there exists an action of a countable group Γ with R = R Γ,X Question: (Feldman-Moore) Can one always find an essentially free action of some group? Theorem (F. 1999) There exist II 1 (and II ) relations that cannot be generated by a free action of any group. Idea: Cripple a very rigid relation: e.g. R := R SL3 (Z),T 3 A where A T 3 with µ(a) Q

43 Rigidity for Products of Hyperboli-like groups Theorem (Monod-Shalom 2005) Let Γ = n i Γ i (X, µ) free n 2, where Γ i are hyperbolic-like and Γ i (X, µ) erg.

44 Rigidity for Products of Hyperboli-like groups Theorem (Monod-Shalom 2005) Let Γ = n i Γ i (X, µ) free n 2, where Γ i are hyperbolic-like and Γ i (X, µ) erg. Then R X,Γ remembers the number of factors: n.

45 Rigidity for Products of Hyperboli-like groups Theorem (Monod-Shalom 2005) Let Γ = n i Γ i (X, µ) free n 2, where Γ i are hyperbolic-like and Γ i (X, µ) erg. Then R X,Γ remembers the number of factors: n. If Λ (Y, ν) is a free and mildly mixing, and R X,Γ R Y,ν then Γ = Λ and Γ X = Λ Y.

46 Rigidity for Products of Hyperboli-like groups Theorem (Monod-Shalom 2005) Let Γ = n i Γ i (X, µ) free n 2, where Γ i are hyperbolic-like and Γ i (X, µ) erg. Then R X,Γ remembers the number of factors: n. If Λ (Y, ν) is a free and mildly mixing, and R X,Γ R Y,ν then Γ = Λ and Γ X = Λ Y. Theorem (Monod-Shalom 2005, Hjorth-Kechris 2004) Let Γ = Γ 1 Γ 2 acts on (X, µ) with both Γ 1, Γ 2 erg, α : Γ X Γ a non-elementary cocycle into a hyp-like Γ. Then α is cohom to a hom ρ : Γ Γ i Γ for i = 1 or 2.

47 Computations of Out (R) = Aut (R)/Inn (R) Theorem (Gefter ) There exist equivalence relations without outer automorphisms.

48 Computations of Out (R) = Aut (R)/Inn (R) Theorem (Gefter ) There exist equivalence relations without outer automorphisms. Theorem (F. 2005) One can explicitly compute Out (R Γ,X ) for all the standard algebraic actions of higher rank lattices.

49 Computations of Out (R) = Aut (R)/Inn (R) Theorem (Gefter ) There exist equivalence relations without outer automorphisms. Theorem (F. 2005) One can explicitly compute Out (R Γ,X ) for all the standard algebraic actions of higher rank lattices. SL n (Z) T n gives Out (R Γ,X ) = {Id, x x} Γ G/Γ gives Out (R Γ,G/Γ ) = Z/2Z Γ K with K cpct connected, Out (R Γ,K ) K Γ K/K 0 with K cpct connected, Out (R Γ,K ) N K (K 0 )/K 0 SL n (Z) SL n (Z p ) gives Out (R Γ,X ) = SL n (Q p )

50 Computations of Out (R) = Aut (R)/Inn (R) Theorem (Gefter ) There exist equivalence relations without outer automorphisms. Theorem (F. 2005) One can explicitly compute Out (R Γ,X ) for all the standard algebraic actions of higher rank lattices. SL n (Z) T n gives Out (R Γ,X ) = {Id, x x} Γ G/Γ gives Out (R Γ,G/Γ ) = Z/2Z Γ K with K cpct connected, Out (R Γ,K ) K Γ K/K 0 with K cpct connected, Out (R Γ,K ) N K (K 0 )/K 0 SL n (Z) SL n (Z p ) gives Out (R Γ,X ) = SL n (Q p ) Ingredients Rigidity: Out (R Γ,X ) comes from Aut (Γ X ) and quotients X G/Γ uses Ratner s theorem...

51 Cost, L 2 -Betti numbers etc. after D.Gaboriau Definition (From groups to II 1 relations, or groupoids...) cost(r) in analogy to d(γ) = min{n F n Γ} (Levitt)

52 Cost, L 2 -Betti numbers etc. after D.Gaboriau Definition (From groups to II 1 relations, or groupoids...) cost(r) in analogy to d(γ) = min{n F n Γ} β (2) i (R) in analogy to β (2) i (Γ) (Gaboriau) (Levitt)

53 Cost, L 2 -Betti numbers etc. after D.Gaboriau Definition (From groups to II 1 relations, or groupoids...) cost(r) in analogy to d(γ) = min{n F n Γ} β (2) i (R) in analogy to β (2) i (Γ) (Gaboriau) (Levitt) Basic Facts: cost(r A ) = µ(b) µ(a) cost(r B)

54 Cost, L 2 -Betti numbers etc. after D.Gaboriau Definition (From groups to II 1 relations, or groupoids...) cost(r) in analogy to d(γ) = min{n F n Γ} β (2) i (R) in analogy to β (2) i (Γ) (Gaboriau) (Levitt) Basic Facts: cost(r A ) = µ(b) µ(a) cost(r B) β (2) i (R A ) = µ(b) µ(a) β(2) i (R B ) ( i 1)

55 Cost, L 2 -Betti numbers etc. after D.Gaboriau Definition (From groups to II 1 relations, or groupoids...) cost(r) in analogy to d(γ) = min{n F n Γ} β (2) i (R) in analogy to β (2) i (Γ) (Gaboriau) (Levitt) Basic Facts: cost(r A ) = µ(b) µ(a) cost(r B) β (2) i (R A ) = µ(b) µ(a) β(2) i (R B ) ( i 1) β (2) 1 (R) cost(r) 1 (no known examples with <)

56 Cost, L 2 -Betti numbers etc. after D.Gaboriau Definition (From groups to II 1 relations, or groupoids...) cost(r) in analogy to d(γ) = min{n F n Γ} β (2) i (R) in analogy to β (2) i (Γ) (Gaboriau) (Levitt) Basic Facts: cost(r A ) = µ(b) µ(a) cost(r B) β (2) i (R A ) = µ(b) µ(a) β(2) i (R B ) ( i 1) β (2) 1 (R) cost(r) 1 (no known examples with <) Theorem (Gaboriau 1998) For free actions of free groups: cost(r Fn,X ) = n. Corollary F n and F k with n k do not have free OE actions.

57 Cost, L 2 -Betti numbers etc. after D.Gaboriau (cont.) If cost(r Γ,X ) depends only on Γ set =: price(γ).

58 Cost, L 2 -Betti numbers etc. after D.Gaboriau (cont.) If cost(r Γ,X ) depends only on Γ set =: price(γ). Examples: price > 1: for F n, surface groups,... price = 1: for amenable, Γ 1 Γ 2, SL n (Z) (n 3)...

59 Cost, L 2 -Betti numbers etc. after D.Gaboriau (cont.) If cost(r Γ,X ) depends only on Γ set =: price(γ). Examples: price > 1: for F n, surface groups,... price = 1: for amenable, Γ 1 Γ 2, SL n (Z) (n 3)... Theorem (Gaboriau 2002) For Γ (X, µ) free erg: β (2) i (Γ) = β (2) i (R Γ,X )

60 Cost, L 2 -Betti numbers etc. after D.Gaboriau (cont.) If cost(r Γ,X ) depends only on Γ set =: price(γ). Examples: price > 1: for F n, surface groups,... price = 1: for amenable, Γ 1 Γ 2, SL n (Z) (n 3)... Theorem (Gaboriau 2002) For Γ (X, µ) free erg: β (2) i (Γ) = β (2) i (R Γ,X ) Corollary (Gaboriau) For free actions R Γ,X remembers the Euler characteristic χ(γ)

61 Cost, L 2 -Betti numbers etc. after D.Gaboriau (cont.) If cost(r Γ,X ) depends only on Γ set =: price(γ). Examples: price > 1: for F n, surface groups,... price = 1: for amenable, Γ 1 Γ 2, SL n (Z) (n 3)... Theorem (Gaboriau 2002) For Γ (X, µ) free erg: β (2) i (Γ) = β (2) i (R Γ,X ) Corollary (Gaboriau) For free actions R Γ,X remembers the Euler characteristic χ(γ) Theorem (Gaboriau) For f.g. Γ with an infinite normal amenable subgroup β (2) 1 (Γ) = 0

62 After S.Popa... Theorem (S.Popa 2006) Let Γ have (T) and Γ X = (X 0, µ 0 ) Γ be a Bernoulli action. Then for any discrete Λ every cocycle α : Γ X Λ is cohomologous to a homomorphism ρ : Γ Λ.

63 After S.Popa... Theorem (S.Popa 2006) Let Γ have (T) and Γ X = (X 0, µ 0 ) Γ be a Bernoulli action. Then for any discrete Λ every cocycle α : Γ X Λ is cohomologous to a homomorphism ρ : Γ Λ. A gold mine of applications/related results (Popa, Popa-Sasyk, Popa-Vaes,...): Rigidity: R Γ,X determines Γ and Γ X R not generated by free actions Prescribed countable F(R) Prescribed H 1 (R, T) Prescribed Out (R) any Countable Compact Many non-oe actions for many groups von Neumann rigidity! F(factor),...!

64 Many Orbit Structures for non-amenable groups Question Given Γ how big is OrbStr Γ = {R Γ,X Γ (X, µ) free erg}?

65 Many Orbit Structures for non-amenable groups Question Given Γ how big is Conjecture Γ Amen = OrbStr Γ = 2 ℵ 0 OrbStr Γ = {R Γ,X Γ (X, µ) free erg}?

66 Many Orbit Structures for non-amenable groups Question Given Γ how big is Conjecture Γ Amen = OrbStr Γ = 2 ℵ 0 OrbStr Γ = {R Γ,X Γ (X, µ) free erg}? Ornstein-Weiss: Γ Amen = OrbStr Γ = {R hyp.fin }

67 Many Orbit Structures for non-amenable groups Question Given Γ how big is OrbStr Γ = {R Γ,X Γ (X, µ) free erg}? Conjecture Γ Amen = OrbStr Γ = 2 ℵ 0 Ornstein-Weiss: Γ Amen = OrbStr Γ = {R hyp.fin } Connes-Weiss: Γ Amen (T) = OrbStr Γ 2.

68 Many Orbit Structures for non-amenable groups Question Given Γ how big is OrbStr Γ = {R Γ,X Γ (X, µ) free erg}? Conjecture Γ Amen = OrbStr Γ = 2 ℵ 0 Ornstein-Weiss: Γ Amen = OrbStr Γ = {R hyp.fin } Connes-Weiss: Γ Amen (T) = OrbStr Γ 2. Theorem (Hjorth 2005) For Γ (T) the map Acts Γ OrbStr Γ is countable to one.

69 Many Orbit Structures for non-amenable groups Question Given Γ how big is OrbStr Γ = {R Γ,X Γ (X, µ) free erg}? Conjecture Γ Amen = OrbStr Γ = 2 ℵ 0 Ornstein-Weiss: Γ Amen = OrbStr Γ = {R hyp.fin } Connes-Weiss: Γ Amen (T) = OrbStr Γ 2. Theorem (Hjorth 2005) For Γ (T) the map Acts Γ OrbStr Γ is countable to one. Theorem (Gaboriau-Popa 2005) The Conjecture is true for F n

70 Many Orbit Structures for non-amenable groups Question Given Γ how big is OrbStr Γ = {R Γ,X Γ (X, µ) free erg}? Conjecture Γ Amen = OrbStr Γ = 2 ℵ 0 Ornstein-Weiss: Γ Amen = OrbStr Γ = {R hyp.fin } Connes-Weiss: Γ Amen (T) = OrbStr Γ 2. Theorem (Hjorth 2005) For Γ (T) the map Acts Γ OrbStr Γ is countable to one. Theorem (Gaboriau-Popa 2005) The Conjecture is true for F n Theorem (A.Ionna 2007) The Conjecture is true for Γ containing F 2.