Cohomology of Measurable Cocycles Alex Furman (University of - - PowerPoint PPT Presentation

Cohomology of Measurable Cocycles

Alex Furman (University of Illinois at Chicago) January 7, 2007

The Category PMP(G) - Prob Meas-Preserving G-actions

G is a fixed lcsc group (e.g. any countable discrete)

◮ Objects: X = (X, B, µ, G), where

(X, B) std Borel, µ(X) = 1, g∗µ = µ (∀g ∈ G)

The Category PMP(G) - Prob Meas-Preserving G-actions

G is a fixed lcsc group (e.g. any countable discrete)

◮ Objects: X = (X, B, µ, G), where

(X, B) std Borel, µ(X) = 1, g∗µ = µ (∀g ∈ G)

◮ Morphisms: X = (X, B, µ, G) p

− → Y = (Y , C, ν, G) p : X → Y , p∗µ = ν, p(g.x) = g.p(x) (∀g ∈ G)

The Category PMP(G) - Prob Meas-Preserving G-actions

G is a fixed lcsc group (e.g. any countable discrete)

◮ Objects: X = (X, B, µ, G), where

(X, B) std Borel, µ(X) = 1, g∗µ = µ (∀g ∈ G)

◮ Morphisms: X = (X, B, µ, G) p

− → Y = (Y , C, ν, G) p : X → Y , p∗µ = ν, p(g.x) = g.p(x) (∀g ∈ G)

◮ Hereafter implicitly: everything is modulo null sets ◮ Will usually assume ergodicity

Measurable Cocycles

Let L be a discrete countable (lcsc, Polish) group A cocycle α : X → L is α : G × X → L measurable α(gh, x) = α(g, h.x)α(h, x) (g, h ∈ G)

Measurable Cocycles

Let L be a discrete countable (lcsc, Polish) group A cocycle α : X → L is α : G × X → L measurable α(gh, x) = α(g, h.x)α(h, x) (g, h ∈ G) Cocycles α, β : X → L are cohomologous, denoted α ∼ β, if β(g, x) = φ(gx) α(g, x) φ(x)−1 (g ∈ G) for some measurable φ : X → L.

Measurable Cocycles

Let L be a discrete countable (lcsc, Polish) group A cocycle α : X → L is α : G × X → L measurable α(gh, x) = α(g, h.x)α(h, x) (g, h ∈ G) Cocycles α, β : X → L are cohomologous, denoted α ∼ β, if β(g, x) = φ(gx) α(g, x) φ(x)−1 (g ∈ G) for some measurable φ : X → L. Cocycles appear

◮ G (X, µ) θ

− →(Y , ν) L with θ(Gx) = Lθ(x) gives α : G × X → L by θ(gx) = α(g, x)θ(x) (ass. L Y ess free).

Measurable Cocycles

Let L be a discrete countable (lcsc, Polish) group A cocycle α : X → L is α : G × X → L measurable α(gh, x) = α(g, h.x)α(h, x) (g, h ∈ G) Cocycles α, β : X → L are cohomologous, denoted α ∼ β, if β(g, x) = φ(gx) α(g, x) φ(x)−1 (g ∈ G) for some measurable φ : X → L. Cocycles appear

◮ G (X, µ) θ

− →(Y , ν) L with θ(Gx) = Lθ(x) gives α : G × X → L by θ(gx) = α(g, x)θ(x) (ass. L Y ess free).

◮ G → Diff(M, vol) gives α : G × M → SLd(R).

The Functor H1(−, L) : PMP(G) → Sets

Z 1(X, L) = {measurable cocycles α : G × X → L} H1(X, L) = Z 1(X, L)/ ∼.

The Functor H1(−, L) : PMP(G) → Sets

Z 1(X, L) = {measurable cocycles α : G × X → L} H1(X, L) = Z 1(X, L)/ ∼.

Problem

Study X → H1(X, L) as a functor PMP(G)− →SETS.

The Functor H1(−, L) : PMP(G) → Sets

Z 1(X, L) = {measurable cocycles α : G × X → L} H1(X, L) = Z 1(X, L)/ ∼.

Problem

Study X → H1(X, L) as a functor PMP(G)− →SETS. H1(−, L) is contravariant: X

p

− →Y defines H1(X, L)

p

← −H1(Y, L)

The Functor H1(−, L) : PMP(G) → Sets

Z 1(X, L) = {measurable cocycles α : G × X → L} H1(X, L) = Z 1(X, L)/ ∼.

Problem

Study X → H1(X, L) as a functor PMP(G)− →SETS. H1(−, L) is contravariant: X

p

− →Y defines H1(X, L)

p

← −H1(Y, L) Indeed, if α ∼ β : Y → L i.e. α(g, y) = φ(gy) β(g, y) φ(y)−1,

The Functor H1(−, L) : PMP(G) → Sets

Z 1(X, L) = {measurable cocycles α : G × X → L} H1(X, L) = Z 1(X, L)/ ∼.

Problem

Study X → H1(X, L) as a functor PMP(G)− →SETS. H1(−, L) is contravariant: X

p

− →Y defines H1(X, L)

p

← −H1(Y, L) Indeed, if α ∼ β : Y → L i.e. α(g, y) = φ(gy) β(g, y) φ(y)−1, then p∗α ∼ p∗β : X → L, because α(g, p(x)) = φ(p(gx)) β(g, p(x)) φ(p(x))−1

Injectivity of the pull-back

Remarks

◮ H1({pt}, L) = Hom(G, L)

Injectivity of the pull-back

Remarks

◮ H1({pt}, L) = Hom(G, L)

α : X → L is cohom to a hom ρ : G → L iff [α] ∈ H1(X, L) is in the image of H1({pt}, L)

Injectivity of the pull-back

Remarks

◮ H1({pt}, L) = Hom(G, L) ◮ H1(X, L) p∗

← −H1(Y, L) need not be injective.

Injectivity of the pull-back

Remarks

◮ H1({pt}, L) = Hom(G, L) ◮ H1(X, L) p∗

← −H1(Y, L) need not be injective. Easy counter examples with compact extensions X = Y × K.

Injectivity of the pull-back

Remarks

◮ H1({pt}, L) = Hom(G, L) ◮ H1(X, L) p∗

← −H1(Y, L) need not be injective.

Lemma

Let X

p

− →Y be a relatively Weakly Mixing morphism, and L be a discrete countable group. Then H1(X, L)

p∗

← −H1(Y, L) is injective.

Injectivity of the pull-back

Remarks

◮ H1({pt}, L) = Hom(G, L) ◮ H1(X, L) p∗

← −H1(Y, L) need not be injective.

Lemma

Let X

p

− →Y be a relatively Weakly Mixing morphism, and L be a discrete countable group. Then H1(X, L)

p∗

← −H1(Y, L) is injective.

Definition (Furstenberg, Zimmer)

X

p

− →Y is rel WM if X ×Y X is ergodic. Equivalently, X− →Y′ p′ − →Y with Y′ p′ − →Y rel cpct only trivially: Y′ = Y.

Push-outs to Pull-backs

Theorem

Let Y

p

← −X

q

− →Z be relatively WM morphisms of G-actions,

Push-outs to Pull-backs

Theorem

Let Y

p

← −X

q

− →Z be relatively WM morphisms of G-actions, and X− →Y ∧ Z be the push-out. X

p

• q
• Y
• Z
• Y ∧ Z

Push-outs to Pull-backs

Theorem

Let Y

p

← −X

q

− →Z be relatively WM morphisms of G-actions, and X− →Y ∧ Z be the push-out. Let L be a discrete countable group. Then H1(X, L) H1(Y, L)

p∗

• H1(Z, L)

q∗

• H1(Y ∧ Z, L)

Push-outs to Pull-backs

Theorem

Let Y

p

← −X

q

− →Z be relatively WM morphisms of G-actions, and X− →Y ∧ Z be the push-out. Let L be a discrete countable group. Then H1(X, L) H1(Y, L)

p∗

• H1(Z, L)

q∗

• H1(Y ∧ Z, L)
• So, if α : X → L descends to Y and to Z, up to meas cohom, then

(up to meas cohom) α descends to Y ∧ Z.

A special Case of the Main Thm

Proposition (Popa, Furstenberg-Weiss)

Let Y and Z be WM G-actions, and X = Y × Z. Suppose α : Y → L, β : Z → L are cohom on X: α(g, y) = f (gy, gz)β(g, z)f (y, z)−1

A special Case of the Main Thm

Proposition (Popa, Furstenberg-Weiss)

Let Y and Z be WM G-actions, and X = Y × Z. Suppose α : Y → L, β : Z → L are cohom on X: α(g, y) = f (gy, gz)β(g, z)f (y, z)−1 Then ∃ γ : G → L and φ : X → L, ψ : Y → L with α(g, y) = φ(gy)γ(g)φ(y)−1 β(g, z) = ψ(gz)γ(g)ψ(z)−1

Main motivation of the Theorem is:

Problem

Define characteristic factors of a given action G associated with a family of cocycles {αi : X → Li}i∈I.

Main motivation of the Theorem is:

Problem

Define characteristic factors of a given action G associated with a family of cocycles {αi : X → Li}i∈I. Potential difficulty: X− →X1− →X2− → . . . − →Y and α : X → L with α(g, x) = φi(g.x)αi(g, pi(x))φi(x)−1 for some φi : X → L and αi : Xi → L, but no descent to Y = lim Xi.

The class of target groups

The results apply as stated to Polish groups which admit a compatible bi-invariant metric. Includes:

◮ all discrete countable groups ◮ all compact metrizable groups ◮ all separable Abelian groups

The class of target groups

The results apply as stated to Polish groups which admit a compatible bi-invariant metric. Includes:

◮ all discrete countable groups ◮ all compact metrizable groups ◮ all separable Abelian groups

The results apply to all semi-simple algebraic groups under the assumption that the cocycles in question are Zariski dense.

The class of target groups

The results apply as stated to Polish groups which admit a compatible bi-invariant metric. Includes:

◮ all discrete countable groups ◮ all compact metrizable groups ◮ all separable Abelian groups

The results apply to all semi-simple algebraic groups under the assumption that the cocycles in question are Zariski dense. More groups, such as Homeo(S1).

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 .

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 Claim: distL(f (x1), f (x2)) is a G-inv function on X ×Y X.

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 Claim: distL(f (x1), f (x2)) is a G-inv function on X ×Y X. Proof: For (x1, x2) ∈ X ×Y X let y = p(xi) and f (gxi) = α(g, y)f (xi)β(g, y)−1 denoting a = α(g, y), b = β(g, y). distL(f (gx1), f (gx2)) = distL(af (x1)b−1, af (x2)b−1) = distL(f (x1), f (x2))

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 Claim: distL(f (x1), f (x2)) is a G-inv function on X ×Y X. = ⇒ distL(f (x1), f (x2)) = d0 by erg of X ×Y X

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 Claim: distL(f (x1), f (x2)) is a G-inv function on X ×Y X. = ⇒ distL(f (x1), f (x2)) = d0 by erg of X ×Y X Claim: d0 = 0

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 Claim: distL(f (x1), f (x2)) is a G-inv function on X ×Y X. = ⇒ distL(f (x1), f (x2)) = d0 by erg of X ×Y X Claim: d0 = 0 Proof: For ν-a.e. y: we have distL(f (x1), f (x2)) = d0 for µy × µy-a.e. x1, x2.

Proof of the Injectivity Lemma

Given: X

p

− →Y is rel WM, α, β ∈ Z 1(Y, L), and f : X → L with α(g, p(x)) = f (gx)β(g, p(x))f (x)−1 Claim: distL(f (x1), f (x2)) is a G-inv function on X ×Y X. = ⇒ distL(f (x1), f (x2)) = d0 by erg of X ×Y X Claim: d0 = 0 = ⇒ f (x) = φ(y) = ⇒ α(g, y) = φ(gy)β(g, y)φ(y)−1.

About the proof of the Main Theorem

Given α : Y → L and β : Z → L and f : X → L with α(g, p(x)) = f (x)β(g, q(x))f (x)−1

About the proof of the Main Theorem

Given α : Y → L and β : Z → L and f : X → L with α(g, p(x)) = f (x)β(g, q(x))f (x)−1 want to split f as f = (f1 ◦ p) · (f2 ◦ q), f (x) = f1(p(x)) · f2(q(x)) for some f1 : Y → L and f2 : Z → L.

About the proof of the Main Theorem

Given α : Y → L and β : Z → L and f : X → L with α(g, p(x)) = f (x)β(g, q(x))f (x)−1 want to split f as f = (f1 ◦ p) · (f2 ◦ q), f (x) = f1(p(x)) · f2(q(x)) for some f1 : Y → L and f2 : Z → L. Work on the following diagram: X ×Z X

π1

• p×p
• X

p

• q
• YY∧ZY

π1

Y

• Z
• Y ∧ Z