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 ) p ◮ Morphisms: X = ( X , B , µ, G ) − → 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 ) p ◮ Morphisms: X = ( X , B , µ, G ) − → 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 θ ◮ G � ( X , µ ) − → ( Y , ν ) � L with θ ( Gx ) = L θ ( 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 θ ◮ G � ( X , µ ) − → ( Y , ν ) � L with θ ( Gx ) = L θ ( 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 → SL d ( R ).

H 1 ( − , L ) : PMP( G ) → Sets The Functor Z 1 ( X , L ) = { measurable cocycles α : G × X → L } H 1 ( X , L ) = Z 1 ( X , L ) / ∼ .

H 1 ( − , L ) : PMP( G ) → Sets The Functor Z 1 ( X , L ) = { measurable cocycles α : G × X → L } H 1 ( X , L ) = Z 1 ( X , L ) / ∼ . Problem Study X → H 1 ( X , L ) as a functor PMP( G ) − → SETS.

H 1 ( − , L ) : PMP( G ) → Sets The Functor Z 1 ( X , L ) = { measurable cocycles α : G × X → L } H 1 ( X , L ) = Z 1 ( X , L ) / ∼ . Problem Study X → H 1 ( X , L ) as a functor PMP( G ) − → SETS. p p H 1 ( − , L ) is contravariant : X → Y defines H 1 ( X , L ) − H 1 ( Y , L ) − ←

H 1 ( − , L ) : PMP( G ) → Sets The Functor Z 1 ( X , L ) = { measurable cocycles α : G × X → L } H 1 ( X , L ) = Z 1 ( X , L ) / ∼ . Problem Study X → H 1 ( X , L ) as a functor PMP( G ) − → SETS. p p H 1 ( − , L ) is contravariant : X → Y defines H 1 ( X , L ) − H 1 ( Y , L ) − ← Indeed, if α ∼ β : Y → L i.e. α ( g , y ) = φ ( gy ) β ( g , y ) φ ( y ) − 1 ,

H 1 ( − , L ) : PMP( G ) → Sets The Functor Z 1 ( X , L ) = { measurable cocycles α : G × X → L } H 1 ( X , L ) = Z 1 ( X , L ) / ∼ . Problem Study X → H 1 ( X , L ) as a functor PMP( G ) − → SETS. p p H 1 ( − , L ) is contravariant : X → Y defines H 1 ( X , L ) − H 1 ( 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 ◮ H 1 ( { pt } , L ) = Hom ( G , L )

Injectivity of the pull-back Remarks ◮ H 1 ( { pt } , L ) = Hom ( G , L ) α : X → L is cohom to a hom ρ : G → L iff [ α ] ∈ H 1 ( X , L ) is in the image of H 1 ( { pt } , L )

Injectivity of the pull-back Remarks ◮ H 1 ( { pt } , L ) = Hom ( G , L ) p ∗ ◮ H 1 ( X , L ) − H 1 ( Y , L ) need not be injective. ←

Injectivity of the pull-back Remarks ◮ H 1 ( { pt } , L ) = Hom ( G , L ) p ∗ ◮ H 1 ( X , L ) − H 1 ( Y , L ) need not be injective. ← Easy counter examples with compact extensions X = Y × K .

Injectivity of the pull-back Remarks ◮ H 1 ( { pt } , L ) = Hom ( G , L ) p ∗ ◮ H 1 ( X , L ) − H 1 ( Y , L ) need not be injective. ← Lemma p Let X − → Y be a relatively Weakly Mixing morphism, and L be a p ∗ discrete countable group. Then H 1 ( X , L ) − H 1 ( Y , L ) is injective . ←

Injectivity of the pull-back Remarks ◮ H 1 ( { pt } , L ) = Hom ( G , L ) p ∗ ◮ H 1 ( X , L ) − H 1 ( Y , L ) need not be injective. ← Lemma p Let X − → Y be a relatively Weakly Mixing morphism, and L be a p ∗ discrete countable group. Then H 1 ( X , L ) − H 1 ( Y , L ) is injective . ← Definition (Furstenberg, Zimmer) p X − → Y is rel WM if X × Y X is ergodic. Equivalently, → Y ′ p ′ → Y with Y ′ p ′ → Y rel cpct only trivially: Y ′ = Y . X − − −

Push-outs to Pull-backs Theorem p q Let Y ← − X − → Z be relatively WM morphisms of G-actions,

� � Push-outs to Pull-backs Theorem p q Let Y ← − X − → 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 p q Let Y ← − X − → Z be relatively WM morphisms of G-actions, and X − → Y ∧ Z be the push-out. Let L be a discrete countable group. Then H 1 ( X , L ) � ���������� � p ∗ � q ∗ � � � � � � � � H 1 ( Y , L ) H 1 ( Z , L ) � ���������� � � � � � � � � � � � H 1 ( Y ∧ Z , L )

� � Push-outs to Pull-backs Theorem p q Let Y ← − X − → Z be relatively WM morphisms of G-actions, and X − → Y ∧ Z be the push-out. Let L be a discrete countable group. Then H 1 ( X , L ) � ���������� � p ∗ q ∗ � � � � � � � � � H 1 ( Y , L ) H 1 ( Z , L ) � ���������� � � � � � � � � � � � H 1 ( 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 { α i : X → L i } i ∈ I .

Main motivation of the Theorem is: Problem Define characteristic factors of a given action G associated with a family of cocycles { α i : X → L i } i ∈ I . Potential difficulty: X − → X 1 − → X 2 − → . . . − → Y and α : X → L with α ( g , x ) = φ i ( g . x ) α i ( g , p i ( x )) φ i ( x ) − 1 for some φ i : X → L and α i : X i → L , but no descent to Y = lim X i .

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 ( S 1 ).