RIGIDITY OF GROUP ACTIONS I. Introduction to Super-Rigidity Alex - PowerPoint PPT Presentation

RIGIDITY OF GROUP ACTIONS I. Introduction to Super-Rigidity Alex Furman (University of Illinois at Chicago) February 28, 2007

� � The Super-rigidity Phenomenon For some Γ < G representations ρ : Γ − → H extend to G : ρ � H Γ � ¯ � ρ � � � � � G

� � The Super-rigidity Phenomenon For some Γ < G representations ρ : Γ − → H extend to G : ρ � H Γ � ¯ � ρ � � � � � G provided G is a “higher rank” lcsc group

� � The Super-rigidity Phenomenon For some Γ < G representations ρ : Γ − → H extend to G : ρ � H Γ � ¯ � ρ � � � � � G provided G is a “higher rank” lcsc group Γ < G – an (irreducible) lattice

� � The Super-rigidity Phenomenon For some Γ < G representations ρ : Γ − → H extend to G : ρ � H Γ � ¯ � ρ � � � � � G provided G is a “higher rank” lcsc group Γ < G – an (irreducible) lattice ρ : Γ − → H with ρ (Γ) “non-elemntary” in H .

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ .

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ . Γ < G = � n i =1 G i is irreducible if pr i (Γ) dense in G i .

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ . Γ < G = � n i =1 G i is irreducible if pr i (Γ) dense in G i . Examples (Arithmetic) ◮ Γ = Z n in G = R n

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ . Γ < G = � n i =1 G i is irreducible if pr i (Γ) dense in G i . Examples (Arithmetic) ◮ Γ = Z n in G = R n ◮ Γ = SL n ( Z ) in G = SL n ( R )

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ . Γ < G = � n i =1 G i is irreducible if pr i (Γ) dense in G i . Examples (Arithmetic) ◮ Γ = Z n in G = R n ◮ Γ = SL n ( Z ) in G = SL n ( R ) √ √ √ 2) in G = R 2 with ( a + b ◮ Γ = Z ( 2 , a − b 2)

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ . Γ < G = � n i =1 G i is irreducible if pr i (Γ) dense in G i . Examples (Arithmetic) ◮ Γ = Z n in G = R n ◮ Γ = SL n ( Z ) in G = SL n ( R ) √ √ √ 2) in G = R 2 with ( a + b ◮ Γ = Z ( 2 , a − b 2) ◮ “similar” construction of Γ in G = SL 2 ( R ) × SL 2 ( R )

Lattices Definition Γ < G is a lattice if Γ is discrete and Haar ( G / Γ) < ∞ . Γ < G = � n i =1 G i is irreducible if pr i (Γ) dense in G i . Examples (Arithmetic) ◮ Γ = Z n in G = R n ◮ Γ = SL n ( Z ) in G = SL n ( R ) √ √ √ 2) in G = R 2 with ( a + b ◮ Γ = Z ( 2 , a − b 2) ◮ “similar” construction of Γ in G = SL 2 ( R ) × SL 2 ( R ) Example (Geometric) Γ = π 1 ( M ) for M – loc. symmetric, compact (or vol ( M ) < ∞ ) is a lattice in G = Isom ( � M ).

Margulis’ Higher rank Super-rigidity Theorem (Superrigidity, Margulis 1970s) Assume G = � G i – semi-simple Lie group with rk ( G ) ≥ 2 H – simple and center free Γ < G – an irreducible lattice ρ : Γ − → H with ρ (Γ) Zariski dense in H.

Margulis’ Higher rank Super-rigidity Theorem (Superrigidity, Margulis 1970s) Assume G = � G i – semi-simple Lie group with rk ( G ) ≥ 2 H – simple and center free Γ < G – an irreducible lattice ρ : Γ − → H with ρ (Γ) Zariski dense in H. Then ◮ either ρ (Γ) precompact in H ¯ ρ ◮ or ρ : Γ − → H extends to G − → H.

Margulis’ Higher rank Super-rigidity Theorem (Superrigidity, Margulis 1970s) Assume G = � G i – semi-simple Lie group with rk ( G ) ≥ 2 H – simple and center free Γ < G – an irreducible lattice ρ : Γ − → H with ρ (Γ) Zariski dense in H. Then ◮ either ρ (Γ) precompact in H ¯ ρ ◮ or ρ : Γ − → H extends to G − → H. Theorem (Arithmeticity, Margulis 1970s) In higher rank all irreducible lattices are arithmetic !

Measurable Cocycles G , H – lcsc groups, G � ( X , µ ) – prob. m.p. action

Measurable Cocycles G , H – lcsc groups, G � ( X , µ ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀ g 1 , g 2 ∈ G : c ( g 1 g 2 , x ) = c ( g 1 , g 2 . x ) · c ( g 2 , x )

Measurable Cocycles G , H – lcsc groups, G � ( X , µ ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀ g 1 , g 2 ∈ G : c ( g 1 g 2 , x ) = c ( g 1 , g 2 . x ) · c ( g 2 , x ) Cohomologous cocycles: c ∼ c ′ if ∃ f : X → H s.t. c ′ ( g , x ) = f ( g . x ) c ( g , x ) f ( x ) − 1

Measurable Cocycles G , H – lcsc groups, G � ( X , µ ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀ g 1 , g 2 ∈ G : c ( g 1 g 2 , x ) = c ( g 1 , g 2 . x ) · c ( g 2 , x ) Cohomologous cocycles: c ∼ c ′ if ∃ f : X → H s.t. c ′ ( g , x ) = f ( g . x ) c ( g , x ) f ( x ) − 1 Examples ◮ c ( g , x ) = ρ ( g ) for some hom ρ : G → H .

Measurable Cocycles G , H – lcsc groups, G � ( X , µ ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀ g 1 , g 2 ∈ G : c ( g 1 g 2 , x ) = c ( g 1 , g 2 . x ) · c ( g 2 , x ) Cohomologous cocycles: c ∼ c ′ if ∃ f : X → H s.t. c ′ ( g , x ) = f ( g . x ) c ( g , x ) f ( x ) − 1 Examples ◮ c ( g , x ) = ρ ( g ) for some hom ρ : G → H . ◮ σ : G × G / Γ − → Γ – the “canonical” cocycle

Measurable Cocycles G , H – lcsc groups, G � ( X , µ ) – prob. m.p. action Cocycles: measurable maps c : G × X → H s.t. ∀ g 1 , g 2 ∈ G : c ( g 1 g 2 , x ) = c ( g 1 , g 2 . x ) · c ( g 2 , x ) Cohomologous cocycles: c ∼ c ′ if ∃ f : X → H s.t. c ′ ( g , x ) = f ( g . x ) c ( g , x ) f ( x ) − 1 Examples ◮ c ( g , x ) = ρ ( g ) for some hom ρ : G → H . ◮ σ : G × G / Γ − → Γ – the “canonical” cocycle Observation ∼ { ρ : Γ − → H } / conj { c : G × G / Γ → H } / ∼ . =

Zimmer’s Cocycle Super-rigidity Theorem (Cocycle Super-rigidty, Zimmer 1981) Let G = � G i be a semi-simple Lie group with rk ( G ) ≥ 2 . G � ( X , µ ) a prob. m.p. action with each G i ergodic.

Zimmer’s Cocycle Super-rigidity Theorem (Cocycle Super-rigidty, Zimmer 1981) Let G = � G i be a semi-simple Lie group with rk ( G ) ≥ 2 . G � ( X , µ ) a prob. m.p. action with each G i ergodic. H – simple center free, c : G × X → H Zariski dense cocycle.

Zimmer’s Cocycle Super-rigidity Theorem (Cocycle Super-rigidty, Zimmer 1981) Let G = � G i be a semi-simple Lie group with rk ( G ) ≥ 2 . G � ( X , µ ) a prob. m.p. action with each G i ergodic. H – simple center free, c : G × X → H Zariski dense cocycle. Then ◮ either c ∼ c 0 : G × X → K with K – compact subgrp in H ◮ or c ∼ ρ : G → H: c ( g , x ) = f ( g . x ) ρ ( g ) f ( x ) − 1 .

Zimmer’s Cocycle Super-rigidity Theorem (Cocycle Super-rigidty, Zimmer 1981) Let G = � G i be a semi-simple Lie group with rk ( G ) ≥ 2 . G � ( X , µ ) a prob. m.p. action with each G i ergodic. H – simple center free, c : G × X → H Zariski dense cocycle. Then ◮ either c ∼ c 0 : G × X → K with K – compact subgrp in H ◮ or c ∼ ρ : G → H: c ( g , x ) = f ( g . x ) ρ ( g ) f ( x ) − 1 . Remarks ◮ Margulis’ super-rigidity corresponds to X = G / Γ

Zimmer’s Cocycle Super-rigidity Theorem (Cocycle Super-rigidty, Zimmer 1981) Let G = � G i be a semi-simple Lie group with rk ( G ) ≥ 2 . G � ( X , µ ) a prob. m.p. action with each G i ergodic. H – simple center free, c : G × X → H Zariski dense cocycle. Then ◮ either c ∼ c 0 : G × X → K with K – compact subgrp in H ◮ or c ∼ ρ : G → H: c ( g , x ) = f ( g . x ) ρ ( g ) f ( x ) − 1 . Remarks ◮ Margulis’ super-rigidity corresponds to X = G / Γ ◮ Proofs combine Algebraic groups with Ergodic Theory G -boundary ( B , ν ) = ( G / P , Haar ) plays a key role

Cocycles: where from and what for ? ◮ Volume preserving Actions on Manifolds ρ : Γ − → Diff ( M , vol )

Cocycles: where from and what for ? ◮ Volume preserving Actions on Manifolds ρ : Γ − → Diff ( M , vol ) = R d × M where d = dim M Γ � TM ∼ α : Γ × M → GL d ( R ) or α : Γ × M → SL d ( R ). �

Cocycles: where from and what for ? ◮ Volume preserving Actions on Manifolds ρ : Γ − → Diff ( M , vol ) = R d × M where d = dim M Γ � TM ∼ α : Γ × M → GL d ( R ) or α : Γ × M → SL d ( R ). � ◮ Orbit Equivalence in Ergodic Theory Γ � ( X , µ ) and Λ � ( Y , ν ) free erg. actions

Cocycles: where from and what for ? ◮ Volume preserving Actions on Manifolds ρ : Γ − → Diff ( M , vol ) = R d × M where d = dim M Γ � TM ∼ α : Γ × M → GL d ( R ) or α : Γ × M → SL d ( R ). � ◮ Orbit Equivalence in Ergodic Theory Γ � ( X , µ ) and Λ � ( Y , ν ) free erg. actions OE is T : ( X , µ ) ∼ = ( Y , ν ) with T (Γ . x ) = Λ . T ( x )