Fundamental groups of II 1 factors and equivalence relations (joint - PowerPoint PPT Presentation

Fundamental groups of II 1 factors and equivalence relations (joint work with Sorin Popa) Oberwolfach, August 2008. Stefaan Vaes 1/19

Plan of the talk ◮ Introduction to fundamental groups. ◮ The first examples of II 1 factors having uncountable fundamental group different from R + . ◮ Given a countable group Γ , what are the possible fundamental groups • of II 1 factors given as L ∞ ( X , µ) ⋊ Γ ? • of II 1 equivalence relations given by the orbits of Γ ↷ ( X , µ) ? For certain groups Γ , both are always trivial. For other groups Γ , there are a wealth of uncountable fundamental groups. Two related results : ◮ An example of a II 1 equivalence relation with property (T) but nevertheless R + as a fundamental group. ◮ An example of a II 1 factor M with F ( M ) = R + , but no trace scaling action of R + on M ⊗ B (ℓ 2 ) . 2/19

II 1 factors and equivalence relations Definition A factor of type II 1 is a factor with finite trace and non-isomorphic with M n ( C ) . Definition A type II 1 equivalence relation on ( X , µ) is a measurable equivalence relation R on X , with countable equivalence classes and • ergodic : every saturated subset of X has measure 0 or 1, • preserving the probability measure µ : ... Our interest lies in II 1 factors and equivalence relations arising from group actions. 3/19

Group actions (measurable group theory) We are interested in Γ ↷ ( X , µ) • ( X , µ) is a probability space and Γ acts by probability measure preserving (p.m.p.) transformations. • The action is ergodic : if Y ⊂ X is measurable and globally Γ -invariant, then µ( Y ) = 0 or µ( Y ) = 1. • The action is free : almost every x ∈ X has a trivial stabilizer. L ∞ ( X ) ⋊ Γ . Factor of type II 1 given as x ∼ y iff Γ · x = Γ · y . Orbit equivalence relation given by Examples of free ergodic p.m.p. actions SL ( n , Z ) ↷ R n / Z n . ◮ Z ↷ T by irrational rotation, ◮ The Bernoulli action Γ ↷ [ 0 , 1 ] Γ . ◮ If Γ ⊂ K is a dense embedding in a compact group K , consider Γ ↷ ( K , Haar ) by left multiplication. 4/19

Group measure space construction of Murray/von Neumann Let Γ ↷ ( X , µ) be free, ergodic and probability measure preserving. The II 1 factor L ∞ ( X ) ⋊ Γ (and its trace τ ) is generated by • a copy of L ∞ ( X ) , • unitary operators ( u g ) g ∈ Γ satisfying u g u h = u gh , such that for all F ∈ L ∞ ( X ) and g ∈ Γ , • u ∗ g F u g = F g F g ( x ) = F ( g · x ) , where � • τ( F ) = X F d µ τ( Fu g ) = 0 for all g ≠ e . and In fact, Group action orbit equivalence relation II 1 factor L ∞ ( X ) ⋊ Γ Γ ↷ ( X , µ) R ( Γ ↷ X ) Some terminology. Actions Γ ↷ ( X , µ) and Λ ↷ ( Y , η) are called ◮ orbit equivalent if R ( Γ ↷ X ) ≅ R ( Λ ↷ Y ) , ◮ von Neumann equivalent if L ∞ ( X ) ⋊ Γ ≅ L ∞ ( Y ) ⋊ Λ . 5/19

Fundamental group The fundamental group of Murray and von Neumann It is a subgroup of R + , ◮ for a II 1 factor M with trace τ given by F ( M ) = { τ( p )/τ( q ) | pMp ≅ qMq } , ◮ for a II 1 equivalence relation R on ( X , µ) given by F ( R ) = { µ( U )/µ( V ) | R| U ≅ R| V } . Extremely hard to compute in concrete examples. Singer, Feldman/Moore : Γ ↷ ( X , µ) and Λ ↷ ( Y , η) are orbit equivalent iff there exists an isomorphism L ∞ ( X ) ⋊ Γ → L ∞ ( Y ) ⋊ Λ sending L ∞ ( X ) onto L ∞ ( Y ) . Consequence. We have F ( R ( Γ ↷ X )) ⊂ F ( L ∞ ( X )⋊ Γ ) (can be strict). In our results : first determine F ( R ( Γ ↷ X )) and next prove ‘automatic Cartan preservation’. 6/19

Known results about the fundamental group ◮ (Murray and von Neumann, 1943) Let M = R , the hyperfinite II 1 factor. Then, F ( R ) = R + . Explanation. For all projections p , q , both pRp and qRq are the unique hyperfinite II 1 factor. ◮ (Connes, 1980) Whenever Γ is an ICC property (T) group, F ( L ( Γ )) is countable. First ‘restriction’ on F ( M ) , but unexplicit. ◮ (Voiculescu 1989, R˘ adulescu 1991) We have F ( L ( F ∞ )) = R + . • (Voiculescu) If τ( p ) = 1 / k , then p L ( F n ) p ≅ L ( F 1 + k 2 ( n − 1 ) ) . • (Dykema, R˘ adulescu) Same for non-integer k , n : interpolated free group factors. 7/19

Known results about the fundamental group Trivial fundamental group for equivalence relations : ◮ (Gefter, Golodets, 1987) For Γ = SL ( n , Z ) , n ≥ 3 and any free ergodic p.m.p. Γ ↷ ( X , µ) , we have F ( R ( Γ ↷ X )) = { 1 } . Explanation. Orbit equivalence R| U ≅ R| V Zimmer 1-cocycle Γ × X → Γ Zimmer’s cocycle superrigidity theorem. ◮ (Gaboriau, 2001) For Γ = F n , 2 ≤ n < ∞ and any free ergodic p.m.p. Γ ↷ ( X , µ) , we have F ( R ( Γ ↷ X )) = { 1 } . Explanation. Gaboriau introduces cost and L 2 -Betti numbers for equivalence relations. These are scaled by restriction to U . So, the same conclusion holds for every group Γ with 0 < β ( 2 ) n ( Γ ) < ∞ for at least one n . 8/19

Popa’s theory of HT factors The first examples of II 1 factors with trivial fundamental group Remember : in order to prove equality in F ( R ( Γ ↷ X )) ⊂ F ( L ∞ ( X ) ⋊ Γ ) , we need ‘automatic Cartan preservation’. Theorem (Popa, 2001) Let Γ ↷ ( X , µ) be free ergodic p.m.p. Suppose that Γ has the Haagerup property and that Γ ↷ X is rigid. � � θ : L ∞ ( X ) ⋊ Γ → p L ∞ ( X ) ⋊ Γ If p is an isomorphism, the Cartan subalgebras θ( L ∞ ( X )) and p L ∞ ( X ) are unitarily conjugate. Corollary. The II 1 factor M = L ( Z 2 ⋊ SL ( 2 , Z )) = L ∞ ( T 2 ) ⋊ SL ( 2 , Z ) has trivial fundamental group. Popa proves more : uniqueness of HT Cartan subalgebras. 9/19

Rigid actions Definition (Kazhdan, Margulis) The pair Γ ⊂ Λ has the relative property (T) if any unitary rep. of Λ having almost invariant vectors, actually has Γ -invariant vectors. Typical example : Z 2 ⊂ Z 2 ⋊ SL ( 2 , Z ) Z 2 ⊂ Z 2 ⋊ Γ Γ ⊂ SL ( 2 , Z ) non-amenable for ◮ Connes-Jones : property (T) of a II 1 factor M . Every M - M -bimodule H admitting a sequence ξ n of unit vec- tors with � a · ξ n − ξ n · a � → 0 for all a ∈ M , actually has a non-zero vector ξ satisfying a · ξ = ξ · a for all a ∈ M . ◮ Popa : relative property (T) for inclusion A ⊂ M . The free ergodic p.m.p. Γ ↷ ( X , µ) is called rigid, if the inclusion L ∞ ( X ) ⊂ L ∞ ( X ) ⋊ Γ has the relative property (T). Basic example : Γ ↷ ( � H , Haar ) when H ⊂ H ⋊ Γ has relative (T). 10/19

Connes-Størmer Bernoulli actions and fundamental groups (from a point of view of equivalence relations) Let Γ be a countable group and ( X 0 , µ 0 ) an atomic prob. space. Let ( X , µ) = ( X 0 , µ 0 ) Γ and the equivalence relation R generated by ◮ Bernoulli shift : x ∼ g · x for all g ∈ Γ , where ( g · x ) h = x hg , ◮ and also x ∼ y when x g = y g for all g outside a finite set � � I ⊂ Γ and µ 0 ( x g ) = µ 0 ( y g ) . g ∈ I g ∈ I For every a ∈ X 0 , let U a : = { x ∈ X | x e = a } and consider U a → U b . F ( R ) contains µ 0 ( a )/µ 0 ( b ) for all a , b ∈ X 0 . Theorem (Popa, 2003) Let Γ = SL ( 2 , Z ) ⋉ Z 2 . Consider R and the associated II 1 factor M . Then, F ( R ) = F ( M ) and is generated by µ 0 ( a )/µ 0 ( b ) for a , b ∈ X 0 . Prescribed countable fundamental group. These R and M cannot be implemented by a free action. 11/19

Questions that remained open ◮ Can F ( M ), F ( R ) be uncountable without being R + ? (Of course, staying with separable II 1 factors.) ◮ What are the possibilities if M = L ∞ ( X ) ⋊ Γ for Γ ↷ ( X , µ) free ergodic p.m.p. ? Does this force F ( M ) ⊂ Q + ? The F ( L ∞ ( X ) ⋊ F ∞ ) cover a Theorem (Popa - V, 2008). large class of uncountable subgroups of R + . Actually, the same holds for other groups than F ∞ . 12/19

Scheme of the construction ◮ Γ ∗ Λ ↷ ( X , µ) , free, probability measure preserving, with the Γ -action being rigid and ergodic. ◮ Λ ↷ ( Y , η) , infinite measure preserving, free, ergodic. We study the action Γ ∗ Λ ↷ X × Y given by γ · ( x , y ) = (γ · x , y ) if γ ∈ Γ , λ · ( x , y ) = (λ · x , λ · y ) if λ ∈ Λ . II ∞ factor N = L ∞ ( X × Y ) ⋊ ( Γ ∗ Λ ) . Every automorphism of N is Cartan preserving (using results of Ioana-Peterson-Popa). Under the correct assumptions, every automorphism of the II ∞ equivalence relation R ( Γ ∗ Λ ↷ X × Y ) is, modulo inners, given by Centr Aut Y ( Λ ) . � � • We get F ( pNp ) = mod Centr Aut Y ( Λ ) . • If Λ is amenable and Σ any infinite amenable group, pNp ≅ L ∞ ( Z ) ⋊ ( Γ ∗∞ ∗ Σ ) , for some Γ ∗∞ ∗ Σ ↷ Z . 13/19

Adding the correct assumptions Remember : Γ ∗ Λ ↷ X × Y by γ · ( x , y ) = (γ · x , y ) if γ ∈ Γ , λ · ( x , y ) = (λ · x , λ · y ) if λ ∈ Λ . Assumptions. ◮ Γ ↷ X is rigid and ergodic. ◮ Λ is amenable. ◮ Absence of symmetry : every non-singular partial automorphism φ of ( X , µ) sending Γ -orbits into ( Γ ∗ Λ ) -orbits, satisfies φ( x ) ∈ ( Γ ∗ Λ ) · x for almost all x ∈ X . Exists if Γ is itself a free product. To prove : whenever ∆ ∈ Aut ( X × Y ) preserves ( Γ ∗ Λ ) -orbits, we have ∆ ( x , y ) ∈ ( Γ ∗ Λ ) · ( x , ∆ 0 ( y )) for some ∆ 0 ∈ Centr Aut Y ( Λ ) . Crucial step in the proof : rigidity of Γ ↷ X versus amenability of Λ , ensures that ∆ ( x , y ) = (φ( x ), · · · ) on a non-negligible part of X × Y . Then, use the absence of symmetry. 14/19