Fundamental groups of II 1 factors and equivalence relations (joint work with Sorin Popa) Journ´ ees Trans-Couesnon Groupes et Alg` ebres d’0p´ erateurs, Caen, 2008 Stefaan Vaes 1/20
Topics of the talk 1 Introduction to • countable equivalence coming from discrete groups and relations, their actions on prob. spaces. • von Neumann algebras, • fundamental group of II 1 factors and equivalence relations, (subgroup of R + , terminology is ‘gift’ of von Neumann). 2 Popa’s rigidity theorems. 3 Uncountable fundamental groups ≠ R + . 2/20
Measurable group theory ◮ Measurable transformations • of the interval [ 0 , 1 ] (or a standard probability space ( X , µ) ), • preserving Lebesgue measure, • invertible, bimeasurable. ◮ Actions of discrete groups by such transformations. Examples ◮ Action of Z on the circle S 1 , given by rotation over angle α . pour z ∈ S 1 , n ∈ Z . n · z = exp ( in α) z ◮ Action of SL ( 2 , Z ) on the torus T 2 = R 2 / Z 2 , � � � � � � y a z b a b y for all y , z ∈ S 1 . · = y c z d c d z 3/20
Ergodic actions All transformations are invertible and prob. measure preserving. Ergodicity: undecomposability as ‘a sum of two’. Formal definition of ergodicity ◮ The transformation T of ( X , µ) is called ergodic if a globally T -invariant measurable subset Y ⊂ X has measure 0 or 1. ◮ The action Γ ↷ ( X , µ) is called ergodic if a globally Γ -invariant measurable subset Y ⊂ X has measure 0 or measure 1. Examples Rotation over angle α is ergodic iff α/ 2 π is irrational. Action SL ( n , Z ) ↷ T n is ergodic. Take compact group K and countable dense subgroup Γ ⊂ K . Then, Γ ↷ K by left translation, is ergodic. 4/20
Three different points of view Standing assumptions : measure preserving, ergodic, free For almost every x ∈ X , the stabilizer Stab x = { e } . A free ergodic measure preserving action Γ ↷ ( X , µ) . 1 The associated orbit equivalence relation on X 2 x ∼ y iff ∃ g ∈ Γ , x = g · y . The associated von Neumann algebra L ∞ ( X ) ⋊ Γ . 3 Three degrees of precision, three types of isomorphisms: 1 conjugacy 2 orbit equivalence 3 von Neumann equivalence. Basic question: how different are these points of view? 5/20
Conjugacy vs. orbit equivalence Free ergodic actions Γ ↷ ( X , µ) and Λ ↷ ( Y , η) are called conjugate, if there exists ◮ a measure preserving bijection ∆ : X → Y , ◮ a group isomorphism δ : Γ → Λ such that ∆ ( g · x ) = δ( g ) · ∆ ( x ) almost everywhere. orbit equivalent, if there exists ◮ a measure preserving bijection ∆ : X → Y such that ∆ ( Γ · x ) = Λ · ∆ ( x ) almost everywhere. Surprising theorem All free ergodic actions of all the following groups are orbit equiv. ◮ Abelian groups (Dye, 1963). ◮ Amenable groups (Ornstein et Weiss, 1980). see next slide. 6/20
Amenable groups Unitary representation of a group Γ : π : Γ → operators on a Hilbert space ◮ π( g ) is a unitary operator for every g ∈ Γ , ◮ π( gh ) = π( g )π( h ) for all g , h ∈ Γ . Regular representation of Γ : λ : Γ → operators on ℓ 2 ( Γ ) : λ g e h = e gh is the standard orthonormal basis of ℓ 2 ( Γ ) . ( e g ) g ∈ Γ where Almost invariant vectors for a unitary representation π : Sequence ξ n of norm one vectors satisfying � π( g )ξ n − ξ n � → 0 for all g ∈ Γ . Example. The regular representation of Z has 1 √ ξ n = 2 n + 1 χ [ − n , n ] as a sequence of almost invariant vectors. 7/20
Amenability vs. property (T) Definition A countable group Γ is said ◮ to be amenable if its regular representation admits a sequence of almost invariant vectors. ◮ to have Kazhdan’s property (T) if every unitary rep. with a sequence of almost invariant vectors, must have a non-zero invariant vector. amenable and property (T) = finite group. Amenable groups : ◮ abelian groups, solvable groups, ◮ stable under extensions/subgroups/direct limits, but there are more. Property (T) groups : ◮ SL ( n , Z ) for n ≥ 3, lattices in higher rank simple Lie groups, ◮ certain random groups ` a la Gromov. 8/20
Popa’s orbit equivalence superrigidity theorem Recall: all free ergodic actions of all amen. groups are orbit equiv. Construction of the Bernoulli action of Γ ( X , µ) = [ 0 , 1 ] Γ Probability space with the product measure, Γ ↷ ( X , µ) ( g · x ) h = x g − 1 h . Action by shifting Theorem (Popa, 2005) Let Γ be a property (T) group (without finite normal subgroups). If the Bernoulli action Γ ↷ X = [ 0 , 1 ] Γ is orbit equivalent with the free ergodic action Λ ↷ ( Y , η) , then Γ ≅ Λ and the actions are conjugate. The orbit equivalence relation entirely remembers the group and the action. 9/20
More about orbit equivalence rigidity A very quick bird’s eye view : ◮ Zimmer’s seminal work (1980) : e.g. the groups SL ( n , Z ) do not admit orbit equivalent actions for different values of n . ◮ Furman’s superrigidity (1999) : e.g. the action SL ( n , Z ) ↷ T n for n odd is orbit equivalence superrigid. ◮ Gaboriau’s cost and ℓ 2 -invariants (1999-2001) : e.g. the free groups F n do not admit orbit equivalent actions for different values of n . ◮ The action SL ( n , Z ) ↷ R n for n ≥ 5 is orbit equivalence superrigid (Popa – V, 2008). ◮ Much more : Monod-Shalom, Kida, Ioana, ... But it’s time to turn to von Neumann algebras. 10/20
Von Neumann algebras The uninteresting examples : M n ( C ) , B ( H ) , L ∞ ( X ) . Weak topology on B ( H ) given by T ֏ � ξ, T η � . Group von Neumann algebra Let Γ be a countable group and g ֏ λ g its regular rep. on ℓ 2 ( Γ ) . ◮ span { λ g | g ∈ Γ } is the group algebra C Γ . ◮ Define L ( Γ ) as the weak closure of C Γ . Definition : a von Neumann algebra is a weakly closed unital ∗ -subalgebra of B ( H ) . Extremely difficult problem : when is L ( Γ ) ≅ L ( Λ ) ? • (Connes, 1975) All L ( Γ ) for Γ amenable and ICC are isomorphic. • (Open problem) Are L ( F n ) ≅ L ( F m ) ? • (Connes’ conjecture) If Γ has property (T) and L ( Γ ) ≅ L ( Λ ) , then Γ ≅ Λ (virtually). 11/20
Factors of type II 1 Factor M : von Neumann algebra indecomposable ‘as a sum of two’. Equivalent condition : the center of M is trivial. Replaces the ergodicity assumption. τ : M → C , τ( 1 ) = 1, τ( xy ) = τ( yx ) . Trace on M : Replaces ‘measure preserving’. Definition A type II 1 factor is a factor that admits a trace τ , but which is different from M n ( C ) . Example : L ( Γ ) always has a trace, given by τ(λ g ) = δ g , e , and is a factor iff Γ has infinite conjugacy classes (ICC). Crucial for us : If Γ ↷ ( X , µ) is free ergodic, we will build a II 1 factor L ∞ ( X ) ⋊ Γ . (The group measure space construction of Murray and von Neumann) 12/20
Group measure space construction (M-vN 1943) Let Γ ↷ ( X , µ) be a free ergodic prob. measure preserving action. The II 1 factor L ∞ ( X ) ⋊ Γ is generated by ◮ the subalgebra L ∞ ( X ) , ◮ the subalgebra L ( Γ ) ∋ λ g , and, for F ∈ L ∞ ( X ) and g ∈ Γ , λ ∗ g F λ g = F g with F g ( x ) = F ( g · x ) . Let Γ ↷ ( X , µ) and Λ ↷ ( Y , η) . Study Orbit equivalence relations vs. von Neumann algebras. Theorem (Singer 1955, Feldman-Moore 1977) An isomorphism L ∞ ( X ) → L ∞ ( Y ) : F ֏ F ◦ ∆ − 1 extends to an isomorphism L ∞ ( X ) ⋊ Γ → L ∞ ( Y ) ⋊ Λ if and only if ∆ is an orbit equivalence (i.e. ∆ ( Γ · x ) = Λ · ∆ ( x ) a.e.). Orbit equivalence is the same as isomorphism L ∞ ( X ) ⋊ Γ ≅ L ∞ ( Y ) ⋊ Λ sending L ∞ ( X ) onto L ∞ ( Y ) . 13/20
Distinguishing between II 1 factors Extremely hard to prove that II 1 factors are non-isomorphic. All L ( Λ ) for Λ amenable ICC and all L ∞ ( X ) ⋊ Γ for Γ amenable and Γ ↷ ( X , µ) free ergodic, are isomorphic. First idea : find some invariants! Murray and von Neumann: ◮ a qualitative invariant : property ( Γ ), ◮ a quantitative invariant : fundamental group F ( M ) : = { τ( p )/τ( q ) | pMp ≅ qMq } . Murray and von Neumann proved in 1943 that L ( S ∞ ) �≅ L ( F 2 ) , and ... that was it, until the 1960’s Nowadays : II 1 factors are unclassifiable in all meanings of the word (and strictly more difficult than countable groups!) 14/20
Popa’s von Neumann rigidity theorem Theorem (Popa, 2005) Let Γ be a property (T) group and Γ ↷ ( X , µ) free ergodic. Let Λ be an ICC group and Λ ↷ ( Y , η) = [ 0 , 1 ] Λ Bernoulli action. If L ∞ ( X ) ⋊ Γ ≅ L ∞ ( Y ) ⋉ Λ , then the groups Γ and Λ are isomorphic and their actions conjugate. First theorem in the literature deducing conjugacy out of isomorphism of von Neumann algebras. Note the asymmetry in the assumptions. No superrigidity theorem exists for the moment. It suffices that Γ has an infinite normal subgroup with the relative property (T), e.g. Γ = SL ( 3 , Z ) × G for an arbitrary countable group G . 15/20
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