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

Fundamental groups of II1 factors and equivalence relations

(joint work with Sorin Popa) Journ´ ees Trans-Couesnon Groupes et Alg` ebres d’0p´ erateurs, Caen, 2008 Stefaan Vaes

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Topics of the talk

1 Introduction to

• countable equivalence

relations,

• von Neumann algebras,

    

coming from discrete groups and their actions on prob. spaces.

• fundamental group of II1 factors and equivalence relations,

(subgroup of R+, terminology is ‘gift’ of von Neumann).

2 Popa’s rigidity theorems. 3 Uncountable fundamental groups ≠ R+.

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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 S1, given by rotation over angle α.

n · z = exp(inα)z pour z ∈ S1, n ∈ Z.

◮ Action of SL(2, Z) on the torus T2 = R2/Z2,

• a

b c d

• ·
• y

z

• =
• yazb

yczd

• for all y, z ∈ S1.

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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/2π is irrational. Action SL(n, Z) ↷ Tn is ergodic. Take compact group K and countable dense subgroup Γ ⊂ K. Then, Γ ↷ K by left translation, is ergodic.

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Three different points of view

Standing assumptions : measure preserving, ergodic, free For almost every x ∈ X, the stabilizer Stab x = {e}.

1

A free ergodic measure preserving action Γ ↷ (X, µ).

2

The associated orbit equivalence relation on X x ∼ y iff ∃g ∈ Γ, x = g · y.

3

The associated von Neumann algebra L∞(X) ⋊ Γ. 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?

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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.

• rbit 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.

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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 ℓ2(Γ) : λgeh = egh where (eg)g∈Γ is the standard orthonormal basis of ℓ2(Γ) . Almost invariant vectors for a unitary representation π : Sequence ξn of norm one vectors satisfying π(g)ξn − ξn → 0 for all g ∈ Γ .

• Example. The regular representation of Z has

ξn = 1 √ 2n + 1χ[−n,n] as a sequence of almost invariant vectors.

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Amenability vs. property (T)

Definition A countable group Γ is said

◮ to be amenable if its regular representation admits a sequence

• f 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.

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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 Probability space (X, µ) = [0, 1]Γ with the product measure, Action Γ ↷ (X, µ) by shifting (g · x)h = xg−1h . 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.

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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) ↷ Tn for n odd is orbit equivalence superrigid.

◮ Gaboriau’s cost and ℓ2-invariants (1999-2001) :

e.g. the free groups Fn do not admit orbit equivalent actions for different values of n.

◮ The action SL(n, Z) ↷ Rn 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.

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Von Neumann algebras

The uninteresting examples : Mn(C), B(H), L∞(X). Weak topology on B(H) given by T ֏ ξ, Tη. Group von Neumann algebra Let Γ be a countable group and g ֏ λg its regular rep. on ℓ2(Γ).

◮ span{λg | g ∈ Γ} is the group algebra CΓ. ◮ Define L(Γ) as the weak closure of CΓ.

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(Fn) ≅ L(Fm) ?
• (Connes’ conjecture)

If Γ has property (T) and L(Γ) ≅ L(Λ), then Γ ≅ Λ (virtually).

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Factors of type II1

Factor M : von Neumann algebra indecomposable ‘as a sum of two’. Equivalent condition : the center of M is trivial. Replaces the ergodicity assumption. Trace on M : τ : M → C, τ(1) = 1, τ(xy) = τ(yx). Replaces ‘measure preserving’. Definition A type II1 factor is a factor that admits a trace τ, but which is different from Mn(C). Example : L(Γ) always has a trace, given by τ(λg) = δg,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 II1 factor L∞(X) ⋊ Γ.

(The group measure space construction of Murray and von Neumann)

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Group measure space construction (M-vN 1943)

Let Γ ↷ (X, µ) be a free ergodic prob. measure preserving action. The II1 factor L∞(X) ⋊ Γ is generated by

◮ the subalgebra L∞(X), ◮ the subalgebra L(Γ) ∋ λg,

and, for F ∈ L∞(X) and g ∈ Γ, λ∗

g F λg = Fg with Fg(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).

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Distinguishing between II1 factors

Extremely hard to prove that II1 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(F2), and ... that was it, until the 1960’s Nowadays : II1 factors are unclassifiable in all meanings of the word (and strictly more difficult than countable groups!)

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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.

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Fundamental groups of II1 factors and equivalence relations

◮ For a II1 factor : F(M) = {τ(p)/τ(q) | pMp ≅ qMq}. ◮ For a II1 equiv. rel. : F(R) = {µ(U)/µ(V ) | R|U ≅ R|V }.

(E.g. R = R(Γ ↷ X), the orbit equivalence relation of Γ ↷ (X, µ).)

For R and M given by Γ ↷ (X, µ) free ergodic : F(R) ⊂ F(M).

1 We have F(L(S∞)) = R+. (Murray, von Neumann, 1943) 2 If Γ is ICC property (T), F(L(Γ)) is countable. (Connes, 1980) 3 If n ≥ 3, always F(R(SL(n, Z) ↷ X)) = {1}. (Gefter-Golodets, 1987) 4 Always F(R(Fn ↷ X)) = {1}, for n < ∞. (Gaboriau, 2001) 5 We have F(L∞(T2) ⋊ SL(2, Z)) = {1}. (Popa, 2001) 6 All countable subgr. of R+ arise as F(R), F(M). (Popa, 2003)

These M and R are not given by a free ergodic action.

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Uncountable fundamental groups

Questions :

• Can F(M), F(R) be uncountable without being R+ ?
• Can F(M), F(R) be different from {1} and R+,

when M, R given by Γ ↷ (X, µ) free ergodic ? Consider the set of subgroups of R+ : Scentr :=

• mod
• CentrAut(Y)(Λ)
• Λ amenable, Λ ↷ Y ergodic and

infinite measure preserving

• log(F), F ∈ Scentr can have any Hausdorff dim. in [0, 1].

Theorem (Popa – V, 2008) Let G = Γ ∗∞ ∗ Σ for Γ ≠ 1 and Σ infinite amenable (e.g. G = F∞). For every group F ∈ Scentr, there exists a free ergodic p.m.p. action G ↷ (X, µ) such that F(L∞(X) ⋊ G) = F(R(G ↷ X)) = F.

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Trivial fundamental groups

Let us shift from Γ ∗∞ ∗ Σ to simply Γ ∗ Σ. Theorem (Popa – V, 2008) Let G = Γ ∗ Σ be a free product of finitely generated, infinite groups. Assume

◮ either, Γ an ICC group with property (T), ◮ or, Γ = Γ1 × Γ2 a non-amenable product of ICC groups.

Then, F(M) = {1} for all M = L∞(X) ⋊ G given by a free ergodic p.m.p. action G ↷ (X, µ).

◮ R(G ↷ X) has trivial fundamental group by Gaboriau’s L2-Betti

numbers.

◮ We prove F(L∞(X) ⋊ G) = F(R(G ↷ X)) by tough von

Neumann algebra methods.

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Some speculations

(staying on separable Hilbert spaces)

Conjecture : every F(M) is realizable as F(L∞(X) ⋊ F∞). Open problem : characterize intrinsically which subgroups of R+ are of the form F(M).

◮ Every F(M) is a Borel subset of R+. ◮ Every F(M) is Polishable, i.e. carries a (unique) Polish topology

that generates the Borel σ-algebra inherited from R+. Remember : we realize all groups in Scentr as F(M). Every F ∈ Scentr that we are able to construct, is

• a countable union of compact subsets of R+,
• such that log(F) = T(N), for some type III factor N,
• r, if you prefer, log(F) is the eigenvalue group of a

non-singular flow R ↷ (Y, η). We expect that none of both properties holds for all F(M).

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Thank you very much, Emmanuel and Gilbert ! We really enjoyed it !

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