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) Oberwolfach, August 2008. Stefaan Vaes

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Plan of the talk

◮ Introduction to fundamental groups. ◮ The first examples of II1 factors having uncountable

fundamental group different from R+.

◮ Given a countable group Γ, what are the possible fundamental

groups

• of II1 factors given as L∞(X, µ) ⋊ Γ ?
• of II1 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 II1 equivalence relation with property (T) but

nevertheless R+ as a fundamental group.

◮ An example of a II1 factor M with F(M) = R+, but no trace

scaling action of R+ on M ⊗ B(ℓ2).

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II1 factors and equivalence relations

Definition A factor of type II1 is a factor with finite trace and non-isomorphic with Mn(C). Definition A type II1 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 II1 factors and equivalence relations arising from group actions.

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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.

Factor of type II1 given as L∞(X) ⋊ Γ. Orbit equivalence relation given by x ∼ y iff Γ · x = Γ · y. Examples of free ergodic p.m.p. actions

◮ Z ↷ T by irrational rotation,

SL(n, Z) ↷ Rn/Zn.

◮ 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.

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Group measure space construction of Murray/von Neumann

Let Γ ↷ (X, µ) be free, ergodic and probability measure preserving. The II1 factor L∞(X) ⋊ Γ (and its trace τ) is generated by

• a copy of L∞(X),
• unitary operators (ug)g∈Γ satisfying uguh = ugh,

such that for all F ∈ L∞(X) and g ∈ Γ,

• u∗

g F ug = Fg

where Fg(x) = F(g · x) ,

• τ(F) =
• X F dµ

and τ(Fug) = 0 for all g ≠ e. In fact, Group action Γ ↷ (X, µ)

• rbit equivalence relation

R(Γ ↷ X) II1 factor L∞(X) ⋊ Γ 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.

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Fundamental group

The fundamental group of Murray and von Neumann It is a subgroup of R+,

◮ for a II1 factor M with trace τ given by

F(M) = {τ(p)/τ(q) | pMp ≅ qMq},

◮ for a II1 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’.

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Known results about the fundamental group

◮ (Murray and von Neumann, 1943)

Let M = R, the hyperfinite II1 factor. Then, F(R) = R+.

• Explanation. For all projections p, q, both pRp and qRq are the

unique hyperfinite II1 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 pL(Fn)p ≅ L(F1+k2(n−1)).
• (Dykema, R˘

adulescu) Same for non-integer k, n : interpolated free group factors.

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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 = Fn, 2 ≤ n < ∞ and any free ergodic

p.m.p. Γ ↷ (X, µ), we have F(R(Γ ↷ X)) = {1}.

• Explanation. Gaboriau introduces cost and L2-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.

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Popa’s theory of HT factors

The first examples of II1 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. If θ : L∞(X) ⋊ Γ → p

• L∞(X) ⋊ Γ
• p

is an isomorphism, the Cartan subalgebras θ(L∞(X)) and p L∞(X) are unitarily conjugate.

• Corollary. The II1 factor M = L(Z2 ⋊ SL(2, Z)) = L∞(T2) ⋊ SL(2, Z)

has trivial fundamental group. Popa proves more : uniqueness of HT Cartan subalgebras.

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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 : Z2 ⊂ Z2 ⋊ SL(2, Z) Z2 ⊂ Z2 ⋊ Γ for Γ ⊂ SL(2, Z) non-amenable

◮ Connes-Jones : property (T) of a II1 factor M.

Every M-M-bimodule H admitting a sequence ξn of unit vec- tors with a · ξn − ξn · 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).

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Connes-Størmer Bernoulli actions and fundamental groups

(from a point of view of equivalence relations)

Let Γ be a countable group and (X0, µ0) an atomic prob. space. Let (X, µ) = (X0, µ0)Γ and the equivalence relation R generated by

◮ Bernoulli shift :

x ∼ g · x for all g ∈ Γ, where (g · x)h = xhg,

◮ and also

x ∼ y when xg = yg for all g outside a finite set I ⊂ Γ and

• g∈I

µ0(xg) =

• g∈I

µ0(yg). For every a ∈ X0, let Ua := {x ∈ X | xe = a} and consider Ua → Ub. F(R) contains µ0(a)/µ0(b) for all a, b ∈ X0. Theorem (Popa, 2003) Let Γ = SL(2, Z) ⋉ Z2. Consider R and the associated II1 factor M. Then, F(R) = F(M) and is generated by µ0(a)/µ0(b) for a, b ∈ X0. Prescribed countable fundamental group. These R and M cannot be implemented by a free action.

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Questions that remained open

◮ Can F(M), F(R) be uncountable without being R+ ?

(Of course, staying with separable II1 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+ ? Theorem (Popa - V, 2008). The F(L∞(X) ⋊ F∞) cover a large class of uncountable subgroups of R+. Actually, the same holds for other groups than F∞.

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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 CentrAut Y(Λ).

• We get F(pNp) = mod
• CentrAut Y(Λ)
• .
• If Λ is amenable and Σ any infinite amenable group,

pNp ≅ L∞(Z) ⋊ (Γ ∗∞ ∗ Σ), for some Γ ∗∞ ∗ Σ ↷ Z.

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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 ∈ CentrAut 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.

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II1 factors with uncountable fundamental group

Let Scentr = {H ⊂ R | exp(H) arises as mod(CentrAut Y(Λ)) for some amenable group Λ acting freely and ergodically on (Y, η)} Theorem (Popa - V, 2008) Let Γ, Σ be infinite groups with Σ amenable and H ∈ Scentr. Then, there exist uncountably many von Neumann inequivalent, free ergodic p.m.p. actions Γ ∗∞ ∗ Σ ↷ (X, µ) such that F

• L∞(X) ⋊ (Γ ∗∞ ∗ Σ)
• = exp(H) .

Uncountable groups contained in S S Scentr :

◮ A σ-finite measure ν on the Borel sets of R is called ergodic if

• for all x ∈ R, the translation νx of ν satisfies νx = ν or νx ⊥ ν,
• Hν = {x ∈ R | νx = ν} is dense in R and Hν ↷ (R, ν) is ergodic.

◮ We have Hν ∈ Scentr and Hν can have prescribed Hausdorff

dimension (Aaronson, Nadkarni).

◮ With x = d(x, Z), certain

• x ∈ R
• ∞
• n=1

γn αnx < ∞

• arise as Hν.

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A new invariant

Set Sfactor(G) = {H ⊂ R | exp(H) arises as F(L∞(X) ⋊ G)}.

◮ Previous theorem : Scentr ⊂ Sfactor(Γ ∗∞ ∗ Σ). ◮ General statement : if G is ICC with property (T), Sfactor(G) only

contains countable groups. Theorem (Popa - V, 2008) Let Γ, Σ be infinite groups where Γ is ICC property (T) and Σ is finitely generated. Then, Sfactor(Γ ∗ Σ) = {{0}}. In words : L∞(X) ⋊ (Γ ∗ Σ) has trivial fundamental group for any free ergodic p.m.p. action of Γ ∗ Σ. Idea of the proof : automatic preservation of Cartan follows from work of Ioana-Peterson-Popa. Next, use 0 < β(2)

1 (Γ ∗ Σ) < ∞.

More generally, we may take Γ a finitely generated ICC group that is

◮ either a non-amenable direct product of non-trivial groups, ◮ or H ⊳ Γ with relative property (T) and H non virtually abelian.

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Trace scaling actions of R+ R+ R+

Let M be a II1 factor. Then, F(M) = mod(Aut(M ⊗ B(ℓ2))).

• Definition. A trace scaling action of R+ on a II∞ factor N is a

strongly continuous action of R+ by automorphisms αt of N such that Tr ◦αt = t Tr for all t ∈ R+. Theorem (Popa - V, 2008) Consider Γ ∗ Λ ↷ X × Y as previously and N = L∞(X × Y) ⋊ (Γ ∗ Λ). The II∞ factor N admits a trace scaling action of R+ if and only if the homomorphism mod : CentrAut Y(Λ) → R+ is surjective and splits. As a consequence, there exist II1 factors M such that F(M) = R+, but M ⊗ B(ℓ2) does not admit a trace scaling action of R+. Take for i = 1, 2, Λi ↷ (Yi, ηi) such that

◮ mod(CentrAut Yi(Λi)) ≠ R+, ◮ the product of both subgroups of R+ is R+.

Next, consider Λ1 × Λ2 ↷ Y1 × Y2.

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Property (T) for equivalence relations

Zimmer : definition of property (T) for II1 equivalence relations.

◮ A unitary representation of a II1 equivalence relation R on

(X, µ) is a Borel map c : R → U(H) satisfying c(x, z) = c(x, y)c(y, z) for almost all (x, y, z) ∈ R(2).

◮ An unit invariant vector is a Borel map ξ : X → (H)1 satisfying

ξ(x) = c(x, y)ξ(y) for almost all (x, y) ∈ R.

◮ A sequence of almost invariant unit vectors is a sequence of

Borel maps ξn : X → (H)1 satisfying ξn(x) − c(x, y)ξn(y) → 0 for almost all (x, y) ∈ R.

◮ The II1 equivalence relation R has property (T) if and only if

every representation that admits a sequence of almost invariant unit vectors, actually admits a unit invariant vector.

• Proposition. If Γ ↷ (X, µ) is free ergodic p.m.p., then R(Γ ↷ X) has

property (T) if and only if Γ has property (T).

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II1 equivalence relations with property (T)

Theorem (Popa - V, 2008) Let n ≥ 4. Denote by R the restriction of R(SL(n, Z) ↷ Rn) to a subset of finite measure. Then, R is a II1 equivalence relation with the following properties.

◮ R has property (T) in the sense of Zimmer. ◮ R has fundamental group R+. ◮ Hence, neither R, nor any of its amplifications Rt ≅ R, can be

implemented by a free action of a group. Idea :

• Anantharaman-Delaroche : property (T) for measured groupoids

and stability under Morita equivalence.

• The transformation groupoid of SL(n, Z) ↷ Rn is Morita

equivalent with the transformation groupoid of SL(n − 1, R) ⋉ Rn−1 ↷ SL(n, R)/ SL(n, Z), which has property (T) because the group has.

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