On the structural theory of II1 factors ofnegatively curved groups

IONUT CHIFAN(joint with THOMAS SINCLAIR)

Vanderbilt University

Workshop “II1 factors: rigidity, symmetry and classification”IHP, Paris, May, 2011

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Summary

I Cocycles, quasi-cocycles, arrays associated with grouprepresentations π : Γ→ O(H) and the “small cancellation”property

I Structural results for von Neumann algebras arising from suchgroups {LΓ, L∞(X )o Γ}; structure of normalizers for certainsubalgebras, uniqueness of Cartan subalgebra and applicationsto W ∗-superrigidity; some structural results for the orbitequivalence class of such groups

I Brief outline of our approach

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Cocycles

Let π : Γ→ O(H) be an orthogonal representation.

DefinitionA cocycle is a map c : Γ→ H satisfying the cocycle identity

c(γ1γ2) = π(γ1)c(γ2) + c(γ1),

for all γ1, γ2 ∈ Γ.

A cocycle is proper if the map γ → ‖c(γ)‖ is proper, i.e.,for all K > 0,

{γ ∈ Γ : ‖c(γ)‖ ≤ K} <∞.

Examples: large classes of amalgamated free products, HNNextensions, etc

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Quasi-cocycles

DefinitionLet (π,H) be a representation of Γ. A quasi-cocycle is a mapq : Γ→ H satisfying the cocycle identity up to bounded error, i.e.,there exists D ≥ 0 such that

‖q(γ1γ2)− πγ1(q(γ2))− q(γ1)‖ ≤ D,

for all γ1, γ2 ∈ Γ.

Examples: Gromov hyperbolic groups admit properquasi-cocycles into left regular representation (Mineyev, Monod,Shalom ’04)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Arrays

DefinitionLet (π,H) be a representation of Γ. An array is a map q : Γ→ Hsatisfying the following properties :

1. πγq(γ−1) = −q(γ) (anti-symmetry);

2. supλ∈Γ ‖πγq(λ)− q(γλ)‖ <∞ for all γ ∈ Γ (boundedequivariance);

3. γ → ‖q(γ)‖ is proper.

proper cocycle

⇓proper quasi-cocycle

⇓array

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Examples

I If Γ is a lattice in Sp(n, 1), n ≥ 2 then Γ has no proper (evenunbounded) cocycle into any representation. (Delorme,Guichardet)

I Z2 o SL2(Z) has no proper quasi-cocycle into anyrepresentation (Burger - Monod), but has an array into arepresentation weakly contained in the left regularrepresentation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Arrays “prevent” commutation: A group Γ that admits an arrayinto the left regular representation does not have non-amenablesubgroups Λ < Γ with infinite centralizer CΓ(Λ).

Proof.Λ is non-amenable there exist K > 0 and F ⊂ Λ finite s.t. for allξ ∈ `2(Γ) we have ‖ξ‖ ≤ K

∑s∈F ‖πs(ξ)− ξ‖. For all λ ∈ CΓ(Λ):

‖q(λ)‖ ≤ K1

∑s∈F‖πs(q(λ)− q(λ)‖

≤ K1

∑s∈F‖q(sλ)− q(λ)‖+ K 2

1 |F|

= K1

∑s∈F‖q(λs)− q(λ)‖+ K 2

1 |F|

= K1

∑s∈F‖ − πλsq(s−1λ−1) + πλ(q(λ−1))‖+ K 2

1 |F|

= K1

∑s∈F‖ − πs−1q(λ−1)− (q(s−1λ−1))‖+ K 2

1 |F| ≤ 2K 21 |F|

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Solidity and strong solidity

Definition (Ozawa)

A II1 factor M is called solid if for any diffuse A ⊂ M the relativecommutant A′ ∩M is amenable.

Any solid, non-amenable factor M is prime, i.e., M � M1⊗M2,for any M1, M2 diffuse factors.

Definition (Ozawa-Popa)

A II1 factor M is called strongly solid if for any amenablesubalgebra A ⊂ M its normalizing algebra NM(A)′′ is amenable.

Any strongly solid, non-amenable factor M does not haveCartan subalgebra. In particular, it cannot be decomposed asgroup measure space construction M ∼= L∞(X )o Γ.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

History

I Popa (’81) - LFS is prime and has no Cartan subalgebras forany S uncountable

I Voiculescu (’96) - LFn, n ≥ 2 have no Cartan subalgebras

I Ge (’98) - LFn, n ≥ 2 is prime

I Ozawa (’03) - LΓ solid, Γ Gromov hyperbolic

I Popa (’06) M solid, M admits“free malleable” deformation

I Peterson (’06) LΓ solid, b : Γ→ `2Γ, proper cocycle

I Ozawa - Popa (’07) LFn, n ≥ 2 strongly solid

I Ozawa - Popa (’08) LΓ strongly solid, Γ a lattice in SO(2, 1),SO(3, 1), or SU(1, 1)

I Ozawa (’08) L(Z2 o SL2(Z)) is solid

I Sinclair (’10) LΓ strongly solid, Γ a lattice in SO(n, 1) orSU(n, 1)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

History

I Popa (’81) - LFS is prime and has no Cartan subalgebras forany S uncountable

I Voiculescu (’96) - LFn, n ≥ 2 have no Cartan subalgebras

I Ge (’98) - LFn, n ≥ 2 is prime

I Ozawa (’03) - LΓ solid, Γ Gromov hyperbolic

I Popa (’06) - M solid, M admits“free malleable” deformation

I Peterson (’06) - LΓ solid, b : Γ→ `2Γ, proper cocycle

I Ozawa - Popa (’07) - LFn, n ≥ 2 strongly solid

I Ozawa - Popa (’08) - LΓ strongly solid, Γ a lattice inSO(2, 1), SO(3, 1), or SU(1, 1)

I Ozawa (’08) - L(Z2 o SL2(Z)) is solid

I Sinclair (’10) - LΓ strongly solid, Γ a lattice in SO(n, 1) orSU(n, 1)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: solidity and strong solidity

Theorem (C - Sinclair, ’11)

Let Γ be an icc, exact group that admits an array into the leftregular representation. Then LΓ is solid.

It recovers some of the earlier results of Ozawa ’03 andPeterson ’06.

Theorem (C - Sinclair, ’11)

Let Γ be an icc, weakly amenable, exact group that admits aproper quasi-cocycle into the left regular representation. Then LΓis strongly solid.

Examples: all hyperbolic groups (by De Canniere-Haagerup,Cowling-Haagerup, Ozawa)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique prime decomposition

Theorem (C - Sinclair ’11)

Let {Γi}ni=1 be exact, icc groups that admit an array into the leftregular representation. If LΓ1⊗LΓ2⊗ · · · ⊗LΓn

∼= N1⊗N2⊗ · · · ⊗Nm

then n=m and there exist t1 · · · tn = 1 such that after apermutation of indices (LΓi )

ti ∼= Ni for all 1 ≤ i ≤ n.

This result was proven by Ozawa - Popa ’03 for Γi hyperbolic ora lattice is a rank one, connected, simple, Lie group, and byPeterson ’06, for Γi admitting a proper 1-cocycle into the leftregular representation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Theorem (C - Sinclair - Udrea ’11)

Let {Γi}ni=1 be an icc, weakly amenable, exact group that admits aproper quasi-cocycle into the left regular representation. IfA ⊂ LΓ1⊗LΓ2⊗ · · · ⊗LΓn = M is an amenable subalgebra suchthat A′ ∩M is amenable (e.g. A is either a MASA or an irreduciblesubfactor of M) then NM(A)′′ is amenable.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique Cartan subalgebra

Theorem (Ozawa - Popa ’07)

For any Fn y X free, ergodic, p.m.p. weakly compact actionL∞(X ) is the unique Cartan subalgebra of L∞(X )o Fn, up tounitary conjugation.

Theorem (C - Sinclair ’11)

Let Γ be an weakly amenable, exact group that admits a properquasi-cocycle into the left regular representation. For any Γ y Xp.m.p. weakly compact action L∞(X ) is the unique Cartansubalgebra of L∞(X )o Γ, up to unitary conjugation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique Cartan subalgebra, cont.

Theorem (C - Sinclair - Udrea ’11)

Let Γi be an icc, weakly amenable, exact group that admits aproper quasi-cocycle into the left regular representation. For anyΓ1 × Γ2 × · · · × Γn y X p.m.p. weakly compact action L∞(X ) isthe unique Cartan subalgebra of L∞(X )o (Γ1 × Γ2 × · · · × Γn), upto unitary conjugation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: W ∗-superrigidity results

Theorem (Ioana, ’08)

Let Γ be a property (T) group and Γ y X be a profinite, free,ergodic, p.m.p. action. If Λ y Y is any p.m.p. action that is orbitequivalent to Γ y X then the two actions are virtually conjugate.

Theorem (C - Sinclair ’11)

Let Γ be an icc, property (T), hyperbolic group (e.g. Γ a lattice inSp(n, 1) with n ≥ 2). Then any p.m.p. compact action Γ y X isvirtually W ∗-superrigid.

Applying the previous theorems we obtain the same result foractions by products of such groups.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: Structural results for orbit equivalence classof hyperbolic groups

Theorem (C - Sinclair ’11)

Let Γ be a countable discrete group which is orbit equivalent to ahyperbolic group. If Σ < Γ is an infinite, amenable subgroup, thenits normalizer NΓ(Σ) is amenable.

The method we employ is purely von Neumann algebraic. This theorem recovers and extends upon early results of Adams’94 and Monod-Shalom ’04.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique measure-equivalence decompositionresults for product groups

Theorem (C - Sinclair ’11)

Let Γ = Γ1 × · · · × Γn be a product of countable discrete groupswhich admit arrays in the left regular representation and letΛ = Λ1 × · · · × Λm be a product of arbitrary countable discretegroups. If Γ ∼=ME Λ then n = m and after permuting indices thenΓi∼=ME Λi for all 1 ≤ i ≤ n.

This should be compared with earlier results Monod - Shalom’04 and Sako ’09.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Ideas behind the proofs of main results

General strategy: combines the full strength from the followingmethods of approach:

I topological (Ozawa) - C ∗-algebraic techniques (exactness,local reflexivity)

I cohomological (Peterson, Popa, Ozawa - Popa, Sinclair) -proper cocycles into good representations; von Neumannalgebraic techniques (Popa’s deformation/rigidity theory,closable derivations, etc)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: “small action” by normalizers

Theorem (Ozawa - Popa ’07, Ozawa ’10)

Let Γ be an i.c.c. hyperbolic group and let A ⊂ LΓ = M be adiffuse amenable subalgebra. Then the natural action

NM(A)ad(u)y A is weakly compact:

There exists a net of vectors ηn ∈ L2(A⊗A)+ such that

1. ‖(v ⊗ v)ηn − ηn‖ → 0 for all v ∈ U(A);

2. ‖[u ⊗ u, ηn]‖ → 0, for all u ∈ NM(A);

3. 〈(x ⊗ 1)ηn, ηn〉 = τ(x) = 〈(1⊗ x)ηn, ηn〉, for all x ∈ M.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: a family of deformations

I. Exponentiate quasi-cocycle q : Γ→ `2(Γ) to obtain a“deformation” on LΓ: Γ y X gaussian action; H = L2(X )⊗ `2(Γ)For t ∈ R consider υt : Γ→ U(L∞(X )) defined by

υt(γ)(x) = exp(itq(γ)(x))

Vt ∈ B(H) unitaries by letting Vt(ξ ⊗ δγ) = (υt(γ)ξ)⊗ δγ

αt(x) = VtxV ∗t ∈ Aut(B(H))

Let P : H → `2(Γ) is the orthogonal projection.

Note: in general αt moves elements from LΓ ⊂ B(H) outside LΓ;however it moves elements from C ∗r (Γ) into the extended uniformRoe algebra C ∗u (Γ y X ).

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: a family of deformations

Proposition

LΓ ⊂ B(H); for every t > 0 the operator mt = P · αt · P is“compact”; it transforms WOT -convergent sequences of LΓ intoSOT -convergent sequences.

properness, quasi-cocycle

Proposition

The family αt well behaved on C ∗r (Γ) ⊂ LΓ as t → 0

‖(αt(x)− x) · P‖∞ → 0 as t → 0, for all x ∈ C ∗r (Γ).

bounded equivariance is essentially used!

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: deformation/rigidity arguments a la Ozawa-Popa

Let A ⊂ LΓ = M and assume that N = NM(A)′′ is irreducible.From Ozawa -Popa result let

ηn,t = (Vt ⊗ 1)ηn

ζn,t = (P ⊗ 1)ηn,t

ξn,t = (P⊥ ⊗ 1)ηn,t ∈ L20(X )⊗ `2(Γ)

Lemma

limn‖ξn,t‖ ≥

5

12.

Compactness of mt = P · αt · P together with the fact that ηn“converges to the diagonal” is essentially used! traciality of ηn’s isnot necessary in this case.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: deformation/rigidity arguments a la Ozawa-Popa

Define a state Φt on P = B(H) ∩ ρ(Mop)′ by lettingΦt(x) = 1

‖Limnξn,t‖2 〈(x ⊗ 1)ξn,t , ξn,t〉 for every x ∈ P.

LemmaFor every ε > 0 and every finite set K ⊂ C ∗r (Γ) withdist‖·‖2

(y , (N)1) ≤ ε for all y ∈ K one can find tε > 0 and a finiteset LK ,ε ⊂ NM(A) such that

〈((yx − xy)⊗ 1)ξn,t , ξn,t〉 ≤ 4ε+ 2∑

v∈LK ,ε

‖[v ⊗ v , ηn]‖,

for all y ∈ K , ‖x‖∞ ≤ 1,tε > t > 0, and n.

Use successively triangle inequality, deformation property andtraciality.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

LemmaFor every ε > 0 and any finite set F0 ⊂ U(N) there exist a finiteset F0 ⊂ F ⊂ M, a c.c.p. map ΨF ,ε : span(F)→ C ∗r (Γ), andtε > 0 such that

‖Φtε(ΨF ,ε(u)∗xΨF ,ε(u))− Φtε(x)‖ ≤ 47ε,

for all u ∈ F0 and ‖x‖∞ ≤ 1.

local reflexivity (implied by exactness) to locally approximate LΓthrough C ∗r (Γ) completely contractively: ∀ finite subset F ⊂ LΓand ε > 0, there exists φ : span{F ∪ F∗} → C ∗r (Γ) c. c.p. , suchthat ‖φ(x)− x‖2 < ε for all x ∈ F . use Haagerup criterion to show that N is amenable.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Open Problems

ProblemDescribe the class of countable discrete groups which are orbitequivalent to some hyperbolic group.

ProblemIf Γ admits an array into the left regular representation, then doesZ o Γ admit an array into a representation weakly contained in theleft regular?

Conjecture

If Γ is a hyperbolic group, then any free, ergodic p.m.p. actionΓ y X gives rise to a von Neumann algebra with unique Cartansubalgebra. Open even for Γ = Fn with n ≥ 2!

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Thank you for listening!

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups