L2 -Betti numbers for groups, equivalence relations,
and von Neumann algebras?
Discrete Groups and Geometric
Structures
Stefaan Vaes
1/19
From classical to L2 -Betti numbers
Let X be a closed manifold :
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n (X ) .
βn (X ) = dim HdR
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χ(X ) =
∞
X
(−1)n βn (X ) .
n=0
Consider: X a closed Riemannian manifold and a Γ-covering p : X → X .
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n
Build dim H(2),dR
(X ) using L2 -differential forms.
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Typically infinite dimensional, with an action of Γ.
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Atiyah (1976), using von Neumann algebras :
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n
βn (Γ, X ) as the “Γ-dimension” of dim H(2),dR
(X ) .
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Von Neumann algebras
Definition
A von Neumann algebra is a weakly closed, self-adjoint algebra of
operators on a Hilbert space.
Note: weak topology = topology making T 7→ hT ξ, ηi continuous.
Trivial examples:
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B(H) : all bounded operators on H.
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L∞ (X ) acting by multiplication operators on L2 (X ).
Von Neumann’s bicommutant theorem (1929)
Let M ⊂ B(H) be a self-adjoint algebra of operators with 1 ∈ M. TFAE
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Analytic property : M is weakly closed.
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Algebraic property : M = M 00 , where A0 is the commutant of A.
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Group von Neumann algebras
A crucial ingredient to define Γ-dimension.
Let Γ be a countable group.
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A unitary representation of Γ on a Hilbert space H is a map
π : Γ → B(H) : g 7→ πg satisfying
• all πg are unitary operators,
• πg πh = πgh for all g , h ∈ Γ.
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The regular representation of Γ is the unitary representation λ on
`2 (Γ) given by λg δh = δgh ,
where (δh )h∈Γ is the natural orthonormal basis of `2 (Γ).
Definition (Murray - von Neumann, 1943)
The group von Neumann algebra LΓ is defined as the weakly closed
linear span of {λg | g ∈ Γ}.
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Von Neumann’s continuous dimension
Vector subspace V ⊂ Cn
←→ projection p ∈ Mn (C) .
Then: dim V = Tr(p) .
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The functional τ : LΓ → C : τ (x) = hxδe , δe i is a trace :
τ (xy ) = τ (yx) .
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(assuming Γ infinite)
Every t ∈ [0, 1] is the trace of a projection p ∈ LΓ.
Murray-von Neumann : dimension function for LΓ-modules.
• dimLΓ (pLΓ) = τ (p) for every projection p ∈ LΓ.
• dimLΓ (p(Cn ⊗ LΓ)) = (Tr ⊗τ )(p) for p ∈ Mn (C) ⊗ LΓ.
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L2 -Betti numbers of Γ-coverings
Dimension of finitely generated projective LΓ-modules :
dimLΓ (p(Cn ⊗ LΓ)) = (Tr ⊗τ )(p).
Lück: for an arbitrary LΓ-module V :
dimLΓ (V ) = sup dimLΓ (W ) W ⊂ V is finitely generated projective .
Definition
The L2 -Betti numbers of the Γ-covering p : X → X are defined as
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n
(X ) .
βn (Γ, X ) = dimLΓ H(2),dR
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e ) can be irrational !
Possible values : βn (π1 (X ), X
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Invariant under homotopy.
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So: an invariant of Γ if X is contractible.
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Cheeger-Gromov (1986): definition βn (Γ) for arbitrary countable Γ.
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L2 -Betti numbers of countable groups
Definition (Cheeger-Gromov, 1986)
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In terms of Hochschild cohomology : βn (Γ) = dimLΓ H n (Γ, `2 (Γ)) .
More concretely, for the first L2 -Betti number :
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Consider `2 (Γ) as an LΓ-bimodule : λg · δh · λk = δghk .
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Z 1 (Γ, `2 (Γ)) is the set of 1-cocycles c : Γ → `2 (Γ) : cgh = cg + λg · ch
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Coboundaries: cg = ξ − λg · ξ for some ξ ∈ `2 (Γ).
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H 1 (Γ, `2 (Γ)) becomes a right LΓ-module under “pointwise
multiplication”.
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We can thus define β1 (Γ) = dimLΓ H 1 (Γ, `2 (Γ)).
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Rest of this talk : examples, applications and open problems around
L2 -Betti numbers.
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Some computations
Amenable group : existence of a left invariant mean on Γ.
Examples : finite groups, abelian groups, stable under direct limits and
extensions.
Nonamenable groups : groups containing F2 , but also others...
Theorem (Cheeger-Gromov)
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If Γ has an infinite normal amenable subgroup, then βn (Γ) = 0 for all n.
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We have β1 (Fk ) = k − 1 and all other are zero.
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We have β2 (Fk × Fl ) = (k − 1)(l − 1) and all other are zero.
In general: Künneth formula.
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Surface group Γ of genus g : β1 (Γ) = 2g − 2, others 0.
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Computations for lattices in Lie groups.
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Application: an orbit equivalence invariant
Consider actions Γ y (X , µ)
• of countable groups on probability spaces,
• by probability measure preserving (pmp) transformations,
• that are essentially free : {x ∈ X | g · x = x} has measure zero.
Example (Bernoulli action): Γ y [0, 1]Γ by (g · x)h = xhg .
Orbit equivalence
The actions Γ y (X , µ) and Λ y (Y , η) are called orbit equivalent if
there exists an isomorphism of probability spaces ∆ : X → Y such that
∆(Γ · x) = Λ · ∆(x) for a.e. x ∈ X .
Ornstein-Weiss (1980) : all free ergodic pmp actions of all amenable
groups are orbit equivalent !
Gaboriau (1999) : free ergodic pmp actions of F2 and F3 are never orbit
equivalent.
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Application: an orbit equivalence invariant
Gaboriau’s theorem : if Γ and Λ admit orbit equivalent free pmp actions,
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then βn (Γ) = βn (Λ) for all n.
One of the rare invariants of groups up to orbit equivalence !
Given Γ y (X , µ), we have the orbit equivalence relation R(Γ y X ) on
(X , µ) where x ∼ y iff x ∈ Γ · y .
Abstract definition: “countable pmp equivalence relation” R on (X , µ).
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Gaboriau: definition of βn (R) and proof of βn (R(Γ y X )) = βn (Γ).
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(Feldman-Moore) R
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One can define βn (R) = dimL(R) (H n (R, L2 (R))).
tracial von Neumann algebra L(R).
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10/19
Cost of an equivalence relation (Levitt/Gaboriau)
Let R be a countable pmp equivalence relation on (X , µ).
A graphing of R is a family (ϕn ) of partial measure preserving
transformations with domain D(ϕn ) ⊂ X and range R(ϕn ) ⊂ X satisfying :
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(x, ϕn (x)) ∈ R for all n and a.e. x ∈ D(ϕn ),
up to measure zero, R is the smallest equivalence relation that
contains the graphs of all the ϕn .
P
The cost of a graphing is defined as n µ(D(ϕn )).
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The cost of the equivalence relation R is defined as the infimum of the
costs of all graphings.
Observe : cost(R(Γ y X )) ≤ d(Γ), with d(Γ) the minimal number of
elements needed to generate the group Γ.
Gaboriau : cost(R(Fn y X )) = n if the action is free.
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Open problems: cost versus β1 , fixed price
Let Γ y (X , µ) be a free pmp action.
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We have β1 (Γ) ≤ cost R(Γ y X ) − 1,
because 1-cocycles are determined by their values on generators.
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Is the inequality ever strict ?
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Does cost R(Γ y X ) depend on the choice of the action ?
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Note that β1 (Γ) does not.
Known as the fixed price problem.
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Example : k − 1 = β1 (Fk ) ≤ cost R(Fk y X ) − 1
≤ d(Fk ) − 1 = k − 1.
The free group Fk has fixed price k.
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Connection to Lackenby’s rank gradient
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Observation: if [Γ : Λ] < ∞, then β1 (Γ) =
β1 (Λ)
.
[Γ : Λ]
Let Γ be residually finite:
T ∃Γn C Γ, a decreasing sequence of finite index
normal subgroups with n Γn = {e}.
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Lackenby’s rank gradient :
d(Γn ) − 1
rg(Γ, (Γn )) = lim
(limit of a decreasing sequence)
n
[Γ : Γn ]
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Open problem : independent of the choice of (Γn ) ?
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We have β1 (Γ) ≤ rg(Γ, (Γn )).
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Open problem : can the inequality ever be strict ?
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Abert-Nikolov : rg(Γ, (Γn )) = cost(R(Γ y X )) − 1, where X = lim Γ/Γn
←
−
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Compression formula
Let R be a countable ergodic pmp equivalence relation on (X , µ).
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Whenever Y ⊂ X with µ(Y ) > 0, we consider RY = R ∩ (Y × Y ).
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We have : βn (RY ) = µ(Y )−1 βn (R).
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In general : one defines Rt for t > 0 and has βn (Rt ) = t −1 βn (R).
Lattices in the same locally compact group
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βn (Γ)
βn (Λ)
Let Γ, Λ be lattices in G . Then
=
.
covol Γ
covol Λ
Remark : “fundamental group” of R consists of the t > 0 with Rt ∼
= R.
• R(Fn y X ) always has trivial fundamental group (Gaboriau).
• R(F∞ y X ) can have wild fundamental groups (Popa-V).
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L2 -Betti numbers for locally compact groups
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The formula
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βn (Γ)
βn (Λ)
=
for lattices Λ, Γ < G suggests that
covol Γ
covol Λ
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this should be equal to βn (G ).
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Unimodular lc group G : von Neumann algebra LG with infinite trace.
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(H.D. Petersen, 2012) Dimension theory works.
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Definition: βn (G ) = dimL(G ) H n (G , L2 (G )).
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βn (Γ) = covol Γ · βn (G ) if Γ < G is a cocompact lattice.
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(Kyed, Petersen, V, 2013) The same holds for arbitrary lattices.
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L2 -Betti numbers for locally compact groups
Approach of Kyed-Petersen-V, 2013 :
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Choose any free ergodic pmp action G y (X , µ).
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Choose “Borel cross section” Y ⊂ X . In particular, G · y ∩ Y is
countable for all y ∈ Y .
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This yields the “cross section equivalence relation” RY .
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We have βn (G ) = β (2)
(G ) = covol(Y )−1 βn (RY ).
n
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All expected results can be deduced from known results on
equivalence relations.
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• Example : βn (G ) = 0 if G admits a noncompact amenable normal
subgroup.
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• (Petersen-Valette, 2013) Computations of βn (G ) for semi-simple Lie
groups, and more.
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Application: classification of von Neumann algebras
We have seen:
Free pmp action Γ y (X , µ)
orbit equivalence relation R(Γ y X ).
But also: countable pmp equivalence relation R on (X , µ)
von Neumann algebra L(R) containing L∞ (X ).
Construction of L(R)
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R ⊂ X × X carries a natural measure ν.
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Every ϕ : D(ϕ) → R(ϕ) with graph in R defines an operator uϕ on
L2 (R) given by (uϕ∗ ξ)(x, y ) = ξ(ϕ(x), y ) whenever x ∈ D(ϕ).
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L(R) is the von Neumann algebra generated by the uϕ .
We have L∞ (X ) ⊂ L(R) as a Cartan subalgebra.
The inclusion L∞ (X ) ⊂ L(R) encodes the equivalence relation R.
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Application: classification of von Neumann algebras
Equivalence relation R
inclusion L∞ (X ) ⊂ L(R) that encodes R.
Basic question: can we retrieve R from the von Neumann algebra L(R) ?
More precisely: when does L(R) have a unique Cartan subalgebra (up to
unitary conjugacy) ?
Conjecture
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If βn (R) > 0 for some n, then L(R) has a unique Cartan subalgebra.
Theorems
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Popa-V, 2012 : yes, for R(Γ y X ) when β1 (Γ) > 0 and Γ is weakly
amenable. In particular, yes if Γ = Fk .
Ioana, 2013 : yes, for R(Γ y X ) when Γ = Γ1 ∗ Γ2 non-trivially.
Popa-V, 2013 : also yes if Γ is a hyperbolic group.
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Open problem: free group factors
Corollary : if k 6= l, then R(Fk y X ) and R(Fl y Y ) generate
nonisomorphic von Neumann algebras, for any choice of actions.
Indeed : an isomorphism between these von Neumann algebras provides
an isomorphism between the equivalence relations, and this does not exist
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because they have different β1 .
The free group factor problem
Let k 6= l. Is L(Fk ) ∼
= L(Fl ) ?
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Dichotomy: either all Fk (including k = ∞) are isomorphic, or they
are two by two nonisomorphic.
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Obvious idea: develop β1 (M) for tracial von Neumann algebras M.
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And hopefully, β1 (L(Fk )) = k − 1.
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(Popa-V, 2014) A rather pessimistic no-go theorem for β1 (M)...
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