\( \def \curvearrowleft{\downarrow \mspace{-10mu} \cap \mspace{3mu}} \) \( \def \curvearrowright{\mspace{3mu} \cap \mspace{-10mu} \downarrow} \)
This is a transcript of my notes from Uri Bader’s lectures. They may not always accurately reflect the content of the lectures, especially in places I did not put down everything that was said, and some comments or details might have been added by me. In particular the last section does not contain the proofs of the Ornstein–Weiss theorem, Dye’s theorem and Rokhlin lemma that Uri explained in his lecture.
Lecture I
From quasi-isometry to measured equivalence
In all of the following lectures, when no further precision is given all groups will be assumed to belong to the class of locally compact and second countable groups (abbreviated as lcsc groups).
A topological coupling between two groups \( \Gamma, \Lambda \) is an action of \( \Gamma \times \Lambda \) on a locally compact space \( X \) such that both actions \( \Gamma \curvearrowright X \) and \( \Lambda \curvearrowright X \) are properly discontinuous and cocompact.
Theorem (Gromov): The two groups \( \Gamma \) and \( \Lambda \) are quasi-isometric to each other if and only if they admit a topological coupling.
Examples
- If \( \Gamma, \Lambda \) are both cocompact discrete subgroups of a locally compact group \( G \) then \( \Gamma \curvearrowright G \curvearrowleft \Lambda \) (where the actions are by right- and left-translations) is a topological coupling.
- Let \( G = {\mathbb{R}} \) and \( \Gamma = {\mathbb{Z}} \), then the quotient \( \Gamma \backslash G \) is the circle \( \mathbb S^1 \) and a rotation of angle \( \alpha \) gives a coupling of \( \Lambda = \alpha{\mathbb{Z}} \) with \( \Gamma \).
- Let \( U \) be a proper metric space and \( \Gamma, \Lambda \subset \mathrm{Isom}(U) \) act properly discontinuously and cocompactly then they are quasi-isometric by the Milnor–Schwarz lemma. A topological coupling is given by \( G = \mathrm{Isom}(U) \).
Proof of the theorem
Recall that given \( K, C \) the set of \( (K, C) \)-quasi-isometries between two discrete groups \( \Gamma, \Lambda \) is locally compact in the pointwise convergence topology. Moreover the action of \( \Gamma \times \Lambda \) on \( \Lambda^\Gamma \) given by
\[
(\gamma, \lambda) \cdot f = \lambda f(\cdot \gamma)
\]
preserves the subset of \( (K, C) \)-quasi-isometries and the action on both sides is properly discontinuous and cocompact on both sides. If \( \Gamma \) is quasi-isometric to \( \Lambda \) then this set is nonempty for large \( K, C \) and it is a coupling.
Now suppose \( X \) is a topological coupling between \( \Gamma \) and \( \lambda \). Let \( X_0 \) be a relatively compact fundamental domain for the \( \Lambda \)-action. We get a bijection
\[
X_0 \times \Lambda \to X, \, (x, \lambda) \mapsto x \cdot \lambda
\]
and thus for any \( \gamma \in \Gamma \) and \( x \in X \) there exists a unique pair \( c(\gamma, x) \in \Lambda, a(\gamma, x) \in X_0 \) such that
\[
\gamma \cdot x = a(\gamma, x) \cdot c(\gamma, x).
\]
The map
\[
(\gamma, x) = a(\gamma, x)
\]
encodes an action of \( \Gamma \) on \( X / \Lambda \) via the bijection \( X_0 \cong X / \Lambda \). The map \( a \) thus satisfies the relation
\[
a(\gamma_1\gamma_2, x) = a(\gamma_1, \gamma_2 x).
\]
It follows that we have:
\[
c(\gamma_1\gamma_2, x) a(\gamma_1\gamma_2, x) = c(\gamma_2, x) c(\gamma_1, a(\gamma_2, x)) \cdot a(\gamma_1, a(\gamma_2, x)) = c(\gamma_2, x) x(\gamma_1, a(\gamma_2, x)) a(\gamma_1\gamma_2, x)
\]
and it follows that the map \( c : \Gamma \times X / \Lambda \to \Lambda \) satisfies the cocycle relation
\[
c(\gamma_1\gamma_2, \bar x) = c(\gamma_2, \bar x) c(\gamma_1, \gamma_2 \bar x).
\]
To finish the proof of the theorem one can then show that for any two maps \( c( \cdot, x) \) are at bounded distance from each other, and then use the above to see that any of them gives a quasi-isometry \( \Gamma \to \Lambda \).
Advantages of the definition via couplings
- The definition this gives is purely topological and not metric. It can easily be extended to define quasi-isometry between lovally compact groups. (Note that it is a ctually not entirely trivial to check that this yields the natural definition for compactly generated groups).
- The notion of coupling makes for group objects in many categories. For example in the \( \mathbf{Set} \) category, one can define couplings as actions of \( \Gamma \times \Lambda \) which are transitive, and such that both actions of \( \Gamma \) and \( \Lambda \) are cofinite and free (respectively have finite stabilisers). This recovers the notion of commensurability (respectively virtual isomorphism) between discrete groups.
Couplings in the measurable category and measure equivalence
In the measurable category one considers the actions of a locally compact group \( G \) on a measure space \( (X, \mu) \) which preserves the measure \( \mu \) and such that there exists a finite measure space \( X_0 \) and a \( G \) -equivariant measure isomorphism
\[
G \times X_0 \to X
\]
(where \( G \) is endowed with its Haar measure). Such an action is called free-proper-cofinite.
Note that when \( G \) is a discrete group one can view \( X_0 \) as a fundamental domain for \( G \) in \( X \) but not in general (as the measure of \( X_0 \) might have to be zero ; in general it is not possible to find a measurable transversal for the action).
The notion of equivalence obtained in this category is via measured couplings which are defined as follows: a measurable coupling between two locally compact groups \( G, H \) is a measure-preserving action of \( G \times H \) on a measured space \( (X, \mu) \) such that both actions of \( G \) and \( H \) are free-proper-cofinite. Whenever there exists such a coupling the groups \( G \) and \( H \) are said to be measure equivalent.
The fundamental example example of a measurable coupling is when \( \Gamma, \Lambda \) are both lattices in a locally compact group \( G \), and the coupling is given by the right- and left-action of \( \Gamma \) and \( \Lambda \) :
\[
(\gamma, \lambda) \cdot x = \gamma x \lambda.
\]
A particular case of measure equivalence is when the action of \( G \times H \) is transitive. Then one can write \( X = G \times H / \Gamma \) for a discrete group \( \Gamma \subset G \times H \), and both projections of \( \Gamma \) into \( G \) and \( H \) are lattices.
Lecture II
The categorical viewpoint
Measure equivalence between locally compact unimodular groups is an equivalence relation:
- For any such group \( G \) we have \( G \sim G \) via the trivial coupling \( G \curvearrowright G \curvearrowleft G \);
- The relation is trivially reflexive, if \( G \sim H \) via a coupling \( G \curvearrowright X \curvearrowleft H \) then also \( H \sim G \) via the opposite coupling.
- The relation is transitive: if \( G \curvearrowright X \curvearrowleft H \) and \( H \curvearrowright Y \curvearrowleft I \) are two measured couplings then the fibered product:
\[
X \times_H Y := (X \times Y)/ H
\]
(where \( H \) acts by \( h \cdot (x, y) = (x \cdot h, h^{-1} \cdot y) \) is a measurable coupling between \( G \) and \( I \).
Warnings
These operations do not form a monoid:
- If \( G \curvearrowright X \curvearrowleft H \) is a coupling then the coupling \( X \times_G X^{\mathrm{op}} \) is not the trivial \( H \)-coupling.
- The « composition » of couplings via the fibered product is not associative.
Equivalence of couplings
Two couplings \( X, X’ \) between \( G \) and \( H \) are said to be equivalent if there exists a \( G \times H \)-equivariant measure isomorphism between \( X \) and \( X’ \).
More generally, a morphism between two couplings \( X, Y \) between \( G \) and \( H \) is a measure-preserving, \( G \times H \)-equivariant map \( X \to Y \).
Definition: Two couplings \( X, Y \) are weakly equivalent if there exists a third coupling \( Z \) and two morphisms \( Z \to X \) and \( Z \to Y \).
Remark: This can be formalised as a 2-category on the set of free-proper-cofinite actions: a morphism is a coupling, and there is a notion of a « morphism between morphisms ». (Another example of this is given by the category of cobordisms in topology.)
Lemma: Weak equivalence is an equivalence relation on the sets of couplings.
This follows from the construction of fibered products in the category we work in: this gives an equivalence between the conditions
\[
\exists Y_1 \to X \leftarrow Y_2
\]
and
\[
\exists Y_1 \leftarrow Z \to Y_2
\]
for two couplings \( Y_1, Y_2 \).
Construction of fibered products
Let \( \pi: X \to Y \) a measurable map and \( \mu \) a measure on \( X \). Let \( \mu \) a measure on \( X \) and \( \nu = \pi_*\mu \). Disintegration of measures gives a decomposition
\[
\mu = \int_Y \mu_y d\nu(y)
\]
where \( \mu_y \) is a measure on the fiber \( X_y \) above \( y \in Y \). A diagram \( \exists Y_1 \to X \leftarrow Y_2 \) where \( X \) has a measure \( \nu \) thus yields two decompositions
\[
\mu_1 = \int_Y \mu_{1, y} d\nu(y), \, \mu_2 = \int_Y \mu_{2, y} d\nu(y)
\]
and this gives a measure \( \int_D \mu_{1, x} \otimes \mu_{2, x} d\nu(x) \) on the space
\[
Z = \int_X Y_{1, x} \times Y_{2, x} d \nu(x).
\]
One can then check that \( Z \) satisfies the property of a fibered product.
Remark: There is a contravariant functor from the measurable category to the category of von Neumann algebras given by \( X \mapsto L^\infty(X) \). It is in fact an equivalence of categories, and the diagram corresponding to a fibered product
\[
\begin{array}{ccc}
Z & \to & Y_1 \\
\downarrow & & \downarrow \\
Y_2 & \to & X
\end{array}
\]
is the following:
\[
\begin{array}{ccc}
L^\infty(Z) & \supset & L^\infty(Y_1) \\
\cup & & \cup \\
L^\infty(Y_2) & \supset & L^\infty(X)
\end{array}
\]
Observation: A self-coupling \( X \) of a locally compact group \( G \) is weakly equivalent to the trivial coupling \( G \) if and only if there exists a morphism \( X \to G \). Such an \( X \) is called trivial, and a morphism \( X \to G \) is called a Trivialisation.
Example
Let \( \Gamma \) be a discrete group and \( G \) a locally compact group. Let \( \rho_1, \rho_2 \) be two embeddings of \( \Gamma \) into \( G \), whose images \( \rho_i(\Gamma) \) are both lattices in \( G \). Then the fibered product \( X = G \times_\Gamma G \) is trivial if and only if \( \rho_2 \) is conjugated to \( \rho_1 \), that is there exists \( g \in G \) such that
\[
\forall \gamma \in \Gamma : \rho_2(\gamma) = g\rho_1(\gamma)g^{-1}.
\]
Indeed, if is trivial and \( G \times_\Gamma G \to X \) is a trivialisation, denote by \( (g_1, g_2) \mapsto (\overline{g_1, g_2}) \) the composition
\[
G \times G \to G \times_\Gamma G \to X.
\]
Let \( g = (\overline{e, e}) \). Then the definition of \( G \times_\Gamma G \) gives that for any \( \gamma \in \Gamma \) we have:
\[
\rho_0(\gamma)g = (\overline{\rho_0(\gamma)e, e}) = (\overline{e, e\rho_1(\gamma)}) = g\rho_1(\gamma).
\]
Let \( S \) be a closed surface. As an application of the example above it is possible to show that the space
\[
\{ (X, Y) \in \mathrm{Teich}(S) \times \mathrm{Teich}(S) : X \not\cong Y \}
\]
embeds into the space of self-couplings of \( \mathrm{PSL}_2({\mathbb{R}}) \) modulo weak equivalence.
In higher dimensions Mostow rigidity shows that there is no deformation space of lattices. Thus a way to interpret the space of weak equivalence classes of self-coupling of a group is as a measure of failure of rigidity.
ME-rigid groups
Say that a group \( G \) is ME-rigid if any self-coupling of \( G \) admits a unique trivialisation.
Theorem II (Furman): If \( G \) is a measure-rigid group and \( H \) is measure-equivalent to \( G \) then there exists a morphism \( G \to H \) with compact kernel and cofinite, closed image.
The converse of this theorem is trivially true. The next result gives an important family of examples of ME-rigid groups.
Theorem I (Furman): The groups \( \mathrm{PGL}_n({\mathbb{R}}) = \mathrm{Aut}(\mathrm{SL}_n({\mathbb{R}})) \) are rigid for \( n \ge 3 \).
An immediate corollary of Theorems I and II is the following rigidity statement for lattices.
Corollary: Let \( n \ge 3 \) and let \( \Gamma \) be a lattice in \( \mathrm{PGL}_n({\mathbb{R}}) \). If \( \Lambda \) is measure equivalent to \( \Gamma \) then there exists a finite index subgroup in \( \Lambda \) which embeds as a lattice in \( \mathrm{PGL}_n({\mathbb{R}}) \).
Lecture III
Remark: If \( G \) is ME-rigid then \( \mathrm{Aut}(G) = \mathrm{Inn}(G) \) (all automorphisms are inner) and \( Z(G) = \{ e \} \). These conditions amount to the map \( G \to \mathrm{Aut}(G) \) being an isomorphism. The proof of both is very simple:
- Suppose that \( Z(G) \ni z \not= e \). Then the map \( g \mapsto zg \) is \( G \times G \)-equivariant, that is it is a endomorphism of the trivial coupling. Thus trivialisations will not be unique.
- Let \( \varphi \in \mathrm{Aut}(G) \). Then we can get a « twisted » self coupling \( G \curvearrowright G \curvearrowleft G \) defined by
\[
g \cdot x \cdot h = \varphi(g) x h^{-1}.
\]
Then there exists a trivialisation for this coupling if and only if \( \varphi \) is inner (the proof is similar to this one above).
On the unicity of trivialisations in the definition of ME-rigidity
Lemma: Let \( G \) be a lcsc group. The following conditions are equivalent:
- Any trivial self-coupling of \( G \) admits a unique trivialisation.
- The only conjugacy-invariant probability measure on \( G \) is the Dirac mass at the identity:
\[
\mathrm{Prob}(G)^G = \{ \delta_e \}.
\]
Proof: Let \( \mu \in \mathrm{Prob}(G)^G \) and \( X_0 = (G, \mu) \). Then \( X = G \times X_0 \) (where the right factor \( G \) has the Haar measure) is a self-coupling of \( G \) in two ways:
- \( X \curvearrowleft G \) by the diagonal action
\[
(y, x) \cdot g = (yg, g^{-1}xg)
\]
and \( G \curvearrowright X \) by left-multiplication:
\[
h\cdot(y, x) = (hy, x).
\] - \( X \curvearrowleft G \) by right-multiplication
\[
(y, x) \cdot g = (yg, x)
\]
and \( G \curvearrowright X \) by left-multiplication:
\[
h \cdot (y, x) = (y, hxh^{-1}).
\]
Then if \( \mu \neq \delta_{\{e\}} \) the maps \( X \to G \) given respectively by \( (x, y) \mapsto xy \) and by \( (x, y) \mapsto x \) are two distinct trivialisations.
Conversely, suppose that there exists a trivial self coupling \( X \) with two distinct trivialisations
\[
G \overset{\varphi_1}{\leftarrow} X \overset{\varphi_2}{\to}.
\]
Define a map \( X \to G \) by:
\[
\psi(x) = \phi_1(x) \phi_2(x)^{-1}.
\]
Then \( \psi \) is right-\( G \)-invariant and we get a \( G \)-equivariant map \( X/G \to G \). The pushforward of the finite \( G \)-invariant measure on \( X / G \) to \( G \) is a conjugacy-invariant measure on \( G \) with finite mass, and it is not supported on the identity.
Remark
If \( G \) is a discrete group then the condition that \( \mathrm{Prob}(G)^G = \{ \delta_{\{e\}} \} \) is equivalent to the condition that all nontrivial conjugacy classes of \( G \) be infinite (ICC). The former condition is thus sometimes called « strong ICC ».
Invariant probability measures in algebraic actions
Theorem: Let \( G \) be a semisimple \( {\mathbb{R}} \)-algebraic group such that the Lie group \( G_{\mathbb{R}} \) has no compact factor and let \( G \curvearrowright V \) an algebraic action (action by regular automorphisms of the variety \( V \)). Then the natural map
\[
\mathrm{Prob}(V^G) \to \left( \mathrm{Prob}(V) \right)^G
\]
is surjective.
Applied to the action of \[ G \) by conjugation on itself this result has the following consequence.
Corollary: If \( G \) is an adjoint Lie group then \( \mathrm{Prob}(G)^G = \{ \delta_{\{e\}} \).
Demonstration of Theorem I
We will prove that \( G = \mathrm{PGL}_n({\mathbb{R}}) \) is ME-rigid, admitting a deep result due to R. Zimmer.
Since \( G \) is an adjoint group it follows from the corollary above that trivialisations are unique. We need to prove that any self-coupling of \( G \) is trivial. Let \( G \curvearrowright G \curvearrowleft G \) be such a coupling.
As was the case in the metric setting (see above) we can write \( X = G \times X_0 \) where the right-action is by right-multiplication on the \( G \) factor. The left action is given by a measurable cocycle \( c : G \times X_0 \to G \) so that
\[
g \cdot (h, x_0) = (hc(g, x_0), g\cdot x_0).
\]
We apply the following theorem.
Theorem (Cocycle superrigidity, Zimmer): Let \( G \) be a simple, adjoint Lie group of higher rank. If \( c: G \times X_0 \to \mathrm{Gl}_m({\mathbb{C}}) \) is a measurable cocycle whose image is unbounded and has a semisimple Zariski closure, then \( c \) is cohomologous to a morphism \( G \to \mathrm{GL}_m({\mathbb{C}}) \).
Two cocycles \( c, c’ \) are cohomologous if there exists \( \psi : X_0 \to H \) such that for any \( g \in G \) we have \( c'(g, x) = \psi(x)c(g, x)\psi(g\cdot x) \). Thus the theorem above implies that there exists a morphism \( \varphi: G \to G \) such that
\[
g \cdot (h, x_0) = (\varphi(g)h, g \cdot x_0)
\]
and it follows that the projection \( G \times X_0 \to G \) is a morphism, where \( G \) has the twisted coupling structure given by \( g \cdot x \cdot h = \varphi(g) x h \). Since \( G \) is adjoint the automorphism \( \varphi \) is inner, and it follows that the latter coupling admits a trivialisation.
ME-rigidity in rank 1
The group \( \mathrm{PSL}_2({\mathbb{R}}) \) is clearly nonrigid since there are nontrivial deformations of lattices given by the Teichmüller spaces. The ME-rigidity of other rank 1 Lie groups is an open question.
It is however possible to prove rigidity results for a stricter notion of measure equivalence. We will define this notion in analogy with the QI-case seen in the first lecture. Recall that in this setting a cocycle \( c : \Gamma \times X_0 \to \Lambda \) gives a family of quasi-isometries \( \Gamma \to \Lambda \), the maps \( c(\cdot, x_0 \) for \( x_0 \in X_0 \). In the measurable setting, if a coupling between \( G \) and \( H \) gives rise to a cocycle \( c : G \times X_0 \to H \) we say that the former is \( p \)-integrable or \( L^p \) if for any \( g \in G \) the integral
\[
\int_{X_0} |c(g, x)|_H^p dx
\]
is finite (where \( |\cdot|_H \) is a left-invariant metric on \( H \)). Then there are rigidity results in rank 1 for such couplings due to Bader–Furman–Monod.
Lecture IV
Sketch of proof of Theorem II
Suppose that \( G \) is a ME-rigid group and that \( H \) is measure equivalent to \ (G \). We want to construct a morphism \( \rho : G \to H \) which has a compact kernel and a closed cofinite image.
Let \( G \curvearrowright X \curvearrowleft H \) be a measured coupling between \( G \) and \( H \). Then \( X \times X^{\mathrm{op}} \) is a self-coupling of \( G \), and since \( G \) is ME-rigid there exists a unique trivialising morphism
\[
\bar\Phi : X \times X^{\mathrm{op}} \to G,
\]
which can be lifted to an application \( \Phi : X \times X \to G \) which is \( H \)-invariant and \( G \)-equuivariant. Moreover \( \Phi \) is uniquely determined by these properties. For the rest of the proof we supposed that we have chosen a point \( x_0 \in X \) which is « generic », in a sense to be precised later, and we define a map \( \Psi : X \to G \) by \( \Psi(x) = \Phi(x, x_0) \).
We will make use of the following notion.
Definition: Let \( G_i, i=1, 2, 3 \) be lcsc groups. A tripling between them is a measure presering action of \( G_1 \times G_2 \times G_ 3 \) on a space \( Y \) such that for any \( 1 \le i < j \le 3 \) the restriction of the action to \( G_i \times G_j \) is a coupling between \( G_i \) and \( G_j \).
Taking \( G_i = G \) and \( Y = (X \times X \times X) / H \) (where \( H \) acts by the diagonal action) we get a tripling. For any \( \{i, j, k\} = \{1, 2, 3\} \), by ME-rigidity of \( G \) applied to the coupling between \( G_j \) and \( G_k \) we obtain a map \( F_{j, k} : X^3 \to G \) which is \( H \times G_i \)-invariant and \( G_j \times G_k \)-equivariant (for the action \( G_j \curvearrowright G \curvearrowleft G_k \) at the target), and which is uniquely determined by these properties. Now the application given by
\[
X \times X \times X \overset{\pi_{j, k}}{\to} X \times X \overset{\Phi}{\to} G \]
(where \( \pi_{j, k} \) is the projection onto the product of the \( j\)th with the \( k \)th factors) satisfies the same properties as \( F_{j, k} \) and hence
\[
F_{j, k} = \Phi \circ \pi_{j, k}.
\]
In the same way we obtain
\[
F_{j, k} = \iota \circ F_{k, j} =: F_{j, k}^{-1}
\]
(where \( \iota \) is the self-map \( g \mapsto g^{-1} \) of \( G \)) and also
\[
F_{j, k} = F_{j, i} \cdot F_{i, k}
\]
(where \( \cdot \) denotes multiplication in \( G \)). It follows that, for generic \( x_1, x_2, x_3 \in X \), we have:
\[
\begin{array}{rl}
\Phi(x_1, x_2) &= F_{1,2}(x_1, x_2, x_3) \\
&= F_{1,3}(x_1, x_2, x_3) F_{3,2}(x_1, x_2, x_3) \\
&= F_{1,3}(x_1, x_2, x_3) F_{2,3}(x_1, x_2, x_3)^{-1} \\
&= \Phi(x_1, x_3) \Phi(x_2, x_3)^{-1}.
\end{array}
\]
We suppose that \( x_0 \) has been chosen so that the equation above holds for \( x_3 = x_0 \) and we get that:
\[
\Phi(x_1, x_2) = \Psi(x_1) \Psi(x_2)^{-1}.
\]
As \( \Phi \) is \( H \)-invariant it follows that for any \( h \in H \) we have:
\[
\Psi(x_1 \cdot h) \Psi(x_2 \cdot h)^{-1} = \Psi(x_1) \Psi(x_2)^{-1}
\]
which we rewrite as
\[
\Psi(x_1)^{-1} \Psi(x_1 \cdot h) = \Psi(x_2)^{-1} \Psi(x_2 \cdot h).
\]
In other words the element \( \Psi(x)^{-1}\Psi(x \cdot h) \in G \) does not depend on \( x \in X \) in a generic subset. Thus we can define
\[
\rho(h) = \Psi(x)^{-1}\Psi(x \cdot h)
\]
and it is easy to see that this defines a morphism \( H \to G \). It is measurable and hence continuous. It remains to check the compactness of the kernel and properties of the image, which we won’t do here.
A simplicial category
For \( n \ge 1 \) a « \( n \)-upling » between lcsc groups \( G_1, \ldots, G_n \) is defined inductively as follows :
- For \( n= 1 \) it is a probability measure preserving (pmp) action;
- For \( n > 1 \) it is a measure-preserving action of \( G_1 \times \cdots \times G_n \) on a space \( Y \) such that for any \( i = 1, \ldots, n \) the action
\[
\prod_{j \not= i} G_j \curvearrowright X / G_i
\]
is an \( (n-1) \)-upling.
For \( n = 2 \) this recovers the notion of coupling and for \( n = 3 \) that of a tripling in the proof above. This gives the set of pmp actions of lcsc groups the structure of a simplicial category, the study of which could lead to new developments.
Invariants of measure equivalence
Induction from a measured equivalence
Let \( G \curvearrowright X \curvearrowleft H \) be a measured coupling between \( G \) and \( H \) and \( \rho : H \to \mathcal U(V) \) a unitary representation of \( H \) on a Hilbert space \( V \). Then the induced representation \( I(\rho) \) of \( G \) from \( H \) via the coupling is defined on the space:
\[
I(V) = \{ f : X \to V, f \text{ is \( H \)-equivariant and } \int_{X / H} \|f(x)\|_V^2 dx < +\infty \}
\]
which is a Hilbert space with the norm given by \( \|f\|_{I(V)}^2 = \int_{X / H} \|f(x)\|_V^2 dx \), on which \( G \) acts unitarily by
\[
I(\rho)(g)f(x) = f(g \cdot x).
\]
Recall that a representation of \( G \) on a Hilbert space \( W \) is said to have almost invariant vectors if for any compact subset \( S \) of \( G \) and any \( \varepsilon > 0 \) there exists a \( w \in W \) with \( \|w\|_W = 1 \) and
\[
\forall s \in S : \| sw – w \|_W \le \varepsilon.
\]
The following lemma is left as an exercise.
Lemma: The induced representation \( I(\rho) \) has almost invariant vectors if and only if \( \rho \) itself does.
Amenability
Recall that a characterisation of amenability for lcsc groups is as follows: \( G \) is amenable if and only if its left- or right-regular representation on \( L^2(G) \) admits almost-invariant vectors.
Let \( G, H \) be ME-equivalent and \( V = L^2(H) \). Then as a \( G \)-representation we have \( I(V) \cong L^2(X) \). Now the measurable decomposition \( X = G \times X_0 \) gives an isomorphism of \( G \)-representations
\[
L^2(X) \cong L^2(G) \otimes W
\]
where \( W \) is a separable Hilbert space on which \( G \) acts trivially.
Now if \( H \) is amenable, \( L^2(H) \) admits almost invariant and hence so does \( L^2(G) \times W \). But this is easily seen to mean that \( L^2(G) \) itself admits almost invariant vectors, and so \( G \) is amenable as well.
A compilation of invariants
Beyond amenability, the lemma about almost invariant vectors allows to prove that the following properties are ME-invariant:
- Kazhdan’s property (T);
- a-T-menability or Haagerup’s property
In addition a theorem of Gaboriau gives a « proportionality principle » for \( L^2 \)-Betti numbers of ME-equivalent groups. A corollary which is easier to state is the following (see Marc Bourdon’s lectures for the definition of the reduced \( \ell_2 \)-cohomology groups \( \overline H^k(\cdot, \ell_2\cdot) \)).
Theorem: If \( \Gamma, \Lambda \) are two discrete groups which are measure equivalent and \( \overline H^k(\Gamma, \ell_2\Gamma) = 0 \) for some \( k \ge 0 \) then also \( \overline H^k(\Lambda, \ell_2\Lambda) = 0 \).
Measure equivalence and quasi-isometry
At first sight it appears that measure equivalence is a weaker form of equivalence than quasi-isometry. For example two lattices in a lcsc group are always ME-equivalent but if one is uniform and the other not they cannot be QI to each other. (And, as seen in Kapovich’s lectures, Schwartz’s theorem shows that the QI-classification of nonuniform lattices in rank 1 Lie groups is quite fine.)
For general groups there is no hierarchical relation between the two notions. In fact there exists examples of two groups quasi-isometric to each other such that one has property (T) and not the other. The construction is subtle and goes as follows. Let \( \Gamma \) be a uniform lattice in the Lie group \( \mathrm{Sp}(n, 1) \). Then \( \Gamma \) has property (T) since \( \mathrm{Sp}(n, 1) \) does. The cohomology space \( H^2(\Gamma, {\mathbb{Q}}) \) can be nonzero and in this case there exists a nontrivial central \( {\mathbb{Z}} \)-extension \( \Gamma_c \) of \( \Gamma \) corresponding to an infinite-order class \( c \in H^2(\Gamma, {\mathbb{Z}}) \). For a well-chosen \( c \) the groups \( \Gamma_c \) and \( \Gamma \times {\mathbb{Z}} \) are quasi-isometric to each other. On the other hand the first has property (T) but not the second. In particular these two groups cannot be measure equivalent.
Lecture V
We have seen the amenability is ME-invariant. It is easy to prove that a lcsc group is measure equivalent to the trivial group if and only if it is compact. The following result shows that there is only one ME-class of non-compact lcsc amenable groups.
Theorem (Dye, Ornstein–Weiss, Connes–Feldman–Weiss, Furman): A lcsc group \( G \) is ME-equivalent to \( {\mathbb{Z}} \) if and only if it is amenable and non-compact.
Since \( {\mathbb{Z}} \) is amenable a group ME to it will be so as well. We will partly explain how to prove the « only if » statement, which requires the following notion.
Definition: Two groups \( G, H \) are orbit equivalent is there exists a finite measure space \( \Omega \) and pmp actions of \( G \) and \( H \) on \( \Omega \) which have the same measured equivalence relation, that is \( H \cdot x = G \cdot x \) for almost all \( x \in \Omega \).
If we have such an \( \Omega \) then we get « rearrangement cocycles » \( c : G \times \Omega \to H \) and \( c’ : H \times \Omega \to G \). In particular \( G \) and \( H \) are ME-equivalent. Thus the following theorem, due to Orntein–Weiss for discrete groups and Connes–Feldman–Weiss in general, implies the theorem above.
Theorem: Let \( G \) be an amenable non-compact lcsc group and \( G \curvearrowright \Omega \) an essentially free pmp action. Then if \( G \) is countable there exists a pmp action of \( {\mathbb{Z}} \) on \( \Omega \) with the same orbits as that of \( G \). If \( G \) is uncountable then the same statement holds with \( {\mathbb{Z}} \) replaced by \( {\mathbb{R}} \).
For a proof in the case where \( G \) is countable see this previous post.