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.

1. First lecture
2. Second lecture
3. Third lecture
4. Fourth lecture
5. Fifth 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

1. 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).
2. 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:

1. 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.
2. 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:

1. \( X \curvearrowleft G \) by the diagonal action
   \[
   (y, x) \cdot g = (yg, g^{-1}xg)
   \]
   and \( G \curvearrowright X \) by left-multiplication:
   \[
   h\cdot(y, x) = (hy, x).
   \]
2. \( X \curvearrowleft G \) 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:

1. Kazhdan’s property (T);
2. 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.

1. First lecture
2. Second lecture
3. Third lecture
4. Fourth lecture
5. Fifth lecture

Lecture I

Simplicial \( \ell_p \) cohomology

Fix \( p \in ]1, +\infty[ \). Let \( X \) be a simplicial complex of finite dimension and of bounded geometry (this means that the number of simplices adjacent to a vertex of \( X \) is bounded). There is a natural path metric on \( X \) obtained by realising every simplex as a regular Euclidean simplex of edge length 1.

Let \( X^{(k)} \) be the set of \( k \)-simplices of \( X \) and \( C_k^p(X) \) the Banach space of \( p \)-integrable cochains, that is :
\[
C_k^p(X) = \{ f : X^{(k)} \to {\mathbb R} : \sum_{\sigma \in X^{(k)}} |f(\sigma)|^p < +\infty \}.
\]
The differential \( \delta_k : C_k^p(X) \to C_{k+1}^p(X) \) is defined as follows: if \( \sigma = [v_0, \ldots, v_k] \) is a \( k+1 \)-simplex put:
\[
\delta_k f(\sigma) = \sum_{i=0}^k (-1)^i f([v_0, \ldots, \hat v_i, \ldots, v_k]).
\]
The image is a \( \ell_p \)-chain, and this is a bounded linear map because of the hypothesis that \( X \) is of bounded geometry.

The \( \ell_p \)-cohomology \( \ell_pH^*(X) \) of \( X \) is the cohomology of the complex \( (C_*^p, \delta_*) \), in other words
\[
\ell_p H^k(X) = \ker(\delta_k) / \mathrm{Im}(\delta_{k-1}).
\]
The reduced \( \ell_p \)-cohomology of \( X \) is defined by
\[
\ell_p \overline{H}^k(X) = \ker(\delta_k) / \overline{\mathrm{Im}(\delta_{k-1})}
\]
where \( \overline V \) denotes the closure in the Banach space \( C_p^k(X) \) of a subspace \( V \).

Quasi-isometry invariance

A metric space \( X \) is said to be uniformly contractible if there exists \( C > 1 \) such that for every \( R > 0 \) and every \( X \in X \) there is a retraction of \( B(x, R) \) to \( \{ x\} \) whose image is contained in \( B(x, CR) \) at all times.

Theorem (Gromov): Let \( X, Y \) be two uniformly contractible simplicial complexes and \( F : X \to Y \) a quasi-isometry. Then there are naturally defined isomorphisms of topological vector spaces
\[
F^* : \ell_p H^k(Y) \to \ell_p H^k(X)
\]
and
\[
\overline F^* : \ell_p \overline H^k(Y) \to \ell_p \overline H^k(X).
\]

Sketch of proof

The proof proceeds by constructing maps \( \varphi : C_p^*(X) \to C_p^*(Y) \) and \( \psi : C_*(Y) \to C_*(X) \) such that \( \varphi \circ \psi \) and \( \psi \circ \varphi \) are both homotopic to the identity. The desired isomorphism is the map between cohomologies induced by \( \varphi \) (and since it will be easily seen to be continuous it also induces an isomorphism between reduced cohomologies).

The map \( \varphi \) is defined inductively on the sets \( X^{(k)} \). On \( X^{(0)} \) we define it to take \( x \) to the vertex of \( Y \) which is closest to \( F(x) \) (chosen arbitrarily of \( F(x) \) is equidistant from distinct vertices).

On \( X^{(1)} \) it can be defined by sending an edge \( [x_0 x_1] \) to a minimal path between \( \varphi(x_0) \) and \( \varphi(x_1) \). For \( k > 0 \), supposing that \( \varphi \) is defined on \( X^{(k-1)} \) one proceeds as follows. Let \( partial \) denote the usual boundary map on chains. Then for any simplex \( \sigma \in X^{(k)} \) we have \( \partial\varphi(\partial\sigma) = 0 \), and from the uniform contractibility of \( Y \) it follows that there exists \( \tau \in \mathbb Z Y^{(k-1)} \) with support in a uniformly bounded neighbourhood of \( F(\sigma) \) such that \( \varphi(\partial\sigma) = \partial\tau \). We then define \( \varphi(\sigma) = \tau \).

The map \( \psi \) is defined in the same way using a quasi-inverse of \( F \). It is then clear that \( \varphi \circ \psi, \psi \circ \varphi \) are at bounded distance from the identity maps on simplices. It remains to construct a map \( H : C_*(X) \to C_{*+1} \) such that
\[
H \circ \partial + \partial \circ H = \mathrm{Id} – \psi \circ \varphi.
\]
For this it is possible use a process similar to the one above. Here are the details for \( k = 0, 1 \): for \( x \in X^{(0)} \) define \( H(x) \) to be the path from \( x \) to \( \psi \circ \varphi(x) \). For \( [x_0x_1] \) let \( c \) be the cycle defined by going via shortest paths from \( x_0 \) to \( x_1 \) to \( \psi \circ \varphi(x_1 \) to \( \psi \circ \varphi(x_1) \) back to \( x_0 \). By uiform contarctiblity it is the boundary of a \( 2 \)-chain \( \tau \) and \( H([x_0x_1]) \) is defined to be \( \tau \).

Group cohomology

Let \( \Gamma \) be a discrete group. Let \( \pi \) be the right-regular representation of \( \Gamma \) on \( \ell_p \Gamma \), given by
\[
\pi(g) f(x) = f(xg).
\]
Let \( C^k(\Gamma, \ell_p \Gamma) \) be the space of functions \( \Gamma^{k+1} \to \ell_p\Gamma \). This is a \( \Gamma \)-module via the action
\[
\gamma f(x_0, \ldots, x_k) = \pi(g) f(\gamma^{-1}x_0, \ldots, \gamma^{-1}x_k).
\]
The \( \ell_p \)-cohomology of \( \Gamma \) is the cohomology of the complex \( C^k(\Gamma, \ell_p\Gamma)^\Gamma \) (where \( V^\Gamma \) means the subspace of fixed points of \( \Gamma \)). The differential is given by :
\[
df(\gamma_0, \ldots, \gamma_k) = \sum_{i=0}^k (-1)^i f(\gamma_0, \ldots, \hat\gamma_i, \ldots, \gamma_k).
\]
The reduced cohomology is defined as in the case of complexes above.

An action of \( \Gamma \) on a metric space is geometric if it is by isometries, properly discontinuous and cocompact.

Proposition: If \( \Gamma \) acts geometrically on a uniformy contractible simplicial complex of bounded geometry then \( \ell_p H^*(X) = H^*(\Gamma, \ell_p\Gamma) \) and similarly for reduced cohomology.

Lecture II

Here we will consider only the first cohomology space \( \ell_p H^1 \). For this we need \( X \) to be a simply connected simplicial complex of bounded geometry. We will denote by \( \ell_p X^{(k)} \) the space of \( p \)-integrable cochains. Then we have:
\[
\ell_p H^1(X) = \frac{\{\omega \in \ell_p X^{(1)} : \forall c \in Z_1(X) \omega(c) = 0\}}{\{ d\alpha : \alpha \in \ell_p X^{(0)}\}}
\]
and this can be identified with
\[
\frac{\{df : f \in {\mathbb R} X^{(0)}, df \in \ell_p X^{(1)} \}}{\{ d\alpha : \alpha \in \ell_p X^{(0)}\}}.
\]
This follows from the fact that \( X \) is simply connected, hence 1-cocycles are locally integrable.

This shows that the space \( \ell_p H^1(X) \) depends only on the 1-skeleton of \( X \). We can give a alternative definition of \( \ell_p \)-cohomology for graphs as follows: let \( G \) be a graph with bounded valencies and put
\[
\ell_p H^1(G) = \frac{ \{df : f \in {\mathbb R} X^{(0)}, df \in \ell_p G^{(1)} \}}{({\mathbb R} + \ell_p G^{(0)})}.
\]
There is a norm \( \|f\|_{1, p} = \|df\|_p \) on the modded-out space. The reduced cohomology is then defined by:
\[
\ell_p \overline H^1(G) = \frac{\{df : f \in {\mathbb R} X^{(0)}, df \in \ell_p G^{(1)} \}}{\left( {\mathbb R} + \overline{\ell_p G^{(0)}}^{\|\cdot\|_{p, 1}} \right)}.
\]
For a group \( \Gamma \) let \( G \) be a Cayley graph for \( \Gamma \). Then we have a first \( \ell_p \)-cohomology space \( H^1(\Gamma, \ell_p\Gamma ) = \ell_p H^1(G) =: \ell_p H^1(\Gamma) \) and a reduced cohomology \( \ell_p \overline H^1(\Gamma) = \ell_p \overline H^1(G) \).

Proposition: The equality \( \ell_p H^1(\Gamma) = \ell_p \overline H^1(\Gamma) \) holds if and only if \( \Gamma \) is not amenable.

In particular this implies that \( \ell_p H^1(\Gamma) \not= 0 \) if \( \Gamma \) is amenable.

Proof of the proposition

The conclusion that \( \ell_p H^1(\Gamma) = \ell_p \overline H^1(\Gamma) \) is equivalent to \( d_0 \) having closed range in \( \ell_p X^{(1)} \). By general functional analysis this is the case if and only if there exists \( C \) such that
\[
\forall f \in \ell_p\Gamma = \ell_p X^{(0)} : \| f \|_p \le C \| df \|_p.
\]
Applying this condition to the characteristic functions \( 1_A \) for \( A \) a finite set in \( \Gamma \) we get that if \( d \) has not a closed range then there is a sequence \( A_n \) of such sets with \( \|d 1_{A_n}\|_p < \| 1_{A_n} \|_p / n \) and since \( d1_A \) is essentially \( 1_{\partial A} \) we see that \( A_n \) is a Følner sequence for \( \Gamma \).

By a theorem of Mazzia the converse is also true.

Cohomology and harmonic functions

For this simplified theory of \( \ell_p \)-cohomology there is an analogue of Hodge theory.

Proposition: Let \( \Gamma = \langle S \rangle \) a finitely generated group and \( \Delta \) the Laplacian on \( \Gamma \) defined by
\[
\Delta f = |S| \cdot f – \sum_{s \in S} f(\cdot s).
\]
Suppose that \( \Gamma \) is not amenable. Then
\[
\ell_p H^1(\Gamma) = \{ f \in \mathbb C\Gamma : df \in \ell_p \Gamma^{(1)}, \Delta f = 0 \} / {\mathbb R}.
\]

This has the following useful consequences.

Corollary:

1. If \( p \le q \) then there is a natural inclusion \( \ell_p H^(\Gamma) \subset \ell_q H^1(\Gamma) \).
2. If \( f \in \ell_p\Gamma \) then \( [f] = 0 \) in \( \ell_p H^1(\Gamma) \) if and only if \( \lim_{|g|\to+\infty}f(g) \) exists.

The point 1 is an immediate consequence of the proposition , while to deduce 2 the maximum principle for harmonic functions is needed.

Proof of the proposition

The assumption that \( \Gamma \) is not amenable implies that \( \Delta \) is invertible when restrited to \( \ell_p \) is invertible. To see this let:
\[
Mf = \frac 1{|S|} \sum_{s \in S} f(\cdot s).
\]
Kesten’s criterion for amenability states that \( \Gamma \) is not amenable if and only if the operator norm \( \| M \|_{\ell_2} < 1 \). On the other hand it is trivial to see that \( \| M \|_{\ell_\infty} = 1 \), and by an interpolation result it follows that \( \| M \|_{\ell_p} < 1 \) for \( 2 \le p < +\infty \). By a duality argument the same is true for \( 1 < p \le 2 \). Since \( \Delta = \mathrm{Id} – M \) this implies that \( \Delta \) is invertible for these values of \( p \).

Now let \( f \) such that \( df \in \ell_p\Gamma^{(1)} \), we want to find an harmonic function \( h \) such that \( [h] = [f] \) in \( \ell_p H^1\Gamma \). We have \( \Delta f = \partial df \) where \( \partial : \ell_p\Gamma^{(1)} \to \ell_p\Gamma \) is bounded and it follows that \( \Delta_f \in \ell_p\Gamma \). Thus by the previous paragraph there exists \( u \in \ell_p\Gamma \) with \( \Delta u = \Delta f \). Let \( h = f – u \); then \( \Delta h = 0 \) and \( [h] = [f] \) since \( u \in \ell_p\Gamma \).

Questions

Gromov asked whether \( \ell_p \overline H^k(\Gamma) = 0 \) for all \( k \ge 0 \) and \( p \in ]1, +\infty [ \) when \( \Gamma \) is amenable.

This is known to be true for \( p=2 \) and any \( k \), by the work of Cheeger–Gromov. For \( k=1 \) this is known for all \( p \), when the group \( \Gamma \) has the Liouville propetry (there are no nonconstant bounded harmonic functions on \( \Gamma \)), by work of Antoine Gournay.

This is also known for all \( p \) and \( k \) when \( \Gamma \) has an infinite center, for example for nilpotent groups.

Lecture III

Hyperbolic groups

Theorem: Let \( \Gamma \) be a word-hyperbolic group. Then for \( p \) large enough we have \( \ell_p H^1(\Gamma) \neq 0 \).

For example take \( \Gamma = F_2 \), a free group on two generators. Its Cayley graph \( G \) with respect to free generators is a 4-valent tree. Any edge \( e \) of \( G \) separates \( G \) into two subtrees and a function \( f \) which is constant equal to 1 on one of these trees and 0 on the rest of the Cayley graph gives an element in \( \ell_p H^1(\Gamma) \) (its differential is supported on the removed edge). It is nonzero by Corollary 2 above. This shows that \( \ell_p H^1(F_2) \not= 0 \) for any \( p \in ]1, +\infty[ \).

General case: Elek’s construction

Let \( \partial\Gamma \) be the visual boundary of the Cayley graph \( G \). We fix the vertex \( x_0 = 1_\Gamma \in \Gamma \) and some \( a > 1 \) and define the visual distance on \( \partial\Gamma \) by:
\[
d(\xi, \eta) = a^{-L}, \, L = d(x_0, ]\xi, \eta[)
\]
where \( ]\xi, \eta[ \) is the geodesic line with endpoints \( \xi, \eta \). Alternatively, \( L \) is the length of the intersection of the rays representing \( \xi, \eta \) and starting at \( x_0 \). Let:
\[
Q = \inf \left( s > 0 : \sum_{g \in \Gamma} a^{-s|g|} < +\infty \right).
\]
We have \( Q > 0 \) unless \( \Gamma \) is virtually cyclic.

Let \( u : \partial\Gamma \to {\mathbb R} \) be a nonconstant Lipschitz function. For \( g \in \Gamma \) choose (arbitrarily) a point \( \xi_g \in \partial\Gamma \) such that \( g \in [x_0, \xi_g[ \) (for the construction below it suffices that \( g \) belongs to the \( R \)-neighbourhood of \( [x_0, \xi_g[ \) for some \( R \) depending only on \( \Gamma \)). Define \( f \) by :
\[
f(g) = u(\xi_g)
\]
we will first show that \( df \in \ell_p G^{(1)} \) for \( p > Q \). We have :
\[
\| df \|_p^p = \sum_{e \in G^{(1)}} |f(e_+) – f(e_-)|^p = \sum_{e \in G^{(1)}} |u(\xi_{e_+}) – u(\xi_{e_-})|^p
\]
and since \( u \) is Lipschitz we get that there is \( K \) such that:
\[
\| df \|_p^p \le \sum_{e \in G^{(1)}} K^p \cdot d(\xi_{e_+}, \xi_{e_-})^p \ll \sum_{e \in G^{(1)}} a^{-p \cdot L(\xi_{e_+}, \xi_{e_-})}.
\]
By hyperbolicity of \( G \) we have \( L(\xi_{e_+}, \xi_{e_-}) = d(x_0, e) \) up to an additive constant and it follows that
\[
\| df \|_p^p \ll \sum_{e \in G^{(1)}} a^{-p \cdot d(x_0, e)}
\]
and by definition of \( Q \) the right-hand side is finite whenever \( p > Q \).

Thus we get a class \( [f] \in \ell_p H^1(\Gamma) \). It is nonzero by Corollary 2 because the boundary values of \( f \) are given by \( u \) which is nonconstant.

Remark

The number \( Q \) is not a quasi-isometry invariant of \( \Gamma \), for example it depends on the choice of Cayley graph \( G \). It can be shown that the construction above actually gives a class in \( \ell_p H^1(\Gamma) \) for all \( p \) larger than the conformal dimension of the boundary \( \partial\Gamma \) for the visual metric \( d \) (which is a QI-invariant).

Actions on \( L^p \)-spaces

The following question was asked by Guoliang Yu.

Which groups admit proper affine isometric actions on a space \( L^p({\mathbb R}, \mu) \)for some \( p < +\infty \)?

Here proper means metrically proper, i.e. any ball intersects only finitely many of its translates.

It is easy to see that the action of \( \Gamma \) on \( \ell_\infty\Gamma \) given by:
\[
g \cdot f = \pi(g) \cdot f + c(g), c(g) = |xg| – |g|
\]
is always proper. On the other hand there exist groups with no proper action on a \( L^p \) space for \( p < +\infty \).

Theorem (Bader–Furman–Gelander–Monod): Let \( \Gamma = \mathrm{SL}(3, \mathbb Z) \) (or any lattice in a simple higher-rank Lie group) then any isometric affine action of \( \Gamma \) on a \( L^p \)-space has a fixed point.

On the other hand for hyperbolic groups the situation is very different.

Theorem (Yu): If \( \Gamma \) is an hyperbolic group then for \( p \) large enough \( \Gamma \) acts proeprly by isometries on \( \ell_p\Gamma \).

Sketch of proof for Yu’s theorem

Let \( V \) be a Banach space with an isometric \( \Gamma \)-action. Then we can write
\[
g \cdot v = \pi(g)\cdot v + c(g)
\]
where \( \pi \) is a linear representation and \( c : \Gamma \to V \) is a cocycle, that is
\[
\forall g_1, g_2 \in \Gamma : c(g_1g_2) = c(g_1) + \pi(g_1)c(g_2).
\]
It is easy to see that the action is proper if and only if \( \| c \| : \Gamma \to {\mathbb R} \) is proper, that is
\[
\lim_{|g| \to +\infty} \| c(g) \| = +\infty.
\]

Let \( \pi \) be the right-regular representation of \( \Gamma \) on \( \ell_p\Gamma \). Our goal is now to construct a proper cocycle \( c \in Z^1(\Gamma, \pi) \). Let \( u \) be a Lipschitz function on \( \partial\Gamma \) and \( f \) its radial extension constructed above. We have \( f \not\in \ell_p\Gamma \) but the cocycle:
\[
c(g) = f – \pi(g)f
\]
takes values in \( \ell_p\Gamma \) for large enough \( p \), as we will now see. The cocycle rules reduces the problem to prove it for the generating set. But for these elements the computation proving that \( df \in \ell_p G^{(1)} \) also shows that \( c(s) \in \ell_p\Gamma \) for \( p > Q \), which proves the claim.

It remains to show that \( c \) is proper. For this we will make the simplifying assumption that \( |u(\xi) – u(\eta)| \ge 1 \) whenever \( \xi, \eta \) are antipodal (meaning \( x_0 \in ]\xi, \eta[ \))—note that in general it is not possible to find such a function, for example on the sphere \( \mathbb S^2 \) every function has two antipodal points with the same value. But by taking a linear combination of enough cocycles one can extend the argument to the general case.

Let \( g \in \Gamma \) and write \( g = s_1 \cdots s_n, n = |g| \) a shortest decomposition of \( g \) into a product of generators. Put \( g_i = s_1 \cdots s_i \) for \( i = 1, \ldots, n \). Then we have:
\[
\| c(g) \|^p \ge \sum_{i=1}^n |c(g)(g_i^{-1})|^p = \sum_{i=1}^n |f(g_i^{-1}) – f(g_i^{-1}g)|^p
\]
and since \( g_i^{-1} \) and \( g_i^{-1}g \) are antipodal, under our assumption on \( u \) the right-hand side is at least \( |g| \). This shows that \( c \) is proper.

Lecture IV

In the rest of the lectures we will explore cohomology in higher degrees for various groups and manifolds.

\( \ell_p \)-homology

Let \( X \) be a simplicial complex of bounded geometry and \( p \in ]1, +\infty[ \). The complex of \( p \)-integrable chains of \( X \) is \( C_{*, p}(X), \partial_* \) where
\[
C_{k, p}(X) = \left\{ \sum_{\sigma \in X^{(k)}} a_\sigma \sigma : \sum_{\sigma \in X^{(k)}} |a_\sigma|^p < +\infty \right\}
\]
and if \( [v_0 \cdots v_k] \) denites the \( k \)-simplex with vertices \( v_0, \ldots, v_k \):
\[
\partial_k [v_0 \cdots v_k] = \sum_{i=0}^k (-1)^i [v_0 \cdots \hat v_i \cdots v_k].
\]
The \( \ell_p \) homology and reduced \( \ell_p \) homology of \( X \) are then defined by
\[
\ell_p H_k(X) = \ker(\partial_k) / \mathrm{Im}(\partial_{k+1})
\]
and
\[
\ell_p \overline H_k(X) = \ker(\partial_k) / \overline{\mathrm{Im}(\partial_{k+1})}.
\]

Proposition: Let \( p \in ]1, +\infty[ \) and \( q = p^* \) be defined by \( 1/p + 1/q = 1 \). There is a natural isomorphism between \( \ell_p \overline H^k(X) \) and the dual space \( (\ell_q \overline H_k(X))^* \).

The proof is classical functional analysis.

Remark:

In general there is no relation between the nonreduced homology spaces \( \ell_p H^k(X) \) and \( \ell_q H^k(X) \). For example whenever \( X \) is infinite we have \( \ell_p H^0(X) = 0 \) but \( \ell_q H_0(X) = 0 \) if and only if \( \partial_1 \) has a closed range, and this is the case if and only if \( X \) has a linear isoperimetric inequality.

Nilpotent groups

Proposition: Suppose that \( X \) is uniformly contractible and that it admits a geometric action of a group \( \Gamma \) whose center \( Z(\Gamma) \) is infinite. Then \( \ell_p \overline H^k(X) = 0 \) for all \( p, k \).

To prove this we reason by contradiction. Suppose that there exists a nonzero class \( [\omega] \in \ell_p \overline H^k(X) \). By the duality in the previous Proposition it follows that there is a \( k \)-cycle \( c \) with \( [c] \in \ell_q \overline H_k(X) \) and \( \langle \omega, c \rangle = 1\).

Now suppose that \( z \in Z(\Gamma) \). Then its action on \( X \) is at bounded distance from the identity, and it follows that \( z^*[\omega] = [\omega] \). This yields:
\[
1 = \langle \omega, c \rangle = \langle [\omega], [c] \rangle = \langle z^*[\omega], [c] \rangle = \langle z^*\omega, c \rangle
\]
but by applying this to a sequence \( z_n \to +\infty \) we get a contradiction.

As finitely generated nilpotent groups virtually have infinite center teh following result is a direct consequence.

Corollary: If \( \Gamma \) is nilpotent then \( \ell_p \overline H^k(X) = 0 \).

\( L^P \)-cohomology of manifolds

In this paragraph \( X \) is a cellulation of an orientable manifold \( M \). We will assume in addition that \( M \) has a Riemannian metric and that the metric on \[ X \) is quasi-isometric to that on \( M \) (an example is when \( M \) is the universal cover of a compact manifold and \( X \) the lift of a cellulation of this manifold).

The classical proof of Poincaré duality through the dual cellulation carries over to the \( \ell_p \) setting to give the following result.

Proposition (Poincaré duality): Let \( n = \dim(M) \). Then for \( k = 0, \ldots, n \) there is an isomorphism between \( \ell_p H^k(X) \) and \( \ell_p H_{n-k} (X) \), as well as between \( \ell_p \overline H^k(X) \) and \( \ell_p \overline H_{n-k} (X) \).

Together with results above this has the following useful consequences.

Corollary: We have \( \ell_p H^n(X) = 0 \) if and only if \( M \) has a linear isoperimetric inequality.

Corollary: Let \( q = p^* \). If \( \ell_p H^k(X) = 0 \) then \( \ell_q H^{n-k}(X) = 0 \) and moreover \( \ell_q H^{n-k+1}(X) = \ell_q \overline H^{n-k+1}(X) \).

Hyperbolic space

Real hyperbolic \( n \)-space \( {\mathbb H}_{\mathbb R}^n \) is the unique simply connected complete Riemannian manifold of constant sectional curvature -1. Since there exists compact hyperbolic manifolds it admits a cellulation \( X \) which is quasi-isometric to its Riemannian metric, and the \( \ell_p \)-cohomology groups \( \ell_p H^k(X) \) as well as their reduced analogues are well-defined.

Theorem (Pansu): If \( p \frac{n-1}{k-1} \) then \( \ell_p H^k(X) = 0 \). When \( p \in ] \frac{n-1} k, \frac{n-1}{k-1} [ \) we have \( \ell_p H^k(X) = \ell_p \overline H^k(X) \neq 0 \).

Lecture V

Proof of Pansu’s theorem

Let \( X \) be an appropriate cellulation of \( {\mathbb H}_{\mathbb R}^n \). Since
\[
\frac{n-1} k = \left( \frac{n-1}{n-k-1} \right)^*
\]
by Poincaré duality (this Corollary), to prove the vaninshing statement and the fact that \( \ell_p H^k(X) = \ell_p \overline H^k(X) \) in the range \( p \in ] \frac{n-1} k, \frac{n-1}{k-1} [ \) it suffices to show that \( \ell_p H^k(X) = 0 \) for \( p > \frac{n-1}{k-1} \). We will not prove the nonvanishing statement.

For \( k = n, 0 \) this is clear. Let \( k \in [1, n-1] \) and \( p \in ]\frac{n-1}{k-1}, +\infty[ \). Let \( \omega \in C_p^k(X) \) be a cocycle, that is \( d\omega = 0 \). We want to show that \( \omega \) is a coboundary, that is there exists \( \alpha \in C_p^{k-1}(X) \) such that \( d\alpha = \omega \). We will give an explicit construction for such an \( \alpha \).

Let \( \Omega^*({\mathbb H}_{\mathbb R}^n), d \) be the de Rham complex of smooth differential forms on \( {\mathbb H}_{\mathbb R}^n \). Any \( k \)-cochain can be represented by an element of \( \Omega^k({\mathbb H}_{\mathbb R}^n) \), and cocycles correspond to closed forms. Let \( \tilde\omega \in \Omega^k \) represent \( \omega \).

Let \( \sigma \in X^{(k-1)} \). We pick an arbitrary point \( \infty \) on the boundary of \( {\mathbb H}_{\mathbb R}^n \). The cone \( C_\sigma \) is the union of the geodesic rays \( [x, \infty[ \) for \( x \in \sigma \); since there are countably many simplices we may assume it is a smooth \( k \)-subamnifold. This allows us to define:
\[
\alpha(\sigma) = \int_{C_\sigma} \tilde\omega
\]
which defines a \( k-1 \)-cochain. If \( \tilde\alpha \) is adifferential form representing \( \alpha \) then by Stokes’ theorem we have \( d\tilde\alpha = \tilde\omega \). Thus we only need to prove that \( \| \alpha \|_p < +\infty \).

Let \( V \) be the unit vector field on \( {\mathbb H}_{\mathbb R}^n \) which points towards \( \infty \) everywhere and \( \phi_t \) its flow. We have:
\[
\alpha(\sigma) = \int_0^{+\infty} \int_\sigma \phi_t^*(\iota_V \omega) dt
\]
where \( \iota_V\omega \in \Omega^{k-1} \) is the contraction with \( V \), i.e. \( \iota_V\omega(v_1, \ldots, v_{k-1}) = \omega(V, v_1, \ldots, v_{k-1}) \). Let
\[
\alpha_t(\sigma) = \int_\sigma \phi_t^*(\iota_V \omega)
\]
so that \( \alpha_t \in C_p^{k-1}(X) \). We have:
\[
\|\alpha_t\|_p^p = \sum_{\sigma \in X^{(k-1)}} \left| \int_\sigma \phi_t^*(\iota_V \omega) \right|^p \le \sum_{\sigma \in X^{(k-1)}} \int_\sigma |\phi_t^*(\iota_V \omega)|_x^p dv_\sigma(x)
\]
where \( v_\sigma \) is the volume element of \( \sigma \). Now the flow \( \phi_t \) is exponentially contracting in the transverse direction and it follows that
\[
|\phi_t^*(\iota_V \omega)|_x^p \le e^{-p(k-1)t} |\iota_V \omega|_{\phi_t(x)}^p.
\]
In addition, since \( V \) is unit length we have \( |\iota_V \omega| = |\tilde\omega| \) so that
\[
\| \alpha_t \|_p^p \le e^{-p(k-1)t} \int_{X^{(k-1)}} |\tilde\omega|_{\phi_t(x)}^p dv(x)
\]
where \( dv \) is the volume element on the \( (k-1) \)-skeleton, well-defined outside of a codimension-1 subset. The right-hand side above is at most, up to a multiplicative constant:
\[
e^{-p(k-1)t} \int_{{\mathbb H}_{\mathbb R}^n} |\tilde\omega|_{\phi_t(x)}^p d\mathrm{vol}(x) = e^{-p(k-1)t} \int_{{\mathbb H}_{\mathbb R}^n} |\tilde\omega|_*^p d(\phi_t^*\mathrm{vol})(x)
\]
and since \( \phi_t^*\mathrm{vol} = e^{(n-1)t}\mathrm{vol} \) we finally get that
\[
\| \alpha_t \|_p^p \le C e^{-(p(k-1) – (n-1))t} \cdot \|\omega\|_p^p.
\]
It follows that for \( p(k-1) > n-1 \) the integral
\[
\| \alpha \|_p^p \le \int_0^{+\infty} \| \alpha_t \|_p^p dt \le C\|\omega\|_p^p \int_0^{+\infty} e^{-(p(k-1) – (n-1))t} dt
\]
is convergent.

Generalisation

We saw in Tullia Dymarz’s lectures that negatively curved homogeneous spaces are obtained as Heintze groups, for example:
\[
G = {\mathbb R}^{n-1} \times_{\alpha^t} {\mathbb R}
\]
where \( \alpha \) is a diagonal matrix with eigenvalues \( e^{\lambda_i} \), \( 0 {\mathrm{tr}} / w_{k-1} \) and it follows by Poincaré duality that \( \ell_p H^k(G) = 0 \) when \( p < {\mathrm{tr}}/w_k \) as well (note that when all \( \lambda_i = 1 \) we recover exactly the vanishing result for real hyperbolic space). This also shows that in the interval \( ]{\mathrm{tr}} / w_k, {\mathrm{tr}} / w_{k-1}[ \) the reduced and non-reduced \( \ell_p \)-cohomologies are equal. It is also true that in this interval they are both nonzero.

The quasi-isometry invariance of \( \ell_p \)-cohomology allows to deduce the following corollary.

Corollary (Pansu): The Heintze groups \( G, G’ \) are quasi-isometric to each other if and only if \( {\mathrm{tr}}/w_i = {\mathrm{tr}}’/w_i’ \) for all \( i = 1, \ldots, n-1 \).

Open questions

A major open problem for \( L^2 \)-cohomology is the following.

Singer’s conjecture: Let \( M \) be a closed aspherical \( n \)-manifold and \( X = \tilde M \) its universal cover. Then \( \ell_2 H^k(X) = 0 \) if \( k \not= n / 2 \). Moreover, if \( n = 2p \) and \( M \) admits a negatively curved Riemannian metric then \( \ell_2 H^p(X) \neq 0.

Pansu’s theorem shows that this is true when \( M \) is a quotient of a Heintze group.

Question (Gromov): Let \( X \) be a symmetric space with no compact or Euclidean factors. Let \( r \) be its rank, the maximal dimension of a totally geodesic flat submanifold. Does \( \ell_p H^k(X) = 0 \) hold for all \( p \) whenever \( k < r \)?

Question (Hamenstädt): Let \( X \) be a \( n \)-dimensional simply-connected Riemannian manifold with negative curvature. For a \( k \in [1, n-1] \), does there exists a \( p \in ]1, +\infty[ \) such that \( H_k^p(X) \neq 0 \)?

1. First lecture
2. Second lecture
3. Third lecture
4. Fourth lecture
5. Fifth lecture

Lecture I

General Outline and references

This series of lectures will be concerned with the following topics.

1. Quasi-isometric rigidity of uniform lattices in \( \mathbb{H}^{n+1} \) for \( n \ge 2 \) (Tukia’s theorem). The reference here is Drutu–Kapovich’s book Geometric group theory (available here).
2. QI-rigidity of lattices in certain solvable Lie groups, using generalisations of Tukia’s theorem. This is work of Dymarz and Xie, detailed in their paper Day’s fixed point theorem, Group cohomology and Quasi-isometric rigidity.
3. Comparison between bi-Lipschitz and quasi-isometric equivalence for discrete groups, following Dymarz’s paper Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups, which shows that certain lamplighter groups are quasi isometric to each other but not bi-Lipshitz.

Preliminary material on quasi-isometries

Definition of quasi-isometric rigidity

Let \( X \) be a metric space. We say that it is quasi-isometrically rigid (« QI-rigid ») if for any finitely generated group \( \Gamma \) which is quasi-isometric to \( X \) there exists a morphism \( \Gamma \to \mathrm{Isom}(X) \) with finite kernel, and with image acting properly discontinuously and cocompactly.

The quasi-isometry group of a space and quasi-actions

Recall the definition of a \( (K, C) \)-quasi-isometry of \( X \) from Kapovich’s lectures: it is a map \( X \to X \) which is a \( (K, C) \)-quasi-isometruc embedding, that is
\[
\frac 1 K d(x, x’) – C \le d(f(x), f(x’)) \le Kd(x, x’) + C
\]
and in addition has a quasi-inverse \( \bar f \) which is also a \( (K, C) \)-quasi isometry. This means that there exists \( D \) such that \( d(\bar f f(x), x) \le D \) and \( d(f\bar f(x), x) \le D \) for all \( x \in X \). We will also say that \( \bar f f \) (and \( f \bar f \)) are at bounded distance from the identity.

The set of quasi-isometries of \( X \) is not a group because maps in there often fail to have an inverse. Let us introduce the equivalence relation \( \sim \) on the self-maps of \( X \), defined by \( f \sim g \) if and only if \( f \) and \( g \) are at bounded distance of each other, i.e. \( \exists D : \forall x \in X d(f(x), g(x)) \le D \). The quasi-isometry group of \( X \) is then defined to be
\[
\mathrm{QI}(X) = \{ \text{quasi-isometries of } X \} / \sim
\]
which is a group by definition of the quasi-inverse.

It is easy to verify that if \( f: \Gamma \to X \) is a quasi-isometry, the map \( \varphi : \Gamma \to \mathrm{QI}(X) \) defined by \( \phi(\gamma) = f\gamma\bar f \) (where \( \gamma \) is seen as acting on \( \Gamma \) by left-translations) is a group morphism. Such a morphism may be called a quasi-action of \( \Gamma \) on \( X \) (compare with the definition in Kapovich’s lectures).

In fact we can say a bit more about the quasi-action of \( \Gamma \) on \( X \) coming from the quasi-isometry \( f \).

1. Its image is uniform in the obvious sense: there are \( (K, C) \) such that every element \( g \in \varphi(\Gamma) \) can be represented by a \( (K, C) \)-quasi-isometry. (Note: in addition, the representative of \( \varphi(\gamma\eta) \) is at bounded distance from the product of that for \( \varphi(\gamma) \) with that for \( \varphi(\eta) \).)
2. The quasi-action is cobounded, that is there exists \( B \) and \( x_0 \in X \) such that for every \( x \in X \) there exists \( \gamma \in \Gamma \) with \( d(x_0, \varphi(\gamma) x) \le B \) (where by \( \varphi(\gamma) \) we mean a \( (K, C) \)-quasi-isometric representative). (Note: this needs the uniformity of the action to make sense.)

Strong quasi-isometric-rigidity

For any \( X \) there is a natural morphism \( \mathrm{Isom}(X) \to \mathrm{QI}(X) \). The space \( X \) is said to be strongly QI-rigid if this is an isomorphism. This implies almost immediately that if \( X \) is proper then \( X \) is QI-rigid in the sense above: from \( \varphi \) we obtain a proper action on \( X \), and the coboundedness of the quasi-action translates into cocompactness.

In general the morphism \( \mathrm{Isom}(X) \to \mathrm{QI}(X) \) is neither injective nor surjective. For example if \( X = \mathbb{R}^n \) the image of the translations in \( \mathrm{QI}(X) \) is trivial. And \( X = \mathbb{H}^2 \) has many more classes of quasi-isometries than those of isometries (any lift of a diffeomorphism between nonisometric compact hyperbolic surfaces gives an example of a quasi-isometry not at bounded distance from any isometry).

However, when \( \mathrm{Isom}(X) \subset \mathrm{QI}(X) \) it is sometimes possible to conjugate \( \varphi(\Gamma) \) into \( \mathrm{Isom}(X) \).

Quasi-isometric rigidity of hyperbolic space

The proof of quasi-isometric rigidity for the hyperbolic space \( \mathbb{H}^{n+1} \), \( n \ge 2 \) proceeds as follows :

• The visual boundary \( \partial\mathbb{H}^{n+1} \) is naturally identified with the sphere \( \mathbb{S}^n \) with its natural conformal structure. Then quasi-isometries of \( \mathbb{H}^{n+1} \) extend to \( \mathbb{S}^n \) as quasi-conformal homeomorphisms. The isometries are exactly those maps which identify with conformal maps. This gives group isomorphisms:
  \[
  \mathrm{QI}(\mathbb{H}^{n+1}) = \mathrm{QC}(\mathbb{S}^n), \: \mathrm{Isom}(\mathbb{H}^{n+1}) = \mathrm{Conf}(\mathbb{S}^n).
  \]
• Tukia’s theorem can then be restated as follows: if \( \varphi \) is a uniform cobounded quasi-action of a discrete group \( \Gamma \) on \( \mathbb{H}^{n+1} \) then the subgroup \( \partial\varphi(\Gamma) \subset \mathrm{QC}(\mathbb{S}^n) \) can be conjugated into \( \mathrm{Conf}(\mathbb{S}^n) \).

Quasi-isometric rigidity of solvable groups

Let \( S \) be a solvable Lie group with a left-invariant Riemannian metric. Then there is an isometric embedding of \( S \) into a product \( H_1 \times \cdots \times H_r \) where \( H_i \) are homogeneous negatively curved Riemannian manifolds. For example there is an embedding of the 3-dimensional group \( \mathrm{Sol} \) into \( \mathbb{H}^2 \times \mathbb{H}^2 \) (see Whyte’s lectures). A scheme to attempt to prove that \( S \) is QI-rigid then goes roughly as follows:

• There are boundaries \( \partial_i S = \partial H_i \setminus \{ \infty \} \) (where \( \infty \) is a point on \( \partial H_i \) depending on the embedding).
• Ideally this gives an isomorphism
  \[
  \mathrm{QI}(S) = \prod_{i=1}^r \mathrm{Bilip}(\partial_i S)
  \]
  (where \( \partial_i S \) has a natural metric, for example the euclidean metric if \( H_i = \mathbb{H}^n \) so that \( \partial_i S = \mathbb{R}^n \)). This is a technical step which uses the coarse differentiation of Eskin–Fisher–Whyte.
• The isometry group \( \mathrm{Isom}(S) \) is contained in the product of the « similarity groups » \( \mathrm{Sim}(\partial_i S) \) (in the case where \( H_i = \mathbb{H}^n \) this is just the group of Euclidean similarities). One needs to generalise Tukia’s theorem to be able to conclude that the image \( \varphi(\Gamma) \) in \( QI(S) = \prod_i \mathrm{Bilip}(\partial_i S) \) of a discrete group \( \Gamma \) quasi-isometric to \( S \) can be conjugated into \( \prod_i \mathrm{Sim}(\partial_i S) \).
• There is a last step needed to conclude.

Lecture II

quasi-conformal maps

Uniform subgroups of \( \mathrm{QC} \)

An homeomorphism \( f : \mathbb{S}^n \to \mathbb{S}^n \) is said to be \( K \)-quasiconformal if it satisfies the following. For \( x \in \mathbb{S}^n \) and \( \varepsilon > 0 \) let
\[
K_\varepsilon(f)(x) = \sup_{d(y, x) = \varepsilon = d(z, x)} \frac{d(f(x), f(y))}{d(f(x), f(z))}
\]
(this measures « how much \( f \) distorts circles around \( x \) »). Then for all \( x \):
\[
\limsup_{\varepsilon \to 0} (K_\varepsilon(f)(x)) \le K.
\]

The set of all quasi-conformal homeomorphisms is a group which was denites by \( \mathrm{QC}(\mathbb{S}^n) \) in the previous lecture. A subgroup \( G \subset \mathrm{QC}(\mathbb{S}^n) \) is said to be uniform if there is some \( K > 0 \) such that all of its elements are \( K \)-quasiconformal. The proof that the boundary extensions of quasi-isometries of \( \mathbb{H}^{n+1} \) yield quasi-conformal maps of \( \mathbb{S}^n \) (see Kapovich’s lectures) also shows that the boundary action of a uniform group of quasi-isometries yields a uniform subgroup of \( \mathrm{QC}(\mathbb{S}^n) \).

Tukia’s theorem

The statement of Tukia’s theorem that will be proven is the following.

Theorem: Let \( n \ge 2 \) and \( G \subset \mathrm{QC}(\mathbb{S}^n) \) be a uniform subgroup. Suppose in addition that the action of \( G \) on distinct triples \( (a, b, c) \in (\mathbb{S}^n)^3 \) is cocompact. Then there exists \( g \in \mathrm{QC}(\mathbb{S}^n) \) such that \( g G g^{-1} \subset \mathrm{Conf}(\mathbb{S}^n) \).

The hypotheses are satisfied if \( G \) is the boundary extension of a uniform cobounded group of quasi-isometries of \( \mathbb{H}^{n+1} \) (we saw this previously for the uniformity, and the cocompacity follows from the fact that there is a proper quasi-equivariant map from the space \( T(\mathbb{S}^n) \) of distinct triples to \( \mathbb{H}^{n+1} \), defined for example by associating to a triple the center of gravity of the ideal triangle having this triple as vertices).

Characterisation of conformal maps

The following characterisation of conformal maps among merely quasiconformal ones is very natural to state but nontrivial to prove. It follows for example from Gehring’s theorem stated in Kapovich’s lectures.

Theorem: A 1-quasiconformal map is conformal.

Radial points

Let \( G \subset \mathrm{QC}(\mathbb{S}^n) \). We view it as quasi-acting on \( \mathbb{H}^{n+1} \) and we fix a basepoint \( x_0 \in \mathbb{H}^{n+1} \). Let \( x \in \mathbb{S}^n \); it is called a radial point (or sometimes conical limit point) for \( G \) if there exists a geodesic ray \( \rho \) in \( \mathbb{H}^{n+1} \) such that \( \partial\rho = \{ x \} \), and a sequence \( g_i \in G \) such that \( g_i x_0 \) converges to \( x \) in \( \mathbb{H}^{n+1} \cup \partial \mathbb{H}^{n+1} \) and stays within bounded distance of \( \rho \).

If \( G \) acts cocompactly on \( T(\mathbb{S}^n) \) (equivalently its quasi-action on \( \mathbb{H}^{n+1} \) is cobounded) then every point in \( \mathbb{S}^n \) is a radial point.

Measurable conformal structures

Recall that an homeomorphism is quasiconformal if and only if it is differentiable at almost every point in \( \mathbb{S}^n \) and the ratio between the maximal eigenvalues of its derivatives is bounded. To take advantage of this we introduce a space of conformal structures on \( \mathbb{S}^n \).

We will consider objects which are defined only almost everywhere so it is more convenient to work on \( \mathbb{R}^n \) rather than \( \mathbb{S}^n = \mathbb{R}^n \cup \{\infty \} \). A measurable conformal structure on \( \mathbb{R}^n \) is a measurable map
\[
\mu : \mathbb{R}^n \to S := \{ n\times n\text{ symmetric positive definite matrices with determinant } 1 \}.
\]
Some remarks:

• We may identify \( S \) with the Riemannian symmetric space \( \mathrm{SL}(n, \mathbb{R}) / \mathrm{SO}(n) \).
• If \( \mu(x) = \mathrm{Id} \) for almost every \( x \) we say that \( \mu \) is the standard conformal structure.

An application \( f : \mathbb{R}^n \to \mathbb{R}^n \) which is differentible at almost every point is said to be conformal between two structures \( \mu \) and \( \mu’ \) if
\[
\mu(x) = Df_x^T \cdot \mu'(x) \cdot Df_x
\]
for almost every \( x \). For \( \mu = \mu’ \) the standard conformal structure in \( \mathbb{R}^n \) we get back the usual notion of conformal map.

For a matrix \( A \in S \) with largest eigenvalue \( \lambda_+ \) and smallest one \( \lambda_- \) let
\[
K(\mathrm{Id}, A) = \frac n 2 \max(\log(\lambda_+), -\log(\lambda_-))
\]
and extend it to a \( \mathrm{SL}(n, \mathbb{R}) \)-invariant function on \( S \times S \). Then a map is \( K \)-quasiconformal between \( \mu \) and \( \mu’ \) if and only if it is differentiable almost everywhere and
\[
\exp(K(\mu(x), Df_x^T\cdot \mu'(x) \cdot Df_x)) \le K
\]
for almost every \( x \). The notion of quasiconformality does not depend on the conformal structure. The first step in proving Tukia’s theorem will be to find a conformal structure \( \mu \) which is preserved by \( G \).

If \( f \) is quasiconformal let
\[
\mu_f = Df^T \cdot Df
\]
which is a conformal structure on \( \mathbb{R}^n \). We have the following equivariance property :
\[
\forall g \in \mathrm{QC}(\mathbb{R}^n) : \mu_{fg} = Dg^T \cdot (\mu_f \circ g) \cdot Dg.
\]
Now define :
\[
M_x = \{ \mu_f(x) : f \in G \} \subset S.
\]
Because \( G \) is uniform this is a bounded subset of \( S \) (with the distance given by \( K \)). In addition it is \( G \)-invariant under the action given above, and by the Cartan–Tits fixed point theorem it follows that \( G \) has a fixed point in \( S \) (the circumcenter of \( M_x \)). Denote by \( \mu_x \) this fixed point; then \( \mu : x \mapsto \mu_x \) is a \( G \)-invariant conformal structure.

It remains to prove that we can pass from \( G \) preserving \( \mu \) to a conjugate of \( G \) preserving the standard structure. In dimension 2 this follows from Sullivan’s theorem which says that any conformal structure is mapped by a quasiconformal map to the standard one. In higher dimension another argument is needed.

Lecture III

Recall that we have a uniform subgroup \( G \subset \mathrm{QC}(\mathbb{S}^n) \) preserving a measurable conformal structure \( \mu \). In addition we know that \( G \) extends to a cobounded quasi-action on \( \mathbb{H}^{n+1} \), and this implies that all points are radial.

We will make use of the following notion.

Definition: Let \( X, Y \) be topological spaces and \( f : X \to Y \) a Borel map. Let \( m \) be a Borel measure on \( X \). The map \( f \) is said to be \( m \)-approximately continuous at a point \( x \in X \) if the following holds:
\[
\forall \varepsilon > 0 : \lim_{r \to 0} \left( \frac{m(y \in B(x, r) : d_Y(f(x), f(y)) \ge \varepsilon)} {m(B(x, r))} \right) = 0.
\]

Any Borel map is in fact \( m \)-approximately continuous at \( m \)-almost every point. This applies to the map \( \mu : \mathbb{S}^n \to S \) (for the Lebesgue measure on \( \mathbb{S}^n \)), and so Tukia’s theorem can be reformulated as follows.

Theorem: Soit \( G \) a uniform subgroup of \( \mathrm{QC}(\mathbb{S}^n) \). Suppose that there exists a \( G \)-invariant measurable conformal structure \( \mu \) on \( \mathbb{S}^n \), and a point \( x \in \mathbb{S}^n \) at which \( \mu \) is approximately continuous, and which is radial for \( G \). Then there exists \( f \in \mathrm{QC}(\mathbb{S}^n) \) such that \( fGf^{-1} \subset \mathrm{Conf}(\mathbb{S}^n) \).

The proof of this statement is a « zooming » argument similar to what was used in the proof of Schwartz’s theorem in Kapovich’s lectures.

Somme lemmas

For \( A \in S \) let \( K(A) = e^{K(\mathrm{Id}, A)} \); for \( g \in \mathrm{QC}(\mathbb{S}^n) \) let \( K(g, x) = K( (Dg_x)^T \cdot Dg_x) \).

Lemma 1: Let \( K > 0 \).

1. Let \( f_i \) be a sequence of \( K \)-quasiconformal applications. Suppose that \( f_i \to f \) pointwise and:
   \[
   \forall \varepsilon > 0 : \lim_{i \to +\infty} (\mathrm{Leb}(x : K(f_i, x) \ge 1 + \varepsilon)) = 0.
   \]
   Then \( f \) is conformal.
2. Let \( (f_i)_{i \ge 0} \) be relatively compact in \( \mathrm{QC}(\mathbb{S}^n) \) (in the pointwise convergence topology), and let \( E_i \) a sequence of Borel sets with \( \mathrm{Leb}(E_i) \to 0 \). Then we have also \( \mathrm{Leb}(f_i(E_i))\to 0 \).

Lemma 2: Fix \( \lambda, c \) and let \( f_i \in \) be a sequence of \( (\lambda, c) \)-quasi-isometries, such that in addition there exists a point \( y \in \mathbb{H}^{n+1} \) such that \( f_i(y) \) is bounded. Then \( (f_i) \) is relatively compact.

Proof of Tukia’s theorem

Let \( \mu \) be a \( G \)-invariant conformal structure. We may assume that \( \mu \) is approximately continuous at 0 and that \( \mu(0) = \mathrm{Id} \). Let \( y \in \mathbb{H}^{n+1} \). We also know that 0 is radial for \( G \) and it follows that there exists a sequence \( g_i \in G \) such that \( g_iy \to 0 \) and \( g_i y \) stays within bounded distance of the geodesic \( [0, \infty] \) in \( \mathbb{H}^{n+1} \).

It follows that there exists a sequence \( T_i \in \mathrm{Isom}(\mathbb{H}^{n+1}) \) such that \( T_iz = t_i z \) for some \( t_i > 0 \) with \( t_i \to +\infty \) and \( T_i g_iy \) is bounded. By Lemma 2 the sequence \( f_i = T_ig_i \) has an accumulation point \( f \in \mathrm{QC}(\mathbb{S}^n) \). The rest of the proof is dedicated to show that
\[
fGf^{-1} \subset \mathrm{Conf}(\mathbb{S}^n)
\]
which concludes the proof of Tukia’s theorem.

To prove this claim we need to show that for any \( g \in G \) the sequence \( h_i = f_i g f_i^{-1} \) converges to a conformal map. We let
\[
\mu_i = (Df_i^{-1})^T \cdot \mu \cdot Df_i^{-1}
\]
then \( h_i \) is conformal for \( \mu_i \). On the other hand, since \( g_i \in G \) preserves \( \mu \) we have in fact that
\[
(\ast) \qquad \mu_i(x) = ((DT_i^{-1})_x)^T \cdot \mu(x) (DT_i^{-1})_x = \mu(x/t_i^2).
\]
Now it is easy to see that
\[
(\dagger) \qquad K(h_i, x) \le K(\mu_i(x)) K(\mu_i(g_ix)).
\]
The approximate continuity of \( \mu \) at 0 gives:
\[
\frac{\mathrm{Leb}(u \in B(0, r) : K(\mu(y)) \ge 1 + \varepsilon)}{r^n} \underset{r \to 0}{\rightarrow} 0
\]
and it follows for each \( i \) that:
\[
\frac{\mathrm{Leb}(u \in B(0, r) : K(\mu_i(y)) \ge 1 + \varepsilon)}{r^n} \underset{r \to 0}{\rightarrow} 0.
\]
Now using a diagonal argument together with \( (\ast) \) and the fact that \( t_i^{-2} \to 0 \) we obtain that, fixing \( R > 0 \) and putting
\[
A_i = \{ y \in B(0, R) : K(\mu_i(y)) \ge 1 + \varepsilon \}
\]
we have \( \mathrm{Leb}(A_i) \to 0. \). Now putting \( B_i = h_iA_i \) we have
\[
B_i = \{ y \in B(0, R) : K(\mu_i(g_i y)) \ge 1 + \varepsilon \}
\]
On the other hand, by Lemma 1.2 we have that \( \mathrm{Leb}(B_i) \to 0 \). By \( (\dagger) \) we have that
\[
\mathrm{Leb}(y \in B(0, R) : K(h_, y) \ge 1 + \varepsilon) \le \mathrm{Leb}(A_i) + \mathrm{Leb}(B_i)
\]
which implies that the left-hand side goes to 0. Applying Lemma 1.1 yields that \( h_i \) converges to a conformal map on \( B(0, R) \), and since \( R \) was arbitrary this concludes the proof.

Remarks

Other symmetric spaces

Tukia’s theorem can be extended to groups of quasiconformal maps on the boundary of complex hyperbolic space (this is due to Richard Chow, and has a proof similar to the one described above). The other rank 1 symmetric spaces are actually strongly quasi-isometrically rigid, that is \( \mathrm{QI}(X) = \mathrm{Isom}(X) \) (this is a theorem of Pansu).

Bi-Lipschitz homeomorphisms

The group \( \mathrm{Bilip}(\mathbb{R}^n) \) of bi-Lipschitz maps of Euclidean space is contained in \( \mathrm{QC}(\mathbb{S}^n) \). A subgroup \( G \subset \mathrm{Bilip}(\mathbb{R}^n) \) is said to be uniform if there exists a \( R > 0 \) such that:
\[
\forall g \in G : \exists K \ge k > 0 : K/k \le R \text{ and } \forall x, y \in \mathbb{R}^n, k|x – y| \le |gx – gy| \le K|x – y|.
\]
A corollary of Tukia’s theorem is then the following: if \( G \subset \mathrm{Bilip}(\mathbb{R}^n) \) is uniform and every point in \( \mathbb{R}^n \) is radial for \( G \) then there exists \( f \in \mathrm{Bilip}(\mathbb{R}^n) \) such that \( fGf^{-1} \subset \mathrm{Sim}(\mathbb{R}^n) \) (where \( \mathrm{Sim} \) is the group of similarities).

This follows from two simple observations: that \( \mathrm{Sim} = \mathrm{Bilip} \cap \mathrm{Conf} \), and that uniform subsets of \( \mathrm{Bilip} \) are closed in the topology of pointwise convergence on \( \mathrm{QC}(\mathbb{S}^n) \).

Results similar to the above for subgroups of homeomorphisms of the circle or the real line are due to Hinkkanen (in Uniformly quasisymmetric groups) and Farb–Mosher (in Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II).

Lecture IV

Let \( h \) be the function \( (x_1, \ldots, x_{n+1}) \in \mathbb{H}^{n+1} \mapsto x_n \) (« height function » on \( \mathbb{H}^{n+1} \)). Let :
\[
\mathrm{QI}_\mathrm{hr}(\mathbb{H}^{n+1}) = \{ f \in \mathrm{QI}(\mathbb{H}^{n+1}) : \exists C_f \in \mathbb{R}, h \circ f – h = C_f + O(1) \}.
\]
Then the boundary extension of a map in \( \mathrm{QI}_\mathrm{hr} \) is a bi-Lipschitz homeomorphism of \( \mathbb{R}^n \).

A subgroup \( G \subset \mathrm{QI}_\mathrm{hr} \) never acts coboundedly on \( \mathbb{H}^{n+1} \). Rather, the condition to consider on these groups is that their boundary action is cocompact on distinct pairs.

Solvable groups and negatively curved homogeneous spaces

There is a natural identification of the solvable group \( \mathbb{R}^n \times_{e^t} \mathbb{R} \) with the hyperbolic space \( \mathbb{H}^{n+1} \) where it acts simply transitively as a group of isometries fixing \( \infty \). In fact the natural left-invariant metric on this group is the hyperbolic metric.

Similarly, complex hyperbolic space \( \mathbb{C}\mathbb{H}^{n+1} \) is naturally isometric to the group \( N \times_{\phi^t} \mathbb{R} \) where \( N \) is the \( 2n + 1 \)-dimensional Heisenberg group and \( \phi \) is a specific automorphism. For \( n = 1 \) the group \( N \) can be realised as the group of matrices
\[
\left(\begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right), \, x, y, z \in \mathbb{R}
\]
and \( \phi^t \) is given by
\[
\phi^t\left(\begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right) = \left(\begin{array}{ccc} 1 & e^tx & e^{2t}z \\ 0 & 1 & e^ty \\ 0 & 0 & 1 \end{array} \right).
\]
The quaternionic hyperbolic spaces and octonionic hyperbolic plane can be realised in similar ways.

In general a theorem of Heintze states that a negatively curved homogeneous Riemannian manifold must be of the form \( N \times_{\phi^t} \mathbb{R} \) where \( N \) is a nilpotent group and \( \phi \) a « dilating » automorphism.

The simplest example is when \( N = \mathbb{R}^n \) is abelian. Then we must have \( \phi^t = e^{tA} \) where \( A \) is a matrix with all its eigenvalues in \( ]0, +\infty [ \). Let \( H_A \) be the space \( \mathbb{R}^n \times_{e^{tA}} \mathbb{R} \). The following examples are illustrative of the geometry of this space in different situations:

• If \( A \) is scalar, \( A = \lambda\mathrm{Id} \) for somes \( \lambda > 0 \) then \( H_A = \mathbb{H}_c^{n+1} \) is the space of constant curvature \( c < 0 \).
• If \( A \) is diagonal with diagonal terms \( \alpha_1, \ldots, \alpha_r > 0 \) with multiplicities \( m_1, \ldots, m_r \) then \( H_A \) contains totally geodesic copies of the spaces \( \mathbb{H}_{c_i}^{m_i} \).
• If \( A \) has nontrivial Jordan blocks, for example \( A = \left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right) \), then the structure is more complicated.

On every space \( H = N \times \mathbb{R} \) there is a height function \( h \) obtained by projectin into the \( \mathbb{R} \) factor. The goal is to study the group \( \mathrm{QI}_\mathrm{hr}(H) \) of quasi-isometries respecting \( h \), which is defined as above, and the related boundary maps \( N \to N \).

Conjecture (Xie, Cornulier, Carrasco): Let \( H \) be a negatively curved Riemannian homogeneous manifold. If \( H \) is not isometric to a rank-one symmetric space then \( \mathrm{QI}(H) = \mathrm{QI}_\mathrm{hr}(H) \).

This conjecture is known to hold in particular when \( N \) is abelian.

The following conjecture relates the coarse geometry of solvable groups with that of products of homogeneous spaces. If \( H_i \) are homogeneous spaces with height functions \( h_i \) then an horocycle product in \( H_1 \times H_2 \) is the subset \( \{ (x_1, x_2) : h_1(x_1) + h_2(x_2) = 0 \} \).

Conjecture (Eskin–Fisher–Whyte): Let \( H_i \) be negatively curved homgeneous spaces. If \( G \subset H_1 \times H_2 \) is an horocycle product, then
\[
\mathrm{QI}(G) = \mathrm{QI}_\mathrm{hr}(H_1 \times H_2) \cap (\mathrm{QI}(H_1) \times \mathrm{QI}(H_2)).
\]

This is known in some cases by work of Eskin–Fisher–Whyte and Peng.

Boundary and quasiconformal maps

Let \( H = N \times_{\phi^t} \mathbb{R} \) be a negatively curved homogeneous space. The visual boundary \( \partial_\infty H \) has a distinguished point \( \infty \) and there is an identification \( \partial_\infty H = N \cup \{ \infty \} \). We will denote \( \partial H = N \).

Define a metric on \( \partial H \) as follows. Let \( x_1, x_2 \in \partial H \), then there is a unique \( t \in \mathbb{R} \) such that \( d_H((x_1, t), (x_2, t)) = 1 \). Put \( d(x_1, x_2) = e^t \).

On \( \mathbb{H}^{n+1} \) this gives the Euclidean metric on \( \mathbb{R}^n \) since \( d_H((x_1, t), (x_2, t)) = e^{-t}|x_1 – x_2| \). On \( \mathbb{H}_{-2}^{n+1} \) this yields the distance \( d(x_1, x_2) = |x_1 – x_2|^{\frac 1 2} \). In general it is harder (impossible?) to compute exactly. An example is \( H_A \) with \( A = \left( \begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array} \right) \). There the distance on \( \mathbb{R}^2 \) is bi-Lipschitz to (but not equal to)
\[
d(x, y) = \max(|x_1 – y_1|, |x_2 – y_2|^{1/2}).
\]

Interlude

In view of QI-rigidity problem it is a natural question to ask when two homogeneous negatively curved spaces are quasi-isometric to each other. For spaces \( H_A, H_B \) this is the case of and only if, for every eigenvalue \( \lambda_i \) of \( A \), there is an eigenvalue \( \mu_i = e^{i\theta_i}\lambda_i \) of \( B \), and the Jordan blocks for \( A, \lambda_i \) and \( B, \mu_i \) are the same.

Bi-Lipschitz maps

In the case of the space \( H_A \) with \( A = \left(\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right) \), any bi-Lipshitz map of \( H_A = \mathbb{R}^2 \times_{e^{tA}} \mathbb{R} \) decomposes as
\[
f(x, y) = (f_1(x, y), f_2(y)).
\]
The map \( f_2(y) \) must then be bi-Lipschitz. So must be the maps \( f_1(\cdot, y) \) for \( y \in \mathbb{R} \). On the other hand the maps \( f_1(x, \cdot) \) only have to be \( 1/2 \)-Hölder (see the distnce formula above for this space).

Suppose now that \( A \) is an \( n \times n \) diagonalisable matrix, with eigenvalues \( \alpha_1, \ldots, \alpha_r \) and corresponding multiplicities \( m_1, \ldots, m_r \). Let \( d_A \) be the distance defined above on \( \mathbb{R}^n = \partial H_A \). Let \( \pi \) be the map \( \mathbb{R}^n, d_A \to \mathbb{R}^{m_r}, d_{\alpha_r} \). Then, by an application of Tukia’s theorem, if \( G \subset \mathrm{Bilip}(\mathbb{R}^n, d_A) \) is uniform, the subbgroup \( \pi_*(G) \subset \mathrm{Bilip}(\mathbb{R}^{n_r}, d_r) \) is conjugated to a group of similarities.

Returning to the case of \( A = \left(\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right) \) (where 1 and 2 may have multiplicities \( n_1, n_2 \)) we see that we can write a bi-Lipshitz map as
\[
f(x, y) = (f_1(x, y), \lambda_2 A_2(y + b_2) )
\]
where \( \lambda_2 > 0, b_2 \in \mathbb{R}^{n_2} \) and \( A_2 \in O(n_2) \).

A « foliated conformal structure » on \( H_A \) is a map \( \mu : \mathbb{R}^n \to S(n_1) \) where \( S(n_1) = \mathrm{SL}_{n_1}(\mathbb{R}) / \mathrm{SO}(n_1) \). It is possible to adapt Tukia’s arguments explained in the previous lecture to prove that in fact:
\[
f_1(x, y) = \lambda_1(y) A_1(y)(x + b_1(y))
\]
Further analysis shows that one can conjugate to the situation where \( A_1, \lambda_1 \) are constant. The final analogue of Tukia’s theorem that is obtained by these methods is the following statement (the similarity group of a metric space is just the group of maps which change distances by a constant factor).

Theorem (Dymarz–Xie): Let \( A \) be diagonalisable with positive eigenvalues and \( G \subset \mathrm{Bilip}(\mathbb{R}^n, d_A) \) be a uniform subgroup which acts cocompactly on distinct pairs. Suppose in addition that \( G \) is amenable, then there exists \( f \in \mathrm{Bilip}(\mathbb{R}^n, d_A) \) such that \( fGf^{-1} \subset \mathrm{Sim}(\mathbb{R}^n, d_A) \).

Application to QI-rigidity of solvable groups

Let \( \Gamma \) be an amenable discrete group. Suppose that there is a quasi-action \( \Gamma \to \mathrm{QI}(S) \). Then viewing \( S \subset H_1 \times H_2 \) the arguments above show that \( \Gamma \) can be conjugated into \( \mathrm{Sim}(\partial H_1 \times \partial H_2) \). There is still a last step to see that \( \Gamma \) is in fact contained in the isometry group.

Lecture V

Bi-Lipschitz and quasi-isometric equivalence

A quasi-isometry between two discrete groups is a bi-Lipschitz map if and only if it is bijective. The question of distinguishing between bi-Lipshitz and quasi-isometric equivalence thus boils down, in one direction, to finding a bijective map at finite distance from a given quasi-isometry.

Theorem (Whyte): Suppose \( G \) is a non-amenable group. Then any quasi-isometry \( G \to G’ \) from \( G \) to another discrete group \( G’ \) is at finite distance from a bijection.

On the other hand this is not true for amenable groups, for example a proper inclusion \( G’ \subset G \) with \( G \) amenable is never at finite distance from a bijection. On the other hand this does not precludes \( G \) and \’ G’ \) to be bi-Lipschitz to each other.

Exercise Prove this for the inclusion \( \mathbb{Z} \subset Z \times K \) where \( K \) is afinite group, and find a bi-Lipshitz map between the two.

Amenability

A group \( G \) is amenable if and only if there exists a Følner sequence in \( G \), that is a sequence \( F_i \) of finite subsets in \( G \) such that for any fixed finite subset \( K \subset G \) we have
\[
\lim_{i \to +\infty} \frac{|\partial_K S_i|}{|S_i|} = 0
\]
where
\[
\partial_K S_i = \{ g \in S_i : \exists k \in K, gk \not\in S_i \}.
\]
Følner sequences are an obstruction to finding a bijection close to an inclusion map: suppose \( S_i’, S_i \) are Følner sequence in \( G, G’ \) and that \( S_i’ \subset S_i \) has size \( \le (1-\delta)|S_i| \) (this is the situation for \( S_i’ \) coming from a subgroup of finite index) and \( i \) is large enough, there are not enough elements in \[ G’ \) which are close to \( S_i’ \) to fill the holes in \( S_i \) since \( |\partial_K S_i’| \le \varepsilon |S_i \setminus S_i’| \) where \( K \) can be taken arbitrarily large and \( \varepsilon \) arbitrarily small.

In general thre is the following criterion.

Whyte’s criterion: Let (\ X, Y \) be two finetely generated groups and \( \varphi : X \to Y \) a quasi-isometry. Then \( \varphi \) is at bounded distance from a bijection if and only if for any Følner sequence \( S_i \subset Y \) there exists \( r > 0 \) such that
\[
\sum_{x \in S_i} |\varphi^{-1}(\{x\}) – |S_i| = O(_partial_r S_i ).
\]
In general two maps \( \varphi, \psi \) are at bounded distance from each other if and only if
\[
\sum_{x \in S_i} (|\varphi^{-1}(\{x\}) – |\psi^{-1}(\{x\})|) = O(\partial_r S_i).
\]

Lamplighter groups

Recall that the wreath product \( G \wr \mathbb{Z} \) is the semi-direct product \( G^\mathbb{Z} \times \mathbb{Z} \) where a generator of \( \mathbb{Z} \) acts by \( (x_i) \mapsto (x_{i+1}) \).

Theorem (Dymarz): The groups
\[
G’ = (\mathbb{Z}/3 \times \mathbb{Z}/3) \wr \mathbb{Z}
\]
and
\[
G = \mathbb{Z}/3 \wr \mathbb{Z}
\]
are QI to each other (in fact there is an embedding \( G’ \subset G \) whise image has index 2) but there does not exist a bi-Lipschitz map \( G \to G’ \).

The Cayley graphs of \( G, G’ \) can be described as follows. Let \( T_d \) be the regular \( d \)-valent tree. Let \( DL(q, q) \) be the full subgraph of \( T_{q+1} \times T_{q+1} \) whose vertices are the pairs \( (x, y) \) such that \( h(x) + h(y) = 0 \) where \( h \) is a height function on \( T_{q+1} \). Then if \( G \) is a finite group of order \( q \) there is a natural generating set of \( G \wr \mathbb{Z} \) for which the Cayley graph is \( DL(q, q) \) (see e.g. this paper).

Thus, there is an edge between two vertices \( (x, y) \) and \( (x’, y’) \) if and only if \( h(x’) = h(x) + 1 \) and \( h(y’) = h(y) – 1 \) or the reverse.
(DESSIN!)

There is always a quasi-isometrically embedded copy of \( DL(q^2, q^2) \) inside \( DL(q, q) \) : fix a vertex in \( DL(q, q) \) and take all vertices at even distance from this one.
(DESSIN!)
In our case this embedding comes from an index 2 copy of \( G’ \) inside \( G \).

Boxes

A box in \( DL(q, q) \) of height \( L \) is a connected component of a « slice »:
\[
\{ (x, y) \in DL(q, q) : a \le h(x) \le a + L \}
\]
for some \( a \in \mathbb{Z}, L \in \mathbb N \). A quick computations shows that if \( S \) is a box of height \( L \) then \( |S| = (L+1) \cdot 3^L \) and \( |\partial_1 S = 2 \cdot 3^L \). Thus we see that a sequence of boxes with increasing height is a Følner sequence in \( DL(q, q) \).

\( DL(3, 3) \) and \( DL(9, 9) \) are not bi-Lipshitz to each other

Suppose that \( \bar\varphi \) is a bijective quasi-isometry from \( DL(9, 9) \) to \( DL(3, 3) \). We want to derive a contradiction.

Claim 1: \( \bar\varphi \) is at bounded distance from an application of the form \( i \circ \varphi \) where \( i \) is the inclusion \( DL(9, 9) \subset DL(3, 3) \) and \( \varphi \) is a 2-to-1 self-quasi-isometry of \( DL(9, 9) \).

Claim 2: There is no such \( \varphi \).

The goal in the rest of the lecture is to prove the second claim. This requires further study of quasi-isometries of \( DL(9, 9) \).

Structure of quasi-isometries

Let \( \mathbb{Q}_n \) be the set of ends of ends of \( T_{n+1} \), minus \( \infty \) (the end from which the height function \( h \) is taken) (in particular if \( n = p \) is prime \( \mathbb{Q}_n \) is identified with the \( p \)-adics integers, otherwise it is a priori just a commutative ring). There are « positive » and « negative » geodesic rays in \( DL(n, n) \) according to whether \( h \) is increasing or decreasing on them, and this yields an upper and lower boundaries which are both identified with \( \mathbb{Q}_n \).

Theorem (Eskin–Fisher–Whyte): Any quasi-isometry \( \varphi \) of \( DL(n, n) \) extends to a map on the boundary of the form
\[
\varphi(x, y) = ( \varphi_l(x), \varphi_u(y))
\]
where \( \varphi_l, \varphi_u \) are both bi-Lipshitz maps of \( \mathbb{Q}_n \).

On \( \mathbb{Q}_n \) there is a natural basis of clopen sets: a « ball » is given by the set of all points that can be reached from an inside point by a positive geodesic ray (if \( n = p \) these are the cosets of the additive subgroups \( p^m\mathbb{Z}_p \)). There is a natural measure \( \mu \) on the balls, where all balls associated to a point at height \( t \) have measure \( q^{-t} \). Similarly, the distance on \( \mathbb{Q}_n \) is defined by \( d(\xi, \eta) = q^{-t} \) if \(xi, \eta\) can be reached by geodesics rays originating from the same point at height \( t \).

Theorem (Cooper): If \( \varphi : \mathbb{Q}_n \to \mathbb{Q}_n \) is bi-Lipshitz then then there exists a ball \( B \) in \( \mathbb{Q}_n \) such that \( \varphi|_B \) is measure-linear, that is \( \varphi_*\mu|_B = \lambda\mu|_B \) for some \( \lambda > 0 \).

This is not enough to prove Claim 2, for this it will be needed that \( \varphi_u, \varphi_l \) be both measure-linear on the whole of \( \mathbb{Q}_n \). It is possible to get to this situation by using a « zooming » argument similar to the one in the last step of the proof of Tukia’s theorem. This yields \( \bar\varphi_u, \bar\varphi_l \) conjugated to the original maps, and which are measure-linear with coefficients \( \lambda_u, \lambda_l \).

Because of the structure of \( \mathbb{Q}_9 \) (compact-open sets are finite union of balls, and a ball is a disjoint union of 9 smaller balls) both \( \lambda_l \) and \( \lambda_u \) must be powers of 3.

Arguments above yield a quasi-isometry \( \psi \) of \( DL(9, 9) \) associated to \( \bar\varphi_l, \bar\varphi_u \) which is at finite distance from a conugate of \( \varphi \) and such that for any box \( S \) we have :
\[
\sum_{y \in S} |\psi^{-1}(\{y\})| = \lambda_l \lambda_u |S| + O(|\partial S|).
\]
On the other hand we have \( \sum_{y \in S} |\varphi^{-1}(\{y\})| = 2|S| \) and since \( \lambda_l \lambda_u \in 3^\mathbb{Z} \) this implies that
\[
\left| \sum_{y \in S} |\psi^{-1}(\{y\})| – \sum_{y \in S} |\varphi^{-1}(\{y\})| \right| \ge |S| + O(|\partial S|)
\]
which contradicts the fact that \( \psi \) and \( \varphi \) are at finite distance via Whyte’s criterion.

1. First lecture
2. Second lecture
3. Third lecture
4. Fourth lecture

Lecture I

Lecture II

Lecture III

Lecture IV