This is a transcript of my notes from Marc Bourdon’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 be mine.

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 \( H_k^p(X) \neq 0 \)?