# Non-vanishing for group $L^{p}$-cohomology of solvable and semisimple Lie
groups

Marc Bourdon and Bertrand Rémy

###### Abstract.

We obtain non-vanishing of group $L^{p}$-cohomology of Lie groups for $p$
large and when the degree is equal to the rank of the group. This applies both
to semisimple and to some suitable solvable groups. In particular, it confirms
that Gromov’s question on vanishing below the rank is formulated optimally. To
achieve this, some complementary vanishings are combined with the use of
spectral sequences. To deduce the semisimple case from the solvable one, we
also need comparison results between various theories for $L^{p}$-cohomology,
allowing the use of quasi-isometry invariance.

2010 Mathematics Subject Classification: 20J05, 20J06, 22E15, 22E41, 53C35,
55B35, 57T10, 57T15.

Keywords and phrases: $L^{p}$-cohomology, Lie group, symmetric space, quasi-
isometric invariance, spectral sequence, cohomology (non-)vanishing, root
system.

###### Contents

  1. 1 de Rham $L^{p}$-cohomology
  2. 2 The Lie group case
  3. 3 A non-vanishing criterion
  4. 4 First applications: proof of two non-vanishing results
  5. 5 Semisimple Lie groups
  6. 6 Cohomologies of semi-direct products
  7. A On asymptotic and de Rham $L^{p}$-cohomology

## Introduction

This paper deals with several variants of group $L^{p}$-cohomology, where $p$
is a real number in $(1,+\infty)$.

The first one, the _continuous $L^{p}$-cohomology_ of locally compact second
countable groups $G$, is the continuous cohomology [BW00, Chap. IX] with
coefficients in the right-regular representation on $L^{p}(G)$; we denote it
by ${\rm H}^{*}_{\mathrm{ct}}(G,L^{p}(G))$. We also consider the associated
_reduced continuous $L^{p}$-cohomology_, denoted by $\overline{{\rm
H}^{*}_{\mathrm{ct}}}(G,L^{p}(G))$ (it is the largest Hausdorff quotient of
the previous one). It is known that both topolical vector spaces ${\rm
H}^{*}_{\mathrm{ct}}(G,L^{p}(G))$ and $\overline{{\rm
H}^{*}_{\mathrm{ct}}}(G,L^{p}(G))$ are invariant for the equivalence relation
given by quasi-isometries between groups $G$ as before when equipped with a
left-invariant proper metric (see [BR20, Theorem 1.1] and [SS18]).

One variant of $L^{p}$-cohomology makes sense for $C^{\infty}$ manifolds: it
is the _de Rham $L^{p}$-cohomology_, denoted by $L^{p}{\rm H}^{*}_{\rm dR}(M)$
for a $C^{\infty}$ manifold $M$. It was studied thoroughly by P. Pansu and, as
the name suggests, it is defined by imposing $L^{p}$-integrability conditions
on differential forms on the manifold (and on their differentials). There are
also intermediate variants, for instance the _asymptotic $L^{p}$-cohomology_
of suitable metric spaces, which elaborates on simplicial cohomology by adding
$L^{p}$-integrability conditions; it is denoted by $L^{p}{\rm H}^{*}_{\rm
AS}(X)$ for a suitable measured metric space $(X,d,\mu)$. Reduced quotients
are also considered in these contexts.

When it makes sense, comparison results between these cohomologies are often
available (see for instance [BR20], [SS18] and Appendix A in this paper;
again, this part owes a lot to P. Pansu’s work).

Beyond these comparisons, our main goal remains to exhibit some sufficient
conditions for vanishing and non-vanishing of $L^{p}$-cohomologies for
topological groups, taking into account the degree of the cohomology space and
the exponent $p$.

The topological groups we are considering in the present paper for these
(non-)vanishing questions are connected Lie groups. This enables us to use
many differential geometric and combinatorial tools. The most popular Lie
groups are the semisimple ones, and this is the family about which we present
the first results in this introduction, but we will see that we are quickly
led to considering solvable non-unimodular groups. This is explained by the
fact that some contraction arguments are crucial for our purposes, and this is
made possible by the aformentioned quasi-isometric invariance of group
$L^{p}$-cohomology combined with Iwasawa decompositions.

###  0.1. Semisimple groups and quasi-isometry invariance

Let us start with semisimple real Lie groups. Our work is motivated by the
following question, asked by M. Gromov [Gro93, p. 253].

######  Question 0.1.

Let $G$ be a semisimple real Lie group. We assume that $l=\mathrm{rk}_{\bf
R}(G)\geqslant 2$. Let $k$ be an integer $<l$ and $p$ be a real number $>1$.
Do we have: ${\rm H}^{k}_{\mathrm{ct}}\bigl{(}G,L^{p}(G)\bigr{)}=\\{0\\}$?

In degree 1, a general result of Pansu [Pan07] and Cornulier-Tessera [CT11]
shows that ${\rm H}^{1}_{\mathrm{ct}}(G,L^{p}(G))=\\{0\\}$ for every $p>1$ and
every connected Lie group $G$, unless $G$ is Gromov hyperbolic or amenable
unimodular. In a previous paper, we obtained some partial vanishing results
which lead to the latter result when the Lie groups are semisimple of rank
$\geqslant 2$, and admissible in the sense that the solvable radical of some
(maximal) parabolic subgroup is quasi-isometric to a real hyperbolic space
[BR20, Corollary 1.6]. This result in degree 1 can also be proved via the
fixed point property for continuous affine isometric actions of higher rank
semisimple groups on $L^{p}$-spaces [BFGM07]. In general, we show in [BR20]
the existence, for any admissible semisimple Lie group $G$ and any $p>1$, of
an interval of degrees out of which the spaces ${\rm
H}^{k}_{\mathrm{ct}}(G,L^{p}(G))$ vanish.

Our main result in the present paper is complementary to Gromov’s question; it
is the following.

######  Theorem A.

Let $G$ be a semisimple real Lie group with finite center and let
$l=\mathrm{rk}_{\bf R}(G)$.

  * (i)

We have: $\overline{{\rm
H}^{l}_{\mathrm{ct}}}\bigl{(}G,L^{p}(G)\bigr{)}\neq\\{0\\}$ for any large
enough $p>1$.

  * (ii)

For every $k>l$, we have: ${\rm
H}^{k}_{\mathrm{ct}}\bigl{(}G,L^{p}(G)\bigr{)}=\\{0\\}$ for any large enough
$p>1$.

This result is proved thanks to the following line of arguments. We introduce
an Iwasawa decomposition $G=KAN$. Geometrically, if $X$ denotes the symmetric
space of $G$ and if $F$ denotes the maximal flat attached to the maximal ${\bf
R}$-split torus $A$ in $G$, then the subgroup $K$ can be chosen to be the
stabilizer of a point in $F$; the subgroup $N$ consists of the unipotent
elements in a parabolic subgroup defined by a regular element in the boundary
$\partial_{\infty}F$ of $F$. Since $G$ has finite center, the group $K$ is
compact (it is in fact a maximal compact subgroup in $G$). We have the
following identifications:

${\rm H}^{*}_{\mathrm{ct}}\bigl{(}G,L^{p}(G)\bigr{)}\simeq{\rm
H}^{*}_{\mathrm{ct}}\bigl{(}AN,L^{p}(AN)\bigr{)}\simeq L^{p}{\rm H}^{*}_{\rm
AS}(AN)\simeq L^{p}{\rm H}^{*}_{\rm dR}(AN),$

and similar ones for reduced cohomology. The first identification comes from
quasi-isometric invariance [BR20, Theorem 1.1], the second one is a comparison
between continuous $L^{p}$-cohomology and asymptotic $L^{p}$-cohomology proved
in [BR20, Theorem 3.6] and the last one is given here by Theorem A.1 (proved
in Appendix A). This reduction explains why the main part of the paper focuses
on solvable Lie groups. The latter situation is investigated in the remaining
two subsections of this introduction and it leads to the desired vanishing and
non-vanishing results above. See Theorem D below for the conclusion of the
argument.

Since the proof of our main result on semisimple groups is spread all over our
paper, here is a summary of the strategy.

Step 1: quasi-isometry invariance and Iwasawa decomposition.— The
decomposition $G=KAN$ implies that $G$ is quasi-isometric to the solvable
group $A\ltimes N$. Invariance of $L^{p}$-cohomology under quasi-isometric
invariance then gives ${\rm H}^{*}_{\rm ct}\bigl{(}G,L^{p}(G)\bigr{)}={\rm
H}^{*}_{\rm ct}\bigl{(}AN,L^{p}(AN)\bigr{)}$ and $\overline{{\rm H}^{*}_{\rm
ct}}\bigl{(}G,L^{p}(G)\bigr{)}=\overline{{\rm H}^{*}_{\rm
ct}}\bigl{(}AN,L^{p}(AN)\bigr{)}$. We denote $R=AN=A\ltimes N$ so that we have
$R\simeq\mathbf{R}^{D}$ where $D$ is the dimension of the Riemannian symmetric
space $G/K$, as well as $A\simeq\mathbf{R}^{l}$ where $l$ is the real rank of
$G$.

Step 2: Poincaré duality and vanishing (after Pansu).— Poincaré duality
reduces the proof of Theorem A to showing that for $p$ close enough to 1, we
have $L^{p}{\rm H}^{k}_{\rm dR}(R)=\\{0\\}$ for every $k<D-l$ and that
$L^{p}{\rm H}^{D-l}_{\rm dR}(R)$ is Hausdorff and non-zero. Arguments due to
Pansu show that a certain contraction condition on the $A$-action on $N$
(called (nC) in Theorem 0.2 below) implies the assertions about vanishing and
Hausdorff property (see Corollary 2.4).

We concentrate now on the non-vanishing $L^{p}{\rm H}^{D-l}_{\rm dR}(R)\neq\\{
0\\}$.

Step 3: Actions of abelian groups on Heintze groups and spectral sequences.—
We pass now from the decomposition $R=A\ltimes N$ to the decomposition
$R=B\ltimes(\\{e^{t\xi}\\}_{t\in\mathbf{R}}\ltimes N)$ where $\xi$ is given by
condition (nC) as before, $H=\\{e^{t\xi}\\}_{t\in\mathbf{R}}\ltimes N$ is a
Heintze group and $B$ is a suitable (abelian) complement of
$\\{e^{t\xi}\\}_{t\in\mathbf{R}}$ in $A$. The motivation for the decomposition
$R=B\ltimes H$ is the possibility to use a spectral sequences and the already
proved vanishings to obtain the identification: $L^{p}{\rm H}^{D-l}_{\rm
dR}(R)\simeq L^{p}\bigl{(}B,L^{p}{\rm H}^{D-l}_{\rm dR}(H)\bigr{)}^{B}$. We
conclude by exhibiting $L^{p}$ de Rham cohomology classes $c$ on $H$ such that
the map $b\mapsto\|b\cdot c\|_{p}$ belongs to $L^{p}(B)$ for the $B$-action on
$c$ given by (pull-back of) conjugation by $B$ on $H$. This is the content of
the (technical) Proposition 3.2, which uses condition (nT), a condition of
non-triviality required for the action of all elements in the Lie algebra of
$A$ (see Question 0.3 below).

Before we definitely move to the framework of solvable groups, let us mention
the case when $p=2$ for semisimple groups. This situation is more directly
relevant to representation theory and was considered by A. Borel decades ago
[Bor85]. The main results are:

  * •

$\overline{{\rm H}^{k}_{\mathrm{ct}}}\bigl{(}G,L^{2}(G)\bigr{)}=0$ unless
$k=\frac{D}{2}$,

  * •

${\rm H}^{k}_{\mathrm{ct}}\bigl{(}G,L^{2}(G)\bigr{)}\neq 0$ at least for
$k\in\bigl{(}\frac{D}{2}-\frac{l_{0}}{2},\frac{D}{2}+\frac{l_{0}}{2}\bigr{]}$,

where $D$ is the dimension of the Riemannian symmetric space $G/K$ and $l_{0}$
is the difference between the complex rank of $G$ and the complex rank of $K$.
In the case $G={\rm SL}_{n}({\bf R})$, one has $D=\frac{n^{2}+n-2}{2}$ and
$l_{0}=\lfloor\frac{n-1}{2}\rfloor$. See also [BFS14] for related results
about vanishing of the reduced $L^{2}$-cohomology. Since the dimension $D$ is
a quadratic polynomial in the rank of the group $G$, A. Borel’s results show
that the assumption that $p$ should be large enough in our results is
necessary.

###  0.2. Solvable groups and contractions

As explained above, we are henceforth dealing with solvable Lie groups until
the end of the introduction. The following result on cohomology vanishing is a
consequence of [Pan99, Corollaire 53]:

######  Theorem 0.2.

Let $R$ be a connected Lie group of the form $A\ltimes N$ with $A\simeq{\bf
R}^{l}$ and $l\geqslant 1$. Let $\mathfrak{a}$ and $\mathfrak{n}$ be the Lie
algebras of $A$ and $N$, respectively. Suppose that the group $R$ satisfies
the following contraction property:

  * (nC)

there exists an element $\xi\in\mathfrak{a}$ such that all the eigenvalues of
$\mathrm{ad}\xi\mbox{\Large$|$\normalsize}_{\mathfrak{n}}$ have negative real
parts.

Then for $p>1$ large enough and for all $k>l$, we have the vanishings:

$L^{p}\mathrm{H}_{\mathrm{dR}}^{k}(R)=\\{0\\}$.

Note that the assumptions force $N$ to be nilpotent and contractible, and $R$
to be solvable, non-unimodular and diffeomorphic to ${\bf R}^{D}$, where
$D=\dim(R)$ (the notation $D$ is consistent with the previous one on
dimensions of symmetric spaces by Iwasawa decomposition). We also mention the
fact that condition (nC) already appears in N. Varopoulos’ paper [Var96, OV.3
p. 799].

We address here the following question:

######  Question 0.3.

Let $R$ be a solvable connected Lie group as in the previous theorem,
satisfying in particular condition ${\rm(nC)}$. Suppose additionally that $R$
satisfies the following non-triviality condition:

  * (nT)

for every non-trivial $X\in\mathfrak{a}$, the operator
$\mathrm{ad}X\mbox{\Large$|$\normalsize}_{\mathfrak{n}}$ admits an eigenvalue
with non-zero real part.

Do we have $L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{l}}(R)\neq\\{0\\}$ for
$p>1$ large enough?

If we denote by ${\rm sp}(u)$ the spectrum (in the field of complex numbers)
of an endomorphism $u$ of a finite-dimensional real vector space, the above
conditions can be reformulated as follows:

  * (nC)

there exists $\xi\in\mathfrak{a}$ such that ${\rm
sp}(\mathrm{ad}\xi\mbox{\Large$|$\normalsize}_{\mathfrak{n}})\subset{\bf
R}_{-}^{\times}\oplus i{\bf R}$,

  * (nT)

for every $X\in\mathfrak{a}\setminus\\{0\\}$, we have: ${\rm
sp}(\mathrm{ad}X\mbox{\Large$|$\normalsize}_{\mathfrak{n}})\cap({\bf
R}^{\times}\oplus i{\bf R})\neq\varnothing$.

Condition (nC)+(nT) forces $N$ to be the nilpotent radical of $R$. When $l=1$,
condition (nC) implies trivially condition (nT); moreover the groups $R$ that
satisfy (nC) form precisely the class of connected Lie groups that carry a
left-invariant Riemannian metric of negative curvature [He74]. For them, it is
known that Question 0.3 admits a positive answer [Pan07, CT11]. The goal of
the paper is to enlarge the family of groups for which Question 0.3 is known
to have a positive answer. In Section 4, we prove two non-vanishing results
described below. Note that at least since we are using spectral sequences
techniques, vanishing results are also useful in the proof of non-vanishing
ones; this explains why condition (nC) is made before condition (nT).

The first positive result makes a commutativity assumption on the nilpotent
radical of the solvable group.

######  Theorem B.

The answer to Question 0.3 is yes, if one assumes in addition that
$N\simeq{\bf R}^{n}$ with $n\geqslant 1$.

The second positive result makes a rank assumption on the solvable group,
_i.e._ a dimension assumption on the quotient of the group by its nilpotent
radical.

######  Theorem C.

The answer to Question 0.3 is yes, if one assumes in addition that $l=2$,
_i.e._ if $A\simeq{\bf R}^{2}$.

Fundamental examples of groups satisfying conditions (nC) and (nT) are
provided by the groups $AN$ that appear in the Iwasawa decompositions $KAN$ of
the semisimple Lie groups with finite center.

######  Theorem D.

The answer to Question 0.3 is yes for the groups $AN$ that appear in the
Iwasawa decompositions $KAN$ of the semisimple Lie groups with finite center.

Together with the reduction contained in Subsection 0.1, the latter result
proves Theorem A on semisimple groups.

###  0.3. Sketch of proof of the non-vanishing theorems

Recall that $D=\dim(R)$. By Poincaré duality (see Proposition 1.11), proving
vanishing and non-vanishing as stated in Theorem 0.2 and Question 0.3, is
equivalent to showing that for $p>1$ close enough to $1$:

  1. (1)

$L^{p}\mathrm{H}_{\mathrm{dR}}^{k}(R)=\\{0\\}$ for every $k<D-l$,

  2. (2)

$L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(R)$ is Hausdorff and non-zero.

The assumption on the existence of $\xi\in\mathfrak{a}$ satisfying condition
(nC), in combination with Pansu’s results on $L^{p}$-cohomology [Pan99, Pan08,
Pan09], imply that item (1) holds, as well as the Hausdorff property in item
(2) (see Corollary 2.4 for a proof).

It remains to establish that
$L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(R)\neq\\{0\\}$. For that, we use $\xi$ as
in condition (nC) in order to decompose $R$ as follows. Write
$\mathfrak{a}={\bf R}\xi\oplus\mathfrak{b}$, where the second factor is
$\mathfrak{b}:=\\{X\in\mathfrak{a}:\mathrm{trace}(\mathrm{ad}X)=0\\}$, the Lie
algebra of a connected Lie subgroup $B<A$ isomorphic to ${\bf R}^{D-1}$. Then
$R$ can be expressed as $R=B\ltimes H$, with $H=\\{e^{t\xi}\\}_{t\in{\bf
R}}\ltimes N$. Again the assumption on $\xi$ and Pansu’s results on
$L^{p}$-cohomology, give a precise rather simple description of the
$L^{p}$-cohomology of $H$ for $p>1$ close to $1$ – see Corollaries 2.4 and
2.5. In particular the above items (1) and (2) hold for $H$, in other words
$L^{p}\mathrm{H}_{\mathrm{dR}}^{k}(H)$ vanishes for $k<\dim(H)-1=D-l$, and
$L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(H)$ is Hausdorff and non-zero.

Now, by using a spectral sequence argument taken from [BR20] – see Corollary
6.10, we obtain a linear isomorphism:

(0.4) $\displaystyle L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(R)\simeq
L^{p}\bigl{(}B,L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(H)\bigr{)}^{B},$

where $B$ acts by translations on itself and by conjugacy on
$L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(H)$. The $B$-invariance implies that the
right hand side space is isomorphic to

$\Bigl{\\{}\psi\in
L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(H):\int_{B}\bigl{\|}C_{b}^{*}(\psi)\bigr{\|}^{p}db<+\infty\Bigr{\\}},$

where $C_{b}$ denotes the conjugation by $b\in B$. Under the assumptions (nC)
and (nT), we give a criterion to ensure that the latter space is non-zero (see
Proposition 3.2). Finally we show that the groups in Theorems B, C and D
satisfy the criterion.

It is worth mentioning that the above strategy would answer affirmatively
Question 0.3 in full generality, if the following question admitted a positive
answer:

######  Question 0.5.

Let $N$ be a connected simply connected nilpotent Lie group. Let
$X_{1},X_{2},\dots,X_{k}$ be non-trivial left-invariant vector fields on $N$.
Does $N$ admit a non-zero, compactly supported, $C^{1}$ function $f:N\to{\bf
R}$, whose integral along every orbit of $X_{i}$ $(i=1,\dots,k)$ is null?

When the fields commute or when $k=2$, we answer affirmatively Question 0.5 –
see Lemmata 4.2 and 4.1.

###  0.4. Remarks and questions

1) We suspect that Condition (nT) in Question 0.3 is a necessary condition.
For example if $\mathfrak{a}$ contains a non-trivial vector $X_{0}$ such that
$\mathrm{ad}X_{0}\mbox{\Large$|$\normalsize}_{\mathfrak{n}}$ is _semisimple_
with imaginary eigenvalues, then one has
$L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{k}}(R)=\\{0\\}$ for every
$k\geqslant 0$ and $p>1$. To see this, observe that for such an $X_{0}$ the
operator $\mathrm{ad}X_{0}$ acting on ${\rm Lie}(R)$ is skew-symmetric.
Therefore the left-invariant vector field associated to $X_{0}$ is a Killing
vector field on $R$. Moreover its flow acts properly on $R$ (since it does on
$A$). These properties, in combination with Poincaré duality, imply vanishing
of $L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{k}}(R)$ in every degree – see
[Pan08, Proof of Proposition 15].

2) Cornulier told us that conditions (nC) and (nT) should admit geometric
characterizations. He claims that a simply connected solvable Lie group
satisfies condition (nC) if and and only if its asymptotic cone (one or every)
is bilipschitz homeomorphic to a ${\rm CAT}(0)$-space. Moreover he thinks that
condition (nT) should be equivalent to the non-existence of direct
$\mathbf{R}$-factor in the asymptotic cone. In the same vein, we notice that
the rank $l$ of a condition (nC) group is equal to the dimension of its
asymptotic cone, see [C08, Theorem 1.1].

3) Nice examples of groups satisfying condition (nC) are provided by Lie
groups that carry a left-invariant non-positively curved Riemannian metric;
although these two classes of groups do not exactly coincide (see [AW76,
Theorem 7.6]).

4) Cornulier advertised us of the following potentially interesting quasi-
isometric invariant:

$p(R):=\inf\bigl{\\{}p>1~{}|~{}L^{p}\mathrm{H}_{\mathrm{dR}}^{l}(R)\neq\\{0\\}\bigr{\\}},$

for groups $R$ of rank $l$ that satisfy conditions (nC) and (nT). (In case the
right side set is void we set $p(R)=+\infty$). Similarly is defined the
invariant $\overline{p}(R)$ associated to the reduced $L^{p}$-cohomology. When
$l=1$, these invariants have been considered and computed by Pansu [Pan07]
(see also [CT11] for related results).

### Organization of the paper

Section 1 is a brief presentation of results on de Rham $L^{p}$-cohomology of
manifolds mainly due to P. Pansu. Section 2 applies these results to the case
of Lie groups, and Section 3 provides a general non-vanishing criterion in
terms of spectra of adjoint actions on solvable groups. Section 4 then applies
the criterion to prove our first two non-vanishing results, namely Theorems B
and C above. Section 5 is dedicated to semisimple Lie groups: it shows, by
combinatorial arguments using Cartan’s classification of Riemannian symmetric
spaces, that the solvable subgroups arising from Iwasawa decompositions do
satisfy the previous criterion; Theorem D follows. Section 6 deals with semi-
direct products and allows the use of spectral sequences, modulo a comparison
result between cohomologies, available when the involved groups are
diffeomorphic to ${\bf R}^{D}$ and which is proved in Appendix A.

### Acknowledgements

We thank Pierre Pansu: the present paper elaborates on several of his ideas
and results. We thank Yves Cornulier for useful remarks and questions. M.B.
was partially supported by the Labex Cempi.

##  1\. de Rham $L^{p}$-cohomology

Pansu’s work on de Rham $L^{p}$-cohomology is dense and subtle. In this
section, we extract from his papers [Pan08, Pan09] the ideas and results that
are needed in the sequel. Since [Pan09] is not published yet, and because we
only need special cases which require simpler arguments, we include full
proofs of the statements. We hope that this section could also serve as a
gentle introduction to the subject.

###  1.1. Definitions

Let $M$ be a $C^{\infty}$ Riemannian manifold. We denote by $d\mathrm{vol}$
its Riemannian mesure, and by $|v|$ the Riemannian length of a vector $v\in
TM$. For $k\in\bf N$, let $\Omega^{k}(M)$ be the space of $C^{\infty}$
differential $k$-forms on $M$.

Let $p\in(1,+\infty)$. The $L^{p}$-norm of $\omega\in\Omega^{k}(M)$ is

$\|\omega\|_{L^{p}\Omega^{k}}=\bigl{(}\int_{M}|\omega|_{m}^{p}~{}d\mathrm{vol}(m)\bigr{)}^{1/p},$
$\mathrm{where~{}~{}~{}~{}}|\omega|_{m}:=\sup\\{|\omega(m;v_{1},...,v_{k})|:v_{1},\dots,v_{k}\in
T_{m}M,~{}|v_{i}|=1\\}.$

We denote by $L^{p}\Omega^{k}(M)$ the norm completion of the normed space
$\\{\omega\in\Omega^{k}(M):\|\omega\|_{L^{p}\Omega^{k}}<+\infty\\}$, _i.e._
the Banach space of $k$-differential forms with measurable $L^{p}$
coefficients.

Let also

$\|\omega\|_{\Omega^{p,k}}:=\|\omega\|_{L^{p}\Omega^{k}}+\|d\omega\|_{L^{p}\Omega^{k+1}}.$

One defines $\Omega^{p,k}(M)$ to be equal to the norm completion of the normed
space $\\{\omega\in\Omega^{k}(M):\|\omega\|_{\Omega^{p,k}}<+\infty\\}$. By
construction $\Omega^{p,k}(M)$ is a Banach space and the standard differential
operator extends to a bounded operator

$d_{k}:\Omega^{p,k}(M)\to\Omega^{p,k+1}(M),$

which satisfies $d\circ d=0$.

######  Definition 1.1.

The de Rham $L^{p}$-cohomology of $M$ is the cohomology of the complex
$\Omega^{p,0}(M)\stackrel{{\scriptstyle
d_{0}}}{{\to}}\Omega^{p,1}(M)\stackrel{{\scriptstyle
d_{1}}}{{\to}}\Omega^{p,2}(M)\stackrel{{\scriptstyle d_{2}}}{{\to}}\dots$. It
will be denoted by $L^{p}\mathrm{H_{dR}^{*}}(M)$. Its largest Hausdorff
quotient is denoted by $L^{p}\overline{\mathrm{H^{*}_{dR}}}(M)$ and is called
the reduced de Rham $L^{p}$-cohomology of $M$.

######  Remark 1.2.

According to [GT06, Theorem 12.8], the inclusion map

$\\{\omega\in\Omega^{*}(M):\|\omega\|_{\Omega^{p,*}}<+\infty\\}\subset\Omega^{p,*}(M)$

induces a topological isomorphism in cohomology. Therefore every element in
$L^{p}\mathrm{H_{dR}^{*}}(M)$ can be represented by a smooth form. In
particular, when $M$ is a compact manifold, its de Rham $L^{p}$-cohomology
coincides with the standard de Rham cohomology.

###  1.2. Vanishing

Let $\varphi:M\to M$ be a $C^{\infty}$ map and $k\in\bf N$. We denote by
$\varphi^{*}:\Omega^{k}(M)\to\Omega^{k}(M)$,
$\omega\mapsto\varphi^{*}(\omega)$, the associated linear map. In case it
induces a bounded operator $\varphi^{*}:L^{p}\Omega^{k}(M)\to
L^{p}\Omega^{k}(M)$, we denote its operator norm by
$\|\varphi^{*}\|_{L^{p}\Omega^{k}\to L^{p}\Omega^{k}}$. Otherwise we set
$\|\varphi^{*}\|_{L^{p}\Omega^{k}\to L^{p}\Omega^{k}}=+\infty$. We define
similarly $\|\varphi^{*}\|_{\Omega^{p,k}\to\Omega^{p,k}}$.

Let $\xi$ be a $C^{\infty}$ unit complete vector field on $M$, and let
$(\varphi_{t})_{t\in{\bf R}}$ be its flow. We assume that
$\varphi_{t}^{*}:L^{p}\Omega^{k}(M)\to L^{p}\Omega^{k}(M)$ is bounded for
every $t\in\bf R$ and $k\in\bf N$. For $k\in\bf N^{*}$, let
$\iota_{\xi}:\Omega^{k}(M)\to\Omega^{k-1}(M)$ be the inner product with $\xi$.
We observe that
$\varphi_{t}^{*}\circ\iota_{\xi}=\iota_{\xi}\circ\varphi_{t}^{*}$. Moreover
$\iota_{\xi}$ contracts the norms $\|\cdot\|_{L^{p}\Omega^{*}}$; indeed
$|\iota_{\xi}\omega|_{m}\leqslant|\omega|_{m}$, since $|\xi|_{m}=1$ by
assumption.

######  Lemma 1.3.

For $t\geqslant 0$ and $k\in\bf N^{*}$, the linear map:
$B_{t}^{k}:\Omega^{k}(M)\to\Omega^{k-1}(M)$ defined by

$B_{t}^{k}(\omega)=\int_{0}^{t}\varphi_{s}^{*}(\iota_{\xi}\omega)~{}ds,$

induces a bounded linear operator from $\Omega^{p,k}(M)$ to
$\Omega^{p,k-1}(M)$, still denoted by $B_{t}^{k}$, that satisfies the
following homotopy relation

$d\circ B_{t}^{k}+B_{t}^{k+1}\circ d=\varphi_{t}^{*}-\mathrm{id},$

and whose operator norm satisfies

$\|B_{t}^{k}\|_{\Omega^{p,k}\to\Omega^{p,k-1}}\leqslant\|\varphi_{t}^{*}-\mathrm{id}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k}}+\int_{0}^{t}\|\varphi_{s}^{*}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k}}~{}ds.$

###### Proof.

For smooth forms the homotopy relation holds as a standard application of the
classical Cartan formula

$\mathcal{L}_{\xi}=d\circ\iota_{\xi}+\iota_{\xi}\circ d$

(see _e.g._ [GHL04, Proposition 1.121] for this formula). Indeed for
$\omega\in\Omega^{k}(M)$ one has

$\varphi_{t}^{*}(\omega)-\omega=\int_{0}^{t}\frac{d}{ds}(\varphi_{s}^{*}\omega)~{}ds=\int_{0}^{t}\varphi_{s}^{*}(\mathcal{L}_{\xi}\omega)~{}ds=dB_{t}^{k}(\omega)+B_{t}^{k+1}(d\omega).$

We will prove that $B_{t}^{k}$ is a bounded linear operator from
$\Omega^{p,k}(M)$ to $\Omega^{p,k-1}(M)$. Since smooth forms are dense in
$\Omega^{p,*}(M)$ the homotopy formula will remain valid in $\Omega^{p,*}(M)$.

For $\omega\in\Omega^{k}(M)$, one has by definition

$\|B_{t}^{k}(\omega)\|_{\Omega^{p,k-1}}=\|B_{t}^{k}(\omega)\|_{L^{p}\Omega^{k-1}}+\|dB_{t}^{k}(\omega)\|_{L^{p}\Omega^{k}}.$

From the homotopy relations we get that

$\displaystyle\|B_{t}^{k}(\omega)\|_{\Omega^{p,k-1}}$
$\displaystyle\leqslant\|B_{t}^{k}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k-1}}\cdot\|\omega\|_{L^{p}\Omega^{k}}$
$\displaystyle+\|\varphi_{t}^{*}-\mathrm{id}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k}}\cdot\|\omega\|_{L^{p}\Omega^{k}}$
$\displaystyle+\|B_{t}^{k+1}\|_{L^{p}\Omega^{k+1}\to
L^{p}\Omega^{k}}\cdot\|d\omega\|_{L^{p}\Omega^{k+1}}.$

Since $\varphi_{s}^{*}\circ\iota_{\xi}=\iota_{\xi}\circ\varphi_{s}^{*}$, and
since the maps $\iota_{\xi}:L^{p}\Omega^{*}(M)\to L^{p}\Omega^{*-1}(M)$ are
contracting, one has

$\|B_{t}^{*}\|_{L^{p}\Omega^{*}\to
L^{p}\Omega^{*-1}}\leqslant\int_{0}^{t}\min\\{\|\varphi_{s}^{*}\|_{L^{p}\Omega^{*-1}\to
L^{p}\Omega^{*-1}},\|\varphi_{s}^{*}\|_{L^{p}\Omega^{*}\to
L^{p}\Omega^{*}}\\}~{}ds.$

The expected upper bound for $\|B_{t}^{k}\|_{\Omega^{p,k}\to\Omega^{p,k-1}}$
follows easily. ∎

We now establish a vanishing result.

######  Proposition 1.4.

[Pan08, Proposition 10] Let $p\in(1,+\infty)$ and $k\in\bf N^{*}$. Suppose
that there exists $C,\eta>0$, such that for every $t\geqslant 0$:

$\max_{i=k-1,k}\|\varphi_{t}^{*}\|_{L^{p}\Omega^{i}\to
L^{p}\Omega^{i}}\leqslant Ce^{-\eta t}.$

Then $L^{p}\mathrm{H}_{\mathrm{dR}}^{k}(M)=\\{0\\}$.

###### Proof.

First, the assumption (with $i=k$) and the previous lemma imply that the
operators $B_{t}^{k}:\Omega^{p,k}(M)\to\Omega^{p,k-1}(M)$ are bounded
independently of $t\geqslant 0$. Secondly, the assumption (with $i=k-1$)
implies that $\|\varphi_{t}^{*}\|_{\Omega^{p,k-1}\to\Omega^{p,k-1}}\to 0$,
when $t\to+\infty$. We claim that these two observations imply that
$B_{t}^{k}$ converges in norm to an operator
$B_{\infty}^{k}:\Omega^{p,k}(M)\to\Omega^{p,k-1}(M)$, when $t\to+\infty$.
Indeed, by using a change of variable, one has for $0\leqslant t_{1}\leqslant
t_{2}$:

$B_{t_{2}}^{k}-B_{t_{1}}^{k}=\int_{t_{1}}^{t_{2}}\varphi_{s}^{*}\circ\iota_{\xi}~{}ds=\varphi_{t_{1}}^{*}\circ
B_{t_{2}-t_{1}}^{k}.$

Thus the above observations yield:

$\|B_{t_{2}}^{k}-B_{t_{1}}^{k}\|_{\Omega^{p,k}\to\Omega^{p,k-1}}\leqslant\|\varphi_{t_{1}}^{*}\|_{\Omega^{p,k-1}\to\Omega^{p,k-1}}\cdot\|B_{t_{2}-t_{1}}^{k}\|_{\Omega^{p,k}\to\Omega^{p,k-1}}\to
0$

when $t_{1},t_{2}$ tend to $+\infty$. Thus the claim follows from
completeness.

Now let $\omega\in\Omega^{p,k}(M)\cap\ker d$. From the homotopy relations in
the previous lemma, we have for every $t\geqslant 0$

$dB_{t}^{k}(\omega)=\varphi_{t}^{*}(\omega)-\omega.$

By letting $t\to+\infty$, we obtain the following relation in
$\Omega^{p,k}(M)$

$dB_{\infty}^{k}(\omega)=-\omega.$

Therefore $\omega\in d\Omega^{p,k-1}$ and thus
$L^{p}\mathrm{H}_{\mathrm{dR}}^{k}(M)=\\{0\\}$. ∎

###  1.3. Identification

We relate the $L^{p}$-cohomology of $M$ with the cohomology of certain
complexes of currents (Proposition 1.9). In some cases, this will lead to non-
vanishing cohomology. See [DS05] for a nice introduction to the theory of
currents.

Another point of view for $\Omega^{p,*}(M)$. Let $D=\dim(M)$. Let
$\Omega_{c}^{k}(M)$ be the space of compactly supported $C^{\infty}$
differential $k$-forms, endowed with the $C^{\infty}$ topology. A _$k$
-current_ on $M$ is by definition a continuous real valued linear form on
$\Omega_{c}^{D-k}(M)$. We denote by $\mathcal{D}^{\prime k}(M)$ the space of
$k$-currents on $M$ endowed with the weak*-topology.

The differential of a $k$-current $T$ is the $(k+1)$-current $dT$ defined by
$dT(\alpha):=(-1)^{k+1}T(d\alpha)$, for every
$\alpha\in\Omega_{c}^{D-k-1}(M)$. It defines a map $d$ satisfying $d\circ
d=0$.

To every $\omega\in L^{p}\Omega^{k}(M)$, one associates the $k$-current
$T_{\omega}$ defined by $T_{\omega}(\alpha):=\int_{M}\omega\wedge\alpha$. The
differential in the sense of currents of $\omega\in L^{p}\Omega^{k}(M)$ is the
$(k+1)$-current $d\omega:=dT_{\omega}$. One says that $d\omega$ belongs to
$L^{p}\Omega^{k+1}(M)$ if there exits $\theta\in L^{p}\Omega^{k+1}(M)$ such
$d\omega=T_{\theta}$. These definitions are consistent with the Stokes
formula:

$\int_{M}d\omega\wedge\alpha=(-1)^{k+1}\int_{M}\omega\wedge d\alpha.$

We will use the following caracterisation of the space $\Omega^{p,*}(M)$:

######  Lemma 1.5.

The space $\Omega^{p,k}(M)$ is equal to the subspace of $L^{p}\Omega^{k}(M)$
consisting of the $L^{p}$ $k$-forms whose differentials in the sense of
currents belong to $L^{p}\Omega^{k+1}(M)$. Moreover the differential operator
$d$ on $\Omega^{p,*}(M)$ agrees with the differential in the sense of
currents.

###### Proof.

Since convergence in $L^{p}$ implies convergence in the sense of currents,
$\Omega^{p,k}(M)$ is contained in space of $L^{p}$ $k$-forms whose
differentials in the sense of currents belong to $L^{p}$. The proof of the
reverse inclusion is based on a regularization process: according to [GT06,
Theorem 12.5] there exists a family of operators $R_{\varepsilon}$, such that
for every $L^{p}$ $k$-form $\omega$ whose differential in the sense of
currents belongs to $L^{p}$, one has:

  1. (1)

$R_{\varepsilon}\omega$ is a $C^{\infty}$ $k$-form on $M$,

  2. (2)

$dR_{\varepsilon}\omega=R_{\varepsilon}d\omega$,

  3. (3)

$\|R_{\varepsilon}\omega-\omega\|_{L^{p}\Omega^{k}}$ and
$\|R_{\varepsilon}d\omega-d\omega\|_{L^{p}\Omega^{k+1}}$ tends to $0$ when
$\varepsilon\to 0$.

Theses properties imply that $\omega$ belongs to $\Omega^{p,k}(M)$. ∎

The complexes $\Psi^{p,*}(M)$ and $\Psi^{p,*}(M,\xi)$. Following [Pan09], we
introduce two complexes of currents.

######  Definition 1.6.

For $p\in(1,+\infty)$ and $k\in\bf N$, let $\Psi^{p,k}(M)$ be the space of
$k$-currents $\psi\in\mathcal{D}^{\prime k}(M)$ that can be written
$\psi=\beta+d\gamma$, with $\beta\in L^{p}\Omega^{k}(M)$ and $\gamma\in
L^{p}\Omega^{k-1}(M)$. In particular $\Psi^{p,0}(M)=L^{p}(M)$. For
$\psi\in\Psi^{p,k}(M)$, let

$\displaystyle\|\psi\|_{\Psi^{p,k}}=\inf\Bigl{\\{}$
$\displaystyle\|\beta\|_{L^{p}\Omega^{k}}+\|\gamma\|_{L^{p}\Omega^{k-1}}:\psi=\beta+d\gamma,$
$\displaystyle~{}~{}\mathrm{with}~{}~{}\beta\in
L^{p}\Omega^{k}(M)~{}~{}\mathrm{and}~{}~{}\gamma\in
L^{p}\Omega^{k-1}(M)\Bigr{\\}}.$

######  Lemma 1.7.

$\|\cdot\|_{\Psi^{p,k}}$ is a norm on $\Psi^{p,k}(M)$. With this norm
$\Psi^{p,k}(M)$ is a Banach space. The inclusion maps
$L^{p}\Omega^{k}(M)\subset\Psi^{p,k}(M)\subset\mathcal{D}^{\prime k}(M)$ are
continuous. The differentials in the sense of currents induce continuous
operators $d_{k}:\Psi^{p,k}(M)\to\Psi^{p,k+1}(M)$, that satisfy $d\circ d=0$.

###### Proof.

To see that $\|\cdot\|_{\Psi^{p,k}}$ is a norm, we only need to check that
$\|\psi\|_{\Psi^{p,k}}=0$ implies $\psi=0$. This is a consequence of the
following inequality. Let $\psi=\beta+d\gamma\in\Psi^{p,k}(M)$, then by Hölder
inequality one has for every $\alpha\in\Omega_{c}^{D-k-1}(M)$:

$\displaystyle|\psi(\alpha)|$
$\displaystyle=\Bigl{|}\int_{M}\beta\wedge\alpha+(-1)^{k+1}\int_{M}\gamma\wedge
d\alpha\Bigr{|}$
$\displaystyle\leqslant\|\beta\|_{L^{p}\Omega^{k}}\cdot\|\alpha\|_{L^{q}\Omega^{D-k}}+\|\gamma\|_{L^{p}\Omega^{k-1}}\cdot\|d\alpha\|_{L^{q}\Omega^{D-k+1}},$

where $1/p+1/q=1$. It also shows that the inclusion map
$\Psi^{p,k}(M)\subset\mathcal{D}^{\prime k}(M)$ is continuous. The continuity
of $L^{p}\Omega^{k}(M)\subset\Psi^{p,k}(M)$ is obvious. To show the
completeness, one notices that $\Psi^{p,k}(M)$ is isometric to
$L^{p}\Omega^{k}(M)\times L^{p}\Omega^{k-1}(M)/E$, where $E$ is the subspace

$E:=\\{(\beta,\gamma)~{}|~{}\beta+d\gamma=0\\}.$

Since $\|\cdot\|_{\Psi^{p,k}}$ is a norm, $E$ is a closed subspace. Since
$L^{p}\Omega^{k}(M)\times L^{p}\Omega^{k-1}(M)$ is a Banach space, the
quotient space is Banach too. Finally the last statement in the lemma is
obvious. ∎

Let $\xi$ be a $C^{\infty}$ unitary complete vector field on $M$, and denote
its flow by $\varphi_{t}$. We suppose that it induces a bounded linear
operator $\varphi_{t}^{*}:L^{p}\Omega^{k}(M)\to L^{p}\Omega^{k}(M)$, for every
$t\in\bf R$ and $k\in\bf N$. Then it induces an automorphism of the complex
$\Psi^{p,*}(M)$ whose operator norm satisfies

$\|\varphi_{t}^{*}\|_{\Psi^{p,k}\to\Psi^{p,k}}\leqslant\max_{i=k-1,k}\|\varphi_{t}^{*}\|_{L^{p}\Omega^{i}\to
L^{p}\Omega^{i}}.$

######  Definition 1.8.

For $p\in(1,+\infty)$ and $k\in\bf N$, we set

$\Psi^{p,k}(M,\xi)=\\{\psi\in\Psi^{p,k}(M):\varphi_{t}^{*}(\psi)=\psi~{}~{}\mathrm{for~{}~{}every}~{}~{}t\in\bf
R\\}.$

The complex $\Psi^{p,*}(M,\xi)$ is a closed subcomplex of $\Psi^{p,*}(M)$. Let

$\mathcal{Z}^{p,k}(M,\xi)=\operatorname{Ker}\bigl{(}d:\Psi^{p,k}(M,\xi)\to\Psi^{p,k+1}(M,\xi)\bigr{)}$

be the space of $k$-cocycles.

The next proposition elaborates on [Pan08, Proposition 10]. It is also a
special case of [Pan09, Corollary 12].

######  Proposition 1.9.

Let $p\in(1,+\infty)$ and $k\in\bf N$. Suppose that there exists $C,\eta>0$
such that for every $t\geqslant 0$, one has

$\max_{i=k-1,k}\|\varphi_{t}^{*}\|_{L^{p}\Omega^{i}\to
L^{p}\Omega^{i}}\leqslant Ce^{-\eta t}.$

(When $i=-1$, we set $L^{p}\Omega^{-1}:=L^{p}\Omega^{0}$ for convenience.)
Then there is a canonical Banach isomorphism

$L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)\simeq\mathcal{Z}^{p,k+1}(M,\xi),$

in particular $L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)$ is Hausdorff.

###### Proof.

The proof is divided into several steps.

(1) We define a bounded linear map

$P:\Omega^{p,k+1}(M)\cap\operatorname{Ker}d\to\mathcal{Z}^{p,k+1}(M,\xi)$

as follows. Consider the operators $B^{*}_{t}$ defined in Lemma 1.3. We claim
that the operators $B_{t}^{k+1}:L^{p}\Omega^{k+1}(M)\to L^{p}\Omega^{k}(M)$
converge in norm to an operator $B_{\infty}^{k+1}$, when $t$ tends to
$+\infty$. Indeed from its definition one has

$\|B_{t}^{k+1}\|_{L^{p}\Omega^{k+1}\to
L^{p}\Omega^{k}}\leqslant\int_{0}^{t}\|\varphi^{*}_{s}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k}}ds.$

Thus our assumption implies that $B_{t}^{k+1}$ is bounded independently of
$t\geqslant 0$. Moreover one has
$B_{t_{2}}^{k+1}-B_{t_{1}}^{k+1}=\varphi^{*}_{t_{1}}\circ
B_{t_{2}-t_{1}}^{k+1}$ for $0\leqslant t_{1}\leqslant t_{2}$. With our
assumption we get that:

$\displaystyle\|B_{t_{2}}^{k+1}$ $\displaystyle-
B_{t_{1}}^{k+1}\|_{L^{p}\Omega^{k+1}\to L^{p}\Omega^{k}}$
$\displaystyle\leqslant\|\varphi_{t_{1}}^{*}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k}}\cdot\|B_{t_{2}-t_{1}}^{k+1}\|_{L^{p}\Omega^{k+1}\to
L^{p}\Omega^{k}}\to 0$

when $t_{1},t_{2}\to+\infty$. The claim follows from completeness.

Let $\omega\in\Omega^{p,k+1}(M)\cap\operatorname{Ker}d$. For $t\geqslant 0$
one has thanks to Lemma 1.3:

$\varphi_{t}^{*}(\omega)=\omega+dB_{t}^{k+1}(\omega).$

By letting $t\to+\infty$, and by using the fact that
$\Omega^{p,k+1}(M)\cap\operatorname{Ker}d\subset L^{p}\Omega^{k+1}(M)$, the
claim implies that $\varphi_{t}^{*}(\omega)$ converges in $\Psi^{p,k+1}(M)$ to
the current $\omega_{\infty}:=\omega+dB_{\infty}^{k+1}(\omega)$.

We set $P(\omega):=\omega_{\infty}$. Since the maps $d$ and $\varphi^{*}$ are
continuous on $\Psi^{p,k+1}(M)$, one has $d\omega_{\infty}=0$ and
$\varphi_{t}^{*}(\omega_{\infty})=\omega_{\infty}$. Therefore $P(\omega)$
belongs to $\mathcal{Z}^{p,k+1}(M,\xi)$. Moreover $P$ is continuous; indeed

$\|P\|\leqslant 1+\|B^{p,k+1}_{\infty}\|_{L^{p}\Omega^{k+1}\to
L^{p}\Omega^{k}}.$

(2) $P(\omega)=0$ implies $[\omega]=0$ in
$L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)$: If $\omega_{\infty}=0$, one has
$\omega=-dB_{\infty}^{k+1}(\omega)$ with $B_{\infty}^{k+1}(\omega)\in
L^{p}\Omega^{k}(M)$. Therefore by Lemma 1.5 one has $[\omega]=0$.

(3) $P$ is surjective: Let $\psi\in\mathcal{Z}^{p,k+1}(M,\xi)$. Then there
exists $\beta\in L^{p}\Omega^{k+1}(M)$ and $\gamma\in L^{p}\Omega^{k}(M)$ such
that $\psi=\beta+d\gamma$ and $d\beta=0$. By Lemma 1.5, one has
$\beta\in\Omega^{p,k+1}(M)\cap\operatorname{Ker}d$, and:

$\beta_{\infty}=\beta+dB_{\infty}^{k+1}(\beta)=\psi+d\bigl{(}B_{\infty}^{k+1}(\beta)-\gamma\bigr{)}.$

Since $\psi$ and $\beta_{\infty}$ are both $\varphi_{t}^{*}$-invariant, so is
$d(B_{\infty}^{k+1}(\beta)-\gamma)$. But by our assumption, one has
$\|\varphi_{t}^{*}\|_{\Psi^{p,k}\to\Psi^{p,k}}\leqslant Ce^{-\eta t}$, and
thus:

$\varphi_{t}^{*}\bigl{(}d(B_{\infty}^{k+1}(\beta)-\gamma)\bigr{)}=d\bigl{(}\varphi_{t}^{*}(B_{\infty}^{k+1}(\beta)-\gamma)\bigr{)}\to
0$

when $t\to+\infty$. Therefore we get that
$d(B_{\infty}^{k+1}(\beta)-\gamma)=0$, and so
$\psi=\beta_{\infty}\in\operatorname{Im}P$.

(4) $[\omega]=0$ implies $P(\omega)=0$: Suppose that $\omega=d\alpha$ with
$\alpha\in\Omega^{p,k}(M)$. Then

$\omega_{\infty}=d\alpha+dB_{\infty}^{k+1}(\omega)=d\bigl{(}\alpha+B_{\infty}^{k+1}(\omega)\bigr{)},$

with $\alpha+B_{\infty}^{k+1}(\omega)\in\Psi^{p,k}(M)$. The argument in
paragraph (3) yields that $d(\alpha+B_{\infty}^{k+1}(\omega))=0$. Thus
$\omega_{\infty}=0$.

(5) Conclusion: Items (1) and (4) show that $P$ induces a continuous
homomorphism from $L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)$ to
$\mathcal{Z}^{p,k+1}(M,\xi)$; and items (2) and (3) that it is bijective. In
particular $L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)$ is Hausdorff and thus a
Banach space. By Lemma 1.7, the space $\mathcal{Z}^{p,k+1}(M,\xi)$ is Banach
too. Therefore Banach Theorem completes the proof. ∎

###  1.4. Non-vanishing

The next proposition is a non-vanishing result which complements Proposition
1.9. It is a special case of [Pan09, Corollary 32].

Observe that the space $\operatorname{Ker}(\iota_{\xi}:L^{p}\Omega^{*}(M)\to
L^{p}\Omega^{*-1}(M))$ is $\varphi_{t}^{*}$-invariant, since $\iota_{\xi}$ and
$\varphi_{t}^{*}$ commute.

######  Proposition 1.10.

Let $p\in(1,+\infty)$ and $k\in\\{0,\dots,D-2\\}$ where $D=\dim M$. Suppose
that $(\varphi_{t})_{t\in{\bf R}}$ acts properly on $M$ and that there exists
$C,\eta>0$ such that for $t\geqslant 0$:

$\displaystyle\|\varphi^{*}_{t}\|_{L^{p}\Omega^{k}\to
L^{p}\Omega^{k}}\leqslant Ce^{-\eta t},$
$\displaystyle\mathrm{and}~{}~{}~{}~{}~{}~{}$
$\displaystyle\|\varphi^{*}_{-t}\|_{L^{p}\Omega^{k+1}\cap\operatorname{Ker}\iota_{\xi}\to
L^{p}\Omega^{k+1}\cap\operatorname{Ker}\iota_{\xi}}\leqslant Ce^{-\eta t}.$

(Note the opposite signs in front of $t$ in the left side members). Then
$\mathcal{Z}^{p,k+1}(M,\xi)$ is non-trivial.

###### Proof.

Since $(\varphi_{t})_{t\in{\bf R}}$ acts properly on $M$, every orbit admits
an invariant open neighborhood $U\simeq{\bf R}\times V$ on which $\varphi_{t}$
acts like a translation along the $\bf R$ factor. In the sequel we suppose
that such an orbit neighborhood has been choosen.

Pick a non-trivial $\varphi_{t}$-invariant form
$\alpha\in\Omega^{k}(M)\cap\operatorname{Ker}\iota_{\xi}$ which is supported
in $U$. Since $k\in\\{0,\dots,D-2\\}$, we can choose $\alpha$ so that
$d\alpha\neq 0$. Set $\psi:=d\alpha$. We claim that
$\psi\in\mathcal{Z}^{p,k+1}(M,\xi)$.

Let $\chi$ be a $C^{\infty}$ function on $U\simeq{\bf R}\times V$, depending
only on the $\bf R$-variable, and such that $\chi(t)=0$ for $t\leqslant 0$ and
$\chi(t)=1$ for $t\geqslant 1$. One has
$\alpha=\chi\cdot\alpha+(1-\chi)\cdot\alpha$, and thus:

$\psi=\beta+d\gamma~{}~{}\mathrm{with}~{}~{}\beta=d(\chi\cdot\alpha)=d\chi\wedge\alpha+\chi\cdot
d\alpha~{}~{}\mathrm{and}~{}~{}\gamma=(1-\chi)\cdot\alpha.$

The form $d\chi\wedge\alpha$ has compact support. The form $d\alpha$ belongs
to $\operatorname{Ker}\iota_{\xi}$, thanks to the formula
$\mathcal{L}_{\xi}=d\circ\iota_{\xi}+\iota_{\xi}\circ d$. Since the
restriction of $d\alpha$ to $[0,1]\times V$ belongs to $L^{p}$, the assumption
on the norm of $\varphi_{-t}^{*}$ implies that the restriction of $d\alpha$ to
$[0,+\infty)\times V$ belongs to $L^{p}$ too. Therefore $\beta\in
L^{p}\Omega^{k+1}(M)$.

Since the restriction of $\alpha$ to $[0,1]\times V$ belongs to $L^{p}$, the
assumption on the norm of $\varphi_{t}^{*}$ implies that the restriction of
$\alpha$ to $(-\infty,0]\times V$ belongs to $L^{p}$ too. Therefore $\gamma\in
L^{p}\Omega^{k}(M)$. ∎

###  1.5. Poincaré duality

Parts of vanishing and non-vanishing results for $L^{p}$-cohomology rely on
the following version of Poincaré duality.

######  Proposition 1.11.

Let $M$ be a complete oriented Riemannian manifold of dimension $D$. Let
$p\in(1,+\infty)$, $q=p/(p-1)$ and $0\leqslant k\leqslant D$. Then

  * •

$L^{p}{\rm H}_{\mathrm{dR}}^{k}(M)$ is Hausdorff if and only if $L^{q}{\rm
H}_{\mathrm{dR}}^{D-k+1}(M)$ is.

  * •

$L^{p}\overline{{\rm H}^{k}_{\mathrm{dR}}}(M)$ and $L^{q}\overline{{\rm
H}^{D-k}_{\mathrm{dR}}}(M)$ are dual Banach spaces. Thus $L^{p}\overline{{\rm
H}^{k}_{\mathrm{dR}}}(M)=\\{0\\}$ if and only if $L^{q}\overline{{\rm
H}^{D-k}_{\mathrm{dR}}}(M)=\\{0\\}$.

###### Proof.

See [Pan08, Corollaire 14] or [GT10]. ∎

###  1.6. Examples and complements

1)  Let $M$ be a complete connected Riemannian manifold of infinite volume and
dimension $D$, and let $p\in(1,+\infty)$, $q=p/(p-1)$. It is obvious that
$L^{p}{\rm H}_{\mathrm{dR}}^{0}(M)=\\{0\\}$. By Poincaré duality (Proposition
1.11) this implies that $L^{q}\overline{{\rm
H}^{D}_{\mathrm{dR}}}(M)=\\{0\\}.$ In another hand, again by Poincaré duality,
$L^{q}{\rm H}_{\mathrm{dR}}^{D}(M)$ is Hausdorff if and only if $L^{p}{\rm
H}_{\mathrm{dR}}^{1}(M)$ is. When $M$ admits a complete unit vector field
whose flow expands exponentially the volume of $M$, then $L^{p}{\rm
H}_{\mathrm{dR}}^{1}(M)$ is Hausdorff for every $p\in(1,+\infty)$. Indeed this
follows from Proposition 1.9 applied with $k=0$. This shows in particular that
any connected non-unimodular Lie group $G$ of dimension $D$ satisfies
$L^{q}{\rm H}_{\mathrm{dR}}^{D}(G)=\\{0\\}$ for every $q\in(1,+\infty)$. In
fact, Pansu in [Pan07, Théorème 1] shows that for any non-compact connected
Lie group $G$, either $L^{p}{\rm H}_{\mathrm{dR}}^{1}(G)$ is Hausdorff or $G$
is amenable unimodular. See also [Tes09] for generalizations to topological
groups.

2)  Under assumptions of Proposition 1.9, one has
$\|\varphi_{t}^{*}\|_{\Psi^{p,k}\to\Psi^{p,k}}\to 0$ when $t\to+\infty$, and
thus $\Psi^{k,p}(M,\xi)=\\{0\\}$. Therefore the conclusion of Proposition 1.9
can be stated as
$L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)\simeq\mathrm{H}^{k+1}(\Psi^{p,*}(M,\xi))$.
In [Pan09, Theorem 4], Pansu establishes a general result that relates the
$L^{p}$-cohomology of $M$ with the cohomology of $\Psi^{p,*}(M,\xi)$.

3) Suppose $M$ is diffeomorphic to ${\bf R}\times N$ and that
$\xi=\frac{\partial}{\partial t}$ is the unit vector field carried by the
${\bf R}$-factor. Let $\pi:M\to N$ be the projection, and let
$\pi^{*}:\mathcal{D}^{\prime i}(N)\to\mathcal{D}^{\prime i}(M)$ be the
continuous extension of the pull-back map
$\pi^{*}:\Omega^{i}(N)\to\Omega^{i}(M)$. Then, under the assumptions of
Proposition 1.9, one can prove that every $\psi\in\mathcal{Z}^{p,k+1}(M,\xi)$
can be written as $\psi=\pi^{*}(T)$, with $T\in\mathcal{D}^{\prime
k+1}(N)\cap\operatorname{Ker}d$. In other words, one has

$\mathcal{Z}^{p,k+1}(M,\xi)=\pi^{*}\bigl{(}\mathcal{D}^{\prime
k+1}(N)\cap\operatorname{Ker}d\bigr{)}\cap\Psi^{p,k+1}(M).$

Therefore, the isomorphism
$L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)\simeq\mathcal{Z}^{p,k+1}(M,\xi)$ can
be interpreted by viewing $\mathcal{Z}^{p,k+1}(M,\xi)$ as a “boundary values”
space for the cohomology classes in $L^{p}\mathrm{H}_{\mathrm{dR}}^{k+1}(M)$.
See [Pan09] for more information along these lines.

##  2\. The Lie group case

This section collects applications of the previous results to Lie groups. It
will serve as a main tool in the paper.

Let $G$ be a connected Lie group equipped with a left-invariant Riemannian
metric. Let $D=\dim(G)$ and let $\mathfrak{g}$ be its Lie algebra. As usual
the left and right multiplications by $g\in G$ are denoted $L_{g}$ and
$R_{g}$, and we let $C_{g}=L_{g}\circ R_{g^{-1}}$ be the conjugacy by $g$.
Given a unit vector $\xi\in\mathfrak{g}$, we will still denote by $\xi$ the
associated left-invariant vector field on $G$. Its flow is
$\varphi_{t}=R_{\exp t\xi}$.

The following result is implicit in [Pan99], see in particular [Pan99,
Corollaire 53] and [Pan99, Proposition 57].

######  Proposition 2.1.

Let $\xi\in\mathfrak{g}$ and suppose that its flow acts properly on $G$. Let
$\lambda_{1}\leqslant\lambda_{2}\leqslant\dots\leqslant\lambda_{D}$ be the
real parts of the eigenvalues of
$(-\mathrm{ad}\xi)\in\mathrm{End}(\mathfrak{g})$, enumerated with their
multiplicities in their generalized eigenspaces. We denote by
$w_{k}=\sum_{i=1}^{k+1}\lambda_{i}$ the sum of the $(k+1)$ first ones, and by
$W_{k}=\sum_{j=1}^{k}\lambda_{D-j+1}$ the sum of the $k$ last ones. Let
$h=\sum_{i=1}^{D}\lambda_{i}$ be the trace of $(-\mathrm{ad}\xi)$, and suppose
that $h>0$. Let $k\in\\{0,\dots,D-1\\}$.

  1. (1)

If $1<p<\frac{h}{W_{k}}$, then $L^{p}\mathrm{H}^{k}_{\mathrm{dR}}(G)=\\{0\\}$.

  2. (2)

Under the same assumptions, the space $L^{p}\mathrm{H}^{k+1}_{\mathrm{dR}}(G)$
is Hausdorff and Banach isomorphic to $\mathcal{Z}^{p,k+1}(M,\xi)$.

  3. (3)

If $1\leqslant\frac{h}{w_{k+1}}<p<\frac{h}{W_{k}}$, then
$L^{p}\mathrm{H}^{k+1}_{\mathrm{dR}}(G)\neq\\{0\\}$.

(When $k=0$, we set $W_{0}:=0$ and $\frac{h}{W_{0}}:=+\infty$.)

######  Remark 2.2.

The above proposition applies to non-unimodular Lie groups only (but no
solvability or semi-simplicity assumption is made). Indeed the trace of
$\mathrm{ad}\xi$ is supposed to be non-zero.

###### Proof.

(1) and (2). To obtain vanishing and identification of the cohomology from
Propositions 1.4 and 1.9, we need to control the norm of
$\varphi^{*}_{t}:L^{p}\Omega^{*}(G)\to L^{p}\Omega^{*}(G)$, for $t\geqslant
0$. Let $k\in\\{0,\dots,D-1\\}$ and $\omega\in\Omega^{k}(G)$. Suppose first
that $\omega$ is left-invariant. Then $\varphi^{*}(\omega)$ is left-invariant
too. In particular their norms $|\omega|_{g}$ and
$|\varphi_{t}^{*}(\omega)|_{g}$ are independant of $g\in G$. We have

$\varphi_{t}^{*}(\omega)=R_{\exp t\xi}^{*}(\omega)=L^{*}_{\exp-t\xi}(R_{\exp
t\xi}^{*}(\omega))=C_{\exp-t\xi}^{*}(\omega).$

Moreover for $X_{1},\dots,X_{k}\in\mathfrak{g}$, we have

$\displaystyle C_{\exp-t\xi}^{*}(\omega)(X_{1},\dots,X_{k})$
$\displaystyle=\omega\bigl{(}\mathrm{Ad}(\exp-t\xi)X_{1},\dots,\mathrm{Ad}(\exp-t\xi)X_{k}\bigr{)}$
$\displaystyle=\omega(e^{-t\mathrm{ad}\xi}X_{1},\dots,e^{-t\mathrm{ad}\xi}X_{k}).$

In other words, by identifying the space of left-invariant $k$-forms with
$\Lambda^{k}\mathfrak{g}^{*}$, we get that
$C_{\exp-t\xi}^{*}(\omega)=(e^{-t\mathrm{ad}\xi})^{*}(\omega).$

Observe that $W_{k}$ is the largest real part of the eigenvalues of
$(-\mathrm{ad}\xi)^{*}$ acting on $\Lambda^{k}\mathfrak{g}^{*}$. Thus for
every $\varepsilon>0$ there is a constant $C=C(\varepsilon,k)>0$ such that for
all $\omega\in\Lambda^{k}\mathfrak{g}^{*}$ and $t\geq 0$, one has

$|\varphi_{t}^{*}(\omega)|\leqslant Ce^{(W_{k}+\varepsilon)t}|\omega|.$

Now let $\omega$ be any element of $\Omega^{k}(G)$ (it is not supposed to be
left-invariant anymore). Pick a basis $(\theta_{I})$ of
$\Lambda^{k}\mathfrak{g}^{*}$ seen as the space of left-invariant $k$-forms on
$G$. Then $\omega$ decomposes uniquely as $\omega=\sum_{I}f_{I}\theta_{I}$,
where $f_{I}\in\Omega^{0}(G)$. Since the norms on
$\Lambda^{k}\mathfrak{g}^{*}$ are all equivalent, there exists a constant
$D=D(k)>0$ such that for every $\omega\in\Omega^{k}(G)$ and $g\in G$:

$D^{-1}\bigl{(}\sum_{I}|f_{I}(g)|^{p}\bigr{)}^{1/p}\leqslant|\omega|_{g}\leqslant
D\bigl{(}\sum_{I}|f_{I}(g)|^{p}\bigr{)}^{1/p}.$

On the other hand, for $t\geqslant 0$:

$|\varphi_{t}^{*}(\omega)|_{g}=\bigl{|}\sum_{I}f_{I}\circ\varphi_{t}\cdot\varphi_{t}^{*}(\theta_{I})\bigr{|}_{g}\leqslant
C\sum_{I}|(f_{I}\circ\varphi_{t})(g)|e^{(W_{k}+\varepsilon)t}.$

Therefore:

$\displaystyle\|\varphi_{t}^{*}(\omega)\|_{L^{p}\Omega^{k}}^{p}$
$\displaystyle=\int_{G}|\varphi_{t}^{*}(\omega)|_{g}^{p}~{}d\mathrm{vol}(g)$
$\displaystyle\leqslant
C^{p}e^{p(W_{k}+\varepsilon)t}\int_{G}\sum_{I}|(f_{I}\circ\varphi_{t})(g)|^{p}~{}d\mathrm{vol}(g)$
$\displaystyle=C^{p}e^{p(W_{k}+\varepsilon)t}\int_{G}\sum_{I}|f_{I}(h)|^{p}\mathrm{Jac}(\varphi_{t}^{-1})(h)~{}d\mathrm{vol}(h)$
$\displaystyle=C^{p}e^{p(W_{k}+\varepsilon-\frac{h}{p})t}\int_{G}\sum_{I}|f_{I}(h)|^{p}~{}d\mathrm{vol}(h)$
$\displaystyle\leqslant
C^{p}D^{p}e^{p(W_{k}+\varepsilon-\frac{h}{p})t}\|\omega\|_{L^{p}\Omega^{k}}^{p},$

since the Jacobian of $\varphi_{t}$ is $e^{ht}$. Finally, by letting $E:=CD$,
one obtains that

$\|\varphi_{t}^{*}\|_{L^{p}\Omega^{k}\to L^{p}\Omega^{k}}\leqslant
Ee^{(W_{k}+\varepsilon-\frac{h}{p})t}.$

Suppose that $k\geqslant 1$ and $1<\frac{h}{W_{k}}$. Then, just from their
definitions, one sees that $0\leqslant W_{k-1}\leqslant W_{k}<h$ (recall that
$h>0$ by assumption). This implies that $W_{k}+\varepsilon-\frac{h}{p}<0$ for
$1<p<\frac{h}{W_{k}}$ and $\varepsilon>0$ small enough; and that the same
holds with $W_{k-1}$ instead of $W_{k}$. We conclude then by using
Propositions 1.4 and 1.9.

Suppose that $k=0$. Then
$W_{0}+\varepsilon-\frac{h}{p}=\varepsilon-\frac{h}{p}<0$ for $\varepsilon>0$
small enough. Thus we can conclude as well by using Propositions 1.4 and 1.9.

(3) To prove non-vanishing of the cohomology by using Propositions 1.9 and
1.10, we need in addition to control the norm of

$\varphi^{*}_{-t}:L^{p}\Omega^{k+1}(G)\cap\operatorname{Ker}\iota_{\xi}\to
L^{p}\Omega^{k+1}(G)\cap\operatorname{Ker}\iota_{\xi},$

for $t\geqslant 0$.

The group $(\varphi_{-t})_{t\in{\bf R}}$ is the flow of the vector field
$-\xi$. The real parts of the eigenvalues of $\mathrm{ad}\xi$ are
$-\lambda_{D}\leqslant-\lambda_{D-1}\leqslant\dots\leqslant-\lambda_{1}$. The
sum of the $(k+2)$ last ones is $-w_{k+1}$; and the trace of $\mathrm{ad}\xi$
is $-h$. We notice that $\xi$ is an eigenvector of eigenvalue $0$. By
assumption one has $w_{k+1}>0$; thus the eigenvalue $0$ appears in the list
$\lambda_{1}\leqslant\dots\leqslant\lambda_{k+2}$, whose sum is $w_{k+1}$.
Therefore $-w_{k+1}$ is largest real part of the eigenvalues of
$(\mathrm{ad}\xi)^{*}$ acting on
$(\Lambda^{k+1}\mathfrak{g}^{*})\cap\operatorname{Ker}\iota_{\xi}$.

By the same argument as in the first part of the proof, one obtains that for
every $\varepsilon>0$ there is a constant
$E^{\prime}=E^{\prime}(\varepsilon,k)>0$, such that for every $t\geqslant 0$:

$\|\varphi^{*}_{-t}\|_{L^{p}\Omega^{k+1}\cap\operatorname{Ker}\iota_{\xi}\to
L^{p}\Omega^{k+1}\cap\operatorname{Ker}\iota_{\xi}}\leqslant
E^{\prime}e^{(-w_{k+1}+\epsilon+\frac{h}{p})t}.$

For $p>\frac{h}{w_{k+1}}$ and $\varepsilon>0$ small enough, one has
$-w_{k+1}+\epsilon+\frac{h}{p}<0$. We conclude by using Propositions 1.9 and
1.10. ∎

######  Example 2.3.

Let $R={\bf R}\ltimes_{\alpha}{\bf R}^{D-1}$ with $\alpha:{\bf
R}\to\mathrm{GL}({\bf R}^{D-1})$ defined by $\alpha(t)=e^{t}\cdot\mathrm{id}$.
Then $R$ is isometric to the real hyperbolic space $\mathbb{H}^{D}_{\bf R}$.
Consider the vector field $\frac{\partial}{\partial t}$. The linear map
$\mathrm{ad}(\frac{\partial}{\partial t})$ is the projection on the ${\bf
R}^{D-1}$ factor. Put $\xi=-\frac{\partial}{\partial t}$. The list of the
eigenvalues of $(-\mathrm{ad}\xi)$ is
$0=\lambda_{1}<\lambda_{2}=\dots=\lambda_{D}=1$. For every
$k\in\\{0,\dots,D-1\\}$ one has $w_{k}=W_{k}=k$. Moreover $h=D-1$. Therefore
the above proposition applied in combination with Poincaré duality
(Proposition 1.11), shows that for $k\in\\{1,\dots,D-1\\}$:

  * •

$L^{p}\mathrm{H}^{k}_{\mathrm{dR}}(R)=\\{0\\}$ for $1<p<\frac{D-1}{k}$ or
$p>\frac{D-1}{k-1}$.

  * •

If $\frac{D-1}{k}<p<\frac{D-1}{k-1}$, then
$L^{p}\mathrm{H}^{k}_{\mathrm{dR}}(R)\neq\\{0\\}$ and is Banach isomorphic to
$\mathcal{Z}^{p,k}(M,\xi)$.

Moreover, by Subsection 1.6.1, one has
$L^{p}\mathrm{H}^{0}_{\mathrm{dR}}(R)=L^{p}\mathrm{H}^{D}_{\mathrm{dR}}(R)=\\{0\\}$
for every $p\in(1,+\infty)$.

Apart for the above example, and few other ones, the $L^{p}$-cohomology is
only partially known. In the sequel we will use the following consequences of
Proposition 2.1.

######  Corollary 2.4.

Suppose that $R$ is a connected Lie group of the form $R=A\ltimes N$, with
$A\simeq{\bf R}^{l}$ and $l\geqslant 1$, that satisfies the contraction
condition (nC) of the introduction (for a given $\xi$ in the Lie algebra of
$A$). Let $D=\dim(R)$. Then, with the notations of Proposition 2.1, the
numbers $w_{l}$, $W_{D-1-l}$ belongs to $(0,h)$, and we have:

  * •

If $1<p<\frac{h}{W_{D-1-l}}$, then
$L^{p}\mathrm{H}^{k}_{\mathrm{dR}}(R)=\\{0\\}$ for all $k<D-l$.

  * •

Under the same assumption, the space $L^{p}\mathrm{H}^{D-l}_{\mathrm{dR}}(R)$
is Hausdorff and Banach isomorphic to $\mathcal{Z}^{p,D-l}(R,\xi)$.

  * •

If $p>\frac{h}{w_{l}}$, then $L^{p}\mathrm{H}^{k}_{\mathrm{dR}}(R)=\\{0\\}$
for all $k>l$ .

###### Proof.

By condition (nC), the list of the real parts of the eigenvalues of
$(-\mathrm{ad}\xi)$ is of the form

$0=\lambda_{1}=\dots=\lambda_{l}<\lambda_{l+1}\leqslant\dots\leqslant\lambda_{D}.$

Thus, one has $0<W_{k}\leqslant W_{D-1-l}<h$ for every $k<D-l$. By Proposition
2.1 the first two items follow. The last one is a consequence of the previous
ones in combination with Poincaré duality (Proposition 1.11). It holds for $p$
larger than the Hölder conjugate of $\frac{h}{W_{D-l-1}}$, _i.e._ for
$p>\frac{h}{w_{l}}$ (since $W_{D-l-1}+w_{l}=h$). Note that
$w_{l}=\lambda_{l+1}$. ∎

In the special case $l=1$, one gets in addition:

######  Corollary 2.5.

Let $R$ be as in the previous corollary, and assume in addition that $l=1$. If
$1<p<\frac{h}{W_{D-2}}$, then $L^{p}\mathrm{H}^{D-1}_{\mathrm{dR}}(R)$ is non-
trivial.

###### Proof.

With the notations of Proposition 2.1, we have: $0<W_{D-2}<w_{D-1}<h$. Thus
the result follows from Proposition 2.1, applied with $k=D-2$. ∎

##  3\. A non-vanishing criterion

Let $R$ be a connected Lie group of the form $R=A\ltimes N$ with $A\simeq{\bf
R}^{l}$ and $l\geqslant 1$, that satisfies conditions (nC) and (nT) of the
introduction. As explained in Subsection 0.3, it decomposes as $R=B\ltimes H$
and the analysis of $L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{l}}(R)$ – for
large $p$ and $l=\dim(A)$ – reduces to the study of the action of $B$ on
$L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(H)$ for $p$ close to $1$. The goal of
this section is to give a sufficient condition on this action (Proposition
3.2), to ensure non-vanishing of
$L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{l}}(R)$ for large $p$ (Corollary
3.4).

As an application we will prove Theorems B, C and D in the next two Sections.

We keep the objects and notations of Subsection 0.3. Recall that the Lie
algebra of $A$ is denoted by $\mathfrak{a}$, and that the Lie algebra of $B$
is $\mathfrak{b}=\\{X\in\mathfrak{a}:\mathrm{trace}(\mathrm{ad}X)=0\\}$. The
group $H$ is defined by by $H=\\{e^{t\xi}\\}_{t\in\mathbf{R}}\ltimes N$, where
$\xi\in\mathfrak{a}$ is a vector that satisfies condition (nC).

As in the statement of Proposition 2.1, we consider the real parts of the
eigenvalues of $(-\mathrm{ad}\xi)$ acting on the Lie of $R$. Since $\xi$
satisfies condition (nC), one has

$0=\lambda_{1}=\dots=\lambda_{l}<\lambda_{l+1}\leqslant\dots\leqslant\lambda_{D}.$

Recall that $w_{k}:=\sum_{i=1}^{k+1}\lambda_{i}$ denotes the sum of the
$(k+1)$ first ones, and $W_{k}:=\sum_{j=1}^{k}\lambda_{D-j+1}$ the sum of the
$k$ last ones. The trace of $(-\mathrm{ad}\xi)$ is
$h:=\sum_{i=1}^{D}\lambda_{i}$.

We also set $n:=D-l=\dim(N)$, and we denote by $\pi$ the projection $\pi:H\to
N$ and by $\mathrm{vol}$ the Riemannian volume form on $N$.

###  3.1. Interpolation

Our first result is a quantitative version of Proposition 1.10. It shows that
the norm $\|\cdot\|_{\Psi^{p,n}(H)}$ interpolates (in some sense) between
$\|\cdot\|_{L^{p}\Omega^{n}(N)}$ and $\|\cdot\|_{L^{p}\Omega^{n-1}(N)}$.

######  Lemma 3.1.

For every $1<p<\frac{h}{W_{n-1}}$, there exist constants $a,b,C>0$, such that
for every $\theta\in\Omega^{p,n-1}(N)$ the form $\pi^{*}(d\theta)$ belongs to
$\mathcal{Z}^{p,n}(H,\xi)$, and satisfies:

$\|\pi^{*}(d\theta)\|_{\Psi^{p,n}(H)}\leqslant C\inf_{s\geqslant
0}\Bigl{\\{}e^{-as}\|d\theta\|_{L^{p}\Omega^{n}(N)}+e^{bs}\|\theta\|_{L^{p}\Omega^{n-1}(N)}\Bigr{\\}}.$

###### Proof.

Recall that the flow of $\xi$ in $H$ is denoted by $\varphi_{t}$. It is simply
a translation along the ${\bf R}$ factor of $H$. From Corollary 2.5 (and its
proof), for every $1<p<\frac{h}{W_{n-1}}$ there exist $\eta>0$ and $C_{1}>0$
such that for $t\geqslant 0$:

  1. (1)

$\|\varphi^{*}_{t}\|_{L^{p}\Omega^{n-1}\to L^{p}\Omega^{n-1}}\leqslant
C_{1}e^{-\eta t}$,

  2. (2)

$\|\varphi^{*}_{-t}\|_{L^{p}\Omega^{n}\cap\operatorname{Ker}\iota_{\xi}\to
L^{p}\Omega^{n}\cap\operatorname{Ker}\iota_{\xi}}\leqslant C_{1}e^{-\eta t}$.

For $s\in{\bf R}$, let $\chi_{s}\in C^{\infty}(H)$ which depends only on the
first variable, and such that $\chi_{s}(t)=0$ for $t\leqslant s$, and
$\chi_{s}(t)=1$ for $t\geqslant s+1$. Let $\alpha=\pi^{*}(\theta)$. Then
$\alpha$ and $d\alpha$ are $\varphi_{t}^{*}$-invariant. One has

$\pi^{*}(d\theta)=d\alpha=d(\chi_{s}\cdot\alpha+(1-\chi_{s})\cdot\alpha)=\beta+d\gamma,$

with $\beta=d(\chi_{s}\cdot\alpha)=d\chi_{s}\wedge\alpha+\chi_{s}\cdot
d\alpha$, and $\gamma=(1-\chi_{s})\cdot\alpha$. Thus
$\|\pi^{*}(d\theta)\|_{\Psi^{p,n}(H)}\leqslant\|\beta\|_{L^{p}\Omega^{n}(H)}+\|\gamma\|_{L^{p}\Omega^{n-1}(H)}$.

The form $d\alpha$ belongs to $\operatorname{Ker}\iota_{\xi}$ and is
$\varphi_{t}^{*}$-invariant. With the above property (2) we obtain (since
$s\geqslant 0$):

$\displaystyle\|\chi_{s}\cdot d\alpha\|_{L^{p}\Omega^{n}(H)}$
$\displaystyle\leqslant\|(d\alpha)\cdot\mathbf{1}_{t\geqslant
s}\|_{L^{p}\Omega^{n}(H)}$
$\displaystyle=\sum_{i=0}^{\infty}\|(d\alpha)\cdot\mathbf{1}_{t\in[s+i,s+i+1]}\|_{L^{p}\Omega^{n}(H)}$
$\displaystyle\leqslant
C_{1}\sum_{i=0}^{\infty}e^{-\eta(s+i)}\|(d\alpha)\cdot\mathbf{1}_{t\in[0,1]}\|_{L^{p}\Omega^{n}(H)}$
$\displaystyle=\frac{C_{1}e^{-\eta
s}}{1-e^{-\eta}}\|(d\alpha)\cdot\mathbf{1}_{t\in[0,1]}\|_{L^{p}\Omega^{n}(H)}.$

Since the Haar measure on $H$ is the product of those on $\mathbf{R}$ and $N$
times $e^{-th}$, the $L^{p}$-norms of $(d\alpha)\cdot\mathbf{1}_{t\in[0,1]}$
and of $d\theta$ are comparable. Therefore there exists a constant $C_{2}$,
depending only on $C_{1}$, $\eta$ and $H$, such that

$\|\chi_{s}\cdot d\alpha\|_{L^{p}\Omega^{n}(H)}\leqslant C_{2}e^{-\eta
s}\|d\theta\|_{L^{p}\Omega^{n}(N)}.$

It remains to bound by above the $L^{p}$-norms of $d\chi_{s}\wedge\alpha$ and
$\gamma$. One has $d\chi_{s}=\chi_{s}^{\prime}(t)dt$, with $\chi_{s}^{\prime}$
supported on $[s,s+1]$. Thus

$\|d\chi_{s}\wedge\alpha\|_{L^{p}\Omega^{n}(H)}\leqslant\|\chi_{s}^{\prime}\cdot\alpha\|_{L^{p}\Omega^{n-1}(H)}\leqslant
C_{3}\|\alpha\cdot\mathbf{1}_{t\in[0,s+1]}\|_{L^{p}\Omega^{n-1}(H)},$

with $C_{3}=\|\chi_{s}^{\prime}\|_{\infty}$.

Similarly, by writing $\gamma=\alpha\cdot\mathbf{1}_{t\leqslant
0}+(1-\chi_{s})\cdot\alpha\cdot\mathbf{1}_{t\in[0,s+1]}$, one has

$\|\gamma\|_{L^{p}\Omega^{n-1}(H)}\leqslant\|\alpha\cdot\mathbf{1}_{t\leqslant
0}\|_{L^{p}\Omega^{n-1}(H)}+\|\alpha\cdot\mathbf{1}_{t\in[0,s+1]}\|_{L^{p}\Omega^{n-1}(H)}.$

The form $\alpha$ is $\varphi_{t}^{*}$-invariant. By using property (1) and a
similar argument as above, one obtains that the $L^{p}$-norm of
$\alpha\cdot\mathbf{1}_{t\leqslant 0}$ is control by above by a constant times
$\|\theta\|_{L^{p}\Omega^{n-1}(N)}$.

Since $\varphi_{t}$ increases the norms at most exponentially, there exists
$b>0$ and $C_{4}>0$ such that for $t\geqslant 0$ one has

$\|\varphi_{-t}^{*}\|_{L^{p}\Omega^{n-1}\to L^{p}\Omega^{n-1}}\leqslant
C_{4}e^{bt}.$

Thus the $L^{p}$-norm of $\alpha\cdot\mathbf{1}_{t\in[0,s+1]}$ is bounded by
above by a constant times $e^{bs}\|\theta\|_{L^{p}\Omega^{n-1}}$. The
statement follows with $a=\eta$. ∎

###  3.2. The criterion

We now state the criterion that will lead to non-vanishing of cohomology. It
requires some preparations.

Since $\mathfrak{a}$ is abelian, there is a basis of
$\mathfrak{n}\otimes\mathbf{C}$ relative to which the matrices of
$\mathrm{ad}X\mbox{\Large$|$\normalsize}_{\mathfrak{n}}$ ($X\in\mathfrak{a}$)
are upper triangular [Hum72, 4.1. Corollary A]. For $i\in\\{1,\dots,n\\}$,
denote by $\varpi_{i}$ the $i$-th diagonal coefficient; it is a linear form on
$A$ with real or complex values. Since we assume that condition (nT) holds, we
have:

$\bigcap_{i=1}^{n}\operatorname{Ker}\Re(\varpi_{i})=\\{0\\}.$

By definition of the factor $\mathfrak{b}<\mathfrak{a}$, one has
$\mathrm{trace}(\mathrm{ad}X)=0$ for every $X\in\mathfrak{b}$; and thus
$\sum_{i=1}^{n}\Re(\varpi_{i}(X))=0$. In combination with the above trivial
intersection, this implies that for every $X\in\mathfrak{b}\setminus\\{0\\}$,
there exists $i\in\\{1,\dots,n\\}$ such that $\Re(\varpi_{i}(X))>0$.

######  Proposition 3.2.

Suppose that we are given:

  1. (1)

A non-empty subset $J\subset\\{1,\dots,n\\}$, such that for every vector
$X\in\mathfrak{b}\setminus\\{0\\}$ there exists $j\in J$ with
$\Re(\varpi_{j}(X))>0$.

  2. (2)

To every $j\in J$, a vector $Z_{j}\in\mathfrak{n}$ or
$\mathfrak{n}\otimes\mathbf{C}$ depending whether $\varpi_{j}$ takes real or
complex values, such that $\mathrm{ad}(X)\cdot Z_{j}=\varpi_{j}(X)Z_{j}$ for
every $X\in\mathfrak{b}$.

  3. (3)

A non-zero $C^{1}$ function $f:N\to{\bf R}$ with compact support, whose
integral along every orbit of the left-invariant vector fields $Y_{j}:=\Re
Z_{j}$ $(j\in J)$ is null.

Then for $1<p<\frac{h}{W_{n-1}}$, the form $\psi=\pi^{*}(f\mathrm{vol})$
belongs to $\mathcal{Z}^{p,n}(H,\xi)$, and satisfies:

$\int_{X\in\mathfrak{b}}\bigl{\|}C^{*}_{\exp
X}(\psi)\bigr{\|}_{\Psi^{p,n}(H)}^{p}~{}dX<+\infty.$

######  Remark 3.3.

Note that assumption (2) holds for any $J$, since the $w_{i}$’s are also the
weights of the representation
$\mathrm{ad}:A\to\mathrm{End}(\mathfrak{n}\otimes\mathbf{C})$. Moreover, as
explained above, assumption (1) is valid with the maximal choice
$J=\\{1,\dots,n\\}$. In general, condition (1) leads to choosing big subsets
$J$ of $\\{1,\dots,n\\}$ but this balanced by condition (3). The latter
assumption is not a straightforward consequence of our previous hypothesis and
the main part of the next section will be dedicated to sufficient conditions
for it.

######  Corollary 3.4.

Suppose that the conditions _(1), (2), (3)_ above are satisfied. Then
$L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{l}}(R)\neq\\{0\\}$ for
$p>\frac{h}{w_{l}}$.

###### Proof.

Let $1<p<\frac{h}{W_{n-1}}$. According to Corollary 2.4 applied with $R=H$,
the space $L^{p}\mathrm{H}_{\mathrm{dR}}^{k}(H)$ vanishes for $k<n$, and is
Hausdorff and Banach isomorphic to $\mathcal{Z}^{p,n}(H,\xi)$ when $k=n$. As
explained in the sketch of proof in Subsection 0.3, these properties in
combination with a spectral sequence argument taken from [BR20] – see
Corollary 6.10, imply that there exists a linear isomorphism:

$L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(R)\simeq\Bigl{\\{}\psi\in\mathcal{Z}^{p,n}(H,\xi):\int_{X\in\mathfrak{b}}\bigl{\|}C^{*}_{\exp
X}(\psi)\bigr{\|}_{\Psi^{p,n}(H)}^{p}~{}dX<+\infty\Bigr{\\}}.$

Thus by Proposition 3.2, the space $L^{p}\mathrm{H}_{\mathrm{dR}}^{D-l}(R)$ is
non-trivial. Moreover it Hausdorff (see Corollary 2.4). By Poincaré duality
(Proposition 1.11) we obtain that
$L^{p}\overline{\mathrm{H}_{\mathrm{dR}}^{l}}(R)\neq\\{0\\}$ for
$p>\frac{h}{w_{l}}$; indeed $\frac{h}{w_{l}}$ and $\frac{h}{W_{n-1}}$ are
Hölder conjugated (since $w_{l}+W_{n-1}=h$). ∎

###### Proof of Proposition 3.2.

Let $\phi^{t}_{j}:=R_{\exp tY_{j}}$ be the flow of $Y_{j}$, and set
$\omega:=f\mathrm{vol}$ so that $\psi=\pi^{*}(\omega)$. Then for every $j\in
J$ the form
$\theta_{j}:=\int_{-\infty}^{0}(\phi^{t}_{j})^{*}(\iota_{Y_{j}}\omega)~{}dt$
is a primitive of $\omega$. Indeed by applying the Cartan formula
$\mathcal{L}_{Y_{j}}=d\circ\iota_{Y_{j}}+\iota_{Y_{j}}\circ d$ to the
compactly suported closed form $\omega$, one gets

$d\theta_{j}=\int_{-\infty}^{0}(\phi^{t}_{j})^{*}(d\iota_{Y_{j}}\omega)~{}dt=\int_{-\infty}^{0}(\phi^{t}_{j})^{*}(\mathcal{L}_{Y_{j}}\omega)~{}dt=\int_{-\infty}^{0}\frac{d}{dt}\bigl{(}(\phi^{t}_{j})^{*}\omega\bigr{)}~{}dt,$

which is equal to $\omega$.

For $X\in\mathfrak{b}$, we want to estimate the $L^{p}$ norms of $C^{*}_{\exp
X}(\omega)$ and of its primitives $C^{*}_{\exp X}(\theta_{j})$, in order to
apply Lemma 3.1 to the form $C^{*}_{\exp X}(\psi)=\pi^{*}(C^{*}_{\exp
X}(\omega))$.

Observe that

$\bigl{\|}C^{*}_{\exp
X}(\omega)\bigr{\|}_{L^{p}\Omega^{n}(N)}=\bigl{\|}(f\circ C_{\exp X})\cdot
C^{*}_{\exp
X}(\mathrm{vol})\bigr{\|}_{L^{p}\Omega^{n}(N)}=\|\omega\|_{L^{p}\Omega^{n}(N)},$

since $C_{\exp X}$ preserves the Riemannian volume on $N$ (by definition of
the factor $\mathfrak{b}<\mathfrak{a}$).

To estimate the $L^{p}$-norm of $C^{*}_{\exp X}(\theta_{j})$, we proceed as
follows. Define $F_{j}=\int_{-\infty}^{0}f\circ\phi_{j}^{t}~{}dt$. By
assumption (3) the function $F_{j}$ has compact support. Since $\mathrm{vol}$
is bi-invariant on $N$, one has
$\theta_{j}=F_{j}\cdot(\iota_{Y_{j}}\mathrm{vol})$; and thus, since
$\mathrm{vol}$ is $C_{\exp X}$-invariant:

$C^{*}_{\exp X}(\theta_{j})=(F_{j}\circ C_{\exp X})\cdot(\iota_{C_{\exp
X}^{*}(Y_{j})}\mathrm{vol}).$

By assumption (2), one has $\mathrm{Ad}(\exp X)Z_{j}=e^{-\varpi_{j}(X)}Z_{j}$.
Since $Y_{j}=\Re Z_{j}$, and since it is left-invariant, one has $|C_{\exp
X}^{*}(Y_{j})|_{g}=e^{-\Re\varpi_{j}(X)}|Y_{j}|$, for every $g\in N$. Thus for
every $p\in(1,+\infty)$:

$\bigl{\|}C^{*}_{\exp
X}(\theta_{j})\bigr{\|}_{L^{p}\Omega^{n-1}(N)}=e^{-\Re\varpi_{j}(X)}|Y_{j}|\|F_{j}\|_{L^{p}(N)}.$

Note that all the $Y_{j}$’s are non-zero, indeed the operators $\mathrm{ad}X$
are _real_ endomorphisms.

Now assumption (1), in combination with the last equality and compactness of
the unit sphere in $\mathfrak{b}$, implies that there exist constants $c,C>0$
such that for every $p\in(1,+\infty)$ and every $X\in\mathfrak{b}$, one has

$\inf_{j\in J}\bigl{\|}C^{*}_{\exp
X}(\theta_{j})\bigr{\|}_{L^{p}\Omega^{n-1}(N)}\leqslant Ce^{-c|X|}.$

Thus, by Lemma 3.1, the forms $\psi$ and $C^{*}_{\exp X}(\psi)$ belong to
$\mathcal{Z}^{p,n}(H,\xi)$ for $1<p<\frac{h}{W{n-1}}$. Moreover the norm of
$C^{*}_{\exp X}(\psi)$ decreases exponentially fast to $0$ when
$|X|\to+\infty$. The statement follows. ∎

##  4\. First applications: proof of two non-vanishing results

This section is dedicated to the proof of the first two non-vanishing
theorems, namely Theorem B and Theorem C mentioned in the introduction; they
deal with general solvable groups satisfying conditions (nC) and (nT).

###  4.1. Functions with prescribed vanishing integrals

The following lemmas will serve to exhibit functions satisfying condition (3)
of Proposition 3.2; these functions will eventually provide suitable non-zero
forms leading to our targeted non-vanishing results by Corollary 3.4.

######  Lemma 4.1.

Let $Y,T$ be non-trivial left-invariant vector fields on a connected simply
connected nilpotent Lie group $N$. Then $N$ admits a non-zero smooth compactly
supported function, whose integral along every orbit of $Y$ and $T$ is null.

###### Proof.

Let $\mathfrak{n}$ be the Lie abgebra of $N$, and let $\mathfrak{l}$ be the
subalgebra generated by $Y$ and $T$. We will prove the lemma by induction on
the length of the descending central series of $\mathfrak{l}$, _i.e._ on the
smallest integer $k\geqslant 1$ such that $\mathfrak{l}^{k+1}=0$ (where
$\mathfrak{l}^{1}:=\mathfrak{l}$,
$\mathfrak{l}^{2}:=[\mathfrak{l},\mathfrak{l}]$,
$\mathfrak{l}^{i+1}:=[\mathfrak{l},\mathfrak{l}^{i}]$). Let denote the length
by $\mathrm{length}(\mathfrak{l})$.

If $\mathrm{length}(\mathfrak{l})=1$, then $Y$ and $T$ commute, and the
statement follows from (the proof of) Lemma 4.2. Suppose now that
$\mathrm{length}(\mathfrak{l})>1$.

Let $\phi_{Y}^{t}$ and $\phi_{T}^{t}$ be the flows of $Y$ and $T$. Let $f_{0}$
be a non-zero compactly supported smooth function on $N$. The flow of every
non-trivial left-invariant vector field acts properly on such a Lie group $N$;
thus there exists $t_{0}\in{\bf R}$ with

$(\mathrm{support}f_{0})\cap\phi_{T}^{-t_{0}}(\mathrm{support}f_{0})=\varnothing.$

The function $f:=f_{0}-f_{0}\circ\phi_{T}^{t_{0}}$ is non-zero, smooth,
compactly supported, and its integrals along the $T$-orbits are null.

We are looking for a sufficient condition on $f_{0}$ which garanties that the
integrals of $f$ along the $Y$-orbits are also null. For $g\in N$ one has

$\int_{\bf R}(f\circ\phi_{Y}^{t})(g)~{}dt=\int_{\bf
R}(f_{0}\circ\phi_{Y}^{t})(g)~{}dt-\int_{\bf
R}(f_{0}\circ\phi_{T}^{t_{0}}\circ\phi_{Y}^{t}\circ\phi_{T}^{-t_{0}})(\phi_{T}^{t_{0}}(g))~{}dt.$

Thus, for the integrals of $f$ along the $Y$-orbits to be null, it is enough
that the integrals of $f_{0}$ along the orbits of $Y$ and of
$(\phi_{T}^{-t_{0}})^{*}(Y)$ are null.

The vector field $(\phi_{T}^{-t_{0}})^{*}(Y)$ is left-invariant. Its value at
$1_{N}$ is

$Z:=\mathrm{Ad}(-t_{0}T)Y=e^{\mathrm{ad}(-t_{0}T)}Y.$

According to the induction hypothesis, such an $f_{0}$ exists if the
subalgebra $\mathfrak{m}$ generated by $Y$ and $Z$ satisfies
$\mathrm{length}(\mathfrak{m})<\mathrm{length}(\mathfrak{l})$.

Since $Y,Z\in\mathfrak{l}$, one has $\mathfrak{m}\subset\mathfrak{l}$.
Moreover:

$[Y,Z]=[Y,e^{\mathrm{ad}(-t_{0}T)}Y]=[Y,Y]-t_{0}\bigl{[}Y,[T,Y]\bigr{]}+\dots\in\mathfrak{l}^{3}.$

It follows that $\mathfrak{m}^{2}\subset\mathfrak{l}^{3}$, thus
$\mathfrak{m}^{3}=[\mathfrak{m},\mathfrak{m}^{2}]\subset[\mathfrak{l},\mathfrak{l}^{3}]=\mathfrak{l}^{4}$,
…, and so $\mathfrak{m}^{i}\subset\mathfrak{l}^{i+1}$ for every $i\geqslant
2$. Therefore one has
$\mathrm{length}(\mathfrak{m})<\mathrm{length}(\mathfrak{l})$ as expected. ∎

######  Lemma 4.2.

Let $M$ be a smooth manifold and $Y_{1},\dots,Y_{k},T$ be a family of complete
smooth vector fields. Assume that $T$ commutes with all the $Y_{i}$’s, and
that its flow acts properly on $M$. Suppose that the subfamily
$Y_{1},\dots,Y_{k}$ satisfies the following property: there exists a non-zero
$C^{1}$ compactly supported function on $M$, whose integral along every orbit
of the $Y_{i}$’s is null. Then the same property holds for the entire family
$Y_{1},\dots,Y_{k},T$.

###### Proof.

Let $f_{0}$ be a function satisfying the property for the subfamily
$Y_{1},\dots,Y_{k}$. Let $\phi^{t}$ be the flow of $T$. Since it acts properly
on $M$ there exists $t_{0}\in{\bf R}$ such that
$(\mathrm{support}f_{0})\cap\phi^{-t_{0}}(\mathrm{support}f_{0})=\varnothing$.
Define $f=f_{0}-f_{0}\circ\phi^{t_{0}}$. It is a non-zero $C^{1}$ compactly
supported function whose integral along every orbit of $T$ is null. Since $T$
commutes with the $Y_{i}$’s, the integrals of $f$ along their orbits remain
null. ∎

###  4.2. Two non-vanishing results for solvable groups

We can now give the proofs of Theorems B and C.

###### Proof of Theorem B.

We can assume that $N={\bf R}^{n}$. With the notations of Subsection 3.2, let
$J\subset\\{1,\dots,n\\}$ be a minimal subset such that

$\\{\varpi_{j}:j\in J\\}\cup\\{\overline{\varpi_{j}}:j\in
J\\}=\\{\varpi_{1},\dots,\varpi_{n}\\}$

as sets of linear forms on $\mathfrak{a}$. It satisfies assumption (1) of
Proposition 3.2. For every $j\in J$, let $Z_{j}$ be as in assumption (2) of
Proposition 3.2. The orbits in ${\bf R}^{n}$ of the vector field $Y_{j}:=\Re
Z_{j}$ are the lines parallel to $Y_{j}$. They commute. Thus Lemma 4.2
provides a function which satisfies assumption (3) of Proposition 3.2. Now
Theorem B follows from Corollary 3.4. ∎

###### Proof of Theorem C.

Since $A\simeq{\bf R}^{2}$, one has $B\simeq\mathbf{R}$. Let $X_{0}$ be a non-
zero vector in $\mathfrak{b}$. Let $J=\\{j_{1},j_{2}\\}\subset\\{1,\dots,n\\}$
be such that $\Re(\varpi_{j_{1}}(X_{0}))>0$ and
$\Re(\varpi_{j_{2}}(-X_{0}))>0$. It satisfies assumption (1) of Proposition
3.2. For every $j\in J$, let $Z_{j}$ be as in assumption (2) of Proposition
3.2, and let $Y_{j}:=\Re Z_{j}$. By applying Lemma 4.1 to the pair of left-
invariant vector fields $Y_{j_{1}},Y_{j_{2}}$, one obtains a function that
satisfies assumption (3) of Proposition 3.2. Now Theorem C follows from
Corollary 3.4. ∎

##  5\. Semisimple Lie groups

In this section, we are looking for connected solvable Lie subgroups $R$ in
semisimple real Lie groups which are of the form $R=A\ltimes N$ with
$A\simeq{\bf R}^{l}$ and $l\geqslant 1$. Of course these groups will satisfy
the contraction condition (nC) and the non-triviality condition (nT). More
precisely, our goal is to prove that in any semisimple real Lie group with
finite center, the solvable subgroups appearing in Iwasawa decompositions
fulfill the assumptions of our non-vanishing criterion Corollary 3.4. As a
consequence we obtain a proof of Theorem D, hence a proof for Theorem A, in
view of the reduction contained in 0.1. From a technical viewpoint, the proofs
in this section are ultimately relevant to the combinatorics of root systems.

###  5.1. Lie-theoretic notions and notations

Let $G$ be a semisimple real Lie group with finite center. We pick in $G$ a
subgroup $A$ which is maximal for the properties of being connected, abelian
and diagonalizable over the real numbers in the adjoint representation of $G$.
We denote the latter representation by

${\rm Ad}:G\to{\rm GL}({\mathfrak{g}}),$

with ${\mathfrak{g}}={\rm Lie}\,G$ . We have a direct sum decomposition

${\mathfrak{g}}={\mathfrak{g}}_{0}\oplus\bigoplus_{\alpha\in\Phi(G,A)}{\mathfrak{n}}_{\alpha},$

where ${\mathfrak{g}}_{0}$ is the subspace on which the adjoint $A$-action is
trivial and where ${\mathfrak{n}}_{\alpha}$ is the weight space associated
with the character $\alpha$ of $A$:

${\mathfrak{n}}_{\alpha}=\\{X\in\mathfrak{g}:{\rm
Ad}(a)X=\alpha(a)X\,\,\hbox{\rm for all}\,\,a\in A\\}.$

We also pick a minimal parabolic subgroup $P$ containing $A$, which provides a
basis and a positive root subset $\Phi^{+}$ for the root system
$\Phi=\Phi(G,A)$ of $G$. In order to fit with the notation of [Bou68], we set
$V=\mathfrak{a}^{*}$. This real vector space is equipped with the Killing
form, which makes it a Euclidean space; this enables us to identify
$\mathfrak{a}$ and $\mathfrak{a}^{*}$. Finally, we set:

$2\rho_{G}=\sum_{\alpha\in\Phi^{+}}{\rm dim}({\mathfrak{n}}_{\alpha})\alpha,$

and we denote by $N$ the nilpotent group integrating
$\bigoplus_{\alpha\in\Phi^{+}}{\mathfrak{n}}_{\alpha}$. Thus, $R=AN$ is the
solvable part of an Iwasawa decompostion of $G$, and all such solvable
subgroups of $G$ can be obtained by varying the choices of $A$ and $P$ above.
The remaining choice is that of a maximal compact subgroup $K$: the stabilizer
of any point in the maximal flat associated to $A$ in the Riemmanian symmetric
space associated to $G$ does the job.

From a combinatorial viewpoint, in the main part of this section we are
looking for a subset $\Psi\subset\Phi^{+}$ and a family of real numbers
$(m_{\beta})_{\beta\in\Psi}$, with $m_{\beta}>0$ for each $\beta\in\Psi$, such
that

  1. (i)

we have: $2\rho_{G}=\sum_{\beta\in\Psi}m_{\beta}\beta$,

  2. (ii)

we have: $V=\sum_{\beta\in\Psi}{\bf R}\beta$,

  3. (iii)

if $\beta,\beta^{\prime}\in\Psi$ then: $\beta+\beta^{\prime}\not\in\Phi$.

We will sometimes relax condition (iii) to the following one

  1. ${\rm(iii)}^{\prime}$

if $\beta,\beta^{\prime}\in\Psi$ then: $\beta+\beta^{\prime}\not\in\Phi$
except maybe for one pair $\\{\beta;\beta^{\prime}\\}\subset\Psi$.

###  5.2. Relationship with the non-vanishing criterion

Let us explain why the above combinatorial conditions are relevant to the non-
vanishing criterion of the previous section. In 3.2, the set
$\\{1,2,\dots,n\\}$ is determined by the dimension of the nilpotent group $N$:
in the present section, we see it more concretely as the set of positive roots
counted with their multiplicities; accordingly, the subset $J$ of 3.2 is
denoted here by $\Psi$ in view of its interpretation in terms of roots. Note
also that, by the choice of $A$ here (a maximal ${\bf R}$-split torus), we do
not need to extend the scalars to ${\bf C}$ in our context.

First of all, the solvable group $R=AN$ from 5.1 satisfies condition (nC)
since we can choose $-\xi$ to be the sum of the fundamental coweights for the
root system of $G$ (in fact, we can take any vector in $\mathfrak{a}^{+}$,
_i.e._ in the Weyl cone given by $AN$, tangent to a regular geodesic in the
symmetric space). The group $R$ also satisfies condition (nT) by definition of
a root as a _non-trivial_ character of $A$ (or $\mathfrak{a}$). We now
concentrate on checking the conditions of the non-vanishing criterion in the
present situation.

######  Lemma 5.1.

We assume we are given $\Psi$ satisfying (i)-(iii)′. Then the choice $J=\Psi$
fulfills the assumptions of Proposition 3.2.

The lemma will lead to the desired non-vanishing statement, hence most of the
rest of the Section will be dedicated to exhibiting such a subset $\Psi$ for
any semisimple real Lie group $G$.

###### Proof.

As mentioned in Remark 3.3, Condition (2) is automatically satisfied by
definition of the adjoint representation and of the roots.

Let us check Condition (1). By (i), we have:

$\sum_{\alpha\in\Phi^{+}}{\rm
dim}({\mathfrak{n}}_{\alpha})\alpha=2\rho_{G}=\sum_{\beta\in\Psi}m_{\beta}\beta.$

Now let $X\in\mathfrak{b}$. Note first that for any $X\in\mathfrak{a}$, we
have: $\mathrm{trace}(\mathrm{ad}X)=2\rho_{G}(X)$; therefore $\mathfrak{b}$,
defined in 0.3 by
$\mathfrak{b}=\\{X\in\mathfrak{a}:\mathrm{trace}(\mathrm{ad}X)=0\\}$, can be
seen as $\mathfrak{b}=\\{X\in\mathfrak{a}:\rho_{G}(X)=0\\}$. We assume that
$\beta(X)\leqslant 0$ for each $\beta\in\Psi$. Since $m_{\beta}>0$ for each
$\beta\in\Psi$, we deduce that $\beta(X)=0$ for each $\beta\in\Psi$. Finally
this implies that $X=0$ by (ii), proving that (1) is satisfied.

Condition (3) is more delicate to check but it is treated thanks to the lemmas
in 4.1 combined with the fact that if for $\alpha,\alpha^{\prime}\in\Phi^{+}$
we have $\alpha+\alpha^{\prime}\not\in\Phi$ then
$[\mathfrak{n}_{\alpha},\mathfrak{n}_{\alpha^{\prime}}]=\\{0\\}$. Indeed, in
view of this, the function requested by Condition (3) is directly given by an
inductive use of Lemma 4.2 in case $\Psi$ satisfies (iii); if $\Psi$ satisfies
(iii)′ only, then one has to apply first Lemma 4.1Â to the pair of roots in
$\Psi$ whose sum is a root, and then again apply inductively Lemma 4.2. ∎

In the rest of the Section, we are thus reduced to exhibiting subsets $\Psi$
satisfying (i)-(iii)′, or even better satisfying (i)-(iii). Note that in view
of Theorem C we may – and shall – assume that the rank $l={\rm dim}(A)$ of $G$
is at least $3$.

###  5.3. A preliminary result in linear algebra

We use the notation from [Bou68, Planches], and more precisely the concrete
descriptions of root systems in terms of linear algebra. In particular, we
consider the standard Euclidean space ${\bf R}^{l}$ with canonical basis
$(\varepsilon_{i})_{1\leqslant i\leqslant l}$. We are interested in sums of
the form

$\displaystyle\sum_{m=1}^{M}m(\varepsilon_{l-2m+1}+\varepsilon_{l-2m}+$
$\displaystyle~{}\varepsilon_{l-2m}+\varepsilon_{l-2m-1})=$
$\displaystyle=\sum_{m=1}^{M}m(\varepsilon_{l-2m+1}+2\varepsilon_{l-2m}+\varepsilon_{l-2m-1}),$

where $M$ is a suitably chosen integer. We will see below that such linear
combinations are very useful to approximate the sum of positive roots of some
fixed norm in suitable root systems.

In addition, the vectors appearing in these sums are sums of two consecutive
vectors of the form $\varepsilon_{i}+\varepsilon_{i+1}$: in many root systems,
such a vector $\varepsilon_{i}+\varepsilon_{i+1}$ is a positive root and the
sum of two such vectors is not a root, which is useful to achieve condition
(iii), or maybe (iii)′, above.

More precisely, in many root systems the sum of positive roots of norm equal
to $\sqrt{2}$ is the vector $\sigma$ given by:

$\sigma=\sum_{i<j}\varepsilon_{i}\pm\varepsilon_{j}=\sum_{i=1}^{l-1}\bigl{(}\sum_{j>i}(\varepsilon_{i}+\varepsilon_{j}+\varepsilon_{i}-\varepsilon_{j})\bigr{)}=2\sum_{i=1}^{l-1}(l-i)\varepsilon_{i}=2\sum_{i=1}^{l-1}i\varepsilon_{l-i},$

and we want to see $\sigma$ as a linear combination, with positive
coefficients, of vectors $\varepsilon_{i}+\varepsilon_{i+1}$.

######  Lemma 5.2.

Besides the vector $\displaystyle\sigma=\sum_{1\leqslant i<j\leqslant
l}\varepsilon_{i}\pm\varepsilon_{j}=2\sum_{i=1}^{l-1}i\varepsilon_{l-i}$
above, we introduce the sum $\displaystyle S=\sum_{m=1}^{\lfloor{l-2\over
2}\rfloor}m(\varepsilon_{l-2m+1}+\varepsilon_{l-2m}+\varepsilon_{l-2m}+\varepsilon_{l-2m-1})$.
Then:

  * $\bullet$

if $l$ is even, we have: $\sigma=2S+l\varepsilon_{1}$;

  * $\bullet$

if $l$ is odd, we have:
$\sigma=2S+(l-1)(\varepsilon_{1}+\varepsilon_{2})+(l-1)\varepsilon_{1}$.

###### Proof.

In the sum
$S_{M}=\sum_{m=1}^{M}m(\varepsilon_{l-2m+1}+\varepsilon_{l-2m}+\varepsilon_{l-2m}+\varepsilon_{l-2m-1})$,
we consider the sum of two consecutive terms, that is:

$k(\varepsilon_{l-2k+1}+2\varepsilon_{l-2k}+\varepsilon_{l-2k-1})+(k+1)(\varepsilon_{l-2k-2+1}+2\varepsilon_{l-2k-2}+\varepsilon_{l-2k-2-1})$
$=k\varepsilon_{l-2k+1}+2k\varepsilon_{l-2k}+(2k+1)\varepsilon_{l-2k-1}+(2k+2)k\varepsilon_{l-2k-2}+(k+1)\varepsilon_{l-2k-3}.$

The three middle terms also appear in ${1\over 2}\sigma$. The maximal index of
summation $M$ is the biggest integer $m$ satisfying $l-2m-1\geqslant 1$, that
is $m\leqslant{l-2\over 2}$. This explains why we are led to considering:

$S=S_{\lfloor{l-2\over 2}\rfloor}=\sum_{m=1}^{\lfloor{l-2\over
2}\rfloor}m(\varepsilon_{l-2m+1}+\varepsilon_{l-2m}+\varepsilon_{l-2m}+\varepsilon_{l-2m-1}).$

The sum $S=S_{\lfloor{l-2\over 2}\rfloor}$ approximates ${1\over 2}\sigma$
down to the index $l-2\lfloor{l-2\over 2}\rfloor-1$.

$\bullet$ If $l$ is even, we have $\lfloor{l-2\over 2}\rfloor={l-2\over 2}$,
then $l-2\lfloor{l-2\over 2}\rfloor-1=1$ and therefore:

$\displaystyle S=$ $\displaystyle\sum_{m=1}^{\lfloor{l-2\over
2}\rfloor}m(\varepsilon_{l-2m+1}+\varepsilon_{l-2m}+\varepsilon_{l-2m}+\varepsilon_{l-2m-1})$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{l-2}j\varepsilon_{l-j}+({l\over
2}-1)\varepsilon_{1}={1\over 2}\sigma-{l\over 2}\varepsilon_{1},$

so that in this case:

${1\over 2}\sigma=S+{l\over 2}\varepsilon_{1}.$

$\bullet$ If $l$ is odd, we have $\lfloor{l-2\over 2}\rfloor<{l-2\over 2}$,
then $l-2\lfloor{l-2\over 2}\rfloor-1=l-(l-3)-1=2$ and therefore:

$S=\sum_{m=1}^{\lfloor{l-2\over
2}\rfloor}m(\varepsilon_{l-2m+1}+\varepsilon_{l-2m}+\varepsilon_{l-2m}+\varepsilon_{l-2m-1})=\sum_{j=1}^{l-3}j\varepsilon_{l-j}+\lfloor{l-2\over
2}\rfloor\varepsilon_{2}.$

If we denote $l=2r+1$ with $r\geqslant 1$, then $\lfloor{l-2\over
2}\rfloor=\lfloor{2r-1\over 2}\rfloor=\lfloor r-{1\over 2}\rfloor=r-1$ and we
have:

$S=\sum_{j=1}^{l-3}j\varepsilon_{l-j}+(r-1)\varepsilon_{2},$

so that, since the coefficient of $\varepsilon_{2}$ in ${1\over 2}\sigma$ is
$l-2=(2r+1)-2=2r-1$, we have ${1\over
2}\sigma=S+r(\varepsilon_{1}+\varepsilon_{2})+r\varepsilon_{1}$ and finally

${1\over 2}\sigma=S+{l-1\over 2}(\varepsilon_{1}+\varepsilon_{2})+{l-1\over
2}\varepsilon_{1}.$

This concludes the proof. ∎

At this stage, in order to find subsets $\Psi$ of positive roots achieving the
conditions (i)-(iii)′ at the end of Section 5.1, we use the classification of
simple real Lie groups. The parameters of this classification are: a (possibly
non-reduced) root system, and the multiplicities of the roots. According to É.
Cartan’s classification, a simple real Lie group either is a simple complex
Lie group seen as a real one (in which case the root system is reduced and all
multiplicities are equal to 2), or is absolutely simple (_i.e._ its Lie
algebra stays simple after complexification) and belongs to the list given for
instance in [Hel01, Chapter X, Table VI, pp. 532-534]. One useful fact is
that the Weyl group acts transitively on roots of given norm (see [Hum72,
Lemma C, p. 53]), so that not so many possibilities of root multiplicities
appear (one multiplicity if the root system is simply laced _i.e._ if the
edges in the Dynkin diagram are all simple, at most 2 if it is reduced, at
most 3 otherwise). This also implies that
$2\rho_{G}=\sum_{\alpha\in\Phi^{+}}{\rm dim}({\mathfrak{n}}_{\alpha})\alpha$
can be computed as a sum of at most 3 partial sums, namely packets of roots of
given norm multiplied by the corresponding multiplicity.

In the rest of this Section, we check that we can find a suitable root subset
$\Psi$ for each isomorphism class of simple real Lie groups by sorting them
according to their relative root system first and then their multiplicities.
The reader will easily check that all Cartan types appearing in [Hel01,
Chapter X, Table VI, pp. 532-534] are covered.

###  5.4. All types except $\mathrm{A}_{l}$

To be consistent with the notation of [Hel01, Chapter X, Table VI, pp.
532-534], we denote by $l$ the real rank of $G$ and by $r$ its complex rank
(recall that the latter rank is the dimension of a subalgebra in
$\mathfrak{g}\otimes_{\bf R}{\bf C}$ consisting of diagonalizable elements in
the adjoint representation, and maximal for this elementwise property). Recall
that we can restrict our attention to the case $l\geqslant 3$, thanks to
Theorem C.

$\bullet$ Type ${\rm B}_{l}$ [Bou68, Planche II, p. 252]. This is a non
simply laced root system, and indeed some cases with different multiplicities
do appear in Cartan’s classification. This is the root system in the standard
Euclidean space ${\bf R}^{l}$ with canonical basis
$(\varepsilon_{i})_{1\leqslant i\leqslant l}$, where the positive roots are
the vectors of the form $\varepsilon_{i}$ (there are $l$ such roots, of norm
1) or $\varepsilon_{i}\pm\varepsilon_{j}$ with $i<j$ (there are $l(l-1)$ such
roots, of norm $\sqrt{2}$). We still denote the partial sum of roots of norm
$\sqrt{2}$ by

$\sigma=\sum_{i<j}\varepsilon_{i}\pm\varepsilon_{j}=2\sum_{i=1}^{l-1}i\varepsilon_{l-i},$

and we denote the partial sum of roots of norm $1$ by
$\tau=\sum_{i=1}^{l}\varepsilon_{i}$. The Cartan types where this root system
appears are the types ${\rm B}_{\rm I}$ and ${\rm D}_{\rm I}$, in which cases
we have:

$2\rho_{{\rm B}_{\rm I}}=\sigma+\bigl{(}2(r-l)+1\bigr{)}\tau\quad\hbox{\rm
and}\quad 2\rho_{{\rm D}_{\rm I}}=\sigma+2(r-l)\tau.$

Let us treat thoroughly the Cartan type ${\rm B}_{\rm I}$, the type ${\rm
D}_{\rm I}$ being similar. We invoke Lemma 5.2, together with the associated
notation.

If $l$ is even, we have:

$2\rho_{{\rm B}_{\rm
I}}=2S+l\varepsilon_{1}+\bigl{(}2(r-l)+1\bigr{)}\sum_{m=1}^{{l\over
2}}(\varepsilon_{2m-1}+\varepsilon_{2m})$

so that we can take the subset of positive roots $\Psi$ to be consisting of
$\varepsilon_{1}$ and of the roots $\varepsilon_{j}+\varepsilon_{j+1}$ for
$1\leqslant j\leqslant l-1$. This choice of $\Psi$ gives immediately (i) in
view of the definition of the sum $S$ and (ii), _i.e._ the fact that $\Psi$
generates $V$, is easy; at last (iii) follows from the fact that no
coefficient 2 appears in the coordinates of roots and the fact that the
support of roots has cardinality $\leqslant 2$.

If $l$ is odd, we have

$\displaystyle 2\rho_{{\rm B}_{\rm I}}=$
$\displaystyle~{}2S+2(l-1)(\varepsilon_{1}+\varepsilon_{2})+(l-1)\varepsilon_{1}$
$\displaystyle+$ $\displaystyle~{}\bigl{(}2(r-l)+1\bigr{)}\sum_{m=1}^{l-1\over
2}(\varepsilon_{2m}+\varepsilon_{2m+1})+\bigl{(}2(r-l)+1\bigr{)}\varepsilon_{1}.$

The expression is more complicated but it gives (i) and the same arguments as
above work to give (ii) and (iii).

At last, as already mentioned, the Cartan type ${\rm D}_{\rm I}$ is deduced
from this case after replacing the coefficient $\bigl{(}2(r-l)+1\bigr{)}$ by
$2(r-l)$.

$\bullet$ Type ${\rm C}_{l}$ [Bou68, Planche III, p. 254]. This is a non
simply laced root system, and again some cases with different multiplicities
do appear in Cartan’s classification. This is the root system in the standard
Euclidean space ${\bf R}^{l}$ with canonical basis
$(\varepsilon_{i})_{1\leqslant i\leqslant l}$, where the positive roots are
the vectors of the form $2\varepsilon_{i}$ (there are $l$ such roots, of norm
2) or $\varepsilon_{i}\pm\varepsilon_{j}$ with $i<j$ (there are $l(l-1)$ such
roots, of norm $\sqrt{2}$). We still denote the partial sum of roots of norm
$\sqrt{2}$ by

$\sigma=\sum_{i<j}\varepsilon_{i}\pm\varepsilon_{j}=2\sum_{i=1}^{l-1}i\varepsilon_{l-i},$

and we denote the partial sum of roots of norm $2$ by
$\kappa=\sum_{i=1}^{l}2\varepsilon_{i}$. The Cartan types covered by this root
system are: some cases ${\rm A}_{\rm III}$, all cases ${\rm C}_{\rm I}$, some
cases ${\rm C}_{\rm II}$, half of the cases ${\rm D}_{\rm III}$ and the case
${\rm E}_{\rm VII}$. For any such type, say ${\rm G}$, we have:

$2\rho_{\rm G}=m_{\sigma}\sigma+m_{\tau}\kappa$

where $m_{\sigma}$ and $m_{\tau}$ are integers $\geqslant 1$111More precisely,
we have: $2\rho_{{\rm A}_{\rm III}}=2\sigma+\kappa$ in the ${\rm C}_{l}$ case
of ${\rm A}_{\rm III}$, $2\rho_{{\rm C}_{\rm I}}=\sigma+\kappa$, $2\rho_{{\rm
C}_{\rm II}}=4\sigma+3\kappa$ in the ${\rm C}_{l}$ case of ${\rm C}_{\rm II}$,
we have $2\rho_{{\rm D}_{\rm III}}=4\sigma+\kappa$ in the ${\rm C}_{l}$ case
of ${\rm D}_{\rm III}$ and $2\rho_{{\rm E}_{\rm VII}}=8\sigma+\kappa$, the
root system being ${\rm C}_{3}$ in the latter case..

We choose for $\Psi$ the set of roots $2\varepsilon_{j}$ for $1\leqslant
j\leqslant l$. Conditions (i) to (iii) are trivially satisfied, for a choice
of integral coefficients in (i) thanks to the coefficients 2 in the
expressions of $\sigma$ and $\tau$.

$\bullet$ Type ${\rm BC}_{l}$ [Bou68, Chapitre VI.14, p. 222]. This is the
only type of non-reduced root systems: the system ${\rm BC}_{l}$ consists of
the union of the system ${\rm B}_{l}$ and of the system ${\rm C}_{l}$ as
described above. The Cartan types covered by this root system are: the
remaining cases ${\rm A}_{\rm III}$ and ${\rm C}_{\rm II}$, as well as the
remaining half of the cases ${\rm D}_{\rm III}$. More precisely, still
denoting by $r$ the complex rank, we have

$2\rho_{{\rm A}_{\rm III}}=2(r-2l+1)\tau+2\sigma+\kappa$

in the remaining cases of ${\rm A}_{\rm III}$,

$2\rho_{{\rm C}_{\rm II}}=4(r-2l)\tau+4\sigma+3\kappa$

in the remaining cases of ${\rm C}_{\rm II}$, and

$2\rho_{{\rm D}_{\rm III}}=4\tau+4\sigma+\kappa$

in the remaining cases of ${\rm D}_{\rm III}$.

This type is treated by taking $\Psi=\\{2\varepsilon_{j}\\}_{1\leqslant
j\leqslant l}$, as for the previous case. Conditions (i) to (iii) are
trivially satisfied, for a choice of integral coefficients in (i).

$\bullet$ Type ${\rm D}_{l}$ [Bou68, Planche IV, p. 256]. This is a simply
laced type but in fact it occurs only once in Cartan’s classification, namely
for ${\rm D}_{\rm I}$. This is the root system in the standard Euclidean space
${\bf R}^{l}$ with canonical basis $(\varepsilon_{i})_{1\leqslant i\leqslant
l}$, where the positive roots are the vectors of the form
$\varepsilon_{i}\pm\varepsilon_{j}$ with $i<j$ (there are $l(l-1)$ such roots,
of norm $\sqrt{2}$). In this situation, we have

$2\rho_{{\rm D}_{\rm I}}=\sigma=2\sum_{i=1}^{l-1}i\varepsilon_{l-i}.$

We use the subset of positive roots $\Psi$ to be consisting of the roots
$\varepsilon_{j}+\varepsilon_{j+1}$ for $1\leqslant j\leqslant l-1$, together
with the roots $\varepsilon_{1}\pm\varepsilon_{l}$. Condition (ii) is checked
by the fact that $\varepsilon_{1}$ is the average of the last two roots, and
then the other canonical vectors are obtained by an easy induction. Condition
(iii)′ is checked by seeing that the only pair of roots in $\Psi$ whose sum is
a root is
$\\{\varepsilon_{1}-\varepsilon_{l},\varepsilon_{l-1}+\varepsilon_{l}\\}$,
again by considerations of support or coefficient $\geqslant 2$. For condition
(i), we use Lemma 5.2. If $l$ is even we have

$2\rho_{{\rm D}_{\rm I}}=2S+l\varepsilon_{1}=2S+{l\over
2}\bigl{(}(\varepsilon_{1}+\varepsilon_{l})+(\varepsilon_{1}-\varepsilon_{l})\bigr{)},$

and if $l$ is odd we have

$2\rho_{{\rm D}_{\rm I}}=2S+(l-1)(\varepsilon_{1}+\varepsilon_{2})+{l-1\over
2}\bigl{(}(\varepsilon_{1}+\varepsilon_{l})+(\varepsilon_{1}-\varepsilon_{l})\bigr{)}.$

The rest of the root systems consists of exceptional types. In the case of
${\rm E}_{6}$, ${\rm E}_{7}$ and ${\rm E}_{8}$ the root systems are simply
laced and can be realized in ${\bf R}^{8}$. We merely mention the targeted
vector $\rho$, the subset $\Psi$ and the linear combination achieving (i).

$\bullet$ Type ${\rm E}_{6}$ [Bou68, Planche V, p. 260]. It occurs in
Cartan’s classification for ${\rm E}_{\rm I}$ only. The underlying vector
space $V$ is the $6$-dimensional subspace $x_{6}=x_{7}=-x_{8}$ in
$\mathbf{R}^{8}$ endowed with the canonical orthonormal basis
$\varepsilon_{1},\dots,\varepsilon_{8}$. The positive roots are
$\pm\varepsilon_{i}+\varepsilon_{j}$, for $1\leqslant i<j\leqslant 5$, and

$\frac{1}{2}\bigl{(}\varepsilon_{8}-\varepsilon_{7}-\varepsilon_{6}+\sum_{i=1}^{5}(-1)^{\nu(i)}\varepsilon_{i}\bigr{)},~{}~{}~{}\mathrm{~{}~{}with~{}~{}}~{}~{}~{}\sum_{i=1}^{5}\nu(i)~{}~{}\mathrm{even}.$

The targeted vector is

$2\rho_{{\rm E}_{\rm
I}}=2\varepsilon_{2}+4\varepsilon_{3}+6\varepsilon_{4}+8\varepsilon_{5}+8(\varepsilon_{8}-\varepsilon_{7}-\varepsilon_{6}).$

To simplify the notation we let
$v:=\varepsilon_{8}-\varepsilon_{7}-\varepsilon_{6}$, and we consider the
positive roots $\beta_{1},\dots,\beta_{5}$ defined by:

  * $2\beta_{1}=v+\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}$,

  * $2\beta_{2}=v-\varepsilon_{1}-\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}$,

  * $2\beta_{3}=v-\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}-\varepsilon_{5}$,

  * $2\beta_{4}=v+\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}$,

  * $2\beta_{5}=v-\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}+\varepsilon_{5}$.

Then one checks easily that

$2\rho_{{\rm E}_{\rm
I}}=6\beta_{1}+2\beta_{2}+4\beta_{3}+2\beta_{4}+2\beta_{5}+2(-\varepsilon_{1}+\varepsilon_{5})+2(\varepsilon_{1}+\varepsilon_{5}).$

We define $\Psi$ to be the union of $\beta_{1},\dots,\beta_{5}$ with
$\pm\varepsilon_{1}+\varepsilon_{5}$. Condition (i) is then satisfied. To show
that condition (ii) holds, one first observes that the two roots
$\pm\varepsilon_{1}+\varepsilon_{5}$ generates $\varepsilon_{1}$ and
$\varepsilon_{5}$. Moreover one has

  * $\beta_{2}-\beta_{1}=-\varepsilon_{1}-\varepsilon_{2}$,

  * $\beta_{3}-\beta_{1}=-\varepsilon_{1}-\varepsilon_{5}$,

  * $\beta_{4}-\beta_{1}=-\varepsilon_{2}-\varepsilon_{3}$,

  * $\beta_{5}-\beta_{1}=-\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}$,

from which one obtains that $\Psi$ generates
$\varepsilon_{1},\dots,\varepsilon_{5}$. Condition (ii) follows easily.
Moreover $\\{\beta_{3},\varepsilon_{1}+\varepsilon_{5}\\}$ is the only pair of
roots in $\Psi$ whose sum is a root in ${\rm E}_{6}$. Thus condition (iii)′ is
satisfied.

$\bullet$ Type ${\rm E}_{7}$ [Bou68, Planche VI, p. 266]. The targeted vector
is

$2\rho_{{\rm E}_{\rm
V}}=2\varepsilon_{2}+4\varepsilon_{3}+6\varepsilon_{4}+8\varepsilon_{5}+10\varepsilon_{6}-17\varepsilon_{7}+17\varepsilon_{8},$

which we decompose as the sum of
$2\varepsilon_{2}+4\varepsilon_{3}+6\varepsilon_{4}+8\varepsilon_{5}+10\varepsilon_{6}$
and of $17(\varepsilon_{8}-\varepsilon_{7})$. It turns out that
$\varepsilon_{8}-\varepsilon_{7}$ is the longest root in the system; we take
it in $\Psi$ together with the four roots
$\varepsilon_{2}+\varepsilon_{3},\varepsilon_{3}+\varepsilon_{4},\varepsilon_{4}+\varepsilon_{5},\varepsilon_{5}+\varepsilon_{6}$,
as well as the roots $\varepsilon_{6}\pm\varepsilon_{1}$. Then we have:

$\displaystyle 2\rho_{{\rm E}_{\rm V}}$
$\displaystyle=2(\varepsilon_{2}+\varepsilon_{3})+2(\varepsilon_{3}+\varepsilon_{4})+4(\varepsilon_{4}+\varepsilon_{5})+4(\varepsilon_{5}+\varepsilon_{6})$
$\displaystyle+3(\varepsilon_{6}+\varepsilon_{1})+3(\varepsilon_{6}-\varepsilon_{1})+17(\varepsilon_{8}-\varepsilon_{7}).$

The subset $\Psi$ satisfies (i), (ii) and (iii), the latter condition being
easily checked by the concrete description of ${\rm E}_{7}$ in terms of the
canonical vectors $\varepsilon_{i}$.

$\bullet$ Type ${\rm E}_{8}$ [Bou68, Planche VII, p. 268]. The targeted
vector is

$2\rho_{{\rm E}_{\rm
VIII}}=2\varepsilon_{2}+4\varepsilon_{3}+6\varepsilon_{4}+8\varepsilon_{5}+10\varepsilon_{6}+12\varepsilon_{7}+46\varepsilon_{8}.$

We use the subset $\Psi$ consisting of the roots
$\varepsilon_{2}+\varepsilon_{3}$, $\varepsilon_{3}+\varepsilon_{4}$,
$\varepsilon_{4}+\varepsilon_{5}$, $\varepsilon_{5}+\varepsilon_{6}$,
$\varepsilon_{6}+\varepsilon_{7}$, $\varepsilon_{7}+\varepsilon_{8}$ together
with $\varepsilon_{8}\pm\varepsilon_{1}$. Then we have:

$\displaystyle 2\rho_{{\rm E}_{\rm V}}$
$\displaystyle=2(\varepsilon_{2}+\varepsilon_{3})+2(\varepsilon_{3}+\varepsilon_{4})+4(\varepsilon_{4}+\varepsilon_{5})+4(\varepsilon_{5}+\varepsilon_{6})$
$\displaystyle+6(\varepsilon_{6}+\varepsilon_{7})+6(\varepsilon_{7}+\varepsilon_{8})+20(\varepsilon_{8}-\varepsilon_{1})+20(\varepsilon_{8}+\varepsilon_{1}).$

The subset $\Psi$ satisfies (i), (ii) and (iii), the latter condition being
easily checked by the concrete description of ${\rm E}_{8}$ in terms of the
canonical vectors $\varepsilon_{i}$.

$\bullet$ Type ${\rm F}_{4}$ [Bou68, Planche VIII, p. 272]. This is a non
simply laced root system corresponding to four cases in Cartan’s
classification, namely ${\rm E}_{\rm II},{\rm E}_{\rm VII},{\rm E}_{\rm IX}$
and ${\rm F}_{\rm I}$. The sum of the positive roots of norm $\sqrt{2}$ is
$\sigma=2(3\varepsilon_{1}+2\varepsilon_{2}+\varepsilon_{3})$ and the sum of
the positive roots of norm 1 is
$\tau=5\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}$. The
targeted vectors are of the form $\sigma+2^{j}\tau$ with $0\leqslant
j\leqslant 3$, so it is enough to treat separately $\sigma$ and $\tau$. We
choose
$\Psi=\\{\varepsilon_{1},\varepsilon_{1}+\varepsilon_{2},\varepsilon_{2}+\varepsilon_{3},\varepsilon_{3}+\varepsilon_{4}\\}$.
Condition (ii) is clear and (iii) is checked by the concrete description of
${\rm F}_{4}$ in terms of the canonical vectors $\varepsilon_{i}$. For (i), we
use the fact that:
$\sigma=2(\varepsilon_{1}+\varepsilon_{2})+2(\varepsilon_{2}+\varepsilon_{3})+4\varepsilon_{1}$
and
$\tau=(\varepsilon_{1}+\varepsilon_{2})+(\varepsilon_{3}+\varepsilon_{4})+4\varepsilon_{1}$.

$\bullet$ Type ${\rm G}_{2}$ [Bou68, Planche IX, p. 274]. This is a root
system of rank 2, corresponding to only one case in Cartan’s classification,
so we do not need to consider it. Still, we can simply say that using the root
notation we have: $2\rho_{\rm G}=10\alpha_{1}+6\alpha_{2}$, which can be
written as: $2\rho_{\rm G}=\alpha_{1}+3(3\alpha_{1}+2\alpha_{2})$.

###  5.5. The type $\mathrm{A}_{l}$

This type is more delicate because pairs of positive roots more often lead to
a sum providing another root.

According to [Bou68, Planche I, p. 250], type $\mathrm{A}_{l}$ is a simply
laced root system in the subspace $V\subset\mathbf{R}^{l+1}$ defined by
$\sum_{i=1}^{l+1}x_{i}=0$. The positive roots are the
$\varepsilon_{i}-\varepsilon_{j}$, with $1\leqslant i<j\leqslant l+1$.

The Cartan types where it appears are the types ${\rm A}_{\rm I}$ and ${\rm
A}_{\rm II}$. One has

$2\rho_{{\rm A}_{\rm
I}}=l\varepsilon_{1}+(l-2)\varepsilon_{2}+(l-4)\varepsilon_{3}+\dots-(l-2)\varepsilon_{l}-l\varepsilon_{l+1},$

and $\rho_{{\rm A}_{\rm II}}=4\rho_{{\rm A}_{\rm I}}$. The latter relation
shows that the same root subset $\Psi$ holds for both type. We will thus
restrict ourself to the type ${\rm A}_{\rm I}$. The idea here is to write
$2\rho_{{\rm A}_{\rm I}}$ by using long roots.

######  Proposition 5.3.

$\mathrm{(1)}$ If $l$ is odd, write $l=2k-1$ with $k\geqslant 1$. Then:

$2\rho_{{\rm A}_{\rm
I}}=2\sum_{k<j-i}(\varepsilon_{i}-\varepsilon_{j})+\sum_{i=1}^{k}(\varepsilon_{i}-\varepsilon_{i+k}).$

Moreover if $\Psi$ is the set of roots that appear in the right side member,
then $\Psi$ satisfies the conditions $(i),(ii),(iii)$.

$\mathrm{(2)}$ If $l$ is even, write $l=2k$ with $k\geqslant 1$. Then:

$2\rho_{{\rm A}_{\rm
I}}=2\sum_{k<j-i<l}(\varepsilon_{i}-\varepsilon_{j})+2(\varepsilon_{1}-\varepsilon_{k+1})+2(\varepsilon_{k+1}-\varepsilon_{l+1}).$

Moreover if $\Psi$ is the set of roots that appear in the right side member,
then $\Psi$ satisfies the conditions $(i),(ii),(iii)^{\prime}$.

###### Proof.

By counting the number of times $\varepsilon_{i}$ and $\varepsilon_{j}$
appear, one has:

$\sum_{k<j-i}\varepsilon_{i}-\varepsilon_{j}=\sum_{i=1}^{l-k}(l+1-k-i)\varepsilon_{i}-\sum_{j=k+2}^{l+1}(j-k-1)\varepsilon_{j}.$

(1) Suppose $l=2k-1$ with $k\geqslant 1$. Then:

$\sum_{k<j-i}\varepsilon_{i}-\varepsilon_{j}=\sum_{i=1}^{k-1}(k-i)\varepsilon_{i}-\sum_{j=k+2}^{l+1}(j-k-1)\varepsilon_{j}.$

Since $2(k-i)+1=l-2i+2$ and $2(j-k-1)+1=2j-l-2$, one obtains

$2\sum_{k<j-i}(\varepsilon_{i}-\varepsilon_{j})+\sum_{i=1}^{k}(\varepsilon_{i}-\varepsilon_{i+k})=2\rho_{{\rm
A}_{\rm I}}.$

Thus $\Psi$ satisfies (i). The roots contained in $\Psi$ are those of the form
$\varepsilon_{i}-\varepsilon_{j}$ with $j-i\geqslant k$. No root of the system
$\mathrm{A}_{l}$ is the sum of two of them, thus (iii) holds. Moreover the
above roots generate $V$. Indeed:

  * •

for $i<k$, one has
$\varepsilon_{i}-\varepsilon_{i+1}=(\varepsilon_{i}-\varepsilon_{l+1})-(\varepsilon_{i+1}-\varepsilon_{l+1})$,
with $l+1-i\geqslant l+1-(i+1)\geqslant k$,

  * •

for $i>k$, one has
$\varepsilon_{i}-\varepsilon_{i+1}=(\varepsilon_{1}-\varepsilon_{i+1})-(\varepsilon_{1}-\varepsilon_{i})$,
with $i+1-1\geqslant i-1\geqslant k$,

  * •

and for $i=k$:
$\varepsilon_{k}-\varepsilon_{k+1}=(\varepsilon_{1}-\varepsilon_{k+1})+(\varepsilon_{k}-\varepsilon_{l+1})-(\varepsilon_{1}-\varepsilon_{l+1})$.

Therefore (i)-(iii) are satisfied.

(2) Suppose $l=2k$ with $k\geqslant 1$. Then:

$\sum_{k<j-i}\varepsilon_{i}-\varepsilon_{j}=\sum_{i=1}^{k}(k+1-i)\varepsilon_{i}-\sum_{j=k+2}^{l+1}(j-k-1)\varepsilon_{j}.$

Since $2(k+1-i)=l-2i+2$ and $2(j-k-1)=2j-l-2$, one obtains that
$2\sum_{k<j-i}(\varepsilon_{i}-\varepsilon_{j})=2\rho_{{\rm A}_{\rm I}}$.
Therefore

$2\sum_{k<j-i<l}(\varepsilon_{i}-\varepsilon_{j})+2(\varepsilon_{1}-\varepsilon_{k+1})+2(\varepsilon_{k+1}-\varepsilon_{l+1})=2\rho_{{\rm
A}_{\rm I}}.$

Thus $\Psi$ satisfies (i). The roots in $\Psi$ are those of the form
$\varepsilon_{i}-\varepsilon_{j}$, with $k<j-i<l$, and the roots
$\varepsilon_{1}-\varepsilon_{k+1}$, $\varepsilon_{k+1}-\varepsilon_{l+1}$.
Apart $\varepsilon_{1}-\varepsilon_{l+1}$ which is the sum of the last ones,
no root of the system $\mathrm{A}_{l}$ is the sum of two of them. Thus (iii)′
holds. In addition the above roots generate $V$. Indeed:

  * •

$\varepsilon_{1}-\varepsilon_{l+1}=(\varepsilon_{1}-\varepsilon_{k+1})+(\varepsilon_{k+1}-\varepsilon_{l+1})$,

  * •

for $i<k$, one has
$\varepsilon_{i}-\varepsilon_{i+1}=(\varepsilon_{i}-\varepsilon_{l+1})-(\varepsilon_{i+1}-\varepsilon_{l+1})$,
with $l+1-i\geqslant l+1-(i+1)>k$,

  * •

for $i>k+1$, one has
$\varepsilon_{i}-\varepsilon_{i+1}=(\varepsilon_{1}-\varepsilon_{i+1})-(\varepsilon_{1}-\varepsilon_{i})$,
with $i+1-1\geqslant i-1>k$,

  * •

for $i=k$:
$\varepsilon_{k}-\varepsilon_{k+1}=(\varepsilon_{k}-\varepsilon_{l+1})-(\varepsilon_{k+1}-\varepsilon_{l+1})$,

  * •

for $i=k+1$:
$\varepsilon_{k+1}-\varepsilon_{k+2}=(\varepsilon_{1}-\varepsilon_{k+2})-(\varepsilon_{1}-\varepsilon_{k+1})$.

This concludes the proof. ∎

###  5.6. Proof of non-vanishing for semisimple groups

We can finally put things together in order to provide a proof for Theorem D,
hence a proof for Theorem A of the Introduction, in view of the reduction
contained in 0.1.

###### Proof of Theorem D.

By Lemma 5.1, it suffices to exhibit a suitable subset $\Psi$ of positive
roots for any semisimple group. This can be done separately for each connected
component of the Dynkin diagram, which amounts to dealing with simple real Lie
groups. The absolutely simple cases were treated by a case-by-case analysis in
Subsections 5.3 to 5.5. The remaining cases correspond to the simple non
absolutely simple groups, _i.e._ simple complex Lie groups seen as real
groups. In the latter cases, the root multiplicities are all equal to 2 since
the groups are split over ${\bf C}$ and the root groups are all isomorphic to
the real Lie group ${\bf C}$. Therefore the function $\rho_{G}$ in this case
is twice the corresponding function for the split groups over ${\bf R}$ with
the same root system, showing that the same subset $\Psi$ can be chosen, up to
multiplying the coefficients $m_{\beta}$ by $2$ for each $\beta\in\Psi$. ∎

##  6\. Cohomologies of semi-direct products

This section relates the de Rham $L^{p}$-cohomology with the group
$L^{p}$-cohomology. Our goal is to tranfer to the setting of de Rham
$L^{p}$-cohomology, a result issued from [BR20] about the group
$L^{p}$-cohomology of semi-direct products, see Corollary 6.10. It leads to
the key relation (0.4) of the introduction.

The section is also an opportunity to advertise several incarnations of
$L^{p}$-cohomology, and to present their properties in a synthetic way. It
collects results issued from [Pan95, SS18, BR20] (see also [Ele98] for related
results in the discrete group case).

###  6.1. Asymptotic and group $L^{p}$-cohomologies

The asymptotic $L^{p}$-cohomology of a metric space has been defined by Pansu
in [Pan95]. Let $(X,d)$ be a metric space equipped with a Borel measure $\mu$.
Suppose it satisfies the following “bounded geometry” condition. There exist
non-decreasing functions $v,V:(0,+\infty)\to(0,+\infty)$, such for every ball
$B(x,R)\subset X$ one has

$v(R)\leqslant\mu\bigl{(}B(x,R)\bigr{)}\leqslant V(R).$

For $R>0$ and $k\in{\bf N}$, let

$\Delta_{R}^{(k)}=\\{(x_{0},\dots,x_{k})\in
X^{k+1}~{}|~{}d(x_{i},x_{j})\leqslant R\,\,\hbox{\rm for all $i,j$}\\}.$

Let $AS^{p,k}(X)$ be the space of (classes of) measurable functions
$f:X^{k+1}\to{\bf R}$ such that for every $R>0$ one has

$N_{R}(f)^{p}:=\int_{\Delta_{R}^{(k)}}\bigl{|}f(x_{0},\dots,x_{k})\bigr{|}^{p}d\mu(x_{0})\dots
d\mu(x_{k})<+\infty.$

We equip $AS^{p,k}(X)$ with the topology induced by the set of the semi-norms
$N_{R}$ ($R>0$).

######  Definition 6.1.

The _asymptotic $L^{p}$-cohomology_ of $X$ is the cohomology of the complex
$AS^{p,0}(X)\stackrel{{\scriptstyle\delta_{0}}}{{\to}}AS^{p,1}(X)\stackrel{{\scriptstyle\delta_{1}}}{{\to}}AS^{p,2}(X)\stackrel{{\scriptstyle\delta_{2}}}{{\to}}\dots$,
where the $\delta_{k}$’s are defined by:

(6.2)
$(\delta_{k}f)(x_{0},\dots,x_{k+1})=\sum_{i=0}^{k+1}(-1)^{i}f(x_{0},\dots,\hat{x_{i}},\dots,x_{k+1}).$

The _reduced_ asymptotic $L^{p}$-cohomology is defined similarly. They are
denoted by $L^{p}{\rm H}_{\mathrm{AS}}^{*}(X)$ and $L^{p}\overline{{\rm
H}_{\mathrm{AS}}^{*}}(X)$ respectively.

######  Theorem 6.3.

[Pan95, Section 2] Let $X$ and $Y$ be metric spaces. Assume that each of them
admits a Borel measure with respect to which it is of bounded geometry (as
defined above). Let $F:X\to Y$ be a quasi-isometry. Then $F$ induces a
homotopy equivalence222All the maps occuring in homotopies are supposed to be
continuous. between the complexes $AS^{p,*}(Y)$ and $AS^{p,*}(X)$, and a
canonical isomorphism of graded topological vector spaces $F^{*}:L^{p}{\rm
H}_{\mathrm{AS}}^{*}(Y)\to L^{p}{\rm H}_{\mathrm{AS}}^{*}(X)$. In particular
$F^{*}$ depends only on the bounded perturbation class of $F$. The same holds
in reduced cohomology.

See also [Gen14, SS18] for a more detailed proof.

We now turn our attention to the continuous group cohomology, see [BW00,
Chap. IX] or [Gui80] for more details.

Let $G$ be a locally compact second countable group. It admits a left-
invariant proper metric defining its topology, see _e.g._  [CdlH16, Struble
Th. 2.B.4]. Let $(\pi,V)$ be a _topological $G$-module_ _i.e._ a Hausdorff
locally convex vector space over ${\bf R}$ on which $G$ acts via a continuous
representation $\pi$. We denote by $V^{G}\subset V$ the subspace of
$\pi(G)$-invariant vectors. For $k\in{\bf N}$, let $C^{k}(G,V)$ be the space
of continuous maps from $G^{k+1}$ to $V$ equipped with the compact open
topology. Then $C^{k}(G,V)$ is a topological $G$-module by means of the
following action: for $g,x_{0},\dots,x_{k}\in G$,

$(g\cdot
f)(x_{0},\dots,x_{k})=\pi(g)\bigl{(}f(g^{-1}x_{0},\dots,g^{-1}x_{k})\bigr{)}.$

Consider the following complex of invariants:

$C^{0}(G,V)^{G}\stackrel{{\scriptstyle\delta_{0}}}{{\to}}C^{1}(G,V)^{G}\stackrel{{\scriptstyle\delta_{1}}}{{\to}}C^{2}(G,V)^{G}\stackrel{{\scriptstyle\delta_{2}}}{{\to}}\dots$

where the $\delta_{k}$’s are defined as in (6.2). The _continuous cohomology
of $G$ with coefficients in $(\pi,V)$_ is the cohomology of this complex, it
will be denoted by ${\rm H}_{\mathrm{ct}}^{*}(G,V)$. Similarly is defined the
reduced cohomology $\overline{{\rm H}_{\mathrm{ct}}^{*}}(G,V)$.

######  Definition 6.4.

Let $\mathcal{H}$ be a left-invariant Haar measure on $G$. The _group
$L^{p}$-cohomology_ of $G$ is the continuous cohomology of $G$, with
coefficients in the right-regular representation of $G$ on
$L^{p}(G,\mathcal{H})$, _i.e._ the representation defined by

$\bigl{(}\pi(g)u\bigr{)}(x)=u(xg)~{}~{}\mathrm{~{}~{}for~{}~{}}~{}~{}u\in
L^{p}(G,\mathcal{H})~{}~{}\mathrm{~{}~{}and~{}~{}}~{}~{}g,x\in G.$

It will be denoted by ${\rm H}_{\mathrm{ct}}^{*}(G,L^{p}(G))$. The _reduced_
group $L^{p}$-cohomology of $G$ is defined similarly and is denoted by
$\overline{{\rm H}_{\mathrm{ct}}^{*}}(G,L^{p}(G))$.

Observe that the right-regular representation on $L^{p}(G,\mathcal{H})$ is
isometric if and only if $G$ is unimodular.

######  Theorem 6.5.

[SS18, Theorem 10], [BR20, Theorem 3.6] Suppose $G$ is a locally compact
second countable topological group equipped with a left-invariant proper
metric. Then the complex of invariants $C^{*}(G,L^{p}(G))^{G}$ is canonically
homotopy equivalent to $AS^{p,*}(G)$. In consequence, there exists a canonical
isomorphism of gradued topological vector spaces ${\rm
H}_{\mathrm{ct}}^{*}(G,L^{p}(G))\simeq L^{p}{\rm H}_{\mathrm{AS}}^{*}(G)$. The
same holds for the reduced cohomology.

We notice that the above isomorphism admits the following property. Suppose we
are given an isomorphism $\varphi:G_{1}\to G_{2}$ of topological groups as
above. It induces isomorphisms of complexes:

$\varphi_{\mathrm{ct}}^{*}:C^{k}\bigl{(}G_{2},L^{p}(G_{2})\bigr{)}^{G_{2}}\to
C^{k}\bigl{(}G_{1},L^{p}(G_{1})\bigr{)}^{G_{1}}$
$\mathrm{and}~{}~{}~{}~{}\varphi_{\mathrm{AS}}^{*}:AS^{p,k}(G_{2})\to
AS^{p,k}(G_{1}),$
$\mathrm{defined~{}by}~{}~{}~{}~{}\varphi_{\mathrm{ct}}^{*}(f)(x_{0},\dots,x_{k};x)=f\bigl{(}\varphi(x_{0}),\dots,\varphi(x_{k});\varphi(x)\bigr{)},$
$\mathrm{and}~{}~{}~{}~{}\varphi_{\mathrm{AS}}^{*}(f)(x_{0},\dots,x_{k})=f\bigl{(}\varphi(x_{0}),\dots,\varphi(x_{k})\bigr{)}.$

Then the isomorphisms ${\rm H}_{\mathrm{ct}}^{*}(G_{i},L^{p}(G_{i}))\simeq
L^{p}{\rm H}_{\mathrm{AS}}^{*}(G_{i})$, together with the induced isomorphisms
$\varphi_{\mathrm{ct}}^{*}:{\rm
H}_{\mathrm{ct}}^{*}(G_{2},L^{p}(G_{2}))\to{\rm
H}_{\mathrm{ct}}^{*}(G_{1},L^{p}(G_{1}))$,
$\varphi_{\mathrm{AS}}^{*}:L^{p}{\rm H}_{\mathrm{AS}}^{*}(G_{2})\to L^{p}{\rm
H}_{\mathrm{AS}}^{*}(G_{1})$, form a commutative diagram.

###  6.2. Asymptotic and de Rham $L^{p}$-cohomologies

Let $M$ be a $C^{\infty}$ complete Riemannian manifold. Unlike its asymptotic
$L^{p}$-cohomology, its simplicial $L^{p}$-cohomology is not invariant by
quasi-isometry (e.g. for compact manifolds it is isomorphic to the standard
de Rham cohomology). For that reason we restrict ourself to manifolds which
are “uniformly diffeomorphic to ${\bf R}^{D}$”.

######  Definition 6.6.

Let $B=B(0,1)$ be the unit open ball in ${\bf R}^{D}$. A manifold $M$ is said
_uniformly diffeomorphic to ${\bf R}^{D}$_, if there exist functions
$\rho,\lambda:[0,+\infty)\to[1,+\infty)$, and for every $m\in M$ a
$C^{\infty}$-diffeomorphism $\varphi_{m}:B\to M$, such that

  * •

for every $m\in M$ and $R>0$, there exists $r\in(\frac{1}{2},1)$ with

$B(m,R)\subset\varphi_{m}\bigl{(}B(0,r)\bigr{)}\subset
B\bigl{(}m,\rho(R)\bigr{)},$

  * •

and $\varphi_{m}\mbox{\Large$|$\normalsize}_{B(0,r)}$ is $\lambda(R)$-bi-
Lipschitz.

For example a Riemannian manifold which is diffeomorphic to ${\bf R}^{D}$, and
which admits a cocompact group of isometries, is uniformly diffeomorphic to
${\bf R}^{D}$.

######  Theorem 6.7.

[Pan95] Let $M$ be a $C^{\infty}$ Riemannian manifold which is uniformly
diffeomorphic to ${\bf R}^{D}$. There exists a homotopy equivalence between
the complexes $AS^{p,*}(M)$ and $\Omega^{p,*}(M)$. It induces a canonical
isomorphism of gradued topological vector spaces $L^{p}{\rm
H}_{\mathrm{AS}}^{*}(M)\simeq L^{p}{\rm H}_{\mathrm{dR}}^{*}(M)$. The same
holds for reduced cohomology.

The proof that Pansu gives in [Pan95] goes through the simplicial
$\ell^{p}$-cohomology. A more direct proof is presented in Appendix A below.

We notice that the isomorphism in Theorem 6.7 above admits the following
property. Suppose we are given a bi-Lipschitz $C^{\infty}$ diffeomorphism
$\varphi:M_{1}\to M_{2}$ between Riemannian manifolds as above. Then the
isomorphisms $L^{p}{\rm H}_{\mathrm{dR}}^{*}(M_{i})\simeq L^{p}{\rm
H}_{\mathrm{AR}}^{*}(M_{i})$, together with the induced isomorphisms
$\varphi^{*}_{\mathrm{dR}}:L^{p}{\rm H}_{\mathrm{dR}}^{*}(M_{2})\to L^{p}{\rm
H}_{\mathrm{dR}}^{*}(M_{1})$, $\varphi_{\mathrm{AS}}^{*}:L^{p}{\rm
H}_{\mathrm{AS}}^{*}(M_{2})\to L^{p}{\rm H}_{\mathrm{AS}}^{*}(M_{1})$, form a
commutative diagram.

###  6.3. Semi-direct products

We transfer to de Rham $L^{p}$-cohomology a result about group
$L^{p}$-cohomology of semi-direct products, see Corollary 6.10. This leads to
relation (0.4) of the introduction.

Let $(V,\|\cdot\|_{V})$ be a separable normed space, let $X$ be a locally
compact second countable topological space endowed with a Radon measure $\mu$,
and let $p\in(1,+\infty)$. We denote by $L^{p}(X,V)$ the normed space
consisting of the (classes of) mesurable maps $f:X\to V$ such that

$\|f\|_{L^{p}(X,V)}^{p}:=\int_{X}\bigl{\|}f(x)\bigr{\|}_{V}^{p}~{}d\mu(x)<+\infty.$

Let $G$ be a locally compact second countable group. Suppose it decomposes as
a semi-direct product $G=Q\ltimes H$, with $Q$, $H$ closed subgroups and the
standard multiplicative law

$(q_{1},h_{1})\cdot(q_{2},h_{2})=(q_{1}q_{2},q_{2}^{-1}h_{1}q_{2}h_{2}).$

Let $\mathcal{H}_{Q}$, $\mathcal{H}_{H}$ be left-invariant Haar measures on
$Q$, $H$ respectively. Then
$\mathcal{H}_{G}:=\mathcal{H}_{Q}\times\mathcal{H}_{H}$ is a left-invariant
Haar measure on $G$. One has:

######  Theorem 6.8.

[BR20, Corollary 5.5] Let $G=Q\ltimes H$ as above. Assume that the complex of
invariants $C^{*}(H,L^{p}(H))^{H}$ is homotopically equivalent to a complex of
Banach spaces. Suppose also that there exists $n\in{\bf N}$, such that ${\rm
H}_{\mathrm{ct}}^{k}(H,L^{p}(H))=0$ for $0\leqslant k<n$ and such that ${\rm
H}_{\mathrm{ct}}^{n}(H,L^{p}(H))$ is Hausdorff. Then ${\rm
H}_{\mathrm{ct}}^{k}(G,L^{p}(G))=0$ for $0\leqslant k<n$ and there is a linear
isomorphism

(6.9) ${\rm H}_{\mathrm{ct}}^{n}\bigl{(}G,L^{p}(G)\bigr{)}\simeq
L^{p}\Bigl{(}Q,{\rm
H}_{\mathrm{ct}}^{n}\bigl{(}H,L^{p}(H)\bigr{)}\Bigr{)}^{Q},$

where the $Q$-action on $L^{p}(Q,{\rm H}_{\mathrm{ct}}^{n}(H,L^{p}(H)))$ is by
right multiplication on itself and by conjugacy on ${\rm
H}_{\mathrm{ct}}^{n}(H,L^{p}(H))$; in other words it is the action induced by

$(q\cdot
f)(y)(x_{0},\dots,x_{n};x)=f(yq)(q^{-1}x_{0}q,\dots,q^{-1}x_{n}q;q^{-1}xq),$

for every $q,y\in Q$, $f:Q\to C^{n}(H,L^{p}(H))$ and $x_{0},\dots,x_{n},x\in
H$.

For Lie groups diffeomorphic to ${\bf R}^{D}$, one gets the following result
which implies the relation (0.4) in the introduction.

######  Corollary 6.10.

Let $G$ be a Lie group diffeomorphic to ${\bf R}^{D}$. Suppose it decomposes
as $G=Q\ltimes H$ with $Q$, $H$ closed subgroups and $H$ diffeomorphic to
${\bf R}^{d}$ for some $d$. Suppose that there exists $n\in{\bf N}$, such that
$L^{p}{\rm H}_{\mathrm{dR}}^{k}(H)=0$ for $0\leqslant k<n$ and such that
$L^{p}{\rm H}_{\mathrm{dR}}^{n}(H)$ is Hausdorff. Then $L^{p}{\rm
H}_{\mathrm{dR}}^{k}(G)=0$ for $0\leqslant k<n$ and there is a linear
isomorphism

(6.11) $L^{p}{\rm H}_{\mathrm{dR}}^{n}(G)\simeq L^{p}\bigl{(}Q,L^{p}{\rm
H}_{\mathrm{dR}}^{n}(H)\bigr{)}^{Q},$

where the $Q$-action on $L^{p}(Q,L^{p}{\rm H}_{\mathrm{dR}}^{n}(H))$ is by
right multiplication on itself and by conjugacy on $L^{p}{\rm
H}_{\mathrm{dR}}^{n}(H)$; in other words, it is induced by the action

$(q\cdot f)(y)=C^{*}_{q^{-1}}\bigl{(}f(yq)\bigr{)},$

for every $q,y\in Q$, $f:Q\to\Omega^{p,n}(H)$.

###### Proof.

Since $G$ and $H$ are homogeneous and respectively diffeomorphic to ${\bf
R}^{D}$ and ${\bf R}^{d}$, they are uniformly diffeomorphic to ${\bf R}^{D}$
and ${\bf R}^{d}$. By applying successively Theorems 6.5 and 6.7, one sees
that the complexes $C^{*}(G,L^{p}(G))^{G}$ and $\Omega^{p,*}(G)$ are homotopy
equivalent. The same holds for $C^{*}(H,L^{p}(H))^{H}$ and $\Omega^{p,*}(H)$.
In particular $C^{*}(H,L^{p}(H))^{H}$ is homotopy equivalent to a complex of
Banach spaces, and thus Theorem 6.8 applies.

It remains to relate the expressions of the $Q$-action on ${\rm
H}_{\mathrm{ct}}^{n}(H,L^{p}(H))$ and on $L^{p}{\rm H}_{\mathrm{dR}}^{n}(H)$.
For that we need to find the expressions the conjugacy $C_{q}^{*}:L^{p}{\rm
H}_{\mathrm{dR}}^{n}(H)\to L^{p}{\rm H}_{\mathrm{dR}}^{n}(H)$ by $q\in Q$,
when transformed by the successive isomorphisms

$L^{p}{\rm H}_{\mathrm{dR}}^{n}(H)\simeq L^{p}{\rm
H}_{\mathrm{AS}}^{n}(H)\simeq{\rm
H}_{\mathrm{ct}}^{n}\bigl{(}R,L^{p}(H)\bigr{)}.$

It is done by using their functional properties described right after Theorems
6.7 and 6.5. ∎

######  Remark 6.12.

The linear isomorphism (6.9) in Theorem 6.8 is the composition of the
following two isomorphisms

${\rm
H}_{\mathrm{ct}}^{n}\bigl{(}G,L^{p}(G)\bigr{)}\stackrel{{\scriptstyle\varphi_{1}}}{{\simeq}}{\rm
H}_{\mathrm{ct}}^{n}\bigl{(}H,L^{p}(G)\bigr{)}^{Q}\stackrel{{\scriptstyle\varphi_{2}}}{{\simeq}}L^{p}\Bigl{(}Q,{\rm
H}_{\mathrm{ct}}^{n}\bigl{(}H,L^{p}(H)\bigr{)}\Bigr{)}^{Q}.$

The second one is a topological isomorphism [BR20, Proposition 5.2]; it comes
from a Fubini type argument. The first one comes from the Hochschild-Serre
spectral sequence [BW00, Theorem IX.4.3]. When $G$ is countable, the
continuous cohomology ${\rm H}_{\mathrm{ct}}^{n}(G,L^{p}(G))$ coincides with
the standard one ${\rm H}^{n}(G,\ell^{p}(G))$. In this case it is known that
$\varphi_{1}$ equals the restriction map [HS53, Theorem III.2]. We suspect
that the equality between $\varphi_{1}$ and the restriction map holds in
general under the assumptions of Theorem 6.8. Since the restriction map is
continuous, this would imply that ${\rm H}_{\mathrm{ct}}^{n}(G,L^{p}(G))$ is
Hausdorff and that the isomorphisms (6.9) and (6.11) are canonical and Banach.

##  Appendix A On asymptotic and de Rham $L^{p}$-cohomology

The goal of the appendix is to give a direct proof of Pansu’s Theorem 6.7,
that we restate below for commodity (see Definition 6.6 for the notion of
_uniformly diffeomorphic to ${\bf R}^{D}$_):

######  Theorem A.1.

Let $M$ be a $C^{\infty}$ Riemannian manifold which is uniformly diffeomorphic
to ${\bf R}^{D}$. There exists a homotopy equivalence333Recall that all the
maps occuring in homotopies are supposed to be continuous. between the
complexes $AS^{p,*}(M)$ and $\Omega^{p,*}(M)$. It induces a canonical
isomorphism of gradued topological vector spaces $L^{p}{\rm
H}_{\mathrm{AS}}^{*}(M)\simeq L^{p}{\rm H}_{\mathrm{dR}}^{*}(M)$. The same
holds for reduced cohomology.

The following standard notion will serve repeatedly in the sequel.

######  Definition A.2.

Let $(A^{*},d)$ be a complex of topological vector spaces, let
$(B^{*},d)\subset(A^{*},d)$ be a subcomplex, and let denote by $i:B^{*}\to
A^{*}$ the inclusion map. One says that $A^{*}$ retracts by deformation onto
$B^{*}$, if there exists a continuous linear map $r:A^{*}\to B^{*}$ such that
$r\circ d=d\circ r$, $r\circ i=\mathrm{id}$, and $i\circ r$ is homotopic to
$\mathrm{id}$.

When $A^{*}$ retracts by deformation onto $B^{*}$, the inclusion map
$i:B^{*}\to A^{*}$ induces canonically isomorphisms of gradued topological
vector spaces $\mathrm{H}^{*}(B^{*})\simeq\mathrm{H}^{*}(A^{*})$ and
$\overline{\mathrm{H}^{*}}(B^{*})\simeq\overline{\mathrm{H}^{*}}(A^{*})$.

The method for proving Theorem A.1 is a variant of the double complex proof of
the isomorphism between de Rham and Cech cohomologies (see _e.g._  [BT82,
Theorem 8.9 and Proposition 9.5]). This method is used in [Pan95] to show the
equivalence between the de Rham and the simplicial $L^{p}$-cohomologies. It is
based on the following general lemma:

######  Lemma A.3.

Let $(C^{k,\ell},d^{\prime},d^{\prime\prime})_{(k,\ell)\in\mathbf{N}^{2}}$ be
a double complex of topological vector spaces, with $d^{\prime}\circ
d^{\prime\prime}+d^{\prime\prime}\circ d^{\prime}=0$. Suppose that for every
$\ell\in\mathbf{N}$, the complex $(C^{*,\ell},d^{\prime})$ retracts by
deformation onto the subcomplex that is null in degrees at least $1$ and equal
to $E^{\ell}:=\operatorname{Ker}d^{\prime}|_{C^{0,\ell}}$ in degree $0$. Then
the complex $(D^{*},d_{D})$, defined by $D^{m}=\bigoplus_{k+\ell=m}C^{k,\ell}$
and $d_{D}=d^{\prime}+d^{\prime\prime}$, retracts by deformation onto its
subcomplex $(E^{*},d^{\prime\prime})$.

This kind of result is well-known from the specialists (see _e.g._  [BT82,
Proposition 9.5 and Remark p. 104]). It is stated in this form in [Pan95,
Lemme 5], with a sketch of proof. A detailed proof appears in [Seq20, Lemma
2.2.1].

###  A.1. Definition of the double complex $C^{*,*}$

Let $M$ be a $C^{\infty}$ complete Riemannian manifold. We denote by
$\Omega^{p,k}_{\mathrm{loc}}(M)$ the space of (measurable) $k$-differential
forms on $M$ that belong to $\Omega^{p,k}(U)$ for every relatively compact
open subset $U\subset M$. The $\Omega^{p,k}$-norm of the restriction of
$\omega$ to $U$ is denoted by $\|\omega\|_{U}$ for simplicity. Equipped with
the set of semi-norms $\|\cdot\|_{U}$, where $U\subset M$ is open relatively
compact, the space $\Omega^{p,k}_{\mathrm{loc}}(M)$ is a separable Fréchet
space.

For every couple $(k,\ell)\in\mathbf{N}^{2}$, every $R>0$ and every measurable
map $f:M^{\ell+1}\to\Omega^{p,k}_{\mathrm{loc}}(M)$, we define the semi-norm
$N_{R}(f)$ by

$N_{R}(f)^{p}=\int_{\Delta_{R}^{(\ell)}}\bigl{\|}f(m_{0},\dots,m_{\ell})\bigr{\|}_{B(m_{0},R)}^{p}d\mathrm{vol}^{\ell+1}(m_{0},\dots,m_{\ell}),$

where $\mathrm{vol}^{\ell+1}$ denotes the product measure on $M^{\ell+1}$. We
remark that changing $B(m_{0},R)$ by $B(m_{i},R)$ in the definition of
$N_{R}(f)$, leads to an equivalent family of semi-norms; indeed one has
$B(m_{i},R)\subset B(m_{j},2R)$ when
$(m_{0},\dots,m_{\ell})\in\Delta_{R}^{(\ell)}$.

Let $C^{k,\ell}$ be the topological vector spaces of (the classes of) the
measurable maps $f:M^{\ell+1}\to\Omega^{p,k}_{\mathrm{loc}}(M)$ such that
$N_{R}(f)<+\infty$ for every $R>0$. We define $d^{\prime}:C^{k,\ell}\to
C^{k+1,\ell}$ and $d^{\prime\prime}:C^{k,\ell}\to C^{k,\ell+1}$ by

$d^{\prime}f=(-1)^{\ell}d\circ
f~{}~{}~{}\mathrm{~{}~{}and~{}~{}}~{}~{}~{}d^{\prime\prime}f=\delta f,$

where $d$ is the de Rham differential operator and $\delta$ is the discrete
operator defined in (6.2). Then
$C^{*,*}=(C^{k,\ell},d^{\prime},d^{\prime\prime})_{(k,\ell)\in\mathbf{N}^{2}}$
is a double complex of topological vector spaces. It satisfies
$d^{\prime}\circ d^{\prime\prime}+d^{\prime\prime}\circ d^{\prime}=0$.

The complex $C^{*,*}$ interpolates between the complexes $AS^{p,*}(M)$ and
$\Omega^{p,*}$. Indeed:

######  Proposition A.4.

There are canonical isomorphisms of topological complexes
$\operatorname{Ker}d^{\prime}\mbox{\Large$|$\normalsize}_{C^{0,*}}\simeq
AS^{p,*}(M)$ and
$\operatorname{Ker}d^{\prime\prime}\mbox{\Large$|$\normalsize}_{C^{*,0}}\simeq\Omega^{p,*}(M)$.

###### Proof.

(1) One has
$f\in\operatorname{Ker}d^{\prime}\mbox{\Large$|$\normalsize}_{C^{0,\ell}}$ if
and only if $f:M^{\ell+1}\to\Omega_{\mathrm{loc}}^{p,0}(M)$ satisfies $d\circ
f=0$, _i.e._ $f(m_{0},\dots,m_{\ell})$ is a constant function for a.a.
$(m_{0},\dots,m_{\ell})\in M^{\ell+1}$. Moreover when
$f(m_{0},\dots,m_{\ell})$ is a constant function, one has

$\bigl{\|}f(m_{0},\dots,m_{\ell})\bigr{\|}_{B(m_{0},R)}^{p}=\bigl{|}f(m_{0},\dots,m_{\ell})\bigr{|}^{p}\mathrm{vol}\bigl{(}B(m_{0},R)\bigr{)}.$

Therefore
$\operatorname{Ker}d^{\prime}\mbox{\Large$|$\normalsize}_{C^{0,*}}\simeq
AS^{p,*}(M)$.

(2) One has
$f\in\operatorname{Ker}d^{\prime\prime}\mbox{\Large$|$\normalsize}_{C^{k,0}}$
if and only if $f:M\to\Omega_{\mathrm{loc}}^{p,k}(M)$ satisfies $\delta f=0$
_i.e._ $f$ is a constant map. Let $\omega$ be its constant value. Since $M$ is
uniformly diffeomorphic to ${\bf R}^{D}$, it admits bounded geometry in the
sense of Subsection 6.1; there exist functions $v,V:(0,+\infty)\to(0,+\infty)$
such that such for every ball $B(m,R)\subset M$ one has

$v(R)\leqslant\mathrm{vol}\bigl{(}B(m,R)\bigr{)}\leqslant V(R).$

By Fubini one has for every $R>0$

$v(R)\|\omega\|^{p}_{\Omega^{p,k}}\leqslant\int_{M}\|\omega\|^{p}_{B(m,R)}d\mathrm{vol}(m)\leqslant
V(R)\|\omega\|^{p}_{\Omega^{p,k}}.$

Therefore
$\operatorname{Ker}d^{\prime\prime}\mbox{\Large$|$\normalsize}_{C^{*,0}}\simeq\Omega^{p,*}(M)$.
∎

###  A.2. Homotopy type of the columns

######  Proposition A.5.

For every $\ell\in\mathbf{N}$, the complex $(C^{*,\ell},d^{\prime})$ retracts
by deformation onto the subcomplex
$(\operatorname{Ker}d^{\prime}\mbox{\Large$|$\normalsize}_{C^{0,\ell}}\to 0\to
0\to\dots)$.

Its proof is postened at the end of the subsection. A crucial ingredient is
the following lemma issued from [IL93, Section 4].

######  Lemma A.6.