Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings

We show that the continuous $L^p$-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M.~Gromov, suggesting a classical behaviour of continuous $L^p$-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu's vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.


Introduction
Let G be a locally compact second countable group. Then G is metrizable and countable at infinity [Bou07, IX §2.9, Cor. to Prop In this paper we study the group L p -cohomology of G for p > 1. By definition, this is the continuous cohomology of G [BW00, Chap. IX], with coefficients in the right-regular representation on L p (G). It shall be denoted by H * ct G, L p (G) . We also consider the associated reduced cohomology, denoted by H * ct G, L p (G) (they are the largest Hausdorff quotients of the previous spaces -see Definition 2.2).
Our first result is the quasi-isometry invariance of these cohomology spaces (see Corollary 3.8 for details).
Theorem 1.1. Let G 1 , G 2 be locally compact second countable groups, equipped with left-invariant proper metrics. Every quasi-isometry F : G 1 → G 2 induces canonically an isomorphism of graded topological vector spaces F * : H * ct G 2 , L p (G 2 ) → H * ct G 1 , L p (G 1 ) . The same holds for the reduced cohomology.
For example the group L p -cohomology of a group G as above is isomorphic to the one of any of its cocompact lattices. When G is semisimple, its group L p -cohomology is isomorphic to the one of any of its parabolic subgroups; this generalizes in fact to (the points of) any connected real algebraic group. Theorem 1.1 was already known for the finite K(π, 1)-groups by M. Gromov (see [Gro93,p. 219] and Remark 3.4), and then proved for finitely generated groups by G. Elek [Ele98]. In degree 1, the theorem is due to Cornulier-Tessera [CT11]. We prove it (in any degree) by relating the group L p -cohomology with the asymptotic L p -cohomology. The latter has been defined by Pansu, and it is known to be a quasiisometric invariant [Pan95]. The equivalence between the group L pcohomology and the asymptotic L p -cohomology was established by G.
Elek [Ele98] for finitely generated groups, and by R. Tessera [Tes09] in degree 1. We recently learnt from R. Sauer and M. Schrödl that they established the coarse invariance of vanishing of ℓ 2 -Betti numbers for unimodular locally compact second countable groups [SS17]. To prove this result they established a coarse equivalence version of Theorem 1.1, by using the same comparison stategy -see Theorem 12 in [SS17].
Our motivation for Theorem 1.1, which leads to the second main theorem of this paper, is understanding the L p -cohomology of semisimple groups of real rank 2. In fact, we have in mind the following question, asked by M. Gromov [Gro93,p. 253].
Question 1.2. Let G be a simple real Lie group. We assume that l = rk R (G) 2. Let k be an integer < l and p be a real number > 1. Do we have: H k ct G, L p (G) = {0}?
The previous theorem allows one to reduce the study of the group L p -cohomology of a Lie group G as in the question, to the one of any of its parabolic subgroups. Such a parabolic subgroup, say P , admits a Levi decomposition, which is a semi-direct product decomposition P = M ⋉ AN where M is semi-simple and AN is the (solvable) radical of P . The class for which we can fruitfully use Theorem 1.1 consists of the simple Lie groups in which some maximal parabolic subgroup has suitable geometric properties. By lack of better terminology, we will call them admissible. Here is their definition: Definition 1.3. A simple real Lie groups is called admissible if it admits a parabolic subgroup whose radical is quasi-isometric to a real hyperbolic space.
Theorem 1.4. Let G be an admissible simple real Lie group, and let d be the dimension of the corresponding real hyperbolic space. Let also D denote the dimension of the Riemannian symmetric space attached to G.
We also note that the same circle of ideas can be used to prove the following result, which is a particular case of a general result of Pansu [Pan07] and Cornulier-Tessera [CT11], saying that H 1 ct G, L p (G) = {0} for every p > 1 and every connected Lie group G, unless G is Gromov hyperbolic or amenable unimodular.
Corollary 1.6. For every admissible simple Lie group G such that rk R (G) 2 and every p > 1, we have: This result also follows from the fixed point property for continuous affine isometric actions of semi-simple groups of real rank 2 on L pspaces, which is established in [BFGM07].
Let us finish this introduction by describing roughly our method and by discussing its efficiency and its validity.
The main idea is to combine tools from homological algebra, namely the Hochschild-Serre spectral sequence [BW00,IX.4], together with geometric results such as Pansu's description of the L p -cohomology of real hyperbolic spaces [Pan99, Pan08]. We also use resolutions already considered in [Bla79], whose measurable nature makes them flexible enough to describe nicely intermediate spaces involved in the spectral sequences.
The general outcome is a new description of the L p -cohomology of G -see Theorem 6.1 and Remark 6.4.
In order to discuss the efficiency of the method, let us reformulate Theorem 1.4 as follows: for each admissible group G and each exponent p > 1, we exhibit an interval of degrees, whose explicit length does not depend on p and outside of which vanishing of L p -cohomology is garanteed (i.e., this interval is a strip of potential non-vanishing of L p -cohomology). First, by Lie-theoretic considerations, we can provide the full list of admissible simple Lie groups. It contains several infinite sequences of groups, say {G l } l 2 where l = rk R (G). In the best cases, the dimension D l of the symmetric space of G l and the dimension d l of the radical of the suitable parabolic subgroup are quadratic polynomials in l, and we have lim Since the width of the strip of potential non-vanishing is D l − d l + 2, this makes us say that we obtain in the best cases of admissible groups nearly half of the relevant vanishings.
At last, our results can also be compared with Borel's result [Bor85] about the L 2 -cohomology of connected semi-simple Lie groups G, which makes heavy use of representation theory while we use a more geometric approach. It asserts that: where D is the dimension of the Riemannian symmetric space G/K, and where l 0 is the difference between the complex rank of G and the complex rank of K. In the case G = SL n (R), one has D = n 2 +n−2 2 and l 0 = [ n−1 2 ]. See also [BFS14] for related results about vanishing of the reduced L 2 -cohomology.
Remarks and Questions. 1) The definition of an admissible simple Lie group (Definition 1.3) generalizes to semisimple Lie groups. It is straightforward that a semisimple real Lie group is admissible if and only if one of its simple factors is. Theorem 1.4 is still valid for admissible semisimple Lie groups (the proof is the same). However we think that some L p -version of the Künneth formula may be true for products of groups and could give better results. More precisely, suppose that G 1 , G 2 are locally compact second countable groups, such that for some p ∈ (1, +∞) and n 1 , n 2 ∈ N, one has H k ct G i , L p (G i ) = 0 for every k ≤ n i . Is H k ct G 1 × G 2 , L p (G 1 × G 2 ) = 0 for every k ≤ n 1 + n 2 + 1 ? We remark that these problems are discussed in [Gro93,p. 252].
2) Even if a simple real Lie group is non-admissible, the solvable radical of any maximal proper parabolic subgroup decomposes as R ⋉ N, where N is a nilpotent Lie group on which R acts by contracting diffeomorphisms. Such a group belongs to the class of the so-called Heintze groups, i.e. the connected Lie groups which admit a left-invariant Riemannian metric of negative curvature. The L p -cohomology of negatively curved manifolds has been investigated by P. Pansu [Pan99, Pan08] (see also [Bou16]). However, apart from the real hyperbolic spaces, only a partial description is known. To keep the exposition of the paper as simple as possible, we have choosen to restrict ourself to the class of admissible simple Lie groups.
Organization of the paper. Section 2 is a brief presentation of some standard topics in cohomology of groups. Section 3 discusses group L pcohomology: we first consider the discrete group case and recall some classical connections with the simplicial L p -cohomology; in the general case, we state the existence of an isomorphism between the group L pcohomology and the asymptotic L p -cohomology (Theorem 3.7). This result, whose proof is given in Section 4, implies the first quasi-isometric invariance theorem (Theorem 1.1 of the introduction). Section 5 contains some preliminary results on the group L p -cohomology of semidirect products via the Hochschild-Serre spectral sequence. Section 6 focuses on applications of the previous sections to the group L pcohomology of admissible simple real Lie groups (an improved version of Theorem 1.4 of the introduction). Corollary 1.5 is proved in this section. Section 7 provides the list of admissible groups and Section 8 gives tables in which the numerical efficiency of the method can be discussed. Finally, in Section 9 we prove Corollary 1.6 on vanishing in degree 1.
Acknowledgements. We thank Nicolas Monod, whose paper [Mon10] was a valuable source of inspiration. We also thank Pierre Pansu: the present paper elaborates on several of his results. At last, we thank Y. Cornulier for useful comments and suggestions, and R. Sauer and M. Schrödl for drawing our attention to their paper [SS17]. M.B. was partially supported by the Labex Cempi and B.R. by the GeoLie project (ANR-15-CE40-0012). Both authors were supported by the GDSous/GSG project (ANR-12-BS01-0003).

Continuous cohomology
We review some of the basic notions of the continuous cohomology of groups. We refer to [BW00,Chapter IX] or to [Gui80] and their references for more details.
Let X, Y be topological spaces. The set of continuous mappings f : X → Y equipped with the compact open topology is denoted by C(X, Y ).
Let G be a locally compact second countable group, and let (π, V ) be a topological G-module i.e. a Hausdorff locally convex vector space over R on which G acts via a continuous representation π. We denote by V G ⊂ V the subspace of π(G)-invariant vectors.
For k ∈ N, let C k (G, V ) := C(G k+1 , V ) be the set 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 , ..., x k ∈ G, (g · f )(x 0 , ..., x k ) = π(g) f (g −1 x 0 , ..., g −1 x k ) .
(1) The continuous cohomology of G with coefficients in (π, V ) is the cohomology of this complex. It is the collection of groups H (2) The reduced continuous cohomology is the collection H (3) When G is a discrete group, we omit the subscript "ct" and simply write H k (G, V ) and H k (G, V ).
We now recall some useful notions in homological algebra.
• A map f : A → B between topological G-modules is a Gmorphism if it is a continuous G-equivariant linear map. • A complex of topological G-modules and G-morphisms is said to be a strong G-resolution of V if for every k ∈ N, there exists a continuous linear map h k : ..) be complexes of Hausdorff locally convex vector spaces.
Example 2.4. Suppose G is a discrete group that acts properly discontinuously and freely on a contractible locally finite simplicial complex X by simplicial automorphisms. Let X (k) be the set of k-simplices of X and let C k (X, V ) be the set of linear maps from the vector space RX (k+1) to V . Let δ k : C k (X, V ) → C k+1 (X, V ) be defined as follows; for f ∈ C k (X, V ), and σ ∈ X (k+1) , The C k (X, V )'s are G-modules for the following action: for g ∈ G, f ∈ C k (X, V ) and σ ∈ X (k) , The δ k 's are G-morphisms. The complex is a strong injective G-resolution of V . Indeed one defines the h k 's by induction on k by using a retraction of X to a point. To show that C k (X, V ) is strong injective, one considers a fundamental domain D in X, the set X (k) D := {σ ∈ X (k) | the first vertex of σ belongs to D} and the vector space C k D (X, V ) of linear maps from RX . The latter is strong injective (see the discussion in Example 2.3 above).
Strong injective resolutions compute the continuous cohomology: Then the complexes (A * ) G and (B * ) G are homotopy equivalent.
Since homotopy equivalent complexes have the same cohomology, one obtains Corollary 2.6. Suppose 0 → V → A * is a strong injective G-resolution of V . Then, the cohomology and the reduced cohomology of the complex (A * ) G are topologically isomorphic to H * ct (G, V ) and H * ct (G, V ), respectively.
Proof. From Example 2.3 and Proposition 2.5 above, the complexes (A * ) G and C * (G, V ) are homotopy equivalent. Therefore, by standard arguments, their cohomological spaces (equipped with the quotient topology) are topologically isomorphic. By definition, the cohomology of C * (G, V ) is H * ct (G, V ). Thus the cohomology H * (A * ) G of (A * ) G is topologically isomorphic to H * ct (G, V ). The reduced cohomological spaces of (A * ) G and C * (G, V ) are respectively topologically isomorphic

Continuous group L p -cohomology
The group L p -cohomology is defined in this section (Definition 3.1). When the group is a finite K(π, 1), we relate its group L p -cohomology with the simplicial L p -cohomology (Proposition 3.2). For locally compact second countable topological groups, the group L p -cohomology is isomorphic to the asymptotic L p -cohomology (Theorem 3.7). This implies Theorem 1.1 of the introduction.
Definition 3.1. Let G be a locally compact second countable topological group, and H be a left-invariant Haar measure. Let p ∈ (1, +∞). 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, H), i.e. the representation defined by It will be denoted by H * ct G, L p (G) . Similarly is defined the reduced group L p -cohomology of G denoted by H * ct G, L p (G) . When G is discrete we omit the subscript "ct" and simply write H * G, L p (G) and H * G, L p (G) .
Observe that the right regular representation is isometric if and only if H is bi-invariant.
The discrete group case. In this paragraph we relate the group L pcohomology of certain discrete groups with the simplicial L p -cohomology of some simplicial complexes (Proposition 3.2). This is a standard material. We present it for completeness and also because its proof can be seen as a model for the general case.
Let X be a simplicial complex and let X (k) be the set of its ksimplices. We will always assume that X has bounded geometry i.e. that there is an N ∈ N such that each simplex intersects at most N simplices.
Let C k,p (X) be the Banach space of linear maps u : The simplicial L p -cohomology of X is the cohomology of the complex where the δ k 's are defined as in Example 2.4. Similarly is defined its reduced simplicial L p -cohomology. They are denoted by L p H * (X) and L p H * (X) respectively.
Proposition 3.2. Suppose that G acts by simplicial automorphisms, properly discontinuously, freely and cocompactly, on a locally finite contractible simplicial complex X. Then the complexes C * G, L p (G) G and C * ,p (X) are homotopy equivalent. In particular there are topo- Proof. According to Corollary 2.6 and Example 2.4, C * G, L p (G) G is homotopy equivalent to the complex In particular their cohomologies (reduced or not) are topologically isomorphic. To prove the proposition it is enough to show the latter complex is topologically isomorphic to the complex C * ,p (X).
To this end, one considers the map Φ : With the norms expressions, one sees that it is a topological embedding.
. Because G acts properly discontinuously and cocompactly on X, by using the partition of X (k) into orbits, one gets that u ∈ C k,p (X).
Remark 3.4. Equip every simplicial complex X with the length metric obtained by identifying every simplex with the standard Euclidean one. A simplicial complex is called uniformly contractible if it is contractible and if there exists a function ρ : (0, +∞) → (0, +∞) such that every ball B(x, r) ⊂ X can be contracted to a point in B x, ρ(r) . Among bounded geometry uniformly contractible simplicial complexes, the simplicial L p -cohomology (reduced or not) is invariant by quasiisometry. Indeed if X and Y are quasi-isometric bounded geometry uniformly contractible simplicial complexes, then the complexes C * ,p (X) and C * ,p (Y ) are homotopy equivalent (see [Gro93,p. 219], and [BP03] for a detailed proof). As a consequence, the above proposition implies that the group L p -cohomology of fundamental groups of finite aspherical simplicial complexes is invariant by quasi-isometry. We will give below another proof of this fact that applies in a much larger generality.
The general case. Suppose that G is a locally compact second countable topological group. We will relate the group L p -cohomology and the asymptotic L p -cohomology of G. The latter has been considered by Pansu in [Pan95], see also [Gen14]. It is a quasi-isometric invariant (see Theorem 3.6 below).
Following [Pan95], we define the asymptotic L p -cohomology of a metric measure space. Let X = (X, d, µ) be a metric measure space. Suppose it satisfies the following "bounded geometry" condition. There exists increasing functions v, V : (0, +∞) → (0, +∞) such for every For s > 0 and k ∈ N, let ∆ We equip AS k,p (X) with the topology induced by the set of the seminorms N s (s > 0).
where the d k 's are defined as in (2.1). Similarly is defined the reduced asymptotic L p -cohomology. They are denoted by L p H * AS (X) and L p H * AS (X) respectively.
Asymptotic L p -cohomology (reduced or not) is invariant by quasiisometry. In fact one has Theorem 3.6 ([Pan95]). Let X and Y be bounded geometry metric spaces and let F : X → Y be a quasi-isometry. Then F induces a homotopy equivalence between the complexes AS * ,p (X) and AS * ,p (Y ). Moreover the associated isomorphism of graded topological vector spaces See also [Gen14] for a more detailed proof. In the next section we will prove the Theorem 3.7. Suppose G is equipped with a left-invariant proper metric. Then C * G, L p (G) G and AS * ,p (G) are homotopy equivalent complexes. In particular there exist topological isomorphisms: In combination with Theorem 3.6 one obtains the following result, which implies Theorem 1.1 of the introduction: Corollary 3.8. Let G 1 and G 2 be locally compact second countable topological groups, equipped with left-invariant proper metrics and let F : G 1 → G 2 be a quasi-isometry. Then F induces a homotopy equivalence between the complexes C * G 1 , L p (G 1 ) Moreover the associated isomorphism of graded topological vector spaces depends only the bounded perturbation class of F . Remark 3.9. Suppose X is a bounded geometry simplicial complex. In general its asymptotic and simplicial L p -cohomologies are different (for example the asymptotic L p -cohomology of a finite simplicial complex is trivial since it is quasi-isometric to a point). Pansu [Pan95] asked whether the asymptotic and the simplicial L p -cohomologies coincide for uniformly contractile bounded geometry simplicial complexes. He proved that it is indeed the case for those which are in addition nonpositively curved; for these spaces the complexes C * ,p (X) and AS * ,p (X) are homotopy equivalent.

Proof of quasi-isometric invariance
As explained in the previous section the quasi-isometric invariance of the group L p -cohomology is a consequence of Theorem 3.7. We prove this theorem in this section. The proof is inspired by the one of Proposition 3.2. It will use a strong injective G-resolution (Lemma 4.1) that appears in Blanc [Bla79]. As already mentioned in the introduction, Theorem 3.7 was proved by G. Elek [Ele98] for finitely generated groups. In degree 1, it was established by R. Tessera [Tes09]. R. Sauer and M. Schrödl obtained a result very similar to Theorem 3.7 by using the same strategy -see [SS17, Theorem 10]. They applied it to show that vanishing of ℓ 2 -Betti numbers is a coarse equivalence invariant for unimodular locally compact second countable groups.
Suppose that the topological vector space V is Fréchet (i.e. metrizable and complete). Let X be a locally compact second countable topological space endowed with a Borel measure µ. Let p ∈ (1, +∞). We denote by L p loc (X, V ) the set of (classes of) measurable functions f : X → V such that for every compact K ⊂ X and every continuous semi-norm N on V defining its topology, one has Suppose that G acts on X continuously by preserving µ. Equipped with the set of semi-norms · K,N , the vector space L p loc (X, V ) is a Fréchet G-module for the action (g · f )(x) = π(g) f (g −1 x) , see [Bla79]. Let H be a left-invariant Haar measure on G. Denote by G k the cartesian product of k copies of G equipped with the product measure.
In the special case V = L p (G), one can express the spaces L p loc (X, V ) G as follows.
Lemma 4.2. Let X, µ, G, H as above. Let E be the topological vector space that consists of the (classes of the) Borel functions u : X → R such that for every compact subset K ⊂ X one has Then E is topologically isomorphic to the Fréchet space L p loc X, L p (G) G .
Proof. Let u ∈ E. Since u is Borel and since the map An element f ∈ L p loc X, L p (G) is null if and only if f (x)(g) = 0 for almost all (g, x) ∈ G × X. Therefore the above discussion in combination with the semi-norms expressions and the Fubini-Tonelli theorem imply that Φ is a topological embedding. It remains to prove that its image is the subspace for simplicity. One has for h ∈ G and for almost all x ∈ X and g ∈ G: Let f ∈ L p loc X, L p (G) G , we are looking for u ∈ E such that Φ(u) = f . One would like to define u by u(x) = f (x)(1); but this has no meaning in general since f (x) ∈ L p (G). Let (U n ) n∈N be a decreasing sequence of relatively compact open neighborhoods of 1 in G such that n∈N U n = {1}. Define a measurable function u : X → R by Therefore Lebesgue's theorem implies that u(gx) = f (x)(g) for almost all g ∈ G and x ∈ X. By modifying u to a Borel function in the same class if necessary, one obtains that u ∈ E and that Φ(u) = f .
According to Proposition 2.5 and Lemma 4.1, to finish the proof of Theorem 3.7, it is enough to establish: Proof. Denote by H n the product measure on G n . According to Lemma 4.2, it is enough to show that the following two sets of semi-norms (on the space of measurable functions u : G k+1 → R) define the same topology. The first set of semi-norms is the one considered in Lemma 4.2 in the case X = (G k+1 , H k+1 ). These are the N K 's, with K ⊂ G k+1 compact, defined by The second one is the set of semi-norms that appears in the definition of the asymptotic L p -cohomology of (G, d, H). These are the N s 's , with s > 0, defined by N s (u) p = ∆s u(y 0 , ..., y k ) p dH k+1 (y 0 , ..., y k ).

Semi-direct products
We partially relate the L p -cohomology of semi-direct products P = Q⋉R with the L p -cohomology of the normal subgroups R (see Corollaries 5.4 and 5.5). The next section will contain examples of application.
Some generalities. Let V be a Hausdorff locally convex topological vector space, and let X be a locally compact second countable topological space endowed with a Borel measure µ. Let p ∈ (1, +∞). We denote by L p (X, V ) the set of (classes of) measurable functions f : X → V such that for every continuous semi-norm N on V defining the topology of V , one has Equipped with the set of semi-norms · N , the vector space L p (X, V ) is a Hausdorff locally convex vector space. When V is a Fréchet space, or a Banach space, then so is L p (X, V ).
Proposition 5.1. Suppose X, Y are topological spaces as above endowed with Borel measures. Then Proof. Let f : X × Y → V be a measurable function. For a compact subset K ⊂ Y and a continuous semi-norm N on V , denote by N K,N and · N the associated semi-norms on L p loc (Y, V ) and L p (X, V ) respectively. One has by Fubini-Tonelli The proposition follows.
Hochschild-Serre spectral sequence. Let R be a closed normal subgroup of a locally compact second countable group P and let (π, V ) be a topological P -module. The P -action on C * (R, V ), defined by for every g ∈ P , f ∈ C k (R, V ) and In the special case (π, V ) is the right regular representation on L p (P ) and P is a semi-direct product P = Q ⋉ R (where Q and R are closed subgroups) we give in this subsection a more comprehensive description of these cohomological spaces. In the following statement the measures H Q and H R are left-invariant Haar measures on Q and R respectively. We denote by ∆ the modular function for the Q-action by conjugation on R, i.e. the function on Q defined by q −1 H R q = ∆(q) · H R , for every q ∈ Q.
Proposition 5.2. Let k ∈ N. Assume that H k ct R, L p (R) is Hausdorff and that the complex C * R, L p (R) R is homotopically equivalent to a complex of Banach spaces. Then H k ct R, L p (P ) is Hausdorff and there is a canonical isomorphism of topological Q-modules: where, in the latter space, the group Q is equipped with the measure ∆ −1 · H Q , and where the Q-action is induced by for every q, y ∈ Q, f : Q → C k R, L p (R) , and x 0 , ..., x k ∈ R.
Proof. As a consequence of Blanc's lemma (Lemma 4.1) the complex L p loc R * +1 , L p (P ) R retracts to its subcomplex C * R, L p (P ) R . Thus, it is not only that these two complexes have the same cohomology, but the Q-action on the cohomology can be expressed in the same manner.
We claim that L p (P ) ≃ L p Q, L p (R) as topological P -modules, where Q is equipped with the measure ∆ −1 · H Q , and where the expression of the right-regular representation of P on the latter module is π(q, r)f (y, x) = f (yq, xyry −1 ) for every q, y ∈ Q and r, x ∈ R.
Indeed this follows from Fubini, by observing that ∆ −1 · H R × H Q is a left-invariant measure on P = Q ⋉ R. Therefore, with Proposition 5.1, we obtain that L p loc R * +1 , L p (P ) R is isomorphic to L p Q, L p loc R * +1 , L p (R) R . In this representation, one can check that the Q-action on the latter complex can be written as in the statement of the proposition.
It remains to prove that the kth-cohomology space of the complex L p Q, L p loc R * +1 , L p (R) R is isomorphic to L p Q, H k ct R, L p (R) . By assumption and from Lemma 4.1, the complex L p loc R * +1 , L p (R) R is homotopy equivalent to a complex of Banach spaces that we denote by B * . Since every continuous linear map ϕ : V 1 → V 2 between Hausdorff locally convex topological vector spaces, extends to a continuous linear map ϕ * : L p (Q, V 1 ) → L p (Q, V 2 ) defined by ϕ * (f ) = ϕ • f , the above homotopy equivalence induces a homotopy equivalence between the complexes L p Q, L p loc R * +1 , L p (R) R and L p (Q, B * ). Since H k (B * ) ≃ H k ct R, L p (R) , the following lemma ends the proof of the proposition.
Lemma 5.3. Suppose k ∈ N and that B * is a complex of Banach spaces such that H k (B * ) is Hausdorff. Then the kth-cohomology space of the complex L p (Q, B * ) satisfies H k L p (Q, B * ) ≃ L p Q, H k (B * ) canonically and topologically.
Proof of Lemma 5.3. Consider the complex L p (Q, B * ). One has clearly Ker d| L p (Q,B k ) = L p (Q, Ker d| B k ). We claim that The direct inclusion is obvious. For the reverse one, we use the assumption that Im(d : B k−1 → B k ) is a Banach space in combination with the following Michael's theorem [Mic56,Prop. 7.2], that previously appears in [Mon01, p. 93] in the context of bounded cohomology: Suppose ϕ : B → C is a continuous surjective linear map between Banach spaces. Then for every λ > 1 there exists a continuous (non linear) section σ : C → B such that for every c ∈ C one has The natural map L p (Q, Ker d| B k ) → L p Q, H k (B * ) and the above equalities induce an injective continuous map It is surjective, thanks again to Michael's theorem, since the projection map Ker d| B k → H k (B * ) is a surjective continuous linear map between Banach spaces.
Since H k (B * ) is a Banach space, so is L p Q, H k (B * ) . From the previous description of Ker d| L p (Q,B k ) and Im d| L p (Q,B k−1 ) , the cohomological space H k L p (Q, B * ) is also a Banach space (for the quotient norm). Therefore Banach's theorem implies that the above isomorphism is a topological one.
As a consequence of Proposition 5.2, the Hochschild-Serre spectral sequence for L p -cohomology takes the following form: Corollary 5.4. Suppose that P = Q ⋉ R where Q and R are closed subgroups of P . Assume that C * R, L p (R) R is homotopically equivalent to a complex of Banach spaces and that every cohomology space H k ct R, L p (R) is Hausdorff. Then, there exists a spectral sequence (E r ), abutting to H * ct P, L p (P ) , in which Corollary 5.5. Suppose that P = Q ⋉ R where Q and R are closed subgroups of P . Assume that C * R, L p (R) R is homotopically equivalent to a complex of Banach spaces. Suppose also that there exists n ∈ N, such that H k ct R, L p (R) = 0 for 0 k < n and such that H n ct R, L p (R) is Hausdorff. Then H k ct P, L p (P ) = 0 for 0 k < n and there is a linear isomorphism We notice that the latter isomorphism is just a linear one, in particular it does not imply that H n ct P, L p (P ) is Hausdorff.

The (possibly) non-vanishing strip
Let G be a non-compact simple Lie group. As a consequence of Corollary 3.8, the group L p -cohomology of G is isomorphic to the one of any of its parabolic subgroups. Indeed, every parabolic subgroup acts cocompactly on G, and thus is quasi-isometric to G.
Recall that a parabolic subgroup decomposes as P = M ⋉ AN with M semi-simple, A ≃ R r , and N nilpotent. One can expect to use this decomposition to derive, from the Hochschild-Serre spectral sequence, some informations about the group L p -cohomology of P . To do so, and according to Corollaries 5.4 and 5.5, one has to know a bit of the L p -cohomology of the group AN. When the rank r of A is at least 2, very few is known about the L p -cohomology of AN (apart from the vanishing of H 1 ct AN, L p (AN) for all p -see [Pan07, CT11]). In contrast, when the rank of A is one, i.e. when P is a maximal proper parabolic subgroup, the group AN admits an invariant negatively curved Riemannian metric, and the L p -cohomology of negatively curved manifolds has been investigated by Pansu [Pan99, Pan08].
We focus on the class of simple Lie groups G that admit a maximal proper parabolic subgroup P = M ⋉ AN with AN quasi-isometric to a real hyperbolic space H d of constant negative curvature. According to Definition 1.3 such groups G are called admissible. They will be classified in the next section. The elementary case G = SL n (R) is discussed in Example 6.5. We obtain the following partial description of their cohomology: Theorem 6.1. Let G be an admissible simple Lie group. Let P = M ⋉AN be a maximal proper parabolic subgroup with radical AN quasiisometric to H d . Let X = G/K be the associated symmetric space and let D = dim X. One has: (1) H 0 ct G, L p (G) = H k ct G, L p (G) = 0 for k D. In particular one sees that the possible couples (p, k) for which H k ct G, L p (G) = 0 lie in the strip d−1 The numerical computation of the width will occupy Section 8.
A key ingredient in the proof of Theorem 6.1 is the following lemma which is a straightforward consequence of Pansu's results.
Lemma 6.2. Suppose that AN is a Lie group quasi-isometric to a real hyperbolic space H d . Then for p > 1, its group L p -cohomology satisfies  Proof of Lemma 6.2. By Corollary 3.8, the group L p -cohomology of AN is isomorphic to the one of any cocompact lattice in Isom(H d ); which in turn is the same as the simplicial L p -cohomology of H dsee Proposition 3.2. By the simplicial L p -cohomology of a complete Riemannian manifold X, we mean the simplicial L p -cohomology of any bounded geometry quasi-isometric simplicial decomposition of X. Now, the simplicial L p -cohomology of H d has been computed by Pansu [Pan99,Pan08] ( 1 ). The result of this computation is precisely the statement of our lemma.
Parts of Theorem 6.1 rely on the following version of Poincaré duality established in [Pan08] Corollaire 14 ( 2 ). Lemma 6.3. Let X be a complete Riemannian manifold of bounded geometry and of dimension D. Denote by L p H * (X) and L p H * (X) its simplicial L p -cohomology and its reduced simplicial L p -cohomology. Let q = p/(p − 1) and 0 k D. Then Proof of Theorem 6.1. (1). According to Corollary 3.8, the group L pcohomology of G is isomorphic to the one of any of its cocompact lattices; which in turn is isomorphic to the simplicial L p -cohomology of X -see Proposition 3.2. The latter is trivially null in degree 0 and degrees k > dim X. It is also null in degree D = dim X. Indeed, the 1 In [Pan99, Pan08], Pansu computes the de Rham L p -cohomology of H d , i.e. the cohomology of the de Rham complex of L p differential forms with differentials in L p . In [Pan95], he proves an L p -cohomology version of the de Rham theorem, namely the homotopy equivalence between the de Rham L p -complex and the simplicial L p -complex, for complete Riemannian manifolds of bounded geometry. The fact that the simplicial L p -cohomology is Hausdorff for k = d−1 p + 1, and vanishes for k < d−1 p and k > d−1 p + 1, can also be established without the de Rham theorem, as a consequence of Corollary B in [Bou16]. Again, Pansu establishes this result for the de Rham L p -cohomology. One obtains the result for the simplicial L p -cohomology by using the L p -cohomology version of the de Rham theorem. The lemma can also be established more directly without the de Rham theorem, as a consequence of Proposition 1.2 and Theorem 1.3 in [Bou16]. lemma above implies that L p H D (X) = 0. Moreover, since X satisfies a linear isoperimetric inequality, L p H 1 (X) is Hausdorff -see [Pan07] or [Gro93]. Hence the lemma implies that L p H D (X) is Hausdorff, and so it is null.
(2). Since G and P are quasi-isometric, one has H * ct G, L p (G) ≃ H * ct P, L p (P ) by Corollary 3.8. The vanishing of H k ct P, L p (P ) for every k d−1 p , follows directly from the analogous result for AN in Lemma 6.2, and from Corollary 5.5 applied with P = M ⋉AN, Q = M and R = AN. We note that the assumptions of Corollary 5.5 are satisfied; indeed C * AN, L p (AN) AN is homotopy equivalent to the simplicial L p -complex of H d which is a complex of Banach spaces.
To show that H k ct G, L p (G) = 0 for k d−1 p + D − d + 2, we use again the isomorphism between the group L p -cohomology of G and the simplicial L p -cohomology of X. We know from above that the latter is null for k d−1 p . Then Lemma 6.3 implies that L p H k (X) is null for k d−1 p +D−d+1 and that L p H k (X) is Hausdorff for k d−1 p +D−d+2. The proof of the first two items is now complete.
Suppose that d−1 p / ∈ N. Then, according to Lemma 6.2, the only degree ℓ such that H ℓ ct AN, L p (AN) is non-zero, is ℓ = ⌊ d−1 p ⌋ + 1. Moreover H ℓ ct AN, L p (AN) is Hausdorff. By applying Corollary 5.4, one obtains the third item.
Remark 6.4. In view of the last item in Theorem 6.1, to analyse further the case where d−1 p < k < d−1 p + D − d + 2, in particular to decide whether H k ct G, L p (G) vanishes or not, one would like to take advantage of a good description of H ℓ ct AN, L p (AN) and of the Maction on it. In particular one could try to exploit the description of H ℓ ct AN, L p (AN) as a functional space on N ≃ R d−1 ≃ ∂H d \ {∞}, that is established in Section 8.2 of [Pan99] (see also [Pan89] or [Rez08] or [BP03] for the case ℓ = 1). We will follow this idea in Section 9 to prove the vanishing of the first group L p -cohomology of admissible Lie groups of real rank 2, see Corollary 1.6.
Example 6.5. Every maximal parabolic subgroup of SL n (R) is conjugated to a subgroup of the form P = p 1 p 3 0 p 2 ∈ SL n (R) p 1 ∈ M s,s , p 2 ∈ M n−s,n−s , p 3 ∈ M s,n−s , with 0 < s < n. Moreover P = M ⋉ AN with and N = I x 0 I x ∈ M s,n−s ≃ (R s(n−s) , +). Since one has n−s x. Therefore AN is isometric to H s(n−s)+1 and SL n (R) is admissible. Take s = ⌊ n 2 ⌋. Then one has d − 1 = dim N = s(n − s) = n 2 4 if n is even, and n 2 −1 4 otherwise. Since D = dim X = (n−1)(n+2) 2 , Theorem 6.1 gives the following possibly non-vanishing strip for the L p -cohomology of SL n (R) : if n is even, and if n is odd, which can be summarized by: ⌊ n 2 4 ⌋ · 1 p < k < ⌊ n 2 4 ⌋ · 1 p + ⌊ (n+1) 2 4 ⌋.

The admissible simple real Lie groups
We are looking for the simple Lie groups that we called admissible, i.e. that contain a proper parabolic subgroup, say P with Levi decomposition M P ⋉ A P N P where A P is a split torus and N P is a unipotent group normalized by A P , for which the following metric condition holds: ( * ) the underlying Riemannian variety of the radical A P N P is quasi-isometric to a real hyperbolic space.
The list of admissible simple Lie groups is given by the following.
Proposition 7.1. The admissible simple real Lie groups are those whose relative root system is of type A l , B l , C l , D l , E 6 and E 7 . In particular, for classical types the only excluded groups are those with non-reduced root system BC l .
Proof. Let P ∅ be a minimal parabolic subgroup in G containing A, a maximal split torus. The subgroup A defines a set Φ of roots with respect to A and the inclusion A ⊂ P ∅ defines, inside Φ, a subset Φ + of positive roots and a subset ∆ of simple roots. We have: Any parabolic subgroup is conjugated to exactly one of the parabolic subgroups containing P ∅ , and the latter subgroups are parametrized by subsets of ∆. More precisely, they are of the form where M ∆ ′ is the Levi subgroup generated by the roots of Φ in the linear span Z∆ ′ of ∆ ′ ⊂ ∆, the subtorus A ∆ ′ of A lies in the kernels of the roots in ∆ ′ and the Lie algebra of the unipotent group N ∆ ′ is linearly spanned by the weight spaces indexed by the roots in Φ + \ Φ(∆ ′ ) where Φ(∆ ′ ) = Φ ∩ Z∆ ′ . This description is consistent with the notation P ∅ for a minimal parabolic subgroup; note that we also have G = P ∆ .
Let us go back to condition ( * ). By homogeneity, it implies that A P N P = A P ⋉ N P is actually homothetic to a real hyperbolic space, and in terms of roots it implies that A P must act on N P via a single character. The first consequence is that P must be maximal for inclusion among proper parabolic subgroups. Therefore, since we work up to conjugacy, this implies that we may -and shall -assume that there is a simple root γ ∈ ∆ such that P = P ∆\{γ} . To simplify notation, we set: The torus A γ is 1-dimensional and it acts on N γ (more precisely, on the root subspaces of the Lie algebra of N γ ) via powers of γ since it lies in the kernel of any other simple root of the basis ∆. Note that the simple root γ itself always appears as a weight in the root space decomposition of Lie(N γ ). Therefore, for A γ N γ to be homothetic to a real hyperbolic space, it is necessary and sufficient that A γ act on N γ via the single character γ (i.e. we must avoid powers γ k with k 2).
We have obtained this way a combinatorial interpretation of condition ( * ). Indeed, let δ be a root occuring as a weight in the root space decomposition of Lie(N γ ): it is a root in Φ + for which the coordinate along γ in the basis ∆ has a coefficient m 1 (since vanishing amounts to being in the root system Φ(∆ \ {γ}) of M ∆\{γ} ). In this case, any a ∈ A γ acts on the root space of weight δ via the scalar δ(a) m .
The remaining task now is to investigate the descriptions of root systems, and to check for which ones there is a simple root γ ∈ ∆ with respect to which the γ-coordinate in the basis ∆ of each positive root is equal to 0 or 1: the first case says that the positive root belongs to Φ(∆ \ {γ}) and the second one says that it belongs to Φ + hence occurs as a weight for the A γ -action on N γ . We refer now to [Bou68, Planches] and its notation. Let us investigate the root systems by a case-by-case analysis of the formulas given for the linear decompositions of the positive roots according to the given bases (formulas (II) in [loc. cit.]): • for type A l , any simple root can be chosen for γ since no coefficient 2 occurs in formula (II) of [loc. cit., Planche I]; • for type B l , the root γ can be chosen to be α 1 in the notation of [loc. cit., Planche II]; • for type C l , the root γ can be chosen to be α l which occurs only for two kinds of positive roots, each time with coefficient 1, in formula (II) of [loc. cit., Planche III]; • for type D l , we use the formulas in [loc. cit., p.208] instead of Planche IV (where the formulas are not correct), to see that the root γ can be chosen to be α 1 or α l ; • for type E 6 , the root γ can be chosen to be α 1 or α 6 , as inspection of the positive roots with some coefficient 2 shows in [loc. cit., Planche V]; • for type E 7 , the root γ can be chosen to be α 7 , as inspection of the positive roots whose support contains α 7 and with some coefficient 2 shows in [loc. cit., Planche VI].
To exclude the remaining case, it is enough to note that: • for type E 8 , the last positive root described on the first page of [loc. cit., Planche VII] has all its coefficients 2; • for type F 4 (resp. G 2 ), the last root in (II) of [loc. cit., Planche VIII (resp. IX)] has the same property; • for the only non-reduced irreducible root system of rank l, namely BC l , the positive root ε 1 has all its coefficients 2 [loc. cit., p. 222].
This finishes the determinations of the admissible simple Lie groups by means of their relative root systems.
As a complement, we can say that the groups that are not admissible can be listed thanks toÉ. Cartan's classification as stated for instance in [Hel01,Table VI,. The classical groups with nonreduced relative root system correspond to the cases where the value of m 2λ is 1 in this table. To sum up, we can reformulate the previous proposition in more concrete terms: Proposition 7.2. The admissible simple real Lie groups are the split simple real Lie groups of classical types and of types E 6 et E 7 , the complex simple groups of classical types and of types E 6 et E 7 (all seen as real Lie groups), all non-compact orthogonal groups SO p,q (R), all special linear groups SL n (H) = SU * 2n (R) over the quaternions H, the special unitary groups SU n,n (R) with n 2, the groups Sp 2n,2n (R) = SU n,n (H) with n 2, the groups SO * 4n (R) = SO 2n (H) with n 2, the exceptional groups of absolute type E 6 and real rank 2 and of absolute type E 7 and real rank 3.
For all pratical purposes, we also provide the list of non-admissible simple real Lie groups. These are: • all the compact simple groups, • the groups SU p,q (R) and Sp p,q (R) with 0 < p < q, • the simple groups SO * 2r (R) with odd r, • the split groups, or the complex ones seen as real groups, of type E 8 , F 4 , G 2 , • for each of E 6 , E 7 , E 8 , the real form of rank 4, • the outer real form of rank 2 of E 6 , • the real form of rank 1 of F 4 .

Numerical efficiency of the method
Let us turn now to the numerical efficiency of the method, namely the computation of the actual width of the (possibly) non-vanishing strips, i.e. the intervals outside of which our method proves vanishing of continuous group L p -cohomology (Theorem 6.1).
Recall that for a given admissible group G with maximal compact subgroup K, this width is equal to D−d+2 where D is the dimension of the symmetric space X = G/K and d is the dimension of the solvable radical A γ N γ . Since d = 1 + dim N γ , we have: where m α is the multiplicity dimN(α) the root space N(α) attached to α. Now, we have just seen that the root system of an admissible group is reduced, so we can freely use [Hum72] (in which root systems are assumed to be reduced by definition). By [loc. cit., Lemma C p. 53] there are at most two root lengths in Φ and the Weyl group acts transitively on the roots of given length, therefore since this Weyl group action lifts to the action of N G (A) on the root spaces N(α), we deduce that there are at most two multiplicities, say m 1 and m 2 , and with the notation Ψ = Φ + \ Φ(∆ \ {γ}) the previous formula becomes: The case of split groups is easy since all multiplicities m α are equal to 1 (hence in this case d = 1 + |Ψ|), and the case of Weil restrictions (i.e. complex groups seen as real ones) is easy too: all multiplicities are equal to 2, hence in this case d = 1 + 2 · |Ψ|.
For the other cases, where two multiplicities may occur, we proceed as follows. For each admissible group, thanks to the previous section we make the choice (often, but not systematically, unique) of a good root, that is a simple root γ such that A γ N γ is quasi-isometric to a real hyperbolic space. Thanks to [Bou68, Planches], this gives easily |Ψ| since the root system of Φ(∆ \ {γ}) is given by the Dynkin diagram of Φ minus the vertex of type γ and the edges emanating from it. We must then sort the roots of Ψ according to their length, which is done again thanks to the concrete descriptions of [loc. cit.]. It remains then to apply the last formula for d − 1 and to use the multiplicities given in [Hel01,  The rest of this section is dedicated to computing the width d = D − d + 2 of the (possibly) non-vanishing strip, and to determine the minimal one when the choice of a good root γ is not unique for the group G under consideration.
The case of split groups.-Investigating the class of admissible split simple real Lie groups is the opportunity to provide a description of the roots sets Ψ of nilpotent radicals, that will be useful in the more complicated cases; we use the notation of [Bou68, Planches].
• When G = SL n+1 (R), the root system has type A n , any simple root α i (1 i n) is a good root, and we have |Ψ| = i×(n−i); all roots have the same length. • When G = SO n+1,n (R), the root system has type B n , the only good root is α 1 and we have: |Ψ| = |Φ + (B n )| − |Φ + (B n−1 )| = n 2 − (n − 1) 2 = 2n − 1, with one root of length 1 and (n − 1) + (n − 1) = 2n − 2 roots of length √ 2. • When G = Sp 2n (R), the root system has type C n , the only good root is α n and we have: , with n(n−1) 2 roots of length √ 2 and n roots of length 2.
As already mentioned, in this case we can then compute d = 1 + |Ψ| and take w = D − d + 2 to be minimal when several good roots are available.
The case of complex groups seen as real ones.-Then the multiplicities are then 2; in this case we have d = 1 + 2|Ψ| and again we can take w = D − d + 2 to be minimal when several good roots are available.
The remaining admissible simple real Lie groups.-We use here the formula d − 1 = m 1 · |{α ∈ Ψ : m α = m 1 }| + m 2 · |{α ∈ Ψ : m α = m 2 }| and the above partition of Ψ into short and long roots when the root system Ψ is not simply laced, combined with the multiplicities given in [Hel01,  • When G = SL n+1 (H), the root system has type A n , any simple root α i (1 i n) is good, we have |Ψ| = i × (n − i) and all roots have multiplicity 4, so that d − 1 = 4i(n − i) for γ = α i . • When G = SU n,n (R), the relative root system has type C n , the good root is α n , the n(n−1) 2 roots of length √ 2 have multiplicity 2 and the n roots of length 2 have multiplicity 1, so that d−1 = n(n − 1) + n = n 2 .
• When G = SO * 4n (R), the relative root system has type C n , the good root is α n , the n(n−1) 2 roots of length √ 2 have multiplicity 4 and the n roots of length 2 have multiplicity 1, so that d−1 = 2n(n − 1) + n = 2n 2 − n.
• When G = E 2 6 (R), the relative root system has type A 2 , any of two simple roots is good and all roots have multiplicity 8, so that d − 1 = 16.
• When G = E 3 7 (R), the relative root system has type C 3 , the good root is α 3 , there are 3 roots of each length √ 2 and 2, and since the multiplicities are 8 and 1, this gives d − 1 = 27.
At last, in the case of non-split orthogonal groups SO p,q (R) (i.e., where q − 2 p 2), we always obtain a relative root system of type B p , with 2p − 2 roots of length √ 2 and multiplicity 1 and one root of length 1 and multiplicity q − p, so that d − 1 = p + q − 2.
The next two pages contain tables which summarize the vanishing results obtained by our method. In conclusion, apart from the general family of non-split orthogonal groups (which is special since it depends on two parameters), the infinite families of classical admissible groups provide a vanishing proportion of 1 2 asymptotically in the rank. For non-split orthogonal groups, fixing the rank (i.e. the smallest number in the signature) and letting the bigger parameter go to infinity provide a vanishing proportion equal to the inverse of the rank.

Admissible group
Cartan type   We claim that M u • m p B dH(m) = +∞ unless u is constant a.e.; the corollary will follow. The proof relies on three lemmata. The first one gives a sufficient condition for u to be constant. Lemma 9.3. Let u ∈ B (d−1)/p p,p (R d−1 ) that is invariant by a 1-parameter group of translations. Then u is constant a.e.
Proof. Express the Besov norm of u as u p Suppose that u is invariant by a 1-parameter group of translations along some vector line Rv 0 . Then ψ is invariant by the 1-parameter group of translations along R(v 0 , v 0 ). If in addition one has R d−1 ×R d−1 ψ dxdx ′ < +∞, then Fubini-Tonelli implies that ψ is null a.e. This in turn implies that u is constant a.e.
The second lemma collects some simple properties of the function ϕ defined in (9.2).
(2) There exists a positive constant c 0 such that ϕ c 0 .
Proof. The third item follows from the fact that H is a lelf-invariant measure. To obtain the last item, one observes that the first and third items imply that Thus ϕ(v) = +∞.
We now describe the behaviour of ϕ in a neighborhood of an eigenvector.
Lemma 9.5. Let (g t ) t∈R ⊂ M be a non-trivial 1-parameter subgroup which acts diagonally with positive eigenvalues on R d−1 . Let λ max > 1 and λ min < 1 be the maximum and minimum eigenvalues of g. Let v 0 ∈ R n−1 be an eigenvector of g for the eigenvalue λ max . There exist a constant c 2 > 0 and a neighborhood U of v 0 such that for every v ∈ U one has ϕ(v) c 2 v − v 0 α with α = 2(d − 1) log λmax log λmax−log λ min .
Proof. If v is an eigenvector then ϕ(v) = +∞ thanks to Lemma 9.4. Suppose now that v is not an eigenvector. Let W ⊂ R d−1 be a subspace such that R d−1 = Rv 0 ⊕ W is a {g t } t∈R -invariant decomposition. We write every vector v ∈ R d−1 as v = v 1 + v ′ according to this decomposition. For simplicity we will abusively denote by the same symbol · the standard norm on R d−1 and the norm max( v 1 , v ′ ).
Finally we complete the Proof of Corollary 1.6. Let u ∈ B (d−1)/p p,p (R d−1 ) be a function such that M u • m p B dH(m) < ∞. We want to prove that u is constant a.e. According to Lemma 9.3, it is enough to show that there is a nonzero vector v 0 ∈ R d−1 such that u is constant almost everywhere along almost every line directed by v 0 . Since M is a semi-simple non-compact Lie group, it contains a 1-parameter subgroup (g t ) t∈R which acts on R d−1 as in Lemma 9.5. Let λ max , λ min , v 0 , c 2 , U and α be as in Lemma 9.5. By changing g to its inverse, if necessary, we can assume that λ max λ −1 min ; so that we have α d − 1. Since u is measurable, it is approximately continuous a.e. (see [Fed69,2.9.13]). Let a, b be distinct points on a line directed by v 0 , such that u is approximately continuous at a and b. We want to show that u(a) = u(b). By multiplying v 0 by a real number if necessary, we can assume that v 0 = b − a. For every ε > 0, there exists r ε > 0 such that for every 0 < r r ǫ the set where c 3 is a positive constant that depends only on a − b . Since b − a = v 0 , for v close enough to v 0 , one has meas E r (a) ∩ E r (b) − v (3/10)meas{B(a, r)}.
With Lemma 9.5 and the inequality α d − 1 one gets that for every small enough neighbourhood U of v 0 : where c 4 depends only on c 2 , c 3 , r. Since the last integral is equal to +∞, we obtain a contradiction.