arxiv:2001.02490v3 [math.gr] 1 oct 2020

arX

iv:2

001.

0249

0v3

[m

ath.

GR

] 1

Oct

202

0

ISOMETRIC ACTIONS ON Lp-SPACES:

DEPENDENCE ON THE VALUE OF p

AMINE MARRAKCHI AND MIKAEL DE LA SALLE

Abstract. Answering a question by Chatterji–Drutu–Haglund, we prove that,for every locally compact group G, there exists a critical constant pG ∈ [0,∞]such that G admits a continuous affine isometric action on an Lp space (0 <

p < ∞) with unbounded orbits if and only if p ≥ pG. A similar result holds forthe existence of proper continuous affine isometric actions on Lp spaces. Usinga representation of cohomology by harmonic cocycles, we also show that suchunbounded orbits cannot occure when the linear part comes from a measurepreserving action, or more generally a state-preserving action on a von Neu-mann algebra and p > 2. We also prove the stability of this critical constantpG under Lp measure equivalence, answering a question of Fisher. We use thisto show that for every connected semisimple Lie group G and for every latticeΓ < G, we have pΓ = pG.

1. Introduction

The study of affine isometric actions of groups on Banach spaces is an importanttheme in mathematics that is related to many other topics such as group coho-mology, fixed point properties and geometric group theory. The case of actionson Hilbert spaces is very well-studied. For example, it is known that a secondcountable locally compact group G has an affine isometric action on Hilbert spaceswithout fixed points (resp. proper) if and only if G does not have Kazhdan’s prop-erty (T) (resp. has the Haagerup property). On the other hand, while the groupsG = Sp(n, 1) have property (T), Pansu showed in [33] that they admit affine iso-metric actions without fixed points on Lp(G) for all p > 4n + 2 (and actuallyproper actions by [11]). This was generalized by Bourdon and Pajot in [5]. In [43],Yu proved that any hyperbolic group Γ admits a proper affine isometric action onℓp(Γ × Γ) for all p large enough, see also [29]. For more results and references,we refer to [3] where a systematic study of affine isometric actions of groups onLp-spaces was undertaken. As suggested by the previous results, it is very naturalto expect that for a given group G, it should be “easier” to act isometrically on anLp-space when the value of p gets larger. The main result of this paper confirmsthis intuition. In this statement as in the whole paper, Lp space means Lp(X,µ)for a standard measure space (X,µ).

Theorem 1. Let G be a topological group. Take 0 < p ≤ q < ∞. Then for everycontinuous affine isometric action α : G y Lp, there exists a continuous affineisometric action β : Gy Lq such that ‖αg(0)‖

pLp

= ‖βg(0)‖qLq

for all g ∈ G.

Theorem 1 implies in particular that if a group G has a continuous action byisometries on an Lp space with unbounded (respectively metrically proper) orbits,then it has such an action on an Lq space.

1

http://arxiv.org/abs/2001.02490v3

2 MARRAKCHI AND DE LA SALLE

Corollary 2. For every topological group G,

(i) The set of values of p ∈ (0,∞) such that G admits a continuous action byisometries on an Lp space with unbounded orbits is an interval of the form(pG,∞) or [pG,∞) for some pG ∈ 0 ∪ [2,∞].

(ii) The set of values of p ∈ (0,∞) such that G admits a proper continuous actionby isometries on an Lp space is an interval of the form (p′G,∞) or [p′G,∞)for some p′G ∈ 0 ∪ [2,∞].

Recall that for 1 ≤ p < ∞, an action by isometries on an Lp space has a fixedpoint if and only if it has bounded orbits ([3, Lemma 2.14] for p 6= 1, [2] for p = 1).So Corollary 2 answers a question raised in [7], sometimes refered to as Drutu’sconjecture [31, 24]. A partial answer for ℓp spaces had already been obtainedindependantly by Czuron [9] and Lavy–Ollivier [24]. In [24] actions coming fromergodic probability measure preserving actions were also covered. Unlike theseprevious results, it is worth mentionning that in Theorem 1, the linear part of theaction β that we construct is very different from the linear part of the originalaction α. We refer to Theorem 4.3 for a more precise statement.

When G is a second countable locally compact group, it is known that pG > 0if and only if G has property (T) [3], in which case we must have pG ≥ 2 (infact, by an argument of Fisher and Margulis from [15] that appears in [3], onecan even show that pG > 2 and that G admits a continuous action by isometrieson an Lp space without fixed points for p = pG, see also [24, 12, 34]). Similarly,it is known that p′G > 0 if and only if G does not have the Haagerup property,in which case p′G ≥ 2 [30]. This last fact, as well as the fact that pG /∈ (0, 2),if often stated for second countable locally compact groups, but they are true forarbitrary topological groups, see Proposition 3.1 (but the fact that p′G /∈ (0, 2) ismeaningful only for locally compact groups, as p′G = ∞ trivially for non locallycompact groups). The critical constant pG and p′G are different in general. Forexample, if G is a locally compact group that has neither Kazhdan’s property (T)nor Haagerup property, then pG = 0 and p′G ≥ 2. It is also known that, amongGromov-hyperbolic groups, the value of pG is unbounded, and explicit lower boundshave been obtained for random groups [13] (see also [23]).

The linear part the action β constructed in Theorem 1 comes from an actionon (X,µ) preserving an infinite measure. It is not possible to achieve the samewith a finite measure. Indeed, when G has property (T), any affine action on Lp

(1 ≤ p <∞) whose linear part comes from a probability measure preserving actionhas a fixed point. This is known when G is discrete [24] or when G admits afinite Kazhdan set [10]. In general, this is a particular case of the following resultdealing with non-commutative Lp spaces. Its proof relies on a general observation ofindependant interest: under a spectral gap and uniform convexity assumption, anycohomology class with values in an isometric representation has a unique harmonicrepresentent (Lemma 6.1).

Theorem 3. Let G be a locally compact property (T) group. An action α : G y

Lp(M) by affine isometries on the non-commutative Lp space of a von Neumannalgebra M has a fixed point in the following two cases:

• 1 < p ≤ 2 and M = B(H) for a Hilbert space H.• 2 ≤ p <∞ and the linear part of α comes from an action by automorphismsof M preserving a faithful normal state.

ISOMETRIC ACTIONS ON Lp-SPACES: DEPENDENCE ON THE VALUE OF p 3

We insist that the first conclusion of the theorem is not trivial. It is indeed anopen question by Masato Mimura whether a locally compact property (T) groupcan have an unbounded action by isometries on a non-commutative Lp space forp < 2 (as explained above this is not possible for usual Lp spaces), and the previousresult provides a negative answer for Schatten classes.

It is known that the metric space (Lp, ‖·‖pqp ) isometrically embeds into (Lq, ‖·‖q)

when p ≤ q [27, Remark 5.10]. This implies the following inequalities for thecompression exponents [28] of a compactly generated group G

(1.1) ∀0 < p < q <∞, pαp(G) ≤ qαq(G).

As a consequence of Theorem 1, we obtain the same inequalities for the equivariantcompression exponents.

(1.2) ∀0 < p < q <∞, pα#p (G) ≤ qα#

q (G).

This inequality is often strict, see [28].We also note that Theorem 1 can be applied to the whole isometry group of Lp

and this yields the following corollary.

Corollary 4. Take 0 < p ≤ q <∞. Then Isom(Lp) is isomorphic as a topologicalgroup to a closed subgroup of Isom(Lq).

Note that if p > 2, the subgroup of translations Lp ⊂ Isom(Lp) is not unitarilyrepresentable [26, Theorem 3.1]. In particular, it cannot be embedded as a closedsubgroup of Isom(L2), which is unitarily representable (by using the affine Gaussianfunctor [1, Proposition 4.8] for example).

With Corollary 2 in hand, we can rephrase a Problem asked by Gromov in [17,§6.D3] as Given a group Γ, find the value of pG and p′G. The last section of thispaper is devoted to this question, in the specific case of Lie groups and their lattices.We first show that for any connected semisimple Lie group G, the constants pG andp′G depend only on its Lie algebra g and not onG itself (see Theorem 8.3). Moreover,we show how this constants can be in principle computed from the decomposition ofg into simple Lie algebras (see Example 8.2). When G is simple, we have pG = p′G(see Theorem 8.1). Pansu’s result shows that if G = Sp(n, 1) then 2 < pG ≤ 4n+2while if G has real rank ≥ 2, it is known that pG = ∞ [3, Theorem B]. Our mosttechnical result is the following one.

Theorem 5. Let G be a connected semisimple Lie group and let Γ < G be a lattice.Then pΓ = pG and p′Γ = p′G.

The proof of this theorems relies on two ingredients. The first one is the followingresult which shows that the constants pG and p′G behave nicely with respect toLp-measure equivalence (see Theorem 6 and the definitions preceeding it). Thisanswers a question by David Fisher.

Theorem 6. If two compactly generated locally compact groups G1 and G2 are Lp

measure equivalent, then the critical constants defined in Corollary 2 satisfy

min(pG1, p) = min(pG2

, p) and min(p′G1, p) = min(p′G2

, p).

When Γ < G is a lattice, then G and Γ are measure equivalent. But by definition,they are Lp measure equivalent if and only if Γ is Lp integrable. Thus for the proofof Theorem 5, we need the following second ingredient.


Theorem 7. Let G be a connected semisimple Lie group and let Γ < G be a lattice.Then Γ is Lp integrable for all p < pG.

This paper is organized as follows. After some preliminaries in Section 2, The-orem 1 and its Corollary 4 are proved in Section 3 for p = 2 and Section 4 in thegeneral case. Section 5 contains a discussion on the validity of Corollary 2 whenthe linear part of the action of Lp is fixed. Section 6 deals with harmonic cocyclesand the proof of Theorem 3. In Section 7, stability properties of the constants pGand p′G are investigated and in particular, Theorem 6 is proved. The last section isdedicated to the study of pG and p′G for connected semisimple Lie groups.

Acknowledgments. We are grateful to David Fisher for his question on Lp-measure equivalence which led us to Theorem 6. We also thank Piotr Nowak forhis comments which helped us to improve the presentation of this paper. We thankRomain Tessera for providing us with useful references.

2. Preliminaries

Nonsingular actions. Let (X,µ) be a σ-finite standard measure space (we willalways assume that our measure spaces are standard and we omit the σ-algebra).We denote by [µ] the measure class of µ. We denote by Aut(X, [µ]) the groupof all nonsingular (preserving the measure class [µ]) automorphisms of (X,µ) upto equality almost everywhere. It is known that Aut(X, [µ]) is a Polish groupfor the topology of pointwise convergence on probability measures: a sequenceθn ∈ Aut(X, [µ]) converges to the identity if and only if limn ‖(θn)∗ν − ν‖1 = 0 forevery probability measure ν ∈ [µ]. We denote by Aut(X,µ) the group of all measurepreserving automorphisms of (X,µ) up to equality almost everywhere. It is a closedsubgroup of Aut(X, [µ]). A continuous nonsingular action σ : G y (X,µ) of atopological group G is a continuous homomorphism σ : G ∋ g 7→ σg ∈ Aut(X, [µ]).

Cohomology. Let π : G y V be a continuous linear representation of a topo-logical group G on a topological vector space V . We denote by Z1(G, π, V ) theset of all continuous 1-cocycles, i.e. all continuous maps c : G → V such thatc(gh) = c(g)+g · c(h) for all g, h ∈ G. We denote by B1(G, π, V ) ⊂ Z1(G, π, V ) theset of all coboundaries, i.e. cocycles c of the formc(g) = g · v − v for some v ∈ V .

Let σ : G y (X,µ) be a continuous nonsingular action of a topological groupG. Let A be an abelian topological group (here we use the additive notation for Abut this might change sometimes when A = T). We denote by Z1

σ(G,A) the setof all A-valued 1-cocycles of σ, i.e. all continuous functions c : G 7→ L0(X,µ,A)such that c(gh) = c(g) + σg(c(h)). Here for every f ∈ L0(X,µ,A), we use thenotation σg(f) = f σ−1

g . We denote by B1σ(G,A) the set of all 1-coboundaries, i.e.

cocycles of the form g 7→ σg(f) − f for some f ∈ L0(X,µ,A). Finally, we denoteby H1

σ(G,A) = Z1σ(G,A)/B

1σ(G,A) the cohomology group of σ.

Skew-product actions. Let σ : G y (X,µ) be a continuous nonsingular actionof a topological group G. Suppose that A is a locally compact abelian group andlet m be the Haar measure of A. Then for every c ∈ Z1

σ(G,A), we can define a newcontinuous nonsingular action σ ⋊ c of G on (X ×A, µ⊗m) by the formula

(σ ⋊ c)g(x, a) = (gx, a+ c(g−1)(x)).

The action σ ⋊ c is called the skew-product action of σ by c. Define a functionh : X × A → A by h(x, a) = a for all (x, a) ∈ X × A. Then, by construction, we


have c(g) ⊗ 1 = (σ ⋊ c)g(h) − h. Thus the skew-product action σ ⋊ c turns thecocycle c into a coboundary.

The Maharam extension. Let σ : G y (X,µ) be a continuous nonsingularaction of a topological group G. Then we can define the Radon-Nikodym cocycle

D ∈ Z1σ(G,R

∗+) by the formula D(g) =

d(σg)∗µdµ for all g ∈ G. The skew-product

action σ = σ ⋊D : Gy X ×R∗+ is called the Maharam extension of σ. Note that

σ preserves the measure µ⊗ dλ where dλ is the restriction to R∗+ of the Lebesgue

measure of R.

Isometric actions on Lp-spaces. Take p > 0 and let (X,µ) be a σ-finite measurespace. For every θ : Aut(X, [µ]), we define a linear isometry of Lp(X,µ) given by

f 7→

(θ∗µ

µ

)1/p

θ(f).

The group L0(X, [µ],T) also acts by multiplication on Lp(X,µ). We thus obtain acontinuous linear isometric representation

πp,µ : Aut(X, [µ])⋉ L0(X, [µ],T) → O(Lp(X,µ)).

It follows from the Banach-Lamperti theorem that this map is surjective whenp 6= 2. Note that if ν ∈ [µ], the canonical isometry Lp(X,µ) → Lp(X, ν) given by

f 7→(µν

)1/pf is equivariant with respect to the natural actions πp,µ and πp,ν of

Aut(X, [µ])⋉ L0(X, [µ],T).Let σ : G → Aut(X, [µ]) be a continuous nonsingular action of a topological

group G. Then σp,µ = πp,µ σ is a continuous linear isometric representation of Gon Lp(X,µ). Let D ∈ Z1

σ(G,R∗+) be the Radon-Nikodym cocycle. Then for every

p > 0 and every g ∈ G, the isometry σp,µ(g) is given by the formula

σp,µ(g) : Lp(X,µ) ∋ f 7→ D(g)1/pσg(f) ∈ Lp(X,µ).

Take now some cocycle ω ∈ Z1σ(G,T). Then the map g 7→ ω(g)σp

g is again acontinuous linear isometric representation of G on Lp(X,µ). Conversely, if p 6= 2,it follows from the Banach-Lamperti theorem that every continuous linear isometricrepresentation of G on Lp(X,µ) is of the form π : g 7→ ω(g)σp,µ

g for some continuous

nonsingular action σ of G and some cocycle ω ∈ Z1σ(G,T).

Let α be an affine isometric action of G on some Lp-space. Since the affineisometry group Isom(Lp(X,µ)) decomposes as a semi-direct product

Isom(Lp(X,µ)) = O(Lp(X,µ))⋉ Lp(X,µ)

where Lp(X,µ) acts by translations, we see that α is of the form αg(f) = π(g)f +c(g) where π is an isometric linear representation of G on Lp(X,µ) and c ∈Z1(G, π, Lp(X,µ)) is a cocycle. Observe that even when π = σp,µ for some non-singular action σ : G y (X,µ), we do not have Z1(G, σp,µ, Lp(X,µ)) ⊂ Z1

σ(G,C)unless σ preserves the measure µ.

3. The case p = 2

Let G be a topological group and let p > 0. We denote by Kp(G) the set ofall continuous functions ψ : G → R+ of the form ψ(g) = ‖αg(0)‖

pLp

for some

continuous affine isometric action α of G on some Lp-space. Note that K2(G) isthe set of all continuous functions on G that are conditionally of negative type.


By using the Gaussian functor, one has the following (classical) result.

Proposition 3.1. Let G be a topological group. Take ψ ∈ K2(G) and p > 0. Thenthere exists a continuous probability measure preserving action σ : Gy (X,µ) and

a cocycle c ∈ Z1σ(G,R) such that ψ(g)

p2 = ‖c(g)‖pLp

for all g ∈ G. In particular,

ψp2 ∈ Kp(G) for all p > 0.

Proof. By definition, there exists an orthogonal representation π : G → O(H) onsome Hilbert space H and a cocycle c ∈ Z1(G, π,H) such that ψ(g) = ‖c(g)‖2 forall g ∈ G. Let σπ : G y (X,µ) be the Gaussian action associated to σ. This

means that there exists a linear map ξ 7→ ξ ∈ L0(X,µ,R) such that ξ is a centered

Gaussian random variable of variance ‖ξ‖2 for all ξ ∈ H , and that σπ(ξ) = π(g)ξ

for all ξ ∈ H . Let c(g) = c(g) ∈ Lp(X,µ) for all g ∈ G. Then c is a cocycle forσπ and a computation shows that ‖c(g)‖pLp

= Cp‖c(g)‖p for all g ∈ G and some

constant Cp > 0.

Corollary 3.2. For every topological group G, we have K2(G) ⊂ Kp(G) for allp ≥ 2.

Proof. The function x 7→ xα is a Bernstein function for all 0 < α ≤ 1. It followsthat for every ψ ∈ K2(G), we have ψα ∈ K2(G), hence ψα p

2 ∈ Kp(G). Theconclusion follows by taking α = 2

p .

4. Proof of the main theorem

Proposition 4.1. Let G be a topological group. For every p > 0 and every ψ ∈Kp(G), there exists a continuous nonsingular action σ : G y (X,µ) and a cocyclec ∈ Z1(G, σp,µ, Lp(X,µ)) such that ψ(g) = ‖c(g)‖pLp

for all g ∈ G.

Proof. We may assume that p 6= 2 thanks to Proposition 3.1. By definition, thereexists an affine isometric action α : Gy Lp(X,µ) for some probability space (X,µ)such that ψ(g) = ‖αg(0)‖

pLp

for all g ∈ G. Write αg(f) = πg(f)+c(g) where π is an

isometric linear representation of G on Lp(X,µ) and c ∈ Z1(G, π, Lp(X,µ)). Sincep 6= 2, we can write π(g) = ω(g)σp,µ

g where σ : G y (X,µ) is some nonsingular

action and ω ∈ Z1σ(G,T). Consider the skew-product nonsinglar action σ = σ⋊ω :

Gy (X ×T, µ ⊗m) where m is the Haar measure of T. Observe that σg(u)u∗ =

ω(g)⊗1 where u is the function on X×T given by u(x, z) = z for all (x, z) ∈ X×T.It follows that c : g 7→ uc(g) defines an element c ∈ Z1(G, σp, Lp(X × T, µ ⊗m))such that ‖c(g)‖p = ‖c(g)‖p for all g ∈ G, where m is the Haar measure of T.

Lemma 4.2. Take 0 < p < q <∞. Let ϕ : C → R be a nonzero, radial, compactlysupported, Lipschitz function. Then there exists a constant C(q) > 0 such that forall w ∈ C, we have

(4.1)

∫

C

∫ ∞

0

|ϕ(z + λ−1/pw)− ϕ(z)|q dλ dz = C(q)|w|p

Proof. Let S be the Lebesgue measure of the support of ϕ, M = ‖ϕ‖∞ and K theLipschitz constant of ϕ. Then we have |ϕ(z + λ−1/p) − ϕ(z)| ≤ min(2M,Kλ−1/p)for all z ∈ C and all λ ∈ R∗

+. Therefore, we have∫

C

|ϕ(z + λ−1/p)− ϕ(z)|q dz ≤ 2Smin((2M)q,Kqλ−q/p).


Since q > p, the function λ 7→ min((2M)q,Kqλ−q/p) is integrable on R∗+. There-

fore, we can define

C(q) =

∫

C

∫ ∞

0

|ϕ(z + λ−1/p)− ϕ(z)|q dλ dz < +∞.

Since ϕ is not constant, we have C(q) > 0. Finally, the statement for all w ∈ C

follows from the change of variable λ 7→ |w|pλ and the fact that ϕ is radial.

Theorem 4.3. Let G be a topological group. Take 0 < p < q <∞. Then for everyψ ∈ Kp(G), there exists a continuous measure preserving action σ : Gy (Y, ν) anda function h ∈ L∞(Y, ν) such that b(g) = σg(h)−h ∈ Lq(Y, ν) with ψ(g) = ‖b(g)‖qLq

for all g ∈ G. In particular ψ ∈ Kq(G).

Proof. By Proposition 4.1, there exists a nonsingular action σ : G y (X,µ) anda cocycle c ∈ Z1(G, σp,µ, Lp(X,µ)) such that ψ(g) = ‖c(g)‖pLp

for all g ∈ G.

Let σ : G y (X, µ) be the Maharam extension of σ. This means that (X, µ) =(X×R∗

+, µ⊗dλ) where dλ is the restriction toR∗+ of the Lebesgue measure ofR and

σ : Gy (X, µ) is the measure-preserving action given by g(x, λ) = (gx, dg−1µdµ (x)λ).

Define c ∈ Z1σ(G,C) by the formula c(g, x, λ) = λ−

1p c(g, x) (observe that c indeed

satisfies the cocycle relation thanks to the term λ−1/p). Let ρ : G y (Y, ν) be the

skew-product action of σ by c. This is the measure space (Y, ν) = (X ×C, µ⊗ dz)and ρ is the measure preserving action given by g(x, z) = (gx, z + c(g−1, x)). Letϕ : C → R be a nonzero, radial, compactly supported, Lipschitz function. Definea function h ∈ L∞(Y, ν) by h(x, z) = ϕ(z), and let b(g) = ρg(h) − h for all g ∈ G.Then Lemma 4.2 shows that b is a cocycle with values in Lq(Y, ν) that satisfies theconclusion of the theorem up to a constant C(q) > 0.

Remark 4.4. The idea of the proof of Theorem 4.3 and of the cocycle c becomesvery natural if one uses the notion of modular bundle and Haagerup’s canonical Lp-spaces as explained in [1, Sections A.2 and A.3]. Indeed, by viewing the canonicalLp-space Lp(X) as a subspace of L0(Mod(X)), the isometric linear representationσp : G y Lp(X) associated to some nonsingular action σ : G y X is identifiedwith the restriction to Lp(X) of the Maharam extension Mod(σ) : G y Mod(X).We can therefore identify every cocycle c ∈ Z1(G, σp, Lp(X)) with a cocycle c ∈Z1Mod(σ)(G,C). This is crucial in order to be able to use the skew-product con-

struction.

Proof of Corollary 4. Let G = Isom(Lp) with its canonical affine isometric ac-tion on Lp. By applying Theorem 1, we obtain a continuous homomorphismΨ : Isom(Lp) → Isom(Lq) such that

‖g(0)‖pLp= ‖Ψ(g)(0)‖qLq

for all g ∈ Isom(Lp).

We have to show that Ψ is a homeomorphism on its range. Take (gn)n∈N a sequencein Isom(Lp) and suppose that Ψ(gn) → id. We have to show that gn → id. Takef ∈ Lp and let τf ∈ Isom(Lp) be the translation by f . Then

‖gn(f)− f‖pLp= ‖(τ−1

f gn τf )(0)‖pLp

= ‖Ψ(τ−1f gn τf )(0)‖

qLq.


Since Ψ(gn) → id, we also have Ψ(τ−1f gn τf ) = Ψ(τf )

−1 Ψ(gn) Ψ(τf ) → id.This yields

limn

‖gn(f)− f‖pLp= lim

n‖Ψ(τ−1

f gn τf )(0)‖qLq

= 0.

Since this holfs for all f ∈ Lp, we conclude that gn → id as we wanted.

Question 4.5. Is Theorem 4.3 still true when q = p? Can we at least realize ψwith an affine isometric action whose linear part comes from a measure preservingaction of G? One can show that if a measurable function ϕ : R → R satisfies

∫

R

∫ ∞

0

|ϕ(x+ λ−1/p)− ϕ(x)|p dλ dx < +∞

for some p ≥ 1, then ϕ is almost surely constant. Therefore, our method for theproof of Theorem 4.3 cannot work when q = p.

Question 4.6. Take 0 < p, q < ∞. Is it true that Isom(Lp) embeds as a closedsubgroup of Isom(Lq) if and only if p ≤ max(2, q)?

5. Formal coboundaries

By the Banach-Lamperti theorem, an action by linear isometries on Lp(X,µ)(p 6= 2) comes from a non-singular action on (X,µ) and a cocyle with values in T,and in particular it corresponds to an action by linear isometries on Lq(X,µ) forevery other q. However, in the proof of Theorem 1, the linear part of the action βon Lq is very different from the linear part of the original action α. It is naturalto wonder when the same linear part can be chosen. The purpose of this section isto investigate this question; the main result shows that this is the case when thetranslation part is a formal coboundary. This is a variant of [9, 24].

In this section, we will sometimes change the σ-algebra. To fix notation weconsider a standard measure space (X,µ) with σ-algebra A.

5.1. Facts on the Mazur maps. The Mazur map Mp,q is the nonlinear map

given by Mp,q(f)(x) = sgn(f(x))|f(x)|pq for every measurable function f : X → C

(where for z ∈ C \ 0, sgn(z) = z|z| and sgn(0) = 0). It is well-known that, for

every 1 ≤ p, q < ∞, the Mazur map is a uniformly continuous homeomorphismfrom the unit ball of Lp to the unit ball of Lq. This follows from the existence, forevery 1 ≤ p ≤ q <∞, of a constant c = c(p, q) such that for all a, b ∈ C

(5.1) |sgn(a)|a|pq − sgn(b)|b|

pq | ≤ c|a− b|

pq

and

(5.2) |a− b| ≤ c(|sgn(a)|a|

pq − sgn(b)|b|

pq |

qp + |a|1−

pq |sgn(a)|a|

pq − sgn(b)|b|

pq |).

For example, we have

Lemma 5.1. Assume that (X,µ) is a probability space and let f ∈ Lp(X,µ) ofmean 0. Then for every z ∈ C,

‖Mp,q(f)− z‖q ≥ C(p, q)‖f‖pqp

for a number C(p, q) > 0 depending only on 1 ≤ p, q <∞.


Proof. By homogeneity we can assume that ‖f‖p = 1. We can assume that 12 ≤

|z| ≤ 32 as otherwise

‖Mp,q(f)− z‖q ≥ |‖Mp,q(f)‖q − |z|| ≥1

2.

If ωq,p is the modulus of uniform continuity of the restriction of the Mazur mapMq,p to the ball of radius 3

2 in Lq, we obtain

|z|qp ≤ ‖f −Mq,p(z)‖a ≤ ωp,q(‖Mp,q(f)− z‖).

The first inequaliy is just the inequality |∫Fdµ| ≤ ‖F‖p. So we obtain the lemma

with C(p, q) = min(ω−1p,q(2

− qp ), 2−1).

If we go outside of the unit ball, the relative position of p and q matter. Forexample, the next result is not true if p > q.

Lemma 5.2. Let 1 ≤ p ≤ q < ∞. If f, g : X → C are two measurable functionssuch that f − g ∈ Lp, then Mp,q(f)−Mp,q(g) ∈ Lq.

Proof. Replacing a by f(x), b by g(x) in (5.1), raising to the power q and integratingwith respect to x, we get

‖Mp,q(f)−Mp,q(g)‖Lq≤ c‖f − g‖

pq

Lp.

Lemma 5.3. Let 1 ≤ p ≤ q < ∞. Let (X,A, µ) be a σ-finite measure space, andB ⊂ A be a σ-subalgebra. Assume that f ∈ L0(X,A, µ) satisfies that, for everyh1 ∈ L0(X,B, µ), there is h2 ∈ L0(X,B, µ) such that Mp,q(f +h1)−Mp,q(h2) ∈ Lq.Then there is h ∈ L0(X,B, µ) such that f − h ∈ Lp.

Proof. Let us first consider the case when (X,B, µ) is purely infinite, that is everyelement of B has measure 0 or ∞. Applying the hypothesis to h1 = 0, we obtainh0 such that Mp,q(f)−Mp,q(h0) ∈ Lq. Applying now the hypothesis to h1 = −h0,we obtain h2 such Mp,q(f − h0) −Mp,q(h2) ∈ Lq. If h2 6= 0, there is ε > 0 andC > 0 such that |h2| > ε∩|h0| ≤ C is not null, that is has infinite measure as itbelongs to B. This implies that |f−h0| >

ε2∩|h0| ≤ C also has infinite measure,

as it contains the preceding set minus |Mp,q(f − h0) −Mp,q(h2)| > (2pq − 1)ε

pq ,

which has finite measure because Mp,q(f − h0) − Mp,q(h2) ∈ Lq. But by (5.2),|f − h0| >

ε2 ∩ |h0| ≤ C is contained in |Mp,q(f) −Mp,q(h0)| > δ whenever

c(δqp +C1− p

q δ) < ε2. We obtain a contradiction because Mp,q(f)−Mp,q(h0) ∈ Lq.

So our hypothesis that h2 6= 0 is absurd, so we obtain Mp,q(f − h0) ∈ Lq, orequivalently f − h0 ∈ Lp.

Let us now consider the case when (X,B, µ) is semifinite. Replacing µ by ρµ fora positive ρ ∈ L1(B), we can assume that µ is a probability measure. So we can talkabout the µ-preserving conditional expectation EB. As in the previous case, thereis h0 ∈ L0(B) such that Mp,q(f) −Mp,q(h0) ∈ Lq. In particular EB(|f |

q|) < ∞almost everywhere, and we can define h = EB(f). By the assumption for h1 = −h,there is h2 ∈ L0(B) such that Mp,q(f − h)− h2 ∈ Lq. By Lemma 5.1, we obtain

‖Mp,q(f − h)− h2‖qq = E(EB(|Mp,q(f − h)− h2|

q))

≥ C(p, q)E[EB(|f − h|p))

= C(p, q)‖f − h‖pp.


This proves that f − h ∈ Lp. Finally, the general case follows by decomposing(X,B, µ) as a purely infinite part and a semifinite part.

5.2. Formal coboundaries. Let (X,µ) be a σ-finite measure space. Let G →Aut(X, [µ])⋉L0(X,µ,T) be a continous group homomorphism. For every 0 < p <∞, denote by πp,µ : G → O(Lp(X,µ)) the corresponding continuous representa-tion. We also denote by πp,µ the continous extension of this linear representationto L0(X,µ) (by using the same formula for πp,µ as in the preliminaries). We con-sider the two cohomology spaces H1(G, πp,µ, Lp(X,µ)) and H

1(G, πp,µ, L0(X,µ)).Clearly, since Lp(X,µ) ⊂ L0(X,µ), we have a natural homomorphism

η : H1(G, πp,µ, Lp(X,µ)) → H1(G, πp,µ, L0(X,µ)).

We are interested in its kernel H1♯ (G, π

p,µ, Lp(X,µ)) := ker η which corresponds

to cocycles in Lp(X,µ) that are formal coboundaries. Note that both cohomol-ogy groups H1(G, πp,µ, Lp(X,µ)) and H

1♯ (G, π

p,µ, Lp(X,µ)) do not depend on the

choice of µ in the class [µ] because the map f 7→(µν

)1/pf intertwines πp,µ and πp,ν

for ν ∈ [µ].We were unable to determine if the set p > 0 | H1(G, πp,µ, Lp(X,µ)) = 0 is

an interval in general. However, we have the following result.

Theorem 5.4. Let 1 ≤ p < q and G → Aut(X, [µ]) ⋉ L0(X,µ,T) as above. IfH1

♯ (G, πq,µ, Lq(X,µ)) = 0 then H1

♯ (G, πp,µ, Lp(X,µ)) = 0.

Proof. For f ∈ L0(X,µ), denote ∂pf(g) = πp,µ(g)f − f . By assumption, for allf ∈ L0(X,µ), we have

(5.3) ∀g ∈ G, ∂qf(g) ∈ Lq(X,µ) =⇒ ∃h ∈ Lq(X,µ), ∂qf = ∂qh.

We want to prove that for all f ∈ L0(X,µ), we also have

(5.4) ∀g ∈ G, ∂pf(g) ∈ Lp(X,µ) =⇒ ∃h ∈ Lp(X,µ), ∂pf = ∂ph.

In order to prove this, we use the properties of the Mazur map. Indeed, it isobvious from the definition of πq,µ that Mp,q establishes a continuous (nonlinear)

bijection between the spaces L0(X,µ)πp,µ(G) and L0(X,µ)

πq,µ(G), with inverseMq,p.Moreover, the map Mp,q sends Lp(X,µ) onto Lq(X,µ) and it is G-equivariant, inthe sense that it intertwines πp,µ and πq,µ.

Let f ∈ L0(X,µ) such that ∂pf(g) ∈ Lp for every g ∈ G. Denote f1 = Mp,q(f).By Lemma 5.2, ∂qf1(g) = Mp,q(π

p,µ(g)f) − Mp,q(f) belongs to Lq. By our as-

sumption, there exists h1 ∈ L0(X,µ)πq,µ(G) such that f1 − h1 ∈ Lq. The set

X0 = x ∈ X, |h1(x)| = 0 is G-invariant, and the restriction of f1 to X0 be-longs to Lq. Equivalently the restriction of f to X0 belongs to Lp. So replac-ing X by X \ X0, we can assume that h1 is almost everywhere non-zero. Sinceh1 is invariant under the representation πq,µ, this easily implies that |h1|

qµ isan invariant measure and that the cocycle in L0(X,µ,T) is the coboundary ofsgn(h1) = h1/|h1| ∈ L0(X,µ,T). So without loss of generality we can assume thatthe homomorphism G → Aut(X, [µ]) ⋉ L0(X,µ,T) takes its values in Aut(X,µ).Denote by B the σ-algebra of G-invariant sets. Then L0(X,µ)

πq,µ(G) coincides withL0(X,B, µ). The theorem is now just a reformulation of Lemma 5.3.

Question 5.5. Is Theorem 5.4 still true if we replace H1♯ (G, π

p,µ, Lp(X,µ)) by

H1(G, πp,µ, Lp(X,µ))?


We observe that it is not true in general that

H1♯ (G, π

p,µ, Lp(X,µ)) = H1(G, πp,µ, Lp(X,µ)).

For example, let ρ : G→ O(H) be an orthogonal representation and c ∈ Z1(G, ρ,H).Let σ be the Gaussian action associated to ρ and c the cocycle for σ associated toc as in the proof of Proposition 3.1. Then one can show that the cocycle c is nevera formal coboundary, unless c itself is a coboundary.

The following proposition also emphasizes the difference between H1♯ and H1.

Proposition 5.6. Let σ : G y (X,µ) be a probability measure preserving action.Fix p ≥ 1 and let c ∈ Z1(G, π, Lp(X,µ)) where π is the Koopman representationof σ. Suppose that c(g) = σg(f) − f for all g ∈ G and some f ∈ L0(X,µ). Then

c ∈ B1(G, π, Lp(X,µ)).In particular, if π has spectral gap, then H1

♯ (G, πp,µ, Lp(X,µ)) = 0.

Proof. Define a sequence of functions fn = max(−n,min(f, n)). Let cn(g) =σg(fn)− fn. Observe that cn ∈ B1(G, π, Lp(X,µ)) for all n because fn is bounded.Since fn converges in measure to f when n → ∞, we have that cn(g) converges inmeasure to c(g) when n→ ∞. Observe also that |cn(g)| ≤ |c(g)| for all g ∈ G. Thus,by the dominated convergence theorem, we know that cn(g) → c(g) in Lp(X,µ)for all g ∈ G. This proves that the sequence of coboundaries cn converges to c inZ1(G, π, Lp(X,µ)), as we wanted.

6. Harmonic cocycles and state preserving actions

Let E be a uniformly convex Banach space and let π be a continuous representa-tion of a locally compact group G on E. Then we can decompose E as a π-invariantdirect sum E = Eπ ⊕E

π where Eπ is the subspace of π-invariant vectors and Eπ isits natural complement (defined as the orthogonal of (E∗)π

∗

where π∗ is the dualrepresentation of π on E∗), see [3]. We say that π has spectral gap if π|Eπ

hasno almost invariant vectors. By [14, Theorem 1.1], this is equivalent to the exis-tence of a symmetric compactly supported probability measure µ on G such that‖π(µ)|Eπ

‖ < 1.

Lemma 6.1. Let G be a locally compact group and let π be a representation of Gon a uniformly convex Banach space E that has spectral gap. Let µ be a symmetriccompactly supported probability measure µ on G such that ‖π(µ)|Eπ

‖ < 1. Thenevery cohomology class in H1(G, π,E) admits a unique µ-harmonic representent.

Proof. By decomposing E = Eπ⊕Eπ, we can reduce the problem to the case where

π is either trivial or has no invariant vectors.Assume first that π is trivial. ThenB1(G, π,E) = 0 and an element of Z1(G, π,E)

is just a group homomorphism from G to E. Thus, since µ is symmetric, every el-ement of Z1(G, π,E) is µ-harmonic.

Now, assume that π has no invariant vectors, i.e. E = Eπ. Take c ∈ Z1(G, π,E).Let α be the affine isometric action of G on E associated to c. Then the affinemap α(µ) =

∫g∈G αg dµ(g) is k-Lipschitz with k = ‖π(µ)‖ < 1. Therefore, α(µ)

has a fixed point ξ ∈ E for some ξ ∈ E. Let c′(g) = c(g) + π(g)ξ − ξ. Then we get∫G c

′(g) dµ(g) = α(µ)ξ − ξ = 0. We conclude that c′ is a µ-harmonic represententof the cohomology class of c. For the uniqueness part, observe that if we have acoboundary b(g) = π(g)ξ − ξ that is µ-harmonic, then π(µ)ξ = ξ, hence ξ = 0because ‖π(µ)‖ < 1.


Lemma 6.2. Let G be a compactly generated locally compact group and let π1, π2 betwo representations of G on two strictly convex Banach spaces E1 and E2. Supposemoreover that E1 is uniformly convex and that π1 has spectral gap. Then everyinjective continuous G-equivariant linear map ψ : E1 → E2 induces an injectivemap ψ∗ : H1(G, π1, E1) → H1(G, π2, E2).

Proof. Since π1 has spectral gap and E1 is uniformly convex, we can find a sym-metric compactly supported probability measure µ on G such that ‖π1(µ)Eπ1

‖ < 1.Since G is compactly generated, we can assume that the support of µ generates G.

Take ω ∈ H1(G, π1, E1). By Lemma 6.1 ω admits a µ-harmonic represententc ∈ Z1(G, π1, E1). Then ψ∗ω ∈ H1(G, π1, E1) is represented by the cocycle c′ :g 7→ ψ(c(g)). Note that c′ is still µ-harmonic. Suppose that ψ∗ω = 0. This meansthat c′ is a coboundary, i.e. c′(g) = π2(g)ξ − ξ for some ξ ∈ E2 and all g ∈ G.Since c′ is µ-harmonic, we have

∫G π2(g)ξ dµ(g) = ξ. Since E2 is stricly convex,

this implies that π2(g)ξ = ξ for all g in the support of µ hence for all g ∈ G. Weconclude that c′ = 0, hence c = 0 because ψ is injective.

Remark 6.3. The previous lemma applies for example when σ : G y (X,µ) is acontinuous probability measure preserving action of a compactly generated locallycompact group, (πi, Ei) = (σpi,µ, Lpi(X,µ)) for ∞ > p1 ≥ p2 > 1 and ψ is theinclusion map Lp1 → Lp2 . The lemma shows that

∀1 < p1 < p2 <∞, H1(G, σp2,µ) = 0 =⇒ H1(G, σp1,µ) = 0.

Indeed, the assumption that H1(G, σp2,µ) = 0 implies that σp2,µ has spectral gap,which implies that σp1,µ has spectral gap, see [3]. In particular, if G is a locallycompact property (T) group, H1(G, σp,µ) = 0 for every p ∈ [1,∞). This wasalready known when G is discrete [24], or G admits a finite Kazhdan set [10].

More generally, the lemma applies to the actions on non-commutative Lp spacesassociated to state-preserving actions on von Neumann algebras, and also for actionson Schatten p-classes. We start with the latter as it is more elementary.

For a Hilbert spaceH , we denote by Sp(H) the Schatten p-class, that is the space

of operators on H such that ‖T ‖p := (Tr(|T |p))1p < ∞. We say that a group has

FSp if for every H and every continous isometric representation π : G y Sp(H)),H1(G, π, Sp(H)) = 0.

Theorem 6.4. If G is a σ-compact locally compact group, then the set of 1 < p <∞such that G has FSp is empty if G does not have property (T), and is an intervalcontainting (1, 2] otherwise.

Proof. Assume that G does not have property (T). By Guichardet’s theorem, thereis a unitary representation π on a Hilbert space H and an unbounded cocycleb ∈ Z1(G, π). If ξ ∈ H∗ is a unit vector, then the formula g · T = π(g)T + b(g)⊗ ξdefines an unbounded action by isometries on Sp(H) for every 1 ≤ p ≤ ∞. So Gdoes not have FSp.

If G has property (T), then G is compactly generated, and has FS2 by Delorme’stheorem (S2(H) is a Hilbert space). We have to prove that, if 1 < p < q < ∞are such that G has FSq, then G has FSp. We can assume that p 6= 2. Letπ : G → O(Sp(H)) be an orthogonal representation. By [42], πg is of the formπg(T ) = WgJg(T ) for a unitary Wg and a Jordan automorphism Jg of B(H). Inparticular, the same fomula gives rise to an isometric representation on Sq(H) and


the inclusion Sp(H) ⊂ Sq(G) is equivariant. Moreover both representations havespectral gap, [42], and the Schatten spaces are uniformly convex when 1 < p <∞.We conclude by Lemma 6.2 that H1(G,Sp(H)) → H1(G,Sq(H)) = 0 is injectiveand H1(G,Sp(H)) = 0.

To state our result about state-preserving actions, we need to recall Haagerup’sgeneral definition of non-commutative Lp spaces [19]. We follow the approach in[25, §2.1]. IfM is a von Neumann algebra, the core ofM is the unique (up to uniqueisomorphism) tuple (c(M), τ, θ, ι) of a von Neumann algebra c(M), a normal faithfulsemifinite trace τ on c(M), a continuous homomorphism θ : R∗

+ → Aut(M), anda normal injective ∗-homomorphism ι : M → C(M) satisfying τ θs = e−sτ andx ∈ c(M) | ∀s ∈ R, θs(x) = x = ι(M). In the sequel, we will identify Mwith its image in c(M) and write x instead of ι(x). For example, if φ is a normalfaithful weight on M , the core can be realized as c(M) = M ⋊σφ R, the crossedproduct by the modular automorphism group of φ. In that case, θ is given by thedual action and ι the natural inclusion. For details on the construction and theexistence of a trace τ as required, see [39]. Then Lp(M) is defined as the space of

τ -measurable operators affiliated to c(M) such that θs(h) = e−s/ph for all s ∈ R.By [19, Theorem 1.2 and 1.3], for every normal state ϕ on M , there is a uniqueelement of L1(M), that we also denote ϕ, satisfying

ϕ

(∫

R∗

+

θt(x)dt

t

)= τ(ϕx)

for every nonnegative x ∈ c(M). Moreover, this map extends by linearity to anisomorphism M∗ → L1(M). This allows to define, for 1 ≤ p < ∞ and x ∈ Lp(M),

‖h‖p := ‖|h|p‖1p

M∗

. This turns Lp(M) into a Banach space that is uniformly convex

if 1 < p <∞, and the ‖ · ‖p norms satisfy Holder’s inequality.By the uniqueness of the core, any continuous action by automorphisms of G

on M gives rise to a continuous action by isometries on Lp(M), that we call thep-Koopman representation.

Theorem 6.5. Let G be a locally compact group with property (T). Let σ : G →Aut(M) be an action on a von Neumann algebra M that preserves some faithfulnormal state ϕ on M . Take p ≥ 2 and let πp : G → O(Lp(M)) be the p-Koopmanrepresentation associated to σ. Then H1(G, πp, Lp(M)) = 0.

Proof. First, πp has spectral gap by [32].Define a continuous linear map ψ : Lp(M) → L2(M) by the formula

ψ(xϕ1/p) = xϕ1/2

for all x ∈M . This map is well-defined because Mϕ1/p is dense in Lp(M) and forall x ∈M , we have

‖xϕ1/2‖2 = ‖xϕ1/p · ϕ1/q‖2 ≤ ‖xϕ1/p‖p · ‖ϕ1/q‖q = ‖xϕ1/p‖p

where 1p +

1q = 1

2 . Observe that πp(g)(xϕ1/p) = σ(g)(x)ϕ1/p for all x ∈M , because

σ preserves ϕ, and we have the same formula for p = 2. This implies that ψ isG-equivariant with respect to the Koopman representations of σ. Thus it inducesan injective map from H1(G, πp, Lp(M)) into H1(G, π2, L2(M)) by Lemma 6.2.Since G has property (T), we know that H1(G, π2, L2(M)) = 0. We conclude thatH1(G, πp, Lp(M)) = 0.


7. Stability properties of the constants pG and p′G

We start with some elementary stability properties.

Proposition 7.1. Let G be a locally compact group and let H < G be a closedsubgroup. Then p′H ≤ p′G. If H ⊳ G is a closed normal subgroup, then pG/H ≥ pG.

Proof. Any proper action of G on an Lp space restricts to a proper action of H ,so p′H ≤ p′G. If H is normal, any action of G/H with unbounded orbits on an Lp

space can be seen as an action of G, so pG/H ≥ pG.

Proposition 7.2. Let G1, G2 be two locally compact groups. Then

pG1×G2= min(pG1

, pG2) and p′G1×G2

= max(p′G1, p′G2

).

Proof. The inequalities pG1×G2≤ min(pG1

, pG2) and p′G1×G2

≥ max(p′G1, p′G2

) fol-low from Proposition 7.1.

For the inequality pG1×G2≥ min(pG1

, pG2), observe that if G1 × G2 y Lp has

unbounded orbits, then its restriction to either G1 or G2 also has unbounded orbits.For the inequality p′G1×G2

≤ max(p′G1, p′G2

), observe that given two isometricactions G1 y Lp(Ω1) and G1 y Lp(Ω2), one can construct the action of G1 ×G2

on Lp(Ω1 ∪ Ω2), which is proper whenever both actions were proper.

Proposition 7.3. Let G be a locally compact group and let K ⊳ G be a normalcompact subgroup. Then

pG/K = pG and p′G/K = p′G.

Proof. The only thing to notice is that the space of K-invariant vectors for anisometric representation of G on an Lp-space is isometric to an Lp-space. Thiseither follows from the general form of isometries of Lp-space, or from the classicalresult [40] that the range of a norm 1 projection in B(Lp) is isometric to an Lp-space.

We now investigate the stability of the constants pG and p′G under measureequivalence. For this we need to recall some definitions.

Let G be a locally compact group G with left Haar measure mG. A measurepreserving action of G on (X,µ) is called principal if there is a measure preservingconjugacy between G y (X,µ) and an action of the form G y (G × Ω,mG ⊗ ν)where G acts by translation of the left coordinate and (Ω, ν) is a measure spacewith finite measure. In that case, we can identify Ω with X/G and thus equip X/Gwith the finite measure space structure coming from this identification. The finitemeasure on X/G coming from this identification will be denoted µ/G.

Let G1 and G2 be two locally compact groups. The groups G1, G2 are said tobe measure equivalent if there is a measure equivalence coupling, that is a measurespace (X,µ) equipped with two commuting measure preserving principal actionsof G1 and G2. Observe that we then obtain a measure preserving action of G2 on(X/G1, µ/G1) and of G1 on (X/G2, µ/G2). We will use the notations µ1 = µ/G1

and µ2 = µ/G2.Let p1 : X → G1 be a G1-equivariant map (it always exists because the action

of G1 is principal). Then for every g2 ∈ G, the map

X ∋ ω 7→ p1(g2ω)−1p1(ω) ∈ G1


is G1-invariant. Thus, one can define a cocycle c1 : G2×X/G1 → G1 by the formula

c1(g2, ω1) = p1(g2ω)−1p1(ω)

where ω ∈ X is any representant of ω1 ∈ X/G1. Observe that the cocycle c1depends on the section p1.

Now, suppose that G1 is compactly generated. Take p > 0. We say that themeasure equivalence coupling (X,µ) is Lp-integrable over G1 if there exists a G1-equivariant map p1 : X → G1 such that the associated cocycle c1 defined abovesatisfies the following Lp-integrability condition

∫

X/G1

|c1(g2, ω1)|pG1

dµ1(ω1) <∞ ∀g2 ∈ G2

where | · |G1is the word length with respect to any compact generating set of G1

(the integrability condition does not depend on the choice).We say that two compactly generated groups G1 and G2 are Lp-measure equiva-

lent if there exists a measure equivalence coupling that is Lp-integrable over both G1

and G2. Thanks to the construction in [16, Theorem 3.3], we know that Lp-measureequivalence is more general than Lp-orbit equivalence.

More details on measure equivalence of locally compact groups and Lp-measureequivalence can be found in [4, Section 1.2] and the references therein.

Theorem 7.4. Let G1 and G2 be two locally compact groups. Let 1 ≤ p < ∞.Suppose that G1 is compactly generated and that there exists a measure equivalencecoupling between G1 and G2 that is Lp-integrable over G1.

(i) If G1 admits an affine isometric action without fixed points on some Lp-spacethen so does G2.

(ii) If G1 admits a proper affine isometric action on some Lp-space then so doesG2.

Proof. We use the standard tool of induction of Banach-space actions as in [3,Section 8.b]. Let (X,µ) be measure equivalence coupling that is Lp-integrable overG1. Let p1 and c1 be as in the definition.

Let α : G1 y E be an action by affine isometries of G1 on a Banach space E. LetL0(X,µ,E)G1 be the set of all G1-equivariant Bochner-measurable maps from X toE. Note that L0(X,µ,E)G1 is only an affine subspace of L0(X,µ,E) (it does notcontain 0). Observe that we have a natural affine action β : G2 y L0(X,µ,E)G1

given by βg2(f)(x) = f(g−12 x) for all x ∈ X . If f, h ∈ L0(X,µ,E)G1 , then the map

‖f − h‖E : X → R+ is G1-invariant and thus we can define

‖f − h‖p,G1=

(∫

X/G1

‖f − h‖pE dµ1

)1/p

.

We clearly have ‖βg2(f)− βg2(h)‖ = ‖f − h‖ for all g2 ∈ G2 (because the action ofG2 on X/G1 preserves the measure).

Take x ∈ E and define f0 ∈ L0(X,µ,E)G1 by the formula f0(ω) = p1(ω) · x.Now, define an affine subspace

F = f ∈ L0(X,µ,E)G1 | ‖f − f0‖p,G1<∞.

Note that the space F is isometric to Lp(X/G1, µ1, E). In particular, if E is anLp-space than F is also an Lp-space.


Now, a key observation is that βg2(f0) ∈ F for all g2 ∈ G2. Indeed, we have

‖βg2(f0)− f0‖p,G1=

(∫

X/G1

‖c1(g2, ω1) · x− x‖pE dµ1(ω1)

)1/p

and this integral is finite because of the Lp-integrability of c1 and the fact that thefunction g1 7→ ‖g1 · x− x‖E grows sublinearly: there exists a constant C > 0 suchthat ‖g1 · x− x‖E ≤ C|g1|G1

for all g1 ∈ G1.Since βg2(f0) ∈ F for all g2 ∈ G2, it follows that F is globally invariant under

the action β. We conclude that the action β : G2 y L0(X,µ,E)G1 restricts to anaffine isometric action, still denoted β, of G2 on F . We call the affine isometricaction β : G2 y F the induced action of α (note that it depends on the choice ofp1).

To prove item (i), it is enough to check that if β has a fixed point then α alsohas a fixed point. Let f ∈ F be a fixed point for β. This means that f : X → Eis a G1-equivariant measurable map that is also G2-invariant. Thus we can viewf as a G1-equivariant map from X/G2 to E. Since X/G2 admits a G1-invariantprobability measure µ2, we can push it forward by f to obtain a G1-invariantprobability measure on E. We conclude by [3, Lemma 2.14] that α has a fixedpoint.

To prove item (ii), it is enough to check that if α is proper then β is also proper.We have

‖βg2(f0)− f0‖p,G1=

(∫

X/G1

‖c1(g2, ω1) · x− x‖pE dµ1(ω1)

) 1p

.

Take R > 0. Since α is proper, we can choose a compact subset K1 ⊂ G1 such that‖g1x− x‖ ≥ R for all g1 ∈ G1 \K1. Observe that

µ1ω1 ∈ X/G1 | c(g2, ω1) ∈ K1 =

1

mG1(K1)

µω ∈ X | p1(ω) ∈ K1 and p1(g2ω)−1p1(ω) ∈ K1 ≤

1

mG1(K1)

µω ∈ X | p1(ω) ∈ K1 and p1(g2ω) ∈ K−11 K1

Since µ(p−11 (K1 ∪K

−11 K1)) < +∞, there exists a compact subset K2 ⊂ G2 such

that µ(p−11 (K1 ∪K

−11 K1) \ p

−12 (K2)) ≤

14mG1

(K1). Take g2 ∈ G2 \K2K−12 . Then,

we cannot have p2(ω) ∈ K2 p2(g2ω) ∈ K2 at the same time. Therefore, we get

µω ∈ X | p1(ω) ∈ K1 and p1(g2ω) ∈ K−11 K1 ≤ 2µ(p−1

1 (K1 ∪K−11 K1) \ p

−12 (K2))

which yields

µ1ω1 ∈ X/G1 | c(g2, ω1) ∈ K1 ≤1

2.

By the choice of K1, this means that

µ1ω1 ∈ X/G1 | ‖c(g2, ω1) · x− x‖E ≥ R ≥1

2

hence

‖βg2(f0)− f0‖p,G1≥ 2−1/pR.

This holds for all g2 ∈ G2 outside of the compact subset K2K−12 . We conclude that

β is proper.


When G is a locally compact group and Γ < G is a lattice, then (G,mG) ismeasure equivalence coupling between G and Γ. Therefore, Theorem 7.4 covers inparticular the following corollary which is already implicit in [3].

Corollary 7.5 ([3]). Let G be a locally compact compactly generated group and letΓ < G be a lattice. Then pΓ ≤ pG and p′Γ ≤ p′G. If Γ is Lp-integrable for somep ≥ pΓ (resp. p ≥ p′Γ) then pG = pΓ (resp. p′G = p′Γ).

8. Semisimple Lie groups and their lattices

The following result is a combination of [11, Corollary 1.4] and [41].

Theorem 8.1. Let G be a non-compact connected simple Lie group with finitecenter. Let α : G y E be an affine isometric action on some Lp-space E withp > 1. Then α is either proper or has a fixed point. In particular, pG = p′G.

Proof. Write αg(x) = π(g)x + c(g) where π is the linear part of E and c ∈Z1(G, π,E). Decompose E = Eπ ⊕ E0 where Eπ is the subspace of invariantvectors of π and E0 = kerP where P : E → Eπ is the unique π-invariant projec-tion. The map g 7→ Pc(g) is a continuous homomorphism from G into an abeliangroup. Since G is a connected simple Lie group, it must vanish. Thus c(g) ∈ E0 forall g ∈ G and we can restrict α to an affine isometric action on E0. By the Banachspace version of the Howe-Moore property (due to Veech [41], see also [3, Appendix9]), we know that π|E0

is a C0-representation. We conclude by [11, Corollary 1.4],that α is either proper or has a fixed point.

Let g be a real semisimple Lie Algebra. Let Ad(g) denote the adjoint group ofg, that is the connected component of the identity in Aut(g), and write pg = pAd(g)

and p′g = p′Ad(g). By [21, Section I.14], Ad(g) is isomorphic to G/Z(G) for every

connected Lie group with Lie algebra isomorphic to g.

Example 8.2. We can compute pg for most semisimple Lie algebras:

• If g is a semisimple Lie algebra, then it is a direct sum of simple Lie algebrasg = ⊕igi and Ad(g) =

∏iAd(g). Therefore by Proposition 7.2,

pg = minipgi

and p′g = maxip′gi.

• If g is a simple Lie algebra with real rank 0, then pg = ∞ and p′g = 0because Ad(g) is compact, so all its continuous actions are both proper andbounded.

• If g is a simple Lie algebra with real rank ≥ 2, then pg = p′g = ∞ by [3,Theorem B].

• If g is a simple Lie algebra with real rank 1, then pg = p′g < +∞ and wehave

(8.1) pg = p′g

= 0 if g ≃ so(n, 1), n ≥ 2

= 0 if g ≃ su(n, 1), n ≥ 2

∈ (2, 4n+ 2] if g ≃ sp(n, 1), n ≥ 2

∈ (2, 22] if g ≃ f−204 .

Indeed, the equality pg = p′g follows from Theorem 8.1. The equality pg = 0for g ≃ so(n, 1), su(n, 1) comes from the fact that the groups SO(n, 1) andSU(n, 1) do not have property (T). The remaining cases have property (T),


so pg > 2, and it follows from Pansu’s result [33] (see Theorem A.2) thatpsp(n,1) ≤ 4n+ 2 and pf−20

4

≤ 22. That the algebras appearing in (8.1) are

all the rank one real simple Lie algebras is Cartan’s classification [6], seealso [21, Theorem 6.105]. Observe that the list in [21] is apparently slightlydifferent as it includes su(1, 1), sl2(C), sl2(H) but excludes so(n, 1) forn = 2, 3, 5. But this is the same list thanks to the expectional isomorphismssu(1, 1) ≃ so(2, 1), sl(2,C) ≃ so(3, 1) and sl(2,H) ≃ so(5, 1) [21, Formula(6.110)].

It would be very interesting to compute the exact value of pG for G = Sp(n, 1)or F−20

4 . From the preceding and Theorem 8.3, this would allow to compute pG forevery connected semisimple Lie group and any lattice in it. It is not even knownwhether limn pSp(n, 1) = ∞.

Theorem 8.3. Let G be a connected semisimple Lie group with Lie algebra g. Then

(8.2) pG = pg and p′G = p′g.

Proof. The Theorem is clear if G is compact, so we can and will assume that Gis not compact. We can and will even assume (replacing G by its quotient by itscompact factors) that G does not have compact factors.

Consider first the case when G is a connected simple Lie group. If G has finitecenter Z(G), then G/Z(G) is isomorphic to Ad(g) so (8.2) follows from Proposition7.3. If Z(G) is infinite, it follows from [18] that either g is isomorphic to su(n, 1) forsome n ≥ 1 and G is the universal covering group of SU(n, 1), or g has rank ≥ 2.In the first case, G has the Haagerup property [8, Theorem 4.0.1] so pG = p′G = 0.If g has rang ≥ 2, it is also known that pG = p′G = ∞, see [22, Corollary 1.2]. Weobtain (8.2) in both cases.

Consider now the general case when G is semisimple. Denote by G the universal

cover of G, and q : G → G and q : G → G/Z(G) ≃ Ad(G) the quotient maps.Let g = ⊕n

i=1gi be the decomposition of the Lie algebra of G as a direct sum

of simple Lie algebras, and Gi the simply connected Lie group with Lie algebra

gi. We can identify G with∏n

i=1 Gi, and G/Z(G) with∏

iAd(gi). We deducefrom the case of simple Lie groups already covered and from Proposition 7.2 thatpG = pg = pG/Z(G). On the other hand, Proposition 7.1 gives pG ≥ pG ≥ pG/Z(G),

so we deduce pG = pg. Moreover, the image Gi := q(Gi) is a closed subgroup ofG with Lie algebra gi, so we have p′g = maxi p

′gi

= maxi p′Gi

≤ p′G. So it remainsto show that p′G ≤ p′g. This is obvious if p′g = ∞. When p′g = 0, the gi areisomorphic to su(n, 1) or so(n, 1), so we conclude by [8, Theorem 4.0.1] that G hasthe Haagerup property, or equivalently p′G = 0. It remains to consider the casewhen 2 < p′g < ∞. Let gH ⊂ g denote the sum of the gi with p′gi

= 0, and gT

the sum of the rest, that is the subalgebras isomorphic to sp(n, 1) or f−204 (H for

Haagerup and T for property (T)). Denote GH and GT the corresponding analyticsubgroups of G. As explained above, we know from [18] that Z(GT ) is finite, so byProposition 7.3 we can assume that Z(GT ) = 1. This means that G is isomorphicto GH ×Ad(gT ), so p

′G = max(p′GH

, p′gT). This is equal to p′g as we already justified

that p′GH= 0.

We need the following classical fact. It is for example stated for higher-rankgroups in [35, Lemma 5.2], but the proof remains valid without any rank assump-tion.


Lemma 8.4. Let G be a connected semisimple Lie group, and denote q : G →G/Z(G) the quotient map. If Γ < G is a lattice, then q(Γ) < G/Z(G) is a latticeand Γ has finite index in ΓZ(G).

Theorem 8.5. Let G be a connected semisimple Lie group and Γ < G be a lattice.Then Γ is Lp-integrable for all p < pg. Moreover, if q(Γ) < G/Z(G) is an irreduciblelattice, then Γ is Lp-integrable for all p < p′g.

Proof. It is enough to prove the Theorem when G has trivial center. Indeed, in thenotation of Lemma 8.4, by the argument given in [22, Proposition 7.1], a latticeΓ ⊂ G is Lp-integrable if the lattice q(Γ) < G/Z(G) is Lp-integrable. The essentialingredient is the fact that the central extension G→ Z(G) is given by a bounded 2-cocycle. For simple Lie groups, one can justify this by refering to [18] (see also [38]).The general case of semisimple Lie groups follows by decomposing the universalcover into simple parts.

So assume that G has trivial center. We can assume that G has no compactfactor. By decomposing into irreducible factors, we can assume that Γ is irreducible.When G has higher rank, we have more than the Theorem : Γ is Lp-integrable forevery p < ∞. This is what the proof of the L2-integrability in [36, §2] actuallyshows. See also the proof of [35, Lemma 5.6] for a simpler argument, showing thatthe Lp-integrability holds for every quasi-isometrically embedded lattice in a Liegroup. When G has rank 1, the only cases to consider are g = sp(n, 1), n ≥ 2 andf−204 as the other rank one simple Lie algebras satisfy pg = p′g = 0. By Theorem

A.5, we know that every lattice in Sp(n, 1) or F−204 is Lp-integrable for all p < 4n+2

and p < 22 respectively. Since we also know that pSp(n,1) ≤ 4n+2 and pF−20

4

≤ 22,

thanks to Theorem A.2, this ends the proof.

Theorem 8.6. Let G be a connected semisimple Lie group with Lie algebra g andlet Γ < G be a lattice. Then

(8.3) pΓ = pG = pg and p′Γ = p′G = p′g.

Proof. The Theorem is clear if G is compact, so we can and will assume that Gis not compact. We can and will even assume (replacing G by its quotient by its

compact factors) that G does not have compact factors. Denote by G the universal

cover of G, and q : G → G and q : G → G/Z(G) ≃ Ad(G) the quotient maps. Letg = ⊕n

i=1gi be the decomposition of the Lie algebra of G as a direct sum of simpleLie algebras.

Let Λ := q(Γ) < G/Z(G). When Λ is an irreducible lattice, (8.3) follows fromTheorem 8.5. In general, we reduce to the irreducible case. By Lemma 8.4, Γ is alattice in G/Z(G), and Γ has finite index in ΓZ(G) = q−1(Λ). So replacing Γ by

ΓZ(G), we can assume that Z(G) ⊂ Γ and define Γ = q−1(Γ) = (q q)−1(Λ) < G.Decompose Λ into irreducible components. To do so, consider a maximal partition(A1, . . . , Am) with the property that

∏k(Λk := Λ∩Ad(⊕i∈Ak

gi)) is of finite indexin Λ. Then every Λk is an irreducible lattice in Ad(⊕i∈Ak

gi), and replacing Γ be afinite index subgroup, we can assume that Λ =

∏k Λk.

The lattice Γ is not necessarily a product, but this is true in the universal cover:

if Λk denotes the lift of Λk in the universal cover GAkof Ad(⊕i∈Ak

gi), then Γ =∏k Λk. It follows from Proposition 7.2 and from the case of irreducible lattices

that pΛ = mink pΛk= pg and that pΓ = mink pΛk

= pg, so we deduce pΓ = pg by


Proposition 7.1. For the invariant p′Γ, the inequality p′Γ ≤ p′G = p′g is by Proposition

7.1. For the converse, observe that the discrete subgroup q(Λk) < Γ is a lattice in

the quotient of GAkby GAk

∩ker q, so we obtain from the case of irreducible latticesand Proposition 7.1 that maxi∈Ak

pgi= pq(Λk)

≤ p′Γ. Taking the maximum over k

we obtain p′g ≤ p′Γ. This concludes the proof of (8.3) and of the Theorem.

Appendix A. Rank one symmetric spaces and Lp-integrability.

By Elie Cartan’s classification (see Example 8.2 for references), every connectedsimple Lie group of real rank one is locally isomorphic to the isometry group of oneof the following symmetric spaces of rank one :

• The real hyperbolic space RHn for n ≥ 2, its isometry group is SO(n, 1).• The complex hyperbolic spaceCHn for n ≥ 2, its isometry group is SU(n, 1).• The quaternionic hyperbolic space HHn for n ≥ 2, its isometry group isSp(n, 1).

• The octonionic hyperbolic plane OH2, its isometry group is the exceptionalLie group F−20

4 .

Let K = R,C,H and n ≥ 2 or K = O and n = 2. Let d = dimR K ∈ 1, 2, 4, 8.We give a description of the hyperbolic space KHn that will be convenient for ourstudy. Let 〈·, ·〉 : Kn ×Kn → K be the standard sesquilinear form given by

〈x, y〉 = x1y1 + x1y1 + x2y2 + · · ·+ xnyn.

Define a nilpotent Lie group N formally given by N = Kn−1 ×ℑ(K) but with thegroup law

(ξ1, r1) · (ξ2, r2) = (ξ1 + ξ2, r1 + r2 + ℑ〈ξ2, ξ1〉)

where 〈·, ·〉 : Kn−1 ×Kn−1 → K is the standard sesquilinear form and ℑ denotesthe imaginary part. We define an action of R by group automorphisms on N withthe formula

t · (ξ, r) = (etξ, e2tr), t ∈ R, (ξ, r) ∈ N.

We let P = N ⋊R be the corresponding semi-direct product.

Theorem A.1 (Heintze [20]). The hyperbolic space KHn is isometric to the groupP equipped with the unique left invariant riemannian metric on P that is given atthe origin (0, 0, 0) ∈ P by

ds2 =1

2‖dξ‖2 +

1

4|dr|2 + dt2.

Thanks to the description of X = KHn as a group with a left invariant metric,we can directly apply [33, Proposition 24] to get the following result.

Theorem A.2 (Pansu [33]). Let G be the isometry group of X. Then the coho-mology of G with values in Lp(G) (for the left regular representation) is nonzerofor all p > d(n+ 1)− 2 where d = dimR K.

Proof. We have P = N ⋊R and the action of R on N is generated by a derivationon N = Kn−1 × ℑ(K) that has eigenvalue 1 on Kn−1 and eigenvalue 2 on ℑ(K).It follows from [33, Proposition 24] that the Lp cohomology of X is nonzero for all

p > dimRKn−1 + 2dimRℑ(K) = d(n− 1) + 2(d− 1) = d(n+ 1)− 2.

Finally, the Lp-cohomology of G with values in the left regular representation canbe identified with the Lp-cohomology of X , see [11, Section 4].


Let X = KHn. Let ∂X denote its boundary. Fix ω ∈ ∂X . We define thegeodesic flow (πt)t∈R with source ω as follows : for every x ∈ X , the map t 7→ πt(x)is the unit speed geodesic with π0(x) = x and limt→−∞ πt(x) = ω. We define anequivalence relation on X by saying that x, y ∈ X are equivalent if

limt→−∞

d(πt(x), πt(y)) = 0.

The equivalence classes of this equivalence relation are called horospheres centeredat ω.

It is easy to see from the definition of P and the left invariance of its metricthat t 7→ (0, 0, t) ∈ P is a unit speed geodesic path. Fix an isometry τ : P → Xsuch that the geodesic line t 7→ τ(0, 0, t) converges to ω (this is possible becausethe isometry group of X acts transitively on ∂X). We observe the following factswhich can be deduced easily from the left invariance of the metric of P :

(1) We have πt(τ(ξ, r, s)) = τ(ξ, r, s− t) for all (ξ, r, s) ∈ P .(2) For every s ∈ R, the set Hs = τ(ξ, r, s) | (ξ, r) ∈ N is a horosphere

centered at ω.

Of course the map πt restricts to a bijection from Hs onto Hs−t for all s, t ∈ R butit is not an isometry. Instead we have the following property.

Proposition A.3. Fix ω ∈ ∂X and let (πt)t∈R be the associated geodesic flow.Take x ∈ X and let H be the horosphere centered at ω and containing x. Thenthere exists an orthogonal decomposition of the tangent space

THx = V1 ⊕ V2

such that ‖( dπt)x(v)‖ = ekt‖v‖ for all v ∈ Vk, k = 1, 2 and all t ∈ R. The spaceV1 has dimension d(n− 1) and V2 has dimension d− 1.

Proof. We may choose the isometry τ : P → X such that x = τ(0, 0, 0). DefineV1 to be the subspace of THx tangent to τ(ξ, 0, 0) | ξ ∈ Kn−1 and V2 to be thesubspace tangent to τ(0, r, 0) | r ∈ ℑ(K). Then V1 and V2 are orthogonal by thedefinition of the metric on P . By the left invariance of the metric on P , the map

αt : P ∋ τ(ξ, r, s) 7→ τ((0, 0, t) · (ξ, r, s)) = τ(etξ, e2tr, s+ t) ∈ P

is an isometry. Observe that for all (ξ, r, s) ∈ P , we have

(αt πt)(ξ, r, s) = (etξ, e2tr, s).

This shows that d(αt πt)x(v) = ektv for all v ∈ Vk, k = 1, 2. Since αt is anisometry, the conlusion follows.

For every horosphere H ⊂ X , the distribution x 7→ V1(x) ⊂ THx, defined bythe proposition above, is called the Carnot distribution of H . It has codimensiond− 1. A key fact is that any two points x, y ∈ H can be joined by a smooth pathin H that remains tangent to the Carnot distribution at every point. The Carnotdistance dC(x, y) is then defined as the infinimum of the lengths of all such paths.It follows from Proposition A.3 that dC(πt(x), πt(y)) = etdC(x, y) for all x, y ∈ H .

Lemma A.4. Fix ω ∈ ∂X and let (πt)t∈R be the associated geodesic flow. LetH ⊂ X be a horosphere and let D ⊂ H be a bounded open subset.

(1) There exists a constant C > 0 such that the diameter of πt(D) (in the pathmetric of πt(H)) is bounded by C exp(t) for all t ∈ R.


(2) There exists a constant C′ > 0 such that the volume of πt(D) (with respectto the volume form of πt(H)) is equal to C′ exp((d(n + 1) − 2)t) for allt ∈ R.

Proof. (1) For the Carnot distance, the diameter of πt(D) is equal to the diameterof D times exp(t). But by definition, the path distance is always less than theCarnot distance. Therefore the diameter of πt(D) for the the path distance ofπt(H) is bounded by C exp(t) for some C > 0.

(2) Fix t ∈ R. Let m0 be the volume form of H and m be the volume formof πt(H). Then by Proposition A.3, we have (πt)∗m0 = etd(n−1)+2t(d−1)m. Thisimples

m(πt(D)) = e(d(n+1)−2)tm0(D).

The proof of the following theorem is due to Rich Schwartz. The proof can befound in the appendix of [37]. We briefly argue that the proof of Schwartz can beadapted to include the exceptional octonionic case and the case p 6= 2 in a unifiedway.

Theorem A.5 (Schwartz). Let Γ < G = Isom(X) be any lattice. Then it isLp-integrable for all p < d(n+ 1)− 2.

Proof. We follow the notations and the proof of [37, Theorem 3.7] given in the ap-pendix. We have to show that the function f defined there is Lp-integrable on F 0

and as explained in Lemma 2.1, for this it is enough to show that∑

m∈Ng(m)pv(m) <

+∞. Recall that v(m) is the volume of

B(m) =⋃

m≤t≤m+1

π−t(F0).

By Lemma A.4, we know that

v(m) = C′

∫ m+1

m

exp (−(d(n+ 1)− 2)t) dt = A′ exp (−(d(n+ 1)− 2)m)

for some constant A′ > 0. Fix any x ∈ H0 and let αt be the isometry defined inProposition A.3, with αt(x) = π−t(x) for all t ∈ R. Then we can find a boundedopen ball D ⊂ H0 around x such that the unit tubular neighborhood B1(m) iscontained in ⋃

m−1≤t≤m+2

αt(D)

for all m ∈ N. Thus π(B1(m)) is contained⋃

m−1≤t≤m+2

πt(αt(D)) =⋃

m−1≤t≤m+2

αt(πt(D))

Thanks to Lemma A.3 and since αt is an isometry, the diameter of αt(πt(D)) isbounded by C exp(t) for some C > 0. Since x ∈ αt(πt(D)) for all t, it followsthat the diameter of π(B1(m)) is bounded by A exp(m) for some constant A > 0.Finally, we conclude that

∑

m∈N

g(m)pv(m) ≤ ApA′ exp(pm) exp(−(d(n+ 1)− 2)m) < +∞

whenever p < d(n+ 1)− 2.


References

[1] Yuki Arano, Yusuke Isono, and Amine Marrakchi. Ergodic theory of affine isometric actionson hilbert spaces, 2019.

[2] U. Bader, T. Gelander, and N. Monod. A fixed point theorem for L1 spaces. Invent. Math.,189(1):143–148, 2012.

[3] Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod. Property (T) and rigidityfor actions on Banach spaces. Acta Math., 198(1):57–105, 2007.

[4] Uri Bader, Alex Furman, and Roman Sauer. Integrable measure equivalence and rigidity ofhyperbolic lattices. Invent. Math., 194(2):313–379, 2013.

[5] Marc Bourdon and Herve Pajot. Cohomologie lp et espaces de Besov. J. Reine Angew. Math.,558:85–108, 2003.

[6] Elie Cartan. Les groupes reels simples, finis et continus. Ann. Sci. Ecole Norm. Sup. (3),31:263–355, 1914.

[7] Indira Chatterji, Cornelia Drutu, and Frederic Haglund. Kazhdan and Haagerup propertiesfrom the median viewpoint. Adv. Math., 225(2):882–921, 2010.

[8] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette. Groupswith the Haagerup property, volume 197 of Progress in Mathematics. Birkhauser Verlag,Basel, 2001. Gromov’s a-T-menability.

[9] Alan Czuron. Property Fℓq implies property Fℓp for 1 < p < q < ∞. Adv. Math., 307:715–726, 2017.

[10] Alan Czuron and Mehrdad Kalantar. On fixed point property for lp-representations of kazh-dan groups. arXiv:2007.15168, 2020.

[11] Yves De Cornulier, Romain Tessera, and Alain Valette. Isometric group actions on Banachspaces and representations vanishing at infinity. Transform. Groups, 13(1):125–147, 2008.

[12] Cornelia Drutu and Michael Kapovich. Geometric group theory, volume 63 of AmericanMathematical Society Colloquium Publications. American Mathematical Society, Providence,RI, 2018. With an appendix by Bogdan Nica.

[13] Cornelia Drutu and John M. Mackay. Random groups, random graphs and eigenvalues ofp-Laplacians. Adv. Math., 341:188–254, 2019.

[14] Cornelia Drutu and Piotr W. Nowak. Kazhdan projections, random walks and ergodic theo-rems. J. Reine Angew. Math., 754:49–86, 2019.

[15] David Fisher and Gregory Margulis. Almost isometric actions, property (T), and local rigidity.Invent. Math., 162(1):19–80, 2005.

[16] Alex Furman. Orbit equivalence rigidity. Ann. of Math. (2), 150(3):1083–1108, 1999.[17] M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2

(Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1–295. CambridgeUniv. Press, Cambridge, 1993.

[18] A. Guichardet and D. Wigner. Sur la cohomologie reelle des groupes de Lie simples reels.

Ann. Sci. Ecole Norm. Sup. (4), 11(2):277–292, 1978.[19] Uffe Haagerup. Lp-spaces associated with an arbitrary von Neumann algebra. In Algebres

d’operateurs et leurs applications en physique mathematique (Proc. Colloq., Marseille, 1977),volume 274 of Colloq. Internat. CNRS, pages 175–184. CNRS, Paris, 1979.

[20] Ernst Heintze. On homogeneous manifolds of negative curvature. Mathematische Annalen,211(1):23–34, 1974.

[21] Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathe-matics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.

[22] Tim de Laat and Mikael de la Salle. Strong property (T) for higher-rank simple Lie groups.Proc. Lond. Math. Soc. (3), 111(4):936–966, 2015.

[23] Tim de Laat and Mikael de la Salle. Banach space actions and l2-spectral gap. Anal. PDE,in press, 2020.

[24] Omer Lavy and Baptiste Olivier. Fixed-point spectrum for group actions by affine isometrieson Lp-spaces. Ann. Inst. Fourier (Grenoble), to appear, 2020.

[25] Amine Marrakchi. Spectral gap characterization of full type III factors. J. Reine Angew.Math., 753:193–210, 2019.

[26] Michael Megrelishvili. Reflexively representable but not Hilbert representable compact flowsand semitopological semigroups. Colloq. Math., 110(2):383–407, 2008.


[27] Manor Mendel and Assaf Naor. Euclidean quotients of finite metric spaces. Adv. Math.,189(2):451–494, 2004.

[28] Assaf Naor and Yuval Peres. Lp compression, traveling salesmen, and stable walks. DukeMath. J., 157(1):53–108, 2011.

[29] Bogdan Nica. Proper isometric actions of hyperbolic groups on Lp-spaces. Compos. Math.,149(5):773–792, 2013.

[30] Piotr W. Nowak. Group actions on Banach spaces and a geometric characterization of a-T-menability. Topology Appl., 153(18):3409–3412, 2006.

[31] Piotr W. Nowak. Group actions on Banach spaces. In Handbook of group actions. Vol. II,volume 32 of Adv. Lect. Math. (ALM), pages 121–149. Int. Press, Somerville, MA, 2015.

[32] Baptiste Olivier. Kazhdan’s property (T ) with respect to non-commutative Lp-spaces. Proc.Amer. Math. Soc., 140(12):4259–4269, 2012.

[33] Pierre Pansu. Cohomologie Lp des varietes a courbure negative, cas du degre 1. NumberSpecial Issue, pages 95–120 (1990). 1989. Conference on Partial Differential Equations andGeometry (Torino, 1988).

[34] Mikael de la Salle. A local characterization of Kazhdan projections and applications. Com-ment. Math. Helv., 94(3):623–660, 2019.

[35] Mikael de la Salle. Strong property (T) for higher-rank lattices. Acta Math., 223(1):151–193,2019.

[36] Yehuda Shalom. Rigidity of commensurators and irreducible lattices. Invent. Math., 141(1):1–54, 2000.

[37] Yehuda Shalom. Rigidity, unitary representations of semisimple groups, and fundamentalgroups of manifolds with rank one transformation group. Annals of Mathematics, 152(1):113–182, 2000.

[38] A. I. Shtern. Bounded continuous real 2-cocycles on simply connected simple Lie groups andtheir applications. Russ. J. Math. Phys., 8(1):122–133, 2001.

[39] M. Takesaki. Theory of operator algebras. II, volume 125 of Encyclopaedia of MathematicalSciences. Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry,6.

[40] L. Tzafriri. Remarks on contractive projections in Lp-spaces. Israel J. Math., 7:9–15, 1969.[41] William A. Veech. Weakly almost periodic functions on semisimple Lie groups. Monatsh.

Math., 88(1):55–68, 1979.[42] F. J. Yeadon. Isometries of noncommutative Lp-spaces. Math. Proc. Cambridge Philos. Soc.,

90(1):41–50, 1981.[43] Guoliang Yu. Hyperbolic groups admit proper affine isometric actions on lp-spaces. Geom.

Funct. Anal., 15(5):1144–1151, 2005.

UMPA, CNRS ENS de Lyon, Lyon, FRANCE

E-mail address: [email protected]

UMPA, CNRS ENS de Lyon, Lyon, FRANCE

E-mail address: [email protected]

arxiv:2001.02490v3 [math.gr] 1 oct 2020

Documents