FIXED-POINT SPECTRUM FOR GROUP ACTIONS BY AFFINE ISOMETRIES ON L

ANNALES DE

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Omer Lavy & Baptiste Olivier

Fixed-point spectrum for group actions by affine isometries on Lp-spaces

Tome 71, n^o1 (2021), p. 1-26.

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FIXED-POINT SPECTRUM FOR GROUP ACTIONS BY AFFINE ISOMETRIES ON L

-SPACES

by Omer LAVY & Baptiste OLIVIER (*)

Abstract. —The fixed-point spectrum of a locally compact second countable groupGon `pis defined to be the set of p>1 such that every action by affine isometries ofGon`padmits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms: [1, pc[, [1, pc[\{2}for some 16 pc6∞, or [1, pc], [1, pc]\{2}for some 16pc<∞. This result is closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions onLp(0,1).

More generally, we study the topological properties of the fixed-point spectrum on Lp(X, µ) for general measure spaces (X, µ), and show partial results toward the conjecture for actions onLp(0,1). In particular, we prove that the spectrum associated with actions with linear partπis either empty, or an interval of the form [1, pc] (pc>1) or [1,∞[, wheneverπis an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space (X, µ).

Résumé. —Le spectre des points fixes d’un groupe localement compact à base dénombrableGest défini comme l’ensemble desp>1 tels que chaque action par isométries affines deGsur`padmet un point fixe. Nous montrons que cet ensemble est soit vide, soit peut s’écrire sous une des formes suivantes: [1, pc[, [1, pc[\{2}

pour un certain 16pc6∞, ou [1, pc], [1, pc]\{2}pour un certain 16pc<∞. Ce résultat est en lien étroit avec la conjecture de C. Drutu affirmant que le spectre des points fixes est un ensemble connexe pour les actions isométriques surLp(0,1).

Plus généralement, nous étudions les propriétés topologiques du spectre des points fixes surLp(X, µ) pour des espaces mesurés arbitraires (X, µ), et nous mon- trons des résultats partiels dans le sens de la conjecture pour les actions surLp(0,1).

En particulier, nous prouvons que le spectre associé aux actions dont la partie li- néaire estπest soit vide, soit un intervalle de la forme [1, pc] (pc>1) ou [1,∞[, dès queπest une représentation orthogonale associée à une action ergodique préservant la mesure sur un espace mesuré (X, µ) de mesure finie.

Keywords:Groups with property (T), orthogonal representations onLp-spaces.

2020Mathematics Subject Classification:22D10, 22D12, 20CXX.

(*) This work was done while both authors worked at the TECHNION. The second author was supported by ERC Grant no. 306706.

1. Introduction

Group actions on Banach spaces is a large topic related to many areas of mathematics: group cohomology, Kazhdan’s property (T), and fixed- point properties. In [1], Bader, Furman, Gelander and Monod studied group actions by isometries on Banach spaces. The authors of [1] provide several results concerning property (F_Lp(0,1)), the fixed-point property for group actions by affine isometries on the spaceLp([0,1], λ) (abbreviatedLp(0,1)), where p > 1 and λ denotes the Lebesgue measure. In particular, they show that a locally compact second countable group with the fixed-point property (F_Lp(0,1)) has the Kazhdan’s property (T) whenp >1, and that these properties are equivalent when 1< p 62 (see [1, Theorem A and Theorem 1.3]). Later the main result of [2] was that these properties are equivalent when 16p62.

Some groups have property (F_Lp(X,µ)) for all p > 1 and all standard measure space (X, µ): a more general result states that higher rank groups have property (F_Lp(M)) for all p > 1 and all von Neumann algebra M (see [1, Theorem B], [21, Theorem 1.6], and [14] for a stronger result).

See also [16] and [17] where analogous results are established for universal latticesSLn(Z[x1, . . . , xk]) (n>4).

On the other hand, there exist Kazhdan groups which do not have prop- erty (F_Lp(0,1)) for somep >2. For instance, hyperbolic groups (and among them co-compact lattices inSp(n,1) have property (T)) admit a metrically proper action by affine isometries on`p, as well as on Lp(0,1), forplarge enough (see [19] and [24]).

The main motivation of this article is the following conjecture of C. Drutu:

for every topological groupG, there existspc>1 (pc can be also infinite), such thatGhas property (F_Lp(0,1)) for 16p < p_c, andG does not have property (F_Lp(0,1)) for p > pc. further more, if pc > 1, then pc > 2 (see Question 1.8 in the introduction of [9]). It seems that this question has its root in an older question raised by M. Gromov (see [12, D.6 p. 158]). We introduce the following set and we use a similar terminology as in [20].

Definition 1.1. — Let(X, µ)be a standard Borel measure space, and letGbe a topological group. The set

FL_∞(X,µ)(G) ={p>1|Ghas property(FL_p(X,µ))}

is called the fixed-point spectrum of G (for affine isometric actions) on L_p(X, µ)-spaces.

We use the notationF_L∞(X,µ)(G) since it makes sense in a more general context, that is for actions on non-commutative Lp-spaces; in that case

L_∞(X, µ) is replaced by a general von Neumann algebraM, and the fixed- point spectrum is denoted byF_M(G).

The conjecture of C. Drutu can be rephrased as follows: the set F_L∞(0,1)(G) is connected for all groupG. Our first main result is an answer to the same question, when replacing ([0,1], λ) by a discrete measure space.

We denote by`∞ the space of all bounded infinite sequences of complex numbers, and`p the space ofp-summable sequences in`_∞, equipped with thepnorm.

Theorem 1.2. — LetGbe a locally compact second countable group.

Then one of the following equalities holds:

• F`_∞(G) =∅;

• F`∞(G) = [1, pc[for some 1< pc6∞;

• F`∞(G) = [1, pc]for some 16pc<∞;

• F`∞(G) = [1, p_c[\{2}and2< p_c6∞;

• F`∞(G) = [1, pc]\{2}and2< pc<∞.

Hence the spectrum F_`∞(G) can be the empty set, an interval, or the union of two disjoint intervals: we will show that these three situations can occur, depending on the groupGconsidered.

In the special case of countable groups, a similar result as Theorem 1.2 was also proved independently of our work in [11], for the case of discrete countable groups and real`<sub>p</sub>-spaces. Although some ideas in our proof are similar to the ones in [11] (see Section 4.1 for further precisions), our overall approach is different: Theorem 1.2 is obtained by studying the topological properties of the fixed-point spectrum, while the approach in [11] proves relationships with expander graphs theory. As shown by our Theorem 1.3 and results from Section 5 discussed below, the approach of the current paper permits to go beyond`p-spaces, and generalizes to otherLp-spaces.

Let us now discuss the fixed-point spectrum of actions on general Lp(X, µ)-spaces. We refer to Section 2 for basic facts and definitions related to isometric group representations on L_p-spaces. Let 1 6p < ∞, p 6= 2.

Letπ^p:G→O(Lp(X, µ)) be an orthogonal representation. There is a nat- ural family (π^q)q>1 of orthogonal representations π^q : G→ O(Lq(X, µ)) associated toπ^p, namely the conjugate representations ofπ^pby the Mazur maps Mp,q. Referring to a theorem by Banach and Lamperti recalled in Section 2, we will call(BL) representationsthe representations of the form π^q, 1 6q <∞. In the sequel, we will distinguish (BL) orthogonal repre- sentations from other possible classes of orthogonal representations in the caseq= 2. Then we can define the fixed-point spectrum of affine isometric

actions with linear parts (π^q)q>1as

FL_∞(X,µ)(G,(π^q)q>1) ={q>1|H¹(G, π^q) ={0}}

where H¹(G, π^q) is the first cohomology group of G with coefficients in π^q (see Section 2.3 for a definition). For simplicity in notations, we will denote FL∞(X,µ)(G,(π^q)q>1) by FL∞(X,µ)(G, π^p) for p 6= 2 (but this set does not depend on p). In the sequel, we will say that π^p is measure- preserving (resp. ergodic) if the associated action on (X, µ) is measure- preserving (resp. ergodic). We say thatπ^p is positive ifπ^p(g)f >0 for all g∈Gand allf >0,f ∈L_p(X, µ).

The following theorem describes the form of the spectraFL_∞(X,µ)(G, π) relative to measure-preserving ergodic actions.

Theorem 1.3. — LetGbe a second countable locally compact group.

Let(X, µ)be a finite measure space, andπ:G→O(L_p(X, µ))be a (BL) measure-preserving ergodic representation. Then we have:

• The spectrumF_L∞(X,µ)(G, π)is an interval or is empty.

• IfGhas property(T), the spectrumFL_∞(X,µ)(G, π)is an interval of the form[1, pc]or[1,∞[for some pc >2.

Theorem 1.3 is in contrast with the following well-known fact (detailed in Section 5.2): ifG does not have property (T), there exists some (BL) measure-preserving ergodic representationρ:G→O(Lp(X, µ)) such that F_L∞(X,µ)(G, ρ) is empty.

Theorems 1.2 and 1.3 are in the spirit of C. Drutu’s conjecture. Other partial results in that direction are proved in the paper. We study topolog- ical properties of the fixed-point spectrum. We discuss general arguments for closedness (resp. openness) of fixed-point spectra in Section 3 (resp.

Section 5.1). Moreover, we use recent results from [3] about deformation of cohomology to show the connectedness of spectraFL_∞(0,1)(G, π) under some additional assumptions onGandπ.

The paper is organized as follows. In Section 2, we recall general facts and results we will need about isometric group actions onLp-spaces. In Sec- tion 3, we show general results concerning the closedness of the fixed-point spectrum. Section 4 is devoted to the proof of Theorem 1.2. In Section 5, we prove various partial results toward a proof of the conjecture of C. Drutu concerning the case ofLp(0,1).

Acknowledgments

Authors are grateful to V. Lafforgue for many comments that improve the quality of the paper. They also would like to thank U. Bader, B. Bekka, T. Gelander, B. Nica, E. Ricard and the anonymous reviewers for valuable comments and interesting discussions on preliminary versions of this work.

Both authors wish to thank the Technion, where most of this work was achieved, for the good atmosphere and working conditions.

2. Isometric group actions on L_p(X, µ)-spaces

In this section, we recall some general definitions and properties of linear and affine isometric actions on L_p(X, µ)-spaces. Let G be a topological group, and (X, µ) be a standard Borel measure space.

2.1. Orthogonal representations on Lp(X, µ)

Let 16p <∞,p6= 2, and denote byO(L_p(X, µ)) the group of bijective linear isometries of the spaceLp(X, µ). By a theorem of Banach and Lam- perti, elements inO(L_p(X, µ)) are the linear mapsU :L_p(X, µ)→L_p(X, µ) described as follows:

(∗) U f(x) =h(x)

dϕ∗µ dµ (x)

^1/p

f(ϕ(x))

whereh:X →Cis a measurable function of modulus one, andϕ:X →X is aµ-class-preserving bijective transformation. In the particular case where (X, µ) is a discrete measure space, ϕ is a permutation of the countable setX.

An orthogonal representation πof the group Gon Lp(X, µ) is a group homomorphism π : G → O(L_p(X, µ)) such that the maps g 7→ π(g)f are continuous on G for all f ∈ Lp(X, µ). For 1 6 p < ∞ and p 6= 2, each elementπ(g), g ∈ G, is given by formula (∗), and we call π a (BL) representation. Forp= 2, the isometry groupO(L₂(X, µ)) contains much more elements than (BL) representations described by formula (∗) in the Banach–Lamperti theorem. Orthogonal representations onL₂(X, µ) which are (BL) representations will play a central role in our study.

In the current paper, the set of π(G)-invariant vectors is denoted by Lp(X, µ)^π(G). Recall that whenp= 2, the orthogonal complementL⁰₂(X, µ)

to L₂(X, µ)^π(G) is also a π(G) invariant space. Then the space L₂(X, µ) can be written as a direct sum of π(G)-invariant subspaces, L2(X, µ) = L₂(X, µ)^π(G)⊕L⁰₂(X, µ). The authors of [1] described a similar decompo- sition forLp(X, µ),p6= 2, which we recall now.

For p > 1, and p⁰ = p/(p−1), the contragradient representation π^∗ : G→O(Lp⁰(X, µ)) is defined by duality as follows:

hx, π^∗(g)yi=hπ(g⁻¹)x, yifor allg∈G, x∈Lp(X, µ), y∈Lp⁰(X, µ).

Then we define the following π(G)-invariant decomposition of the space L_p(X, µ):

L_p(X, µ) =L^π(G)_p ⊕L⁰_p(π) whereL⁰_p(π) ={f ∈Lp| ∀h∈L^π

∗(G)

p⁰ ,hf, hi= 0}is the annihilator of the G-invariant vectors for the contragradient representationπ^∗:G→O(Lp⁰) of π (more details can be found in [1, Proposition 2.6 and Section 2.c]).

Given such a decomposition, one can consider the restriction π⁰ : G → O(L⁰_p(π)). When p = 1, the analog of the previous representation is the representationπ⁰:G→O(L₁/L^π(G)₁ ), obtained fromπby composing with the quotient projection. If the context is clear, we will keep the notationπ in place ofπ⁰ in the sequel.

Let π:G→O(L_p(X, µ)) be a (BL) orthogonal representation. We will denote|π|: G→ O(Lp(X, µ)) the map defined by the following formula, extended toL_p(X, µ) by linearity:

|π|(g)f =|π(g)f|for allf ∈Lp(X, µ), f >0.

Notice that|π|is a (BL) representation, sinceπis a (BL) representation.

The representation|π|is obtained fromπby simply omitting the functionh appearing in formula (∗) (or equivalently taking it to be constant 1) for each elementπ(g),g∈G. We will say thatπis positive in the case whereπ=|π|.

Also, we will say thatπis measure-preserving if the corresponding action of Gon (X, µ) isµ-preserving, equivalently if the associated Radon–Nikodym derivative is equal to 1 almost everywhere. By Banach’s description of the isometries of `p, all (BL) representations on the space `p are measure- preserving (see [5, Section 2] for details).

2.2. The Mazur map

A very useful tool to study Lp-spaces and their representations is the Mazur map. Here we recall the definition and some well-known properties of this map. The proofs of the basic properties of the Mazur map can be

found in [6, Chapter 9.1]. For f ∈ L_p(X, µ), we will use notations sg(f) and|f|for the polar decomposition components off, so thatf =sg(f)|f| forsg(f) :X →Cof modulus one, and|f|>0.

Let 16p, q <∞. The map

Mp,q: Lp(X, µ)−→Lq(X, µ) f =sg(f)|f| 7−→sg(f)|f|^p/q

is called the Mazur map. It induces a uniformly continuous homeomorphism between the unit spheres of L_p(X, µ) and L_q(X, µ). More precisely, the Mazur map Mp,q is min(1,^p_q)-Hölder on the unit sphere, i.e. for all 1 6 p, q <∞:

(2.1) kMp,q(x)−Mp,q(y)kq6Cp,qkx−yk^θ_p^p,q

for allx, y∈S(L_p(X, µ)), where the constantC_p,qdepends only onp, q, and θp,q= min(1,^p_q) (see [6, Theorem 9.1] and [23]). The following remark 2.2 will be helpful for our proofs, and it relies on the property stated in the following lemma.

Lemma 2.1. — LetQbe some compact interval in[1,∞[. There exists a choice of constantsCp,q >0such that the statements below hold.

(1) Forq∈Qandp∈[1,∞[,(2.1)above holds withC_p,q.

(2) Forp∈[1,∞[, the constantCp= sup_q∈QCp,q>0satisfiesCp<∞.

(3) Forp∈[1,∞[,(2.1)holds with Cp in place ofCp,q for allq∈Q.

Proof. — The proof is the one given in the proof of [6, Theorem 9.1], with some additional observation. So we detail only the part which requires a new argument. Forp > q, the proof in [6] shows that Cp,q =p/q so our statement is clear.

The authors of [6] also shows the existence of some cp,q >0 such that the inequality below holds

kMp,q(x)−Mp,q(y)kq>cp,q× kx−yk^θ_p^p,q,

and it follows inequality (2.1) forp < q sinceM_p,q⁻¹=M_q,p and by taking Cp,q = (cp,q)⁻¹.

To prove our statement, it is sufficient to show that constantcp,q can be made independent ofqon compact subsetsQ⊂[1,∞[. Following the lines of the proof in [6], we show that we have|a^p/q−θb^p/q|>cp|a−θb|^p/qforcp

independent ofqover compact subsets, and for all a>b>0, and|θ|= 1.

First we notice that it is sufficient to prove the existence ofcp forb= 1, and for all 16a6M, whereM >0 is some positive constant. Indeed, the inequality forb= 0 holds with cp= 1. And forb6= 0 we can consider a/b

in place ofaandb= 1. Asa/bgoes to infinity,|(a/b)^p/q−θ|is equivalent to|(a/b)−θ|^p/q. Hence, there exists some constantM >0 such that for all 0< b6awith a/b>M, we have|(a/b)^p/q−θ|>(1/2)× |(a/b)−θ|^p/q. Hence we can takecp= 1/2 for casesb= 0, orb= 1 anda>M.

Now we assumeb= 1, 16a6M for someM >0, and we fix a compact subsetQ ⊂[1,∞[. As noticed in [6], the constant c¹_p = 1 solves the case

|a−θ|61. For the case|a−θ|>1, we find a constant c²_p >0 as follows.

We notice that the set K=

(q, a, θ)∈Q×[1, M]×S¹

|a−θ|>1 is compact. Hence, the continuous map

K−→R^+∗

(q, a, θ)7−→ |a^p/q−θ|

|a−θ|^p/q

attains a minimumc²_p >0 on the compact set K. To finish the proof, we

setc_p= min(1/2, c¹_p, c²_p).

Remark 2.2. — Assume that we have a sequence (pn)n such that pn ∈ [1,∞[ for all n, and limnpn =pfor some p>1. Then by Lemma 2.1 we can find a constantCp>0, which does not depend onn, and such that:

kMp,p_n(x)−Mp,p_n(y)kp_n 6Cpkx−yk^θp^p,pn

for allx, y∈S(Lp(X, µ). In particular, if we have limn kMp_n,p(xn)−Mp_n,p(yn)kp= 0

for somex_n, y_n ∈ S(L_pn(X, µ)), then the equality lim_nkx_n−y_nk_pn = 0 holds as well.

Let x, ybe any positive numbers and assume 16p6q. Then we have the following inequality

|x^p/q−y^p/q|6|x−y|^p/q.

The following inequality is obtained from the inequality above:

(2.2) kMp,q(x)−Mp,q(y)kq6kx−yk^p/q_p

for allx, y>0 s.t. x−y∈Lp(X, µ).

See also [22, Proposition 1.1.4].

Now let p > 1 and let π^p : G → O(L_p(X, µ)) be a (BL) orthogonal representation ofGonLp(X, µ). Sinceπ^pis described by equation (∗), the formula

π^q(g)x= (Mp,q◦π^p(g)◦Mq,p)x for allg∈G, x∈Lq(X, µ) define an orthogonal representationπ^q :G→O(Lq(X, µ)). Note that when the associated action on (X, µ) is measure preserving the representations π^pand π^q coincide on the intersection,Lp(X, µ)∩Lq(X, µ)

A sequence (f_n)_nin a Banach spaceF is said to be a sequence of almost invariant vectors for an orthogonal representationπ:G→O(F) ifkfnkF = 1 and

limn kπ(g)fn−fnkF = 0 uniformly on compact subsets ofG.

The following lemma was first stated in the proof of Theorem A in [1]

(see [1, Remark 4.3]). For the convenience of the reader, we recall the proof of this lemma, given in Proposition 2.5.1 of [22] in a more general setting.

Lemma 2.3. — For a family(π^p)16p<∞of (BL) orthogonal representa- tions, and for all16p <∞, 16q <∞, a group Gadmits a sequence of almost invariant vectors for the representation(π^p)⁰if and only if it admits a sequence of almost invariant vectors for(π^q)⁰

Proof. — Letfn∈S(L⁰_p(π^p)) be a sequence of almost invariant vectors.

Note that for that the following inequalities hold (see [21, Proposition 3.5]):

d(f_n, L^π_p^p(G))>1/2 for alln.

Notice thatL^πq^q(G)=M_p,q(L^πp^p(G)). Defineh_n =M_p,qf_n. By the uniform continuity of the Mazur map then, hn is a sequence of unit vectors inLq

with

d(h_n, L_q(π^q))>δ >0.

Furthermore we have for every compact setQ⊂G, limn sup

g∈Q

khn−π^q(g)hnkq = 0.

Consider first the caseq > 1. Let vn denote the projections ofhn on the complement subspaceL⁰_q(π^q). Sincekvnkq >δ, the uniform convergence on compact subsets holds when replacinghnbyvn/kvnkp. Hence (vn/kvnkp)n

is a sequence of almost invariant vectors forπ^q with values in (π^q)⁰. For q = 1, we consider vn the projection of hn on the quotient space F=L₁/L^π₁^1(G). Thenkv_nk_F >δ, and the sequencev_n/kv_nk_F is a sequence of almost invariant vectors for the representation (π¹)⁰:G→O(F).

2.3. Affine isometric actions on L_p(X, µ)

Denote by Isom(Lp(X, µ)) the group of affine bijective isometries of L_p(X, µ). An affine isometric action of G on L_p(X, µ) is a group homo- morphism α : G → Isom(Lp(X, µ)) such that the maps g 7→ α(g)f are continuous for allf ∈L_p(X, µ). Then we have

α(g)f =π(g)f+b(g) for allg∈G, f∈L_p(X, µ)

whereπ:G→O(L_p(X, µ)) is an orthogonal representation, and b:G→ Lp(X, µ) is a 1-cocycle associated toπ, that is a continuous map satisfying the following relations:

b(gh) =b(g) +π(g)b(h) for allg, h∈G.

Given an orthogonal representationπand a cocycle associated toπ, we will sometimes use the notationα= (π, b) to denote the affine representation whose linear part isπ, and translation part is b. We denote by H¹(G, π) the first cohomology group with coefficients in π, that is the quotient of the space of 1-cocycles associated toπ, by the subspace of 1-coboundaries (a coboundary is a cocycle of the form b(g) = f −π(g)f for some f ∈ Lp(X, µ)).

We recall that an affine isometric action of G onL_p(X, µ) has a fixed- point if and only if the associated cocycleb:G→Lp(X, µ) is a bounded map (see [1, Lemma 2.14] forp >1, and [2] for p= 1). The action is said to be proper if limg→∞kb(g)kp=∞. A topological groupGis said to have the fixed-point property (F_Lp(X,µ)) if every action by affine isometries ofG onL_p(X, µ) admits a fixed-point, that isH¹(G, π) ={0}for all orthogonal representationπ:G→O(Lp(X, µ)).

A locally compact second countable group G is said to have property (TL_p(X,µ)) if for any orthogonal representation π:G→ O(Lp(X, µ)), the restriction ofπonL⁰_p(π) (whenp >1) orL₁/L^π(G)₁ (whenp= 1), has no sequence of almost invariant vectors. The following facts were proved by Guichardet for the case of Hilbert spaces. Later they were proved in the general setting ofLp spaces in [1] (see Section 3.a for proofs):

Proposition 2.4. — If an orthogonal representationπ:G→O(L_p(X, µ))of a second countable groupGhas a sequence of almost invariant vectors inL⁰_p(π), thenH¹(G, π)6={0}.

An important consequence of the previous proposition is the following corollary, whose result holds in a much more general context (see [1, The- orem 1.3]).

Corollary 2.5. — Property(F_Lp(X,µ))implies property(T_Lp(X,µ)).

In the case where the group Gis second countable and generated by a compact subsetQ⊂G, an orthogonal representationπ:G→O(L_p(X, µ)) without any sequence of almost invariant vectors inLp(X, µ)⁰ satisfies the condition: there exists >0 such that for allx∈L_p(X, µ)⁰ (forp >1), or allx∈L1(X, µ)/L^π(G)₁ (X, µ) (forp= 1), the following inequality holds:

kxkp6kx−π(g)xkp for some g∈Q.

Whenp= 2 and the previous condition holds for all orthogonal represen- tationsπ:G→O(L₂(X, µ)), the pair (Q, ) is called a Kazhdan pair.

3. About the closedness of the fixed-point spectrum Our goal is to show that the fixed-point spectrumFL_∞(X,µ)is an interval containing 1. To achieve this, we study the topological properties of the set FL_∞(X,µ) in [1,∞[. More precisely our goal is to show it is connected.

In this section, we show closedness results for F_L∞(X,µ)(G, π). To prove these results, we use a limit-version of a well-known fact about almost invariant vectors for (BL) representations, which is interesting in its own right (Proposition 3.2 below).

We begin with some general facts concerning isometric actions on Lp- spaces.

Lemma 3.1. — Let 1 6 p < ∞, 1 6 pn < ∞, pn 6= 1, be such that lim_np_n=p. LetGbe a topological group. Let(X, µ)be a standard Borel measure space. Let π^p : G → O(Lp(X, µ)) be a (BL) orthogonal repre- sentation. Then there exists C⁰ >0 and N such that for all n > N, we have

d(Mpn,pf, Lp(X, µ)^πp(G))>C⁰ for allf ∈S(Lpn(X, µ)⁰(π^pn)).

Proof. — Forf ∈S(L⁰_p

n(π^pn)), the following inequalities hold (see [21, Proposition 3.5]):

d(f, L^π_pn^pn(G))>1/2 for alln.

Notice thatL^πp^p(G) =M_pn,p(L^πp_n^pn(G)). Assume by contradiction, that the result does not hold. Then there existfn∈S(L⁰_pn(π^pn)) andan∈L^πp_n^pn(G)

such that

(3.1) lim

n kMp_n,pfn−Mp_n,pankp= 0.

In particular, since kankpn =kMpn,pa_nkp, we have limnkankpn = 1. Let an =_ka^an

nk_pn. It follows that limn kM_pn,pf_n−M_pn,pa_nk_p

6lim

n (kMpn,pf_n−M_pn,pa_nkp+kMpn,pa_n−M_pn,pa_nkp).

Askankp_n→1, we conclude from (3.1) that:

limn kMp_n,pfn−Mp_n,pankp= 0.

Applying Remark 2.2 we obtain

limn kfn−ankp_n= 0.

Asfn∈S(L⁰_pn(π^pn)), andan ∈L^π_pn^pn(G) this contradicts the fact that d(fn, L^π_pn^pn(G))>1/2 for alln.

The following proposition is a limit version of Proposition 2.3.

Proposition 3.2. — Let 1 6 p < ∞, 1 6p_n <∞, p_n 6= 1, be such that limnpn = p. Let G be a second countable topological group. Let (X, µ)be a standard measure space. Letπ^p:G→O(L_p(X, µ))be a (BL) orthogonal representation. Assume that there exists a sequence (fn)n ∈ S(Lp_n(X, µ)⁰(π^pn))such that

limn sup

g∈Q

kfn−π^pn(g)fnkp_n= 0

for all compact subsets Q ⊂ G. Then there exists a sequence of almost invariant vectors forπ^p inLp(X, µ)⁰(π^p).

Proof. — Let Q ⊂ G be a compact set. Define hn = Mpn,pfn ∈ S(Lp(X, µ)) for n ∈ N. As in the previous lemma, we can choose C for the estimates of the Mazur mapsM_pn,p, independent ofn. Then we have

limn sup

g∈Q

khn−π^p(g)h_nkp= 0.(∗)

Now by Lemma 3.1, there existsC⁰>0 such thatd(hn, L^πp^p(G))>C⁰ holds for allnlarge enough.

Consider first the case wherep >1. Letvn denote the projections ofhn

on the complement subspaceL⁰_p(π^p). Sincekv_nk_p>C⁰ fornlarge enough, the uniform convergence (∗) on compact subsets holds when replacinghn

byv_n/kv_nk_p. Hence (v_n/kv_nk_p)_n is a sequence of almost invariant vectors forπ^p with values inL⁰_p(π^p).

Forp= 1, we considerv_nthe projection ofh_non the quotient spaceF = L₁/L^π₁^p(G). ThenkvnkF >C⁰, and the sequencev_n/kvnkF is a sequence of almost invariant vectors for the representation (π¹)⁰:G→O(F).

Now we show a closedness property for some sets F_L∞(X,µ)(G, π). Our proof requires two important assumptions: the monoticity of (Lp(X, µ))p

for the inclusion, and the representationπto be measure-preserving.

Proposition 3.3. — Let 1 6 p < ∞, 1 6pn <∞, pn 6= 1, be such that limnpn =p. Assume that Lp(X, µ) ⊂Lp_n(X, µ) for alln, and that lim_nkfkpn = kfkp for all f ∈ L_p(X, µ). Let G be a second countable topological group. Letπ^p:G→O(Lp(X, µ))be a (BL) measure-preserving orthogonal representation. Assume also that

H¹(G, π^pn) ={0}.

Then we have

H¹(G, π^p) ={0}.

Proof. — Letb:G→L_p(X, µ) be a cocycle associated toπ^p. Letn∈N. Since by assumptionLp(X, µ)⊂Lp_n(X, µ) andπ^p is measure-preserving, bdefines also a cocycle for the representationπ^pn. Then there exists fn ∈ L_pn(X, µ) such that

b(g) =f_n−π^pn(g)f_n for allg∈G.

Notice that we can assume thatfn ∈ L⁰_pn(π^pn) for all n, without loss of generality.

Claim 3.4. — The sequence(kfnkpn)n is bounded.

Let us prove this claim now. We assume the contrary and will show that (π^p)⁰ has a sequence of almost invariant vectors, which is not possible by assumption. Assume then that there exist a subsequence of (fn)n, (and use the same notation for the subsequence), with lim_n(kfnkpn)_n=∞Let Q⊂Gbe a compact subset. Since limnkb(g)kp_n =kb(g)kp for allg ∈G, fornlarge enough we have

sup

q∈Q

fn

kf_nk_pn −π^pn(g) fn

kf_nk_{pn p}

n

=sup_g∈Qkb(g)kpn

kf_nk_pn 62sup_g∈Qkb(g)kp

kfnkp_n

and the right-hand side of the inequality tends to 0 asntends to∞. Thus we can apply Proposition 3.2 on the sequence_kf^fn

nk_pn and deduce that there exists a sequence of almost invariant vectors for (π^p)⁰. Hence by Lemma 2.3

the representations (π^pn)⁰ admits a sequence of almost invariant vectors for allnas well. This contradicts the vanishing of H¹(G, π^pn) (by the Lp

version of Guichardet’s Theorem, stated in Proposition 2.4), and Claim 3.4 is proved.

Then there exists C⁰⁰ > 0 such that kfnkp_n 6 C⁰⁰ for all n ∈ N. Let g∈G. We have

kb(g)kp= lim

n kb(g)kp_n62C⁰⁰.

Hence the cocycleb:G→Lp(X, µ) is bounded in Lp(X, µ).

4. The fixed-point spectrum for actions on `p

This section handles the case of actions ofGon`_p-spaces over a discrete spaceX. It is divided in two parts. In the first part, we prove Theorem 1.2.

Some elements in the proofs of our Proposition 4.1 and Lemma 4.2 can be found in [11, Section 3.1], and in particular the crucial use of estimates (2.2).

However, the approach of the current paper requires different statements from those in [11], and we provide relevant results with complete proofs for convenience of the reader. In the second part of the section, we discuss property (F<sub>`p</sub>) and exhibit examples of groupsGillustrating each form of possible fixed-point spectraF`∞(G) listed in the statement of Theorem 1.2.

4.1. Proof of Theorem 1.2

Theorem 1.2 is a direct consequence of Proposition 4.1, which we state and prove after a few useful remarks.

Let π : G → O(`p) be a (BL) orthogonal representation. Write `p =

`p(X) and decompose X =X<sup>f</sup>∪X<sup>i</sup> where X<sup>f</sup> is the union of the finite orbits of theG-action, andX<sup>i</sup>the union of the infinite orbits. In the sequel, we will make use of the decomposition ofπasπ=π<sup>f</sup>⊕π<sup>i</sup> with respect to the decomposition`_p(X) =`<sub>p</sub>(X<sup>f</sup>)⊕`_p(Xⁱ).

We make the following observation:

H¹(G, π) ={0} ⇐⇒H¹(G, π^f) ={0} and H¹(G, πⁱ) ={0}.

HenceF`∞(G, π) is an interval of the form [1, pc] or [1, pc[ for somepc>1 if the two following conditions hold:

(i) F_`∞(G, π^f) is an interval [1, p⁰_c] or [1, p⁰_c[ for somep⁰_c>1;

(ii) F`∞(G, πⁱ) is an interval [1, p⁰⁰_c] or [1, p⁰⁰_c[ for somep⁰⁰_c >1.

For technical reasons, we are not able to prove that F`∞(G, π) is an interval for all representationsπ(but for positive representations, we do).

Nevertheless, the following Proposition 4.1 shows a very close result to the connectedness of all setsF`∞(G, π), and Theorem 1.2 is a straightforward consequence of this result. Notice that when a groupG is not compactly generated, thenF`∞(G) is empty: for any p >1, such a group does not have property (T`<sub>p</sub>) (see [5]), nor property (F`_p) by well-known results.

Proposition 4.1. — LetGbe a second countable group generated by a compact setQ. Letπ:G→O(`_p)be a (BL) orthogonal representation.

Letπ^f, πⁱ be defined as in the previous discussion. Then we have:

(1) F`∞(G, π^f) = [1, p⁰_c]or[1, p⁰_c[;

(2) F`∞(G, π<sup>i</sup>)∩ F`∞(G,|πⁱ|) = [1, p⁰⁰_c]or[1, p⁰⁰_c[.

Before giving the proof of Proposition 4.1, we start with a simple lemma.

Lemma 4.2. — Let16p < q. Letπ^p:G→O(`p)be a (BL) orthogonal representation. Assume thatX^f =∅, and that

H¹(G, π^q) ={0} and H¹(G,|π^p|) ={0}.

Then we also have

H¹(G, π^p) ={0}.

Proof. — Letb:G→`pbe a cocycle forπ. As`p⊂`q,bis also a cocycle forπ<sup>q</sup>. By assumption, there existsx∈`_q such that

b(g) =x−π(g)xfor allg∈G.

For alla, b∈C, we have

|a| − |b|

6|a−b|, and the following inequalities follow for allg∈G:

|x| − |π^p|(g)|x|

_p6kb(g)kp.

Hence the formula c(g) = |x| − |π^p|(g)|x| defines a cocycle for |π^p| with values in`<sub>p</sub>. By assumption, there existsy∈`_p such that

|x| − |π^p|(g)|x|=y− |π^p|(g)y for allg∈G.

It follows then, that|x| −yis a|π^p|invariant vector. Since`p⊂`q, and|π^p| coincides with|π^q| on`<sub>p</sub>, this is a|π<sup>q</sup>| invariant vector as well. From the assumptionX<sup>f</sup> =∅, we deduce that|π<sup>q</sup>|does not have non-zero invariant vectors. Hence|x| −y= 0, i.e.|x|=y, and thenx∈`_p. Remark 4.3. — Note that in`_p(Xⁱ) there are no non-zero invariant vec- tors for bothπ, and|π|.

Now we are able to prove Proposition 4.1.

Proof of Proposition 4.1.

(1). — We will show that for all 1< p < q,H¹(G,(π^f)^q) ={0}implies H¹(G,(π^f)^p) ={0}. If this assertion holds, the setF`∞(G, π^f) is an interval of one the following forms: [1, p⁰_c], [1, p⁰_c[, ]1, p⁰_c] or ]1, p⁰_c] for some p⁰_c >1.

By the closedness property of Proposition 3.3,F`∞(G, π^f) is an interval of the form [1, p⁰_c] or [1, p⁰_c[.

Letb:G→`p(X<sup>f</sup>) be a cocycle associated toπ<sup>f</sup>. By assumption, there existsx∈`q such that

b(g) =x−π^f(g)x for allg∈G.

Notice that we can assume that x ∈ `_q(X^f)⁰ without loss of generality.

We decompose X^f =tj>0Xj in (finite) orbits, and write x = L

jxj in

`_q(X^f) = L

j`<sub>q</sub>(X<sub>j</sub>). From the definition of the complement `⁰_q (see Sec- tion 2.1), it is clear thatxj∈`q(Xj)⁰ for allj.

SinceH¹(G,(π^f)^q) ={0},π^fhas no sequence of almost invariant vectors in `q(X<sup>f</sup>)<sup>0</sup>. As recalled in Lemma 2.3, it is then also true thatπ<sup>f</sup> has no sequence of almost invariant vectors in`p(X^f)⁰. In particular, there exists >0 such that

kukp6sup

g∈Q

ku−π^f(g)ukp for allu∈`⁰_p(π^f).

Define un =L

j6nxj ∈ `⁰_q, and M = sup_g∈Qkb(g)kp. Then we have for alln,

kunkp6sup

g∈Q

kun−π^f(g)unkp

6sup

g∈Q

kb(g)kp

=M.

Hence the sequence (un)n is bounded in`p byM/. Hencex∈`p and the proposition is proved.

(2). — Let 1 6 p < q < ∞. We assume that q ∈ F`<sub>∞</sub>(G, π<sup>i</sup>)∩ F`∞(G,|πⁱ|). Then we need to show thatp∈ F`∞(G, π<sup>i</sup>)∩ F`∞(G,|πⁱ|). By Lemma 4.2, it is sufficient to show thatH¹(G,|((πⁱ)^p)|) ={0}.

Letb:G→`<sub>p</sub>(X<sup>i</sup>) be a cocycle associated to|((π<sup>i</sup>)<sup>p</sup>)|. By assumption, there existsx∈`q such that

b(g) =x− |πⁱ|(g)xfor allg∈G.

On the one hand, we have for allg∈G, |x| − |πⁱ|(g)|x|

_p6kx− |πⁱ|(g)xkp.

On the other hand, estimates (2.2) for the Mazur maps imply the following inequalities for allg∈G:

kMp,q(|x|)− |πⁱ|(g)Mp,q(|x|)kq 6

|x| − |πⁱ|(g)|x|

p/q p .

Hence the formula Mp,q(|x|)− |πⁱ|(g)Mp,q(|x|), g ∈ G, define a cocycle associated to|πⁱ|with values in`q. By assumption, there existsy∈`q such that

Mp,q(|x|)− |πⁱ|(g)Mp,q(|x|) =y− |πⁱ|(g)yfor allg∈G.

It follows then, that Mp,q|x| −y is a |πⁱ| invariant vector. Since Xⁱ has only infinite orbits for theG-action, there is no non-zero|πⁱ|(G)-invariant vector in`q. Hence we have |x| = Mq,py. It follows that x∈ `p and the

proposition is proved.

As a particular case of the previous proofs, we have the following result.

Corollary 4.4. — Let G be a second countable group, and p > 1.

Let π : G → O(`p) be a (BL) orthogonal positive representation. Then F<sub>`∞</sub>(G, π)is empty, or is an interval of the form[1, p_c[(p_c6∞) or [1, p_c] (pc<∞).

4.2. More about property (F`_p)

Now we give examples of groups G for which the fixed-point spectrum F`∞(G) is the union of two intervals [1, pc[\{2} (pc > 2). We recall the following well-known relationships between properties (T), (T<sub>`p</sub>) and (F_`p) (see for example [1, Theorem 1.3]):

(F_{`p</sub>)⇒(T<sub>`p}) for allp>1, (T)⇒(F`p)⇒(T`p) for 16p62.

There are naturally few continuous actions of a connected group on dis- crete spaces. As shown in [5] where property (T<sub>`p</sub>) was studied, the class of connected groups having property (T`_p) is restricted to the groups with compact abelianization (see [5, Corollary 3]). We now show that the latter groups also have property (F`p).

Theorem 4.5. — LetGbe a locally compact second countable group.

Assume thatGis connected. Letp6= 2. Then the following assertions are equivalent:

(1) Ghas property(F_`p);

(2) Ghas property(T`_p);