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Diss. ETH No. 14259 On Property (RD) for Certain Discrete Groups A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Mathematics presented by Indira Lara Chatterji Diplomee de l'Universite de Lausanne born January 25th, 1973 citizen of Lausanne (VD) Thesis committee: Prof. Marc Burger, examiner Prof. Alain Valette, co-examiner Prof. Frederic Paulin, co-examiner 2001

- group
- duits quelconques d'espaces hyperboliques
- property
- baum-connes
- baum-connes conjec- ture

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Diss. ETH No. 14259

On Property (RD) for Certain Discrete Groups

A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Mathematics presented by

Indira Lara Chatterji Diplom´eedel’Universite´deLausanne born January 25th, 1973 citizen of Lausanne (VD)

Thesis committee: Prof.Marc Burger, examiner Prof.Alain Valette, co-examiner Prof.Pcire´deniluar´F, co-examiner

2001

2 R´esume´.pri´aproditeet´ettCesdelpectnsasrtaierecpxolseeete`h deDe´croissanceRapide,proprite´te´(DR),quiestunph´enom`enerele-vantdelage´ome´trienon-commutativedesgroupesdiscrets.Cettepro-pri´ete´are´cemmentconnuunintenseregaind’inte´rˆetduˆauxtravauxde V.Laﬀorguequienfontusagepourd´emontrerlaconjecturedeBaum-Connes dans certains cas. Outre qu’ils constitu t ´vidente motiva-en une e tion,lestravauxdeV.Laﬀorgueontaussi´et´eunesourced’inspiration pourunepartiedupre´senttravail. e na ure geom Notreapprocheestdt´e´trique:nousconsid´eronsdes groupesdiscretsd’isom´etriesdecertainsespacessym´etriques,notam-mentl’espaceassoci´eaugroupeexceptionnelE6(−26), ainsi que de pro-duitsquelconquesd’espaceshyperboliquesa`laGromov,ouencoredes produits mixtes impliquant les deux types d’espaces. Abstract.explore in this dissertation certain aspects of theWe Rapid Decay property, property (RD), which is a phenomenon in the non-commutative geometry of discrete groups. Due to V. Laﬀorgue’s work on the Baum-Connes conjecture, there has recently been a con-siderable interest in this property. On top of this obvious motivation, V. Laﬀorgue’s techniques were also a source of inspiration for part of the present work. Our approach is geometrical in nature: we consider discrete groups of isometries of certain symmetric spaces, notably the space associ-ated to the exceptional groupE6(−26), as well as arbitrary products of Gromov hyperbolic spaces, or mixed products of both types of spaces.

Contents

Introduction 5 Motivations and statement of results 5 Organization of the text 6 Acknowledgments 6 Chapter 1. Around property (RD) 9 Basic deﬁnitions 9 What is property (RD) good for? 21 Chapter 2. Uniform lattices in products of rank one Lie groups 25 Two properties for a metric space 25 Uniform nets in products of hyperbolic spaces 29 Coxeter groups 33 Chapter 3. Triples of points inSL3(H) andE6(−26)35 The case ofSL3(H) 35 The case ofE6(−26)40 Relation with triples inSL3(C) 47 Chapter 4. Uniform lattices in products ofSL3’s and rank one’s 49 Two properties of triples of points 49 How we establish property (RD) 52 Flat of typeA2endowed with a Finsler distance 55 Loose ends 57 Bibliography 59

3

Introduction Motivations and statement of results A discrete group Γ is said to haveproperty (RD) with respect to a length function`if there exists a polynomialPsuch that for any r∈R+andf∈CΓ supported on elements of length shorter thanr the following inequality holds: kfk∗≤P(r)kfk2 wherekfk∗denotes the operator norm offacting by left convolution on`2(Γ), andkfk2the usual`2norm. Property (RD) has been ﬁrst established for free groups by Haagerup in [7], but introduced and stud-ied by P. Jolissaint in [12], who established it for polynomial growth groups and for classical hyperbolic groups. The extension to Gromov hyperbolic groups is due to P. de la Harpe in [8 the ﬁrst]. Providing examples of higher rank groups, J. Ramagge, G. Robertson and T. Ste-ger in [24] proved that property (RD) holds for discrete groups acting ˜ ˜ ˜ freely on the vertices of anA1×A1orA2building and recently V. Laﬀorgue did it for cocompact lattices inSL3(R) andSL3(C) in [14]. Cocompactness is crucial since the only (up to now) known obstruc-tion to property (RD) has been given by P. Jolissaint in [12] and is the presence of an amenable subgroup with exponential growth. This has been turned into a conjecture: Conjecture1 (A. Valette, see [30] or [2]).Property (RD) with respect to the word length holds for any discrete group acting isomet-rically, properly and cocompactly either on a Riemannian symmetric space or on an aﬃne building. Property (RD) is important in the context of Baum-Connes conjec-ture, precisely, V. Laﬀorgue in [15] proved that for “good” groups hav-ing property (RD), the Baum-Connes conjecture without coeﬃcients holds. The main result of this thesis is the following Theorem0.1.Any discrete cocompact subgroupΓof a ﬁnite prod-uct of type Iso(X1)× ∙ ∙ ∙ ×Iso(Xn) has property (RD) with respect to the word length, where theXi’s are ˜ either complete locally compact Gromov hyperbolic spaces,A2-buildings, or symmetric spaces associated toSL3(R),SL3(C),SL3(H)andE6(−26). 5

6 INTRODUCTION This provides many interesting examples of discrete groups having property (RD), such as cocompact lattices inSL2(R)×SL2(R). Notice that combining our result with V. Laﬀorgue’s crucial theorem in [15] yields the following Corollary0.2.The Baum-Connes conjecture without coeﬃcients (see[30]) holds for any cocompact lattice in G=G1× ∙ ∙ ∙ ×Gn where theGi’s are either rank one Lie groups,SL3(R),SL3(C),SL3(H) orE6(−26). Simultaneously and independently, M. Talbi in his Ph.D. thesis (see [28]) proved that property (RD) holds for groups acting on buildings ˜ ˜ of typeAi1× ∙ ∙ ∙ ×Aik, whereij∈ {12}. Organization of the text Chapter 1 of this work is a general exposition of property (RD), essentially based on P. Jolissaint’s results in [12 slightly improve]. We his result on split extensions by considering general length functions instead of the word length, but otherwise the proof is the same. In Chapter 2 we look at the particular case where theXi’s are locally compact Gromov hyperbolic spaces. In case where all theXi’s are trees, we can even drop the local ﬁniteness condition, which allows to establish property (RD) for Coxeter groups. This remark is due to N. Higson. In Chapter 3, the study ofSL3(H) andE6(−26)will allow us to answer (positively) a question posed by V. Laﬀorgue in [14], which was to know whether his Lemmas 3.5 and 3.7 are still true for the groups SL3(H) andE6(−26), whose associated symmetric spaces have also ﬂats of typeA2. Observing that these lemmas are in fact “three points con-ditions” it will be enough to prove that ifXdenotesSL3(H)/SU3(H) orE6(−26)/F4(−52)then for any three points inXthere exists a totally geodesic embedding ofSL3(C)/SU3(C) containing those three points. In Chapter 4 we will explain how to use the techniques used in [24] and [14 The] for the above described products. “loose ends” chapter is about questions to which I haven’t been able to answer. Acknowledgments My warmest thanks go to both my adviser Prof. A. Valette and my co-adviser Prof. M. Burger for the nice time I had writing my dissertation. I am extremely grateful to Alain Valette for introduc-ing me with a very communicative enthusiasm to this Baum-Connes conjecture related topic which is property (RD), for suggesting me a very nice thesis subject, and later for carefully reading my scratches, patiently answering my uncountable e-mails and oﬀering me very im-portantandmotivatingmathematicaldiscussionsinNeuchˆatel.Ithank

ACKNOWLEDGMENTS

7

Marc Burger for reading my dissertation, for all his very helpful com-ments and for his guidance ever since I completed my undergraduate studies in Lausanne. He provided me with ideal working conditions in Zu¨rich,and“suggested”metomakethescriptfortheNachdiplomvor-lesung Alain Valette gave at ETHZ in 1999, which forced me to learn a lot on the Baum-Connes conjecture. IwarmlythankFre´de´ricPaulinforacceptingtobeco-examiner on this dissertation. His friendly criticism and careful reading of a preliminary version have been a valuable help to me. I am indebted to V. Laﬀorgue who spent a great deal of time ex-plaining me the techniques used in [14] and [24], giving me a very clear ˜ and intuitive picture ofA2andA2 also thank D. Allcockstructures. I and G. Prasad for very helpful conversations onE6(−26).

CHAPTER 1

Around property (RD)

Basic deﬁnitions In this section we will give some conditions which are equivalent to property (RD) and study the stability of property (RD) under group extensions. Most of the results given in this section are either simple remarks or slights improvements of results contained in P. Jolissaint’s paper [12]. Definition1.1.Let Γ be a discrete group, alength functionon Γ is a function`: Γ→R+satisfying: •`(e) = 0, whereedenotes the neutral element in Γ, •`(γ) =`(γ−1) for anyγ∈Γ, •`(γµ)≤`(γ) +`(µ) for anyγ µ∈Γ. The functiond(γ µ) =`(γ−1µ) is a left Γ-invariant pseudo-distance on Γ. We will writeB`(γ r) for the ball of centerγ∈Γ and radiusrwith respect to the pseudo-distance`, and simplyB(γ r) when there is no risk of confusion. Example1.2.If Γ is generated by some ﬁnite subsetS, then the algebraic word lengthLS: Γ→Nis a length function on Γ, where, forγ∈Γ,LS(γ) is the minimal length ofγas a word on the alphabet S∪S−1, that is, LS(γ) = min{n∈N|γ=s1. . . sn si∈S∪S−1}. Let Γ act by isometries on a metric space (X d). Pick a point x0∈Xand deﬁne`(γ) =d(γx0 x0), this is a length function on Γ. This last example is general in the sense that any length function` comes from a metric on a spaceXwith respect to which Γ acts by isometries. Indeed, if`is a length function on Γ, deﬁne the subgroup Nof Γ as, N={γ∈Γ|`(γ) = 0} and thenX= Γ/N map. Thed:X×X→R+,d(γN µN) =`(µ−1γ) is a well-deﬁned Γ-invariant metric onXand`(γ) =d(γN N) for any γ∈Γ. LetH <Γ be a subgroup of Γ and`a length on Γ. The restriction of`toHinduces a length onHthat we callinduced length. 9

10 1. AROUND PROPERTY (RD) Definition1.3.Denote byCΓ the set of functionsf: Γ→Cwith ﬁnite support, which is a ring for pointwise addition and convolution: f∗g(γ) =Xf(µ)g(µ−1γ).(f g∈CΓ γ∈Γ) µ∈Γ We denote byR+Γ the subset ofCΓ consisting of functions with target inR+ for. ConsiderfinCΓ (or inR+Γ) the following norms: (a) the usual`2norm, given by kfk2=sγX∈Γ|f(γ)|2 which actually comes from a scalar product on`2Γ, the space of square summable functions on Γ. (b) theoperator norm, given by kfk∗= sup{kf∗gk2| kgk2= 1} which is the norm offinCr∗Γ, the reduced C*-algebra of Γ, obtained by completingCΓ with respect to the operator norm. (c) aweighted`2norm, depending on a parameters >0 and given by kfk`,s=Xs|f(γ)|2(1 +`(γ))2s. γ∈Γ We denote byHs(Γ) the completion ofCΓ with respect to this ` norm. Obviously, forf∈CΓ we have thatkfk2≤ kfk∗, and the following deﬁnition is an attempt to give an upper bound to the operator norm. Definition1.4 (P. Jolissaint, [12]).Let`be a length function on Γ. We say that Γ hasproperty (RD)(standing forRapid Decay) with respect to`(or that it satisﬁes theHaagerup inequality), if there exists C s >0 such that, for eachf∈CΓ one has kfk∗≤Ckfk`,s. Proposition1.5.LetΓbe a discrete group, endowed with a length function`. Then the following are equivalent: 1)The groupΓhas property (RD) with respect to`. 2)There exists a polynomialPsuch that, for anyr >0and any f∈R+Γso thatfvanishes on elements of length greater than r, we have kfk∗≤P(r)kfk2. 3)There exists a polynomialPsuch that, for anyr >0and any two functionsf g∈R+Γso thatfvanishes on elements of length greater thanr, we have kf∗gk2≤P(r)kfk2kgk2.

BASIC DEFINITIONS 11 4)There exists a polynomialPsuch that, for anyr >0and any f g h∈R+Γso thatfvanishes on elements of length greater thanr, we have f∗g∗h(e)≤P(r)kfk2kgk2khk2 5)Any subgroupHinΓhas property (RD) with respect to the induced length. Proof.We start with Takethe equivalence between 1) and 2). f∈CΓ with support contained in a ball of radiusr, we have: k∗≤Ckfk`sγ∈BX(e,r) kf,s=C|f(γ)|2(`(γ) + 1)2s ≤Csγ∈BX(e,r)|f(γ)|2(r+ 1)2s=C(r+ 1)skfk2 and thus 2) is satisﬁed, for the polynomialP(r) =C(r+1)s. Conversely we denote, forn∈N Sn={γ∈Γ|n≤`(γ)< n+ 1} and compute, forf∈R+Γ: ∞ ∞ ∞ kfk∗=kXf|Snk∗≤Xkf|Snk∗≤XP(n+ 1)kf|Snk2 n=0n=0n=0 ∞ ∞ ≤XC(n+ 1)kkf|Snk2=CX(n+ 1)−1(n+ 1)k+1kf|Snk2 n=0n=0 v∞v∞ ≤CuX(n+ 1) tn=0−2tun=X0(n+ 1)2k+2kf|Snk22 v∞ ≤C√π6tuX X|f(γ)|2(`(γ) + 1)2k+2 n=0γ∈Sn =C√π6sγX∈Γ|f(γ)|2(`(γ) + 1)2k+2=C√π6kfk`,k+1 We ﬁnish by noticing that forf∈CΓ if one denotes by|f|the function given byγ7→ |f(γ)|(which is inR+Γ), thenkfk2=k |f| k2and thus kfk∗≤ k |f| k∗≤P(r)k |f| k2=P(r)kfk2. The equivalence between 2) and 3) is rather obvious since forfas before andg∈R+Γ, non zero: kf∗gk2≤ kfk∗≤P(r)kfk2 kgk2