On the distortion of twin building lattices

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Pierre-Emmanuel Caprace, Bertrand Remy

To cite this version:

PIERRE-EMMANUELCAPRACE*ANDBERTRANDRÉMY**

Abstra t. We show that twin building latti es are undistorted intheir ambient group;

equivalently,the orbitmapofthelatti e to theprodu tofthe asso iated twinbuildingsis

a quasi-isometri embedding. As a onsequen e, we provide an estimateof the quasi-at

rankoftheselatti es,whi himpliesthatthereareinnitelymanyquasi-isometry lassesof

nitelypresentedsimplegroups. Inanappendix,wedes ribehownon-distortionoflatti es

isrelatedtotheintegrabilityofthestru tural o y le.

1. Introdu tion

1.1. Distortion. Let

G

bealo ally ompa tgroupand

Γ < G

beanitelygeneratedlatti e. Then

G

is ompa tlygenerated[CM08,Lemma 2.12℄andthereforeboth

G

and

Γ

admitword metri s, whi h are well dened up to quasi-isometry. It is a natural question to understand

the relationbetween the word metri of

Γ

and therestri tionto

Γ

of theword metri on

G

. Inorder toaddressthisissue,let usxsome ompa t generatingset

Σ

b

in

G

anddenoteby

kgk

_Σ

_b

the wordlength of anelement

g ∈ G

withrespe tto

Σ

b

;wedenote by

d

b

Σ

theasso iated word metri . Similarly, we x a nite generating set

Σ

for

Γ

and denote by

|γ|

Σ

the word length of an element

γ ∈ Γ

with respe t to

Σ

, and by

d

Σ

the asso iated word metri . The latti e

Γ

is alledundistortedin

G

if

d

Σ

isquasi-isometri totherestri tionof

d

Σ

to

Γ

. The onditionamountsto sayingthatthe in lusion of

Γ

in

G

denesa quasi-isometri embedding from themetri spa e

(Γ, d

Σ

)

to themetri spa e

(G, d

b

Σ

)

.

Asiswell-known, any o ompa tlatti e isundistorted: this follows fromthe’var Milnor

Lemma[BH99,PropositionI.8.19℄. Thequestionofdistortionthus entresaroundnon-uniform

latti es. Themainresultof[LMR01℄isthatif

G

isaprodu tofhigher-ranksemi-simple alge-brai groupsoverlo alelds(Ar himedeanornot),then anylatti eof

G

isundistorted. This relieson the deep arithmeti ity theorems due to Margulisin hara teristi 0 and

Venkatara-mana in positive hara teristi , and on a detailed analysis of the distortion of unipotent

subgroups.

Besides the higher-ranklatti esinsemi-simple groups, a lassof non-uniform latti esthat

hasattra tedsomeattentioninre entyearsaretheso- alledKa Moodylatti es(see[Rém99℄

or [CG99℄). A more general lass of latti es is thatof twin building latti es[CR09℄: a twin

building latti eis an irredu iblelatti e

Γ < G = G

+

× G

−

ina produ tof two groups

G

+

and

G

−

a tingstrongly transitively on(lo allynite) buildings

X

+

and

X

−

respe tively,and su hthat

Γ

preservesatwinningbetween

X

+

and

X

−

. Re allthat

Γ

isthennitelygenerated and that, inthisgeneral ontext, irredu iblemeans thatea h oftheproje tions of

Γ

to

G

±

isdense.

Date:July20,2009.

Keywordsandphrases. Latti e,lo ally ompa tgroup,Ka Moodygroup,building,distortion.

*SupportedbytheFundforS ienti Resear hF.N.R.S.,Belgium.

Theorem 1.1. Any twin building latti e

Γ < G

+

× G

−

isundistorted.

Itshouldbenotedthatea hindividualgroup

G

+

or

G

−

alsopossessesnon-uniformlatti es, obtainedforinstan ebyinterse ting

Γ

witha ompa t opensubgroup(e.g.,afa etstabilizer) of

G

−

or

G

+

,respe tively. Other non-uniformlatti eshave been onstru tedbyR.Gramli h andB. Mühlherr [GM℄. We emphasize that, beyond theane ase (i.e. when

G

+

is a semi-simple group over a lo al fun tion eld), a non-uniorm latti e in a single irredu ible fa tor

G

+

(or

G

−

) should be expe ted tobeautomati ally distorted (see Se tion 3.3below).

1.2. Quasi-isometry lasses. Non-distortionofalatti e

Γ

in

G

relatestheintrinsi geometry of

Γ

to thegeometryof

G

. Inthe aseoftwin buildinglatti es,thelattergeometryis (quasi-isometri ally) equivalent to the geometry ofthe produ t building

X

+

× X

−

on whi h

G

a ts o ompa tly. Non-distortion isespe iallyrelevant whenstudyingquasi-isometri rigidityof

Γ

(whi his still anopen problem). Asa onsequen e of Theorem1.1,we an estimatea quasi-isometri invariant ofa twinbuilding latti e

Γ

for

X

+

× X

−

,namely themaximal dimension of quasi-isometri allyembeddedat subspa esinto

(Γ, d

Σ

)

. Thisrankisboundedfrombelow bythemaximal dimension ofan isometri ally embedded at in

X

±

and from above bytwi e thesamequantity(3.4);furthermore,thankstoD.Krammer'sthesis[Kra09℄,thismetri rank of

X

±

an be omputed on retely by means of the Coxeter diagram of the Weyl group of

X

±

. Thisenables us todraw thefollowing group-theoreti onsequen e.

Corollary 1.2. There exist innitely many pairwise non-quasi-isometri nitely presented

simplegroups.

This orollary may also be dedu ed from the work of J. Dymara and Th. S hi k [DS07℄,

whi h givesan estimateofanother quasi-isometryinvariant fortwin buildinglatti es,namely

the asymptoti dimension.

Any nite simple group is of ourse quasi-isometri to the trivial group. Moreover any

nitely presented simple group onstru ted by M. Burger and Sh. Mozes [BM01℄ is

quasi-isometri tothe produ toffreegroups

F

2

× F

2

;thisisdueto[Pap95℄andto thefa tthatthe latter groups are onstru ted as suitable (torsion-free) uniform latti es in produ ts of trees.

Furthermore, on erningthe nitely presentedsimple groups onstru ted by G. Higman and

R.Thompson [Hig74℄, aswell astheiravatarsin[Röv99℄, [Bri04℄,[Bro92℄and[S o92℄, weare

not awareof a lassi ation up to quasi-isometryas of today. Howeversome results seem to

indi atethatmanyof them might bequasi-isometri to one another, ompare e.g.[BCS01℄.

1.3. Integrability of the stru tural o y le. Non-distortion of latti es is also relevant,

in a more subtle way, to the theory of unitary representations and its appli ations. More

pre isely, given a latti e

Γ < G

and a unitary

Γ

-representation

π

, one onsiders the indu ed

G

-representation

Ind

G

Γ

π

. For rigidityquestions(at least)andalso be ausethestru ture of

G

isri herthan that of

Γ

,it is desirable thatthe o y les of

Γ

with oe ientsin

π

extend to ontinuous o y lesof

G

with oe ientsin

Ind

G

Γ

π

. Asexplainedin[Sha00,Proposition1.11℄, a su ient onditionfor this to holdis that

Γ

be square-integrable. By denition, for any

p ∈ [1; ∞)

itissaidthat

Γ

(ormore pre iselythein lusion

Γ < G

) is

p

-integrableifthereis a Borelfundamental domain

Ω ⊂ G

for

G/Γ

su hthat, for ea h

g ∈ G

,we have:

Z

Ω

where

α : G × Ω → Γ

is the indu tion o y ledened by

α(g, h) = γ ⇔ ghγ ∈ Ω

. Mimi k-ing Y. Shalom's arguments in [Sha00, Ÿ2℄, the following statement will be established in an

appendix (withthe above notation forgenerating sets).

Theorem 1.3. Let

G

be a totally dis onne ted lo ally ompa t group and let

Γ < G

be a nitely generated latti e. Assume there is a Borel fundamental domain

Ω ⊂ G

for

G/Γ

su h that for some

p ∈ [1; ∞)

we have:

Z

Ω

khk

_Σ

_b

p

dh < ∞.

Then,if

Γ

isnon-distorted, it is

p

-integrable

For

S

-arithmeti groups, the existen eof fundamental domainssatisfying the ondition of Theorem 1.3is established in [Mar91, Proposition VIII.1.2℄bymeans of Siegeldomains. As we shall see, in the ase of twin building latti es the ondition is straightforward to he k

on eafundamentaldomainprovided bythespe i ombinatorial propertiesoftheselatti es

is used. In parti ular, ombining Theorem 1.1 withTheorem 1.3,we re over themain result of [Rém05℄. We nish by mentioning that square-integrability of latti es is also relevant for

lifting

Γ

-a tionsto

G

-a tionsingeometri situationswhi haremu hmoregeneralthanunitary a tions onHilbertspa es, see[Mon06℄and [GKM08℄.

In order to always start from the same situation, in the above introdu tion we stated

results ex lusively dealing with group in lusions. The proof of the non-distortion statement

is of geometri nature: we prove that a twin building latti e is non-distorted in theprodu t

ofthetwo buildings withwhi h itis asso iated.

Thisarti leiswritten asfollows. Se tion2 onsistsofpreliminaries. Se tion3 providesthe aforementioned geometri proofofnon-distortionanddeals withthevariousmetri notionsof

ranksthat anbebetterunderstoodthankstonon-distortion;weapplythistoquasi-isometry

lasses ofnitelygeneratedsimplegroups. AppendixAisindependent oftheprevioussetting of twin building latti es and establishes a relationship between non-distortion and

square-integrabilityof latti esingeneral totally dis onne ted lo ally ompa t groups.

2. Lifting galleries from the buildings to the latti e

Wereferto[AB08℄forbasi denitionsandfa tsonbuildingsandtwinnings,andto[CR09℄

for twin building latti es. In this preliminary se tion, we merely x the notation and re all

onebasi fa ton twinbuildings whi h playsa keyrole at dierent pla es inthis paper.

Let

X = (X

+

, X

−

)

beatwinbuildingwithWeylgroup

W

asso iatedtoagroup

Γ

admitting aroot groupdatum. Inparti ular

Γ

a ts stronglytransitively on

X

. Welet

d

X

+

(resp.

d

X

−

) denotethe ombinatorialdistan eonthesetof hambersof

X

+

(resp.

X

−

). Wefurtherdenote by

S

the anoni algeneratingsetof

W

andby

Opp(X)

thesetofpairsofopposite hambersof

X

. Throughoutthe paper,wexabasepair

(c

+

, c

−

) ∈ Opp(X)

and allitthefundamental opposite pair of hambers. Two opposite pairs

(x

+

, x

−

)

and

(y

+

, y

−

) ∈ Opp(X)

are alled adja ent ifthere issome

s ∈ S

su h that

x

+

is

s

-adja entto

y

+

and

x

−

is

s

-adja ent to

y

−

. Re all that an opposite pair

x ∈ Opp(X)

is ontained in unique twin apartment, whi h we shall denote by

A(x) = A(x

+

, x

−

)

. The positive (resp. negative) half of

A(x)

is denoted by

A(x)

+

(resp.

A(x)

−

).

Lemma 2.1. Let

ε ∈ {+, −}

. Given any gallery

(x

0

, x

1

, . . . , x

n

)

in

X

ε

and any hamber

y

0

∈ X

−

ε

opposite

x

0

,there existsa gallery

(y

0

, y

1

, . . . , y

n

)

in

X

−

ε

su hthat thefollowinghold for all

i = 1, . . . , n

:

(i)

(x

i

, y

i

) ∈ Opp(X)

;

(ii)

(x

i

, y

i

)

is adja ent to

(x

i−1

, y

i−1

)

;

(iii)

y

i

belongs to the twin apartment

A(x

0

, y

0

)

.

Proof. Thedesiredgallery is onstru ted indu tively asfollows. Let

i > 0

. If

y

i−1

is opposite

x

i

,thenset

y

i

= y

i−1

. Otherwise the odistan e

δ

∗

(x

i

, y

i−1

)

is an element

s ∈ S

andthere is a unique hamber inthe twin apartment

A(x

0

, y

0

)

whi h is

s

-adja ent to

y

i−1

. Dene

y

i

to be that hamber. Itfollowsfrom the axiomsof atwinning that

y

i

isopposite

x

i

. Thegallery

(y

0

, y

1

, . . . , y

n

)

onstru ted inthisway satisesallthedesiredproperties.



3. Non-distortion of twinbuilding latti es

In this se tion, we show that a twin building latti e is non-distorted for its natural

diag-onal a tion on its two twinned building. The arguments are elementary and use the basi

ombinatorial geometry of buildings.

3.1. An adapted generating system. Let

Σ

denote the subset of

Γ

onsisting of those elements

γ

su h that

(γ.c

+

, γ.c

−

)

is adja ent to

(c

+

, c

−

)

, where

(c

+

, c

−

) ∈ Opp(X)

denotes the fundamental oppositepair. Noti ethat

max{d

X

+

(c

+

, γ.c

+

); d

X

−(c

−

, γ.c

−

)} 6 1

forall

γ ∈ S

.

The graphstru ture on

Opp(X)

indu ed by theaforementioned adja en y relation is iso-morphi to the Cayley graph asso iated to the pair

(Γ, Σ)

. Lemma 2.1 readily implies that this graphis onne ted. Thus

Σ

is agenerating setfor

Γ

.

Lemma 3.1. Let

z = (z

+

, z

−

)

be a pair of opposite hambers su h that

max{d

X

+(c

+

, z

+

); d

X

−(c

−

, z

−

)} 6 1.

Thenthere exists

σ ∈ Σ

su h that

σ.z = c

.

Proof. Itis enough to dealwiththe ase when

max{d

X

+

(c

+

, z

+

); d

X

−(c

−

, z

−

)} = 1

.

Ifboth

z

−

and

z

+

belong to the twin apartment

A = A

−

⊔ A

+

, we an write

z

+

= w

+

.c

+

and

z

−

= w

−

.c

−

for

w

±

∈ W

uniquely dened by

z

±

. Sin e

z

−

and

z

+

are assumed to be opposite, the odistan e

δ

∗

(z

−

, z

+

)

is by denition equal to

1

W

. Sin e the diago-nal

Γ

-a tion on

X

−

× X

+

preserves odistan es, we dedu e that

w

+

= w

−

. At last sin e

max{d

X

+

(c

+

, z

+

); d

X

−(c

−

, z

−

)} = 1

,we dedu ethat thereexists a anoni al ree tion

s ∈ S

su h that

w

±

= s

and this ree tion is represented by anelement

n

s

∈ Stab

Γ

(A)

;we learly have

n

s

∈ Σ

.

We hen eforth deal withthe ase when at least one of the elements

z

±

does not lie in

A

. Up to swit hing signs, we may  and shall  assume that

z

−

6∈ A

−

. Let

s

be the anoni al ree tion su h that

z

−

is

s

-adja ent to

c

−

. By the Moufang property, the group

U

−

α

s

a ts simplytransitively on the hambers

6= c

−

whi h are

s

-adja ent to

c

−

. By onjugating by an element

n

s

as above and sin e

z

−

6= s.c

−

(be ause

z

−

6∈ A

−

), we on lude that there exists

If

u

+

.z

+

∈ A

+

,then sin e the

Γ

-a tion preservesthe odistan e, the hamber

u

+

.z

+

∈ A

+

is the unique hamber in

A

whi h is opposite

c

−

= u

+

.z

−

, namely

c

+

; we are thus done in this ase be ause we learly have

u

+

∈ Σ

.

We nish by onsidering the ase when

u

+

.z

+

6∈ A

+

. Then there exists some anoni al ree tion

t ∈ S

su h that

u

+

.z

+

is

t

-adja ent to

c

+

and we an nd similarly an element

u

−

∈ U

−

α

t

\ {1}

su h that

u

−

.(u

+

.z

+

) = c

+

. Setting

σ = u

−

u

+

, we obtain an element of

Γ

sending

z

±

to

c

±

. Sin ethe

Γ

-a tionpreservesea hadja en yrelation,hen ethe ombinatorial distan es,wehave

σ ∈ Σ

be ause

d

X

−(c

−

, σ.c

−

) = d

X

−(u

−

1

−

.c

−

, u

+

.c

−

) = d

X

−(c

−

, u

+

.c

−

) = 1

and

d

X

+

(c

+

, σ.c

+

) = d

X

+

(c

+

, u

−

.c

+

) = 1

.



3.2. Proof of non-distortion. We denethe ombinatorial distan e

d

X

of the hamberset of

X

by

d

X

(x

+

, x

−

), (y

+

, y

−

)

= d

X

+

(x

+

, y

+

) + d

X

−(x

−

, y

−

).

Sin e the

G

-a tion on

X

is o ompa t, it follows from the ’var Milnor lemma [BH99, Proposition I.8.19℄ that

G

is quasi-isometri to

X

. Hen e Theorem 1.1 is an immediate onsequen e of the following.

Proposition 3.2. Let

Γ < G = G

+

× G

−

be a twin building latti e asso iated withthe twin building

X = X

+

× X

−

and let

c = (c

+

, c

−

) ∈ X

be a pair of opposite hambers. Then for ea h

γ ∈ Γ

,we have:

1

2

d

X

(c, γ.c) 6 |γ|

Σ

6

2d

X

(c, γ.c).

Proof of Proposition 3.2. Writing

γ ∈ Γ

asa produ tof

|γ|

Σ

elements ofthegeneratingset

Σ

andusing triangleinequalities, weobtain

d

X

(c, γ.c) 6 2|γ|

Σ

bythedenitionof

d

X

and of

Σ

.

It remains to prove the other inequality, whi h says that

Γ

-orbits spread enough in

X

. We set

x = (x

+

, x

−

) = γ

−

1

_.c

. Let us pi k a minimal gallery in

X

−

, from

x

−

to

c

−

. Using auxiliary positive hambers, one opposite for ea h hamber of the latter gallery, a repeated

use ofLemma 3.1shows thatthere exists

γ

−

∈ Γ

su hthat

γ

−

.x

−

= c

−

and

(∗)

|γ

−

|

_Σ

6

d

X

−(c

−

, x

−

)

.

Moreoverasinthe rstparagraph, we have:

(∗∗)

d

X

+

(c

+

, γ

−

.c

+

) 6 |γ

−

|

Σ

,

bythedenitionof

Σ

. We dedu e:

d

X

+

(c

+

, γ

−

.x

+

) 6 d

X

+

(c

+

, γ

−

.c

+

) + d

X

+

(γ

−

.c

+

, γ

−

.x

+

)

6

|γ

−

|

Σ

+ d

X

+

(c

+

, x

+

)

6

d

X

−(c

−

, x

−

) + d

X

+

(c

+

, x

+

),

su essively by the triangle inequality, by

(∗∗)

and thefa t the

Γ

-a tion is isometri for the ombinatorial distan es on hambers, and by

(∗)

. Therefore,bydenition of

d

X

, we already have:

(∗ ∗ ∗)

d

X

+

(c

+

, γ

−

.x

+

) 6 d

X

(c, x)

.

We now onstru t a suitable element

γ

+

∈ Γ

su h that

γ

+

.x

+

= c

+

and

γ

+

.c

−

= c

−

. Let

be thetwin apartment dened by the opposite pair

c = (c

+

, c

−

)

. Let

c

0

= c

−

, c

1

, . . . , c

k

be the gallery ontained in

A

+

and asso iated to

z

0

, z

1

, . . . , z

k

= c

+

as in Lemma 2.1. Noti e that, sin e

c

k

is opposite

z

k

= c

+

and sin e

c

−

is theunique hamber of

A

−

opposite

c

+

,we have

c

k

= c

−

.

By Lemma 3.1, there exists

σ

1

∈ Σ

su h that

σ

1

.z

k−1

= z

k

and

σ

1

.c

k−1

= c

k

. Moreover a straightforward indu tive argument yields for ea h

i ∈ {1, . . . , k}

an element

σ

i

∈ Σ

su h that

σ

i

σ

i−1

. . . σ

1

.z

k−i

= z

k

and

σ

i

σ

i−1

. . . σ

1

.c

k−i

= c

k

. Let now

γ

+

= σ

k

. . . σ

1

, so that

|γ

+

|

Σ

6

k = d

X

+

(c

+

, γ

−

.x

+

)

. By onstru tion, we have

γ

+

.(γ

−

.x

+

) = c

+

and

γ

+

.c

−

= c

−

, that is

(γ

+

γ

−

).x = c

. Therefore

(γ

+

γ

−

γ

−

₁

).c = c

and hen e there is

σ ∈ Σ

su h that

γ = σγ

+

γ

−

. In fa t, sin e

σ

xes

c

,it follows that

σσ

′

∈ Σ

for ea h

σ

′

∈ Σ

. Upon repla ing

σ

k

by

σσ

k

,wemay and shallassume that

γ = γ

+

γ

−

. Therefore we have:

|γ|

Σ

6

|γ

+

|

Σ

+ |γ

−

|

Σ

6

d

X

+

(c

+

, γ

−

.x

+

) + d

X

−(c

−

, x

−

),

the last inequality oming from

|γ

+

|

Σ

6

k = d

X

+

(c

+

, γ

−

.x

+

)

and

(∗)

above. By

(∗ ∗ ∗)

and the denitionof

d

X

,this nallyprovides

|γ|

Σ

6

2 · d

X

(c, γ.c)

,whi h nishestheproof.



3.3. A remark on distortion of latti es in rank one groups. Let

G = G

+

× G

−

be produ t of two totally dis onne ted lo ally ompa t groups, let

π

±

: G → G

±

denote the anoni al proje tions and let

Γ < G

be a nitely generated latti e. Assume that

π

−

(Γ)

is o ompa t in

G

−

(this is automati for example if

Γ

isirredu ible). Letalso

U

−

< G

−

be a ompa t open subgroup and set

Γ

−

= Γ ∩ (G

+

× U

−

)

. Then theproje tion of

Γ

−

to

G

+

is a latti e, and itis straightforward to verifythat, if

Γ

−

is nitely generated and undistorted in

G

−

, then

Γ

isundistorted in

G

.

Weemphasizehoweverthat,inthe aseoftwinbuildinglatti es,thelatti e

Γ

−

shouldnotbe expe tedtobeundistortedin

G

−

beyondtheane ase(whi h orrespondstothe lassi al ase of arithmeti latti esinsemi-simple groups overlo alfun tion elds). Indeed,a typi al

non-ane aseiswhen

G

+

and

G

−

areGromovhyperboli (equivalently,theWeylgroupisGromov hyperboli or, still equivalently, ea h of the buildings

X

+

and

X

−

are Gromov hyperboli ). Then anon-uniform latti e in

G

+

isalways distorted, asfollows fromthefollowing.

Lemma 3.3. Let

G

be a ompa tly generated Gromov hyperboli totally dis onne ted lo ally ompa t group and

Γ < G

be a nitely generated latti e. Then the following assertions are equivalent.

(i)

Γ

isa uniform latti e. (ii)

Γ

isundistorted in

G

.

(iii)

Γ

isa Gromov hyperboli group.

Proof. (i)

⇒

(ii)Follows fromthe ’var Milnor Lemma.

(ii)

⇒

(iii) Follows fromthewell-knownfa tthataquasi-isometri ally embedded subgroupof a Gromovhyperboli groupisquasi- onvex.

(iii)

⇒

(i) By Serre's ovolume formula (see [Ser71℄) a non-uniform latti e in a totally dis- onne ted lo ally ompa t grouppossessesnite subgroups of arbitrarylarge order, and an

3.4. Various notions of rank. Asa onsequen e of Theorem 1.1, we obtain the following estimate for one of the most basi quasi-isometri invariants atta hed to a nitely generated

group.

Corollary 3.4. Let

Γ < G = G

+

× G

−

be a twin building latti e with nite symmetri generating subset

Σ

. Let

r

denote the quasi-at rankof

(Γ, d

Σ

)

andlet

R

denotethe at rank of the building

X

±

. Thenwe have:

R 6 r 6 2R

.

Re all that by denition, the at rank (resp. quasi-at rank) of a metri spa e is

the maximal rank of a at (resp. quasi-at), i.e. an isometri ally embedded (resp.

quasi-isometri ally embedded) opy of

R

n

. By [CH09℄ the at rank of a building oin ides with

themaximal rankof afreeAbelian subgroupof itsWeylgroup

W

,and thisquantitymaybe omputed expli itlyintermsof theCoxeterdiagram of

W

,see[Kra09,Theorem 6.8.3℄. Proof of Corollary 3.4. Let us rst prove

r 6 2R

. Let

ϕ : (R

r

_{, d}

eucl

) → (Γ, d

Σ

)

denote a quasi-isometri embedding ofa Eu lidean spa eintheCayley graphof

Γ

. Withthenotation of Proposition 3.2, we know that the orbit map

ω

c

: Γ → X

+

× X

−

dened by

γ 7→ γ.c

is a quasi-isometri embedding. Therefore the omposed map

ω

c

◦ ϕ : (R

r

_{, d}

eucl

) → X

+

× X

−

isa quasi-isometri embedding. By[Kle99, Theorem C℄,this implies theexisten e ofats of

dimension

r

inthe produ toftwo spa es ofat rank

R

;hen e

r 6 2R

.

We now turn to the inequality

R 6 r

. Asmentioned above, itis shown in[CH09℄theat rank of a building oin ides with the at rank of any of its apartment. Sin e the standard

twinapartment is ontainedinthe imageof

Γ

undertheorbitmap

Γ → X

+

× X

−

,thedesired inequalityfollowsdire tly fromthenon-distortion of

Γ

establishedinProposition 3.2.



Note that another notion of rank, relevant to G. Willis' general theory of totally

dis on-ne ted lo ally ompa t groups, isdis ussed forthefullautomorphism groups

G

±

= Aut(X

±

)

in[BRW07℄,and turnsout to oin ide withtheabove notions ofrank.

Proof of Corollary 1.2. Sin e there exist twin buildings of arbitrary at rank ( hoose for in-stan e Dynkin diagrams su h that the asso iated Coxeter diagram ontains more and more

ommuting

A

e

2

-diagrams),wededu ethattwinbuildinglatti esfallintoinnitelymany quasi-isometry lasses. Thisobservationmaybe ombinedwiththesimpli itytheoremfrom[CR09℄

toyield thedesiredresult.



Appendix A. Integrability of undistorted latti es

Inthisse tion, wegiveupthespe i settingoftwinbuildinglatti esandprovide asimple

onditionensuringthatnon-distortednitelygeneratedlatti esintotallydis onne tedgroups

are square-integrable.

A.1. S hreiergraphs andlatti ea tions. Letusa onsideratotallydis onne ted,lo ally

ompa t group

G

. As before we assume that

G

ontains a nitely generated latti e, say

Γ

, whi h implies that

G

is ompa tly generated [CM08, Lemma 2.12℄. By [Bou07, III.4.6, Corollaire1℄,we knowthat

G

ontainsa ompa t open subgroup,say

U

. Let

C

bea ompa t generatingsubsetof

G

whi h,upon repla ing

C

by

C ∪ C

−

1

,we mayandshall assumeto

be symmetri :

C = C

−

₁

. We set

Σ = U CU

b

,whi h isstill a symmetri generatingset for

G

. We nowintrodu e theS hreier graph

g

have

g

−

1

_{h ∈ b}

_Σ

[Mon01,Ÿ11.3℄. The natural

G

-a tionon

g

by left translation is proper, and itisisometri wheneverwe endow

g

with themetri

d

g

for whi hall edgeshave length 1. We view theidentity lass

1

G

U

asabasevertex ofthegraph

g

,whi h we denoteby

v

0

.

Denotingby

k · k

b

Σ

theword metri on

G

atta hed to

Σ

b

,wehave:

kgk

b

Σ

= d

Σ

b

(1

G

, g)

for any

g ∈ G

. Noti e that the generating set

Σ

b

of

G

onsistsbydenition of those elements

g ∈ G

su hthat

d

g

(v

0

, g.v

0

) 6 1

. Inparti ular, forall

g, h ∈ G

,we have:

d

g

(g.v

0

, h.v

0

) 6 d

_Σ

b

(g, h) 6 d

g

(g.v

0

, h.v

0

) + 1.

Moreover

d

g

(g.v

0

, h.v

0

) = d

_Σ

b

(g, h)

whenever

g.v

0

6= h.v

0

.

In thepresent setting, using again[Bou07,III.4.6,Corollaire 1℄andthedis reteness ofthe

Γ

-a tion, we may  and shall  work with a S hreier graph

g

dened by a ompa t open subgroup

U

small enough to satisfy

Γ ∩ U = {1

G

}

. Thus we have:

Stab

Γ

(v

0

) = Γ ∩ U = {1

G

}.

Let

V

= {v

0

, v

1

, . . . }

beasetofrepresentativesforthe

Γ

-orbitsofverti es. Theelement

v

0

isthepreviousone,andforea h

i > 0

,we hoose

v

i

insu hawaythat

d

g

(v

i

, v

0

) 6 d

g

(v

i

, γ.v

0

)

forall

γ ∈ Γ

;this ispossible be ause the distan e

d

g

takesintegral values. Weset

g

0

= 1

; for ea h

i > 0

,sin e the

G

-a tion onthe verti es of

g

is transitive,there exists

g

i

∈ G

su h that

g

i

.v

i

= v

0

. Thus for any

g ∈ G

there exists

j > 0

su h that

g.v

0

∈ Γ.v

i

,whi h provides the partition:

G =

G

j>

0

Γg

−

1

j

U.

Furthermore, for ea h

i > 0

, we hoose a Borel subset

V

i

⊂ U

whi h is a se tion of the right

U

-orbitmap

U → Γ \ (Γg

−

1

i

U )

dened by

u 7→ Γg

−

1

i

u

. Setting

F

i

= g

−

1

i

V

i

g

i

,we obtain a subset

F

i

of

Stab

G

(v

i

)

su h that

F

₌

G

i>0

F

v

i

g

−

1

i

isa Borelfundamental domain for

Γ

in

G

. We normalize the Haarmeasure on

G

so that

F

hasvolume

1

.

A.2. Non-distortion implies square-integrability. We an nowturn to theproof ofthe

latterimpli ation,more pre isely Theorem1.3.

Proof of Theorem 1.3. Let

g ∈ G

and

h ∈ F

.

On the one hand, by denition of the indu tion o y le

α : G × F → Γ

, the element

α(g, h) = γ ∈ Γ

is dened by

γhg ∈ F

. Therefore, by onstru tion of the fundamental domain

F

, there exist

i > 0

and

u ∈ F

i

su h that

γhg = ug

−

1

i

. Let us apply the latter element to the origin

v

0

of

g

. We obtain

γhg.v

0

= ug

−

1

i

v

0

= u.v

i

, and sin e

u ∈ F

i

and

F

i

⊂ Stab

G

(v

i

)

, this nallyprovides

γhg.v

0

= v

i

. By this and the hoi e of

v

i

inits

Γ

-orbit, wehave:

(⋆)

d

g

(v

0

, v

i

) 6 d

g

(v

0

, γ

−

₁

.v

i

) = d

g

(v

0

, hg.v

0

)

.

On the other hand, let

Σ

be a nite symmetri generating set for

Γ

and let

d

Σ

be the asso iated word metri ; we set

|γ|

Σ

= d

Σ

(1

G

, γ)

for

γ ∈ Γ

. Sin e the metri spa es

(G, d

b

Σ

)

and

(g, d

fa tthatthe

Γ

-orbitmap

Γ → g

of

v

0

dened by

γ 7→ γ.v

0

isaquasi-isometri embedding. In parti ular, there exist onstants

L > 1

and

M > 0

su hthat

|γ|

Σ

6

L · d

g

(v

0

, γ.v

0

) + C

for all

γ ∈ Γ

. Moreover

d

g

takes integer values and

Stab

Γ

(v

0

) = {1

G

}

, so for all non-trivial

γ ∈ Γ

we have:

L.d

g

(v

0

, γ.v

0

) + C 6 (L + C).d

g

(v

0

, γ.v

0

)

. Therefore, upon repla ing

L

by a larger onstant we may and shallassumethat

C = 0

.

Ouraimistoevaluate

|γ|

Σ

= |α(g, h)|

Σ

intermsof

kgk

b

Σ

and

khk

b

Σ

. Notethat

|γ|

Σ

= |γ

−

₁

|

Σ

sin e

Σ

issymmetri .

First, we dedu e su essively from non-distortion, from the triangle inequality inserting

γ

−

₁

.v

i

,and fromthe fa tthat the

Γ

-a tionon

g

isisometri ,that:

|γ

−

₁

|

Σ

6

L · d

g

(v

0

, γ

−

₁

.v

0

)

6

L · d

g

(v

0

, γ

−

1

_.v

j

) + d

g

(γ

−

1

_.v

0

, γ

−

1

.v

j

)

6

L · d

g

(v

0

, γ

−

1

_.v

j

) + d

g

(v

0

, v

j

)

.

Then,wededu esu essivelyfrom

(⋆)

,fromthetriangleinequalityinserting

h.v

0

,andfrom the fa tthatthe

G

-a tionon

g

is isometri ,that:

|γ

−

1

_|

Σ

6

2L · d

g

(v

0

, hg.v

0

)

6

2L · d

g

(v

0

, h.v

0

) + d

g

(h.v

0

, hg.v

0

)

6

2L · d

g

(v

0

, h.v

0

) + d

g

(v

0

, g.v

0

)

.

Finally, bydenition of the S hreier graph we dedu e that

|γ

−

1

_|

Σ

6

2L · kgk

Σ

b

+ khk

Σ

b

.

Re allthat we want to prove that the fun tion

h 7→ |α(g, h)|

Σ

belongs to

L

p

_{(F , dh)}

. Sin e

Vol(F , dh) = 1

, sodoes the onstant fun tion

h 7→ kgk

b

Σ

, therefore it remains to prove the

lemmabelow.



Lemma A.1. The fun tion

h 7→ khk

b

Σ

belongs to

L

p

_{(F , dh)}

.

Proof. Let

h ∈ F

. By onstru tion ofthe fundamentaldomain

F

,thereexist

i > 0

and

u

i

in

F

i

,hen ein

Stab

G

(v

i

)

,su hthat

h = u

i

g

−

1

i

. Thisimplies

h.v

0

= u

i

.(g

−

1

i

.v

0

) = u

i

.v

i

= v

i

,and also

(γh).v

0

= γ.v

i

for ea h

γ ∈ Γ

. Now the expli it form of the quasi-isometry equivalen e (A.1) between

(g, d

g

)

and

(G, d

b

Σ

)

implies:

d

g

(v

0

, h.v

0

) 6 khk

_Σ

b

6

d

g

(v

0

, h.v

0

) + 1

, and

d

g

(v

0

, (γh).v

0

) 6 kγhk

_Σ

b

6

d

g

(v

0

, (γh).v

0

) + 1

.

Moreover by the hoi e of

v

i

in its

Γ

-orbit, we have

d

g

(v

0

, h.v

0

) 6 d

g

(v

0

, (γh).v

0

)

for any

γ ∈ Γ

. Thisallows us to put togethertheabove two doubleinequalities, and to obtain (after forgetting the extreme upperandlower bounds):

(†)

khk

_Σ

_b

6

kγhk

_Σ

_b

+ 1

.

forany

h ∈ F

and

γ ∈ Γ

.

Butinviewof

(†)

and oftheunimodularity of

G

(whi h ontains a latti e),wehave:

Z

F ∩γ

−1

_Ω

khk

_Σ

_b

p

dh 6

Z

F ∩γ

−1

_Ω

kγhk

_Σ

_b

+ 1

p

dh =

Z

γF ∩Ω

khk

_Σ

_b

+ 1

p

dh

,

whi h nallyprovides

Z

F

khk

_Σ

_b

p

dh 6

X

γ∈Γ

Z

γF ∩Ω

khk

_Σ

_b

+ 1

p

dh =

Z

Ω

khk

_Σ

_b

+ 1

p

dh

.

The on lusionfollows be ause

F

hasHaarvolume equal to 1and be ause byassumption

h 7→ khk

_Σ

_b

belongs to

L

p

_{(Ω, dh)}

.



A.3.

p

-integrability of twin building latti es. Letus nish bymentioningthe following fa t whi h, using Theorem 1.3, allows us to prove the main result of [Rém05℄ in a more on eptualway.

Lemma A.2. Let

Γ

be a twin building latti e and let

G

be the produ t of the automorphism groupsof the asso iated buildings

X

±

. Let

W

be the Weyl group and

P

n>0

c

n

t

n

be the growth seriesof

W

withrespe ttoits anoni alsetofgenerators

S

,i.e.,

c

n

= #{w ∈ W : ℓ

S

(w) = n}

. Let

q

min

denote the minimal order of root groups and assume that

P

n>0

c

n

q

_min

−n

< ∞

. Then

Γ

admits a fundamental domain

F

in

G

, with asso iated indu tion o y le

α

F

, su h that

h 7→ α

F

(g, h)

belongs to

L

p

_{(F , dh)}

for any

g ∈ G

and any

p ∈ [1; +∞)

.

Proof. We freely use the notation of 3.1 and [Rém05℄. We denote by

B

±

the stabilizer of the standard hamber

c

±

in the losure

Γ

Aut(X

±

)

. By [lo . it.℄ there is a fundamental

domain

F

= D =

F

w∈W

D

w

su h that

Vol(D

w

, dh) 6 q

−

ℓ

S

(w)

min

. If we hoose the ompa t generating set

Σ =

b

F

(s

−

,s

+

)∈S×S

B

−

s

−

B

−

× B

+

s

+

B

+

, we see that by denition of

D

w

, whi h his ontained in

B

−

× B

+

w

, we have

khk

b

Σ

6

ℓ

S

(w)

for any

w ∈ W \ {1}

and any

h ∈ D

w

. Therefore for any

p ∈ [1; +∞)

we have:

Z

F

khk

_Σ

_b

p

dh 6

X

n>0

n

p

c

n

q

−

_min

n

,from whi h

the on lusionfollows.



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UniversitéCatholiquedeLouvain,Chemin duCy lotron2,1348Louvain-la-Neuve,Belgium

E-mail address: pierre-emmanuel. apra eu louv ain.b e

UniversitédeLyon,UniversitéLyon 1,InstitutCamilleJordan, UMR5208duCNRS,

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