Geometric Property(T)∗

Rufus WILLETT Guoliang YU

1 Introduction

In[18,Section 7],the current authors introduced geometric property(T):This is a pathological property of metric spaces designed to be an obstruction to the maximal version(see[8])of the coarse Baum-Connes conjecture inK-theory and higher index theory.The treatment in[18]was quite brief;moreover,we expect that geometric property(T)will see some applications outside ofK-theory and higher index theory.It is the purpose of this paper to develop the theory more fully.

Throughout we will work with discrete metric spacesXof bounded geometry:This means that ifB(x;r)denotes the ball of radiusratx∈X,then the quantityis finite for allr.We allow our metrics to take in finite distances.For applications,the two most interesting examples of such spaces are:The vertex set of a graph equipped with the edge metric(for example,the Cayley graph of a finitely generated group);and the disjoint union of a sequence(Xn)of finite graphs,where eachXnhas the edge metric,and the distance between di ff erent graphs is in finity(the coarse geometry of such a space is essentially the “asymptotic geometry”of the sequence).The bounded geometry assumption amounts to the existence of an absolute bound on the degree of all vertices in either case.

Geometric property(T)for a discrete metric spaceXsays that unitary representations of the Gromov-Roe translation algebra Cu[X](see[9,p.262]and[13,Chapter 4])that have almost invariant vectors must have invariant vectors(see Definition 3.4 below).This is a direct analogue of property(T)for a(discrete)group Γ,which says that any unitary representation of the group algebra C[Γ]having almost invariant vectors actually has invariant vectors.In order to make sense of“invariant”and “almost invariant” in the case of metric spaces one has to do a little work,but the basic idea is the same as in the group case.

Geometric property(T)can also be characterised(see Proposition 5.2 below)in terms of a spectral gap property for a Laplacian operator Δ in Cu[X],much as was done by Valette for groups[16,Theorem 3.2].In the case thatXis a graph(or is built from a sequence of graphs),Δ simply is the graph Laplacian.This was our original definition in[18,Section 7],but we found the version based on almost invariant vectors more convenient to work with in the current paper.

In this paper we establish the machinery needed to make rigorous sense of the above definitions.We then prove the following results.

Theorem 1.1(1)Geometric property(T)is a coarse invariant(Theorem4.1).

(2)Geometric property(T)for a sequence of finite graphs implies that the sequence is an expander(Corollary5.2),but is strictly stronger than this(Corollary7.1).

(3)An infinite connected graph X has geometric property(T)if and only ifit is not amenable(Corollary6.1).

(4)LetΓbe a finitely generated discrete group,and···be a sequence of finiteindex normal subgroups such that∩nΓnis the trivial subgroup.ThenΓhas property(T)if and only if the sequence(Γ/Γn)of finite Cayley graphs1Defined with respect to some fixed generating of Γ.has geometric property(T)(Theorem7.1).

(5)A sequence of finite graphs(Xn)with geometric property(T)can not admit a fibered coarse embedding into Hilbert space[5],or have the boundary Haagerup property[7],unlesssupn|Xn|is finite(Theorem8.1).

A few remarks are in order.Points(2)and(3)together suggest that geometric property(T)is not interesting for a single connected graph,but it has serious content for a sequence of finite graphs.Points(1)and(4)have the following consequence,which is perhaps surprising:for a residually finite group,property(T)can be characterised by the geometry of the finite quotients of the group;this contrasts with the well-known fact that property(T)for the group itselfis not a geometric invariant(see[1,Section 3.6]).Property(5)is a strong analogue in coarse geometry of the well-known incompatibility of property(T)and a-T-menability for groups.

OutlineTo facilitate algebraic computations involving Cu[X],we use the language of abstract coarse structures(see[13]),rather than the metric space language of the introduction,throughout the body of this paper;Section 2 recalls the basic definitions of coarse structures and proves some combinatorial lemmas.Section 3 introduces the translation algebra Cu[X],the notion of invariant vectors in its representations,and geometric property(T).Section 4 proves that geometric property(T)is a coarse invariant;we could not find a short proof of this result and this is probably the most technical part of the paper.Section 5 defines general combinatorial Laplacian operators,and characterises invariant vectors and geometric property(T)in terms of them.

Having established the basic properties of geometric property(T),the next three sections discuss the relationship between geometric property(T)and some other properties.Section 6 discusses the relationship of representations of Cu[X]with amenability,and uses this to characterise geometric property(T)for spaces as in part(3)of Theorem 1.1.Section 7 studies the relationship with property(T)for groups,proving part(4)of Theorem 1.1.Section 8 discusses the relationship with the coarse a-T-menability properties in part(5)of Theorem 1.1.

We conclude the paper with some natural open questions in Section 9.

2 Coarse Structures and Some Combinatorics

In this section we first recall the definition of a coarse structure on a setX.We then recall the definition of partial translation,and prove some combinatorial lemmas about decomposing general controlled sets into partial translations.

IfXis a set andE,Fare subsets ofX×X,then the composition ofEandF,denoted byE◦F,is the set

and the inverse ofEis

Forn≥1,we use the shorthand

Finally,we will write diag(E)for the “diagonal part”ofE,that is

Definition 2.1Let X be a set.A coarse structure on X consists of a collection E of subsets of X×X such that

2This condition is not always assumed:coarse structures satisfying this condition are sometimes called unital.

The members of E are called controlled sets for the coarse structure.

A set X equipped with a coarse structure is called a coarse space.

The motivating example comes whenXis a metric space,and a set is controlled if and only ifit is a subset of a“tube”for somer>0.

The following definition lists some additional properties of controlled sets and coarse spaces that we will need.

Definition 2.2Let X be a coarse space,and E be the coarse structure on X.

(1)A controlled set E is called symmetric if E=E−1.

(2)The coarse structure is said to have bounded geometry if for any controlled set E,there is a bound M=M(E)such that for any x∈X,

(3)A controlled set E is said to be generating if for any controlled set F,there exists n suchthat.A coarse space is said to be monogenic if there exists a generating controlled setfor the coarse structure.

(4)Two elements x,y of X are in the same coarse component of X if the set{(x,y)}is controlled.“Being in the same coarse component”defines an equivalence relation on X,and the equivalence classes are called coarse components.If there is only a single coarse component,X is said to be coarsely connected.

(5)If Y is a subset of X,it is itself a coarse space with the controlled sets being the intersection of the controlled sets for X with Y×Y.This is called the induced coarse structure,and will be used implicitly many times below.

Definition 2.3We will say that X is a space as an abbreviation for“X is a bounded geometry,monogenic coarse space,with at most countably many coarse components”.

Note that spaces are automatically countable;this and being monogenic implies that the coarse structure on a space always comes from a metric(see[13,Section 2.5]),with possibly infinite distances.Nonetheless,the language of abstract coarse structures is more convenient for the computations in this paper.

The reader will probably find it useful to keep the following example in mind.

Example 2.1LetXbe the vertex set of an undirected graph,andEbe the set of edges,which we consider as a symmetric subset ofX×X.The coarse structure generated by the setEis monogenic,and is bounded geometry if and only if there is a uniform bound on the degrees of all vertices inX.The coarse components ofXare exactly the(vertex sets of the)connected components of the underlying graph.The coarse structure above is the same as that defined by the edge metric,which sets the distance between two vertices to be the shortest number of edges in a path between them,and infinity if no such path exists.

Particularly important classes of examples are Cayley graphs of discrete groups,and discretisations of Riemannian manifolds.Another important example for us is obtained whenXis a disjoint unionof finite connected graphs:Examples of this form are important in coarse geometry as they are relatively easy to analyse,and as questions about general spaces can often be reduced to questions about spaces of this form.

Remark 2.1Letbe a disjoint union of finite connected graphs as in Example 2.1 above.It is common in coarse geometry(in order to avoid infinite-valued metrics)to metrize such spaces with any metric that restricts to the edge metric on the individualXnand satisfiesd(Xn,XXn)→∞asn→∞.The corresponding coarse structure is not monogenic,but is“weakly monogenic” in the following sense.A controlled setEis said to be a weak generating set for the coarse structure if for any controlled setF,there existsn∈N such thatFE◦nis finite.A coarse space is said to be weakly monogenic if there is a weak generating set for the coarse structure.Most of the results of this paper hold for weakly monogenic spaces,up to minor adjustments(see Remark 3.1 below),so can be applied to such spaces directly.

The following definition is based on[3,Definition 8].

Definition 2.4Let X be a space.A partial translation on X consists of the following data:subsets A and B of X and a bijection t:A→B such that the graph3We have defined the graph of t the “wrong way round” to better match matrix multiplication later.of t

is controlled.The subset A is called the support of t,and B its range.The inverse of a partial translation t:A→B is the partial translation t−1:B→A defined by inverting the bijection t.

A controlled setEis called elementary if there exists a(necessarily unique)partial translationt:A→Bsuch thatE=graph(t).An elementary controlled set is called antisymmetric if the domain and range of the corresponding partial translation do not intersect4Equivalently,the images of the two coordinate projections are disjoint when restricted to E..

In the remainder of this section,we prove some combinatorial lemmas about decomposing controlled sets into partial translations;these will be useful for algebraic computations later in the paper.

The following very general lemma is probably well-known.

Lemma 2.1Let A be a set,and B and C be subsets of A.Let t:B→C be a bijectionsuch that tThen there exists a decomposition

of B into(at most)three disjoint subsets such that t(Bi)∩Bi=∅for all

The exampleA=B=C={1,2,3},tis a cyclic permutation,shows that one can not get away with less than three subsets.

Proof of Lamma 2.1Lets:C→Bbe any bijection which is the identity onReplacingtwiths◦t,it is not difficult to see that it suffices to prove the following statement:IfBis a set andt:B→Bis a bijection such thatfor allthen there exists a decompositionsuch thatfor allWe now prove this.

The bijectiont:B→Bgives rise to an action of Z,which partitionsBinto orbits.Asfor allb∈B,each orbit for this action has one of the following forms.

(1)(going on infinitely in both directions)for someb∈B.

(2)for somen≥1 andb∈Bsuch that

Define subsetsB0,B1andB2ofBas follows.For each orbit,fix once and for all a representation of one of the types above.For an orbit of type(2)withneven andi=n,putti(b)intoB2.In all other cases,putti(b)intoBimod2(whereimod 2 is always construed as 0 or 1).A routine case-by-case analysis shows that this works.

Lemma 2.2Letbe symmetric controlled sets on a space X.Then there existelementary controlled sets E1,···,Ensuch that F is the disjoint union

We may assume moreover that each Eiis antisymmetric.

ProofInductively defineE0=EandEi+1to be any maximal elementary subset of

such that

Assume thatEi+1is not empty,and assume that(x,y)is inEi+1,noting that this forcesThen maximality of eachEjforces the existence of distincty0,···,yisuch that(x,yj)is infor eachj=0,···,i.In particular,

which is impossible forisuitably large by the bounded geometry condition.ThusF=E∪diagfor somen.

Finally,note that Lemma 2.1 applied to the partial translation underlying eachEidecomposesEiinto three antisymmetric parts.Decomposing further,we may thus assume that eachEiis antisymmetric.

Lemma 2.3Let t:A→B be a partial translation on a space X and E be a controlled set for X.Assume that

for some n≥1.

Then there exists a decompositionsuch that if tiis the restriction of t toAithen there exist partial translationssuch that

(1)

(2)graphfor all i=1,···,m and j=1,···,n;

(3)for each i and each j=1,···,n−1,the range ofis equal to the domain of

(4)for each i and j=1,···,n,eitheris the identity map,or the range ofis disjointfrom its support.

ProofFor eachx∈Athe pair(t(x),x)is contained inE◦n,whence we may choose pointsx=r0(x),r1(x),···,rn(x)=t(x)such that for eachj=1,···,n,the pair(rj(x),rj−1(x))is inE.In this way,we define functionsrj:A→X.

Note that the bounded geometry assumption and the fact that the graph ofr1is contained inEimply that there existsN1such thatAdecomposes intosetssuch that the following hold:

•r1is a bijection restricted to each

•eitherr1(x)=r0(x)for allx

Similarly,as the graph ofr2is contained inE◦2,there existsN2such that eachdecomposes into at mostN2setsfor which the restriction ofr2to eachis a bijection,and such that eitherr2(x)=r1(x)for allContinuing in this way,we get a decompositionA=A1,···,Am,wheremis at mostN1N2···Nn,such that

(1)eachrjis a bijection when restricted to eachAi;

(2)for eachi,jeitherrj(x)=rj−1(x)for all

Define for eachi=1,···,m,a functionby the stipulation

as eachrjis injective on eachAi,this is well-defined and bijective.It is not difficult to see that these functionshave the right properties.

3 Translation Algebras and Geometric Property(T)

In this section we introduce translation algebras and define geometric property(T)in terms of their(unitary)representation theory.

Throughout this section,Xdenotes a space in the sense of Definition 2.3.

Definition 3.1The translation algebra,or algebraic uniform Roe algebra of X,denoted byCu[X],is the collection of all X-by-X indexed matrices T=(Txy)x,y∈Xwith entires inCsuch that

is finite,and such that for any T∈Cu[X]the support of T defined by

is a controlled set.The usual matrix operations and adjoint makeinto a∗-algebra.

Note that the collection of matrices insupported on the diagonal constitutes a copy ofinside

Partial translations give rise to operators inin the following way.Lett:A→Bbe a partial translation.Thentgives rise to an operatorvdefined by setting

An operator arising in this way is called a partial translation;note thattandvdetermine each other uniquely,so there should not be any confusion caused by the repeated terminology.It is immediate from the definitions that ifvis a partial translation operator corresponding tot:A→B,thenvis a partial isometry,withv∗the partial translation operator corresponding tot−1.Moreover,the support and range projectionsv∗vandvv∗are the characteristic functions ofA,andBrespectively,considered as elements of the diagonal∗-subalgebraand the support ofvis the graph oft.

Definition 3.2A representation ofis a unital∗-homomorphism π:fromto thealgebra of bounded operators on some Hilbert spaceWe will usuallyleave πimplicit,saying just that H is a representationand writingfor the imageof an elementunder π(T).

The assumption that representations are unital in the above is not very important,and it does not significantly reduce generality but streamlines some arguments slightly.In contrast,the assumption that all representations are∗-preserving is crucial;such representations should be thought of as the analogues of unitary representations of a group.

Definition 3.3Let H be a representationA vectoris said to be invariant,or constant,iffor all partial translations v.

The constant elements form a closed subspace5It is not a subrepresentation in general.of H,which we denote Hc.

Example 3.1is a disjoint union of finite connected graphs as in Example 2.1,then the constant vectors inl2(X)are exactly those square-summable functions onXthat are constant on each coarse componentXn.

Geometric property(T)says that for any representationvectors incan not be “too close” to constants.Here is the formal definition.

Definition 3.4A space X has geometric property(T)if for any controlled generating setE,there exists a constant c=c(E)>0such that for any representation H andthereexists a partial translation v inCu[X]with support in E such that

The reader should compare this to the following definition of property(T)for a discrete group(compare[1,Section 1.1]).For a unitary representation of a finitely generated group Γ,letHcdenote the constant vectors:Thosefor whichfor allA finitely generated6Property(T)forces finite generation on a discrete group,so there is no harm assuming this.group Γ then has property(T),if for any finite generating setEof Γ,there exists a constantc=c(E)>0 such that for any unitary representationand anythere existsg∈Ewith

Remark 3.1A representation ofis called a boundary representation,ifit contains the ideal

in its kernel.Geometric property(T)can be weakened to boundary property(T)by requiring that the property in Definition 3.4 above holds only for all boundary representations.This notion is more appropriate for weakly monogenic coarse spaces as discussed in Remark 2.1.Indeed,the results in this paper all continue to hold for weakly monogenic bounded geometry coarse spaces(with obvious minor variations),if “generating” is replaced by “weakly generating”,“representation” by “boundary representation” and “geometric property(T)” by “boundary property(T)”everywhere.

In the remainder of this section,we give some equivalent formulations of geometric property(T)that will be useful later.

Define a linear map

The map Φ can be used to characterise constant vectors as follows.

Lemma 3.1Letbe a vector in a representationThen the following areequivalent:

(1)For all

(2)For all partial translations v in

Proof For a partial translationv,so clearly(1)implies(2).Assume thatξsatisfies(2),and letTbe an element ofWe may writeTas a finite sum

where eachviis a partial translation,and eachfiis the element ofl∞(X)defined by

Noting thatfor alli,we then have

as required.

Here then is the promised equivalent formulation of geometric property(T).

Proposition 3.1The following are equivalent:

(1)X has geometric property(T).

(2)There exists a controlled generating set E and a constant c>0such that for anyrepresentation H andthere exists a partial translation v inCu[X]with support in Esuch that

(3)For any controlled generating set E,there exists a constant c=c(E)>0such that forany representation H andthere exists an operatorwith support in E suchthat

(4)There exists a controlled generating set E and a constant c>0such that for anyrepresentation H and ξ∈H⊥c,there exists an operatorwith support in E such that

ProofIt is clear that the implication(1)implies(2).For the converse,assume thatEandc>0 are as in(2),and letFbe any controlled generating set for the coarse structure.AsFis generating,there existsnsuch thatF◦ncontainsE,and thus property(2)holds withF◦nreplacingE.Now,letHbe a representation ofandξbe a unit vector inUsing property(2)forF◦n,there exists a partial translationt:A→Bwith graph contained inF◦nsuch that ifvis the corresponding operator,then

Now,using Lemma 2.3,there exist partial translationsv1,···,vnsuch thatv=v1···vn,so that suppFfor eachi,and so thatfor alli=2,···,n.We then havethat

Continuing in this way,we may conclude that

whence for somei=1,···,n,We may thus take

We will now show that(2)and(4)are equivalent;the proof that(1)and(3)are equivalent is analogous.Noting as in the proof of Lemma 3.1 that for a partial translationv,Φ(v)=vv∗,it is clear that(2)implies(4),so it suffices to show that(4)implies(2).

Let thenEandc>0 be as in the statement of(4).LetHbe a representation ofletξbe an element ofand letbe as in the statement of(4)for thisξ.We may writeas in the proof of Lemma 3.1,wherendepends only onE(not onT,and eachfihas norm at mostas an element of As∗-representations of theC∗-algebral∞(X)are contractive,eachfialso has norm at mostwhen considered as an operator onH.We have then that

Hence for somei,asndepends only onE,this implies(2).

4 Coarse Invariance

In this section we show that geometric property(T)is a coarse invariant,i.e.,it is invariant under coarse equivalences as in the following definition.

Definition 4.1A function f:X→Y between two spaces is uniformly expansive if for any controlled set E for X,the set

is controlled for Y.Two functions f,g:X→Y between two spaces are close if the setis controlled for Y.

Two spaces X and Y are coarsely equivalent if there exist uniformly expansive functions

such that the compositions f◦g and g◦f are close to the identities on Y and X respectively.

The following example will be important in the proofs that follow.

Example 4.1A subsetYof a spaceX(with the inherited coarse structure)is coarsely dense if there is a controlled setEforXsuch that the set

is non-empty for allx∈X.It is not difficult to see thatYis coarsely dense if and only if the inclusioni:Y→Xis a coarse equivalence,with “the inverse-up-to-closeness”given by any functionp:X→Ythat takes eachx∈Xto anyyin the set in(4.1)above.

Our main goal in this section then is to prove the following result:We stated that we expected this to be true in[18,Section 7],but did not have a complete proof at that time.

Theorem 4.1Let X and Y be coarsely equivalent spaces.Then X has geometric property(T)if and only if Y does.

We start with a well-known “structural result” about coarse equivalences.

Lemma 4.1Let f:X→Y be a coarse equivalence.Then there exist coarsely densesubspacesof X andof Y such that f restricts to a bijection

In otherwords,for any coarse equivalence f:X→Y,there is a factorization

where p:is an inverse-up-to-closeness of the inclusion ofg is a bijectivecoarse equivalence,and iis the inclusion of a coarsely dense subset.

ProofLetFor eachchooseand defineIt is not difficult to check thathave the required properties.

To prove Theorem 4.1,it will thus suffice to prove the following two results.

Lemma 4.2Let f:X→Y be a bijective coarse equivalence.Then X has geometric property(T)if and only if Y does.

Proposition 4.1Let Y be a coarsely dense subspace of a space X.Then Y has geometric property(T)if and only if X does.

Proof of Lemma 4.2Define a functionIt is not difficult to see thatf∗is a∗-isomorphism that restricts to a bijection between thecollections of partial translations inThe result follows immediately from this.

The proof of Proposition 4.1 is more involved.For the benefit of those readers who know about Morita equivalence,we explain the basic idea as follows.We will define a projectionA∈Cu[X]such that

soAis a full projection,implementing a “∗-algebra Morita equivalence” between Cu[X]and Cu[Y].This Morita equivalence implements a bijective correspondence between the sets of representations of Cu[X]and Cu[Y]roughly defined by

The projectionA=χY,the characteristic function ofYinl∞(X),has the above properties,but it does not behave well with respect to constant vectors.We will thus takeAto be a sort of“averaging operator”:This has the crucial property that the correspondences in(4.2)above almost take constant vectors to constant vectors.

Now for the details.We require some notational preliminaries.Fix a decompositionofXinto subsetsUyparametrized byYsuch that for eachy∈Y,Uycontainsyand so that there is a controlled setEsuch thatfor ally(and in particular,is finite);it is not difficult to see that coarse denseness ofYinXimplies that such a decomposition exists.Forx∈X,writey(x)for the(unique)y∈Ysuch thatxis inUy.Fory∈Y,de fineForxinX,we also de fine

We will think ofNas an(invertible)element of

Define now an operatorAinby the formula

The operatorAcan be thought of as an “averaging operator”:As an operator onl2(X),it is the orthogonal projection onto the subspace of functions that are constant on eachUy.Note thatAcommutes withN.

The proof of Proposition 4.1 now proceeds via a series of(mainly algebraic)lemmas.

Lemma 4.3The following hold for the operator A:

(1)IfΦ :is as in(3.1)above,thenis the constant function1.

(2)If H is any representation ofCu[X],then the constant vectors are a subspace of A·H.

(3)An element T ofis inif and only ifwhenever y(x1)=y(x2)and y

(4)The maps α:and β:defined by

and

are mutually inverse∗-isomorphisms betweenand

(5)The setspans

ProofFor(1),we clearly have

Part(2)follows from part(1)and Lemma 3.1.

For(3),letTbe an element ofandThen

The claim follows from these formulas.We will often implicitly use these formulas from now on.

For(4),note thatαis clearly linear and∗-preserving.Note also that for

where the third equality uses part(3).This,however,is equal to

This implies thatαis a∗-homomorphism.The fact thatβdefines the inverse forαnow follows from more direct computations7From now on in this section,to keep the length controlled,we will leave such matrix coefficient computations to the reader.of matrix coefficients,completing the proof of this part.

Finally,for(5),letTbe an element of Cu[X]such that for eachy∈Y,there is at most onexsuch thatis not{0},and similarly there is at most onezsuch thaty(z)=yandis notDefineby

and

(note that the sum definingDhas at most one non-zero element).A direct computation shows thatT=CAD.As any operator in Cu[X]can be written as a finite sum of operatorsTwith the properties above,this completes the proof.

Lemma 4.4For any non-degenerate representationthere is a canonicallyassociated non-degenerate representationwith the following properties.

(1)The representationidentifies canonically with the representation ofvia the isomorphisms in Lemma4.3(4).

(2)Ifis any non-degenerate representation ofgiving rise to a representationAwe have thatandare canonically isomorphic asrepresentations.

ProofGivenHas in the statement,letdenote the algebraic tensor product ofHandtaken over C.Define a form on this tensor product by the formula

on elementary tensors,and extending tofinite sums of elementary tensors by linearity in the second variable,and conjugate linearity in the first.This form is clearly linear in the second variable,and conjugate linear in the first.It is also positive semi-definite.Indeed,note that for any elementwe have

To show that this is non-negative,it suffices to show that the matrixis equal to a finite sum of matrices of the formwithBin

For eachthen,temporarily write the elements ofFor eachand,de fineby

and note that

Note moreover that for anyi,j∈{1,···,n}andk=l,whence

It thus suffices to show that for eachthe matrixis of the formB∗Bfor some which we will now do.Set

Then one checks thatA,whence the matrixis equal to

inthis is of the desired form.

Now letbe the corresponding separated completion offor the semi-definite inner product in(4.6)above.denote the image ofin this Hilbert space and writefor the class of an elementFordefine an operatorπ(T)onby the formula

A similar argument to that used above for positivity shows thatπ(T)is bounded,and thus extends to all ofHX.The mapis then clearly a unital∗-homomorphism,so this gives the desired representation.We now look at properties(1)and(2).

For property(1),de fine a linear mapby the formula

and note that

This implies thatLas in(4.7)is an isometry fromthus extending to an isometric map,which is clearly onto by non-degeneracy.It is also clear thatLintertwines the representations of

Finally,we look at property(2).Define a mapby the formula

Computing

This implies thatMagain extends to an isometric linear map,and Lemma 4.3(5)and nondegeneracy imply that this is onto.Again,it clearly intertwines the representations of Cu[X],so the proofis complete.

It follows from the lemma above that non-degenerateandrepresentations come canonically in pairssuch thatWe will make these assumptions(and use this notation)throughout the rest of the proof of Proposition 4.1.

Our next task is to study the relationship between the constant vectorsandin the spaces above.Note that Lemma 4.3(1)implies thatis a subspace of(and thatis a subspace ofby definition).Letandbe the linear maps defined in(3.1)above and define

Note that Lemma 3.1 implies that a vectoris inif and only iffor allThe following computations contain the bulk of the rest of the proof of Proposition 4.1.

Lemma 4.5The following hold:

(1)For any partial translation

(2)For any partial translation vsuch that for all

we have the formula

(3)The operator N from(4.3)above on HXrestricts to an isomorphism

(4)For anyif we decomposewhereand ξ2is inthen

(5)For anyif we decomposewhereand ξ2is inthen

ProofFor part(1),direct computations show that for anyx,z∈X,the corresponding matrix coefficients are given by

ifandand zero otherwise.For part(2),a direct computation using the formulas in(3.1),(4.4)–(4.5)shows that for anythe matrix coefficients ofare given by

From here,more direct computation shows that the matrix coefficients of the operators in the statement are given by

ifand there existssuch thatand zero otherwise.

For part(3),assume first thatξis an element ofwe want to show thatis inand thusis a superspace ofLetvbe an arbitrary partial translation,we want to show thatThen using the fact thatξis inwe have

Using part(1),this is equal to

using the fact thatNcommutes withand cancelling theNgives the desired conclusion.

Conversely,assume thatξis an element ofwe want to show thatNξis inand thusis a subspace ofLetvbe a partial translation;we want to showSplittingvup as a finite sum of at mostelements,we may assume thatvsatis fies the conditions in(4.9)for anybe de fined by

(roughly speaking,Ccollapses eachUythat intersects the range ofvinto the single point in which it intersects the range ofv).Note thatNcommutes withC.We then have the formulaCAv=v.Now,

where the last equality uses the fact thatξis inContinuing to use part(2),this is equal to

where the second equality uses the fact thatNcommutes withvv∗by the assumption in(4.9).Hencesoas required.

For part(4),note that ifandis inby part(3).Hence taking the inner product withgives

whence

and so(assuming as we may thatThis in turn implies that

so

This combined with the fact thatforces

from which the claimed inequality follows.Part(5)is analogous,and we are done.

We are now finally ready to complete the proof of Proposition 4.1,and thus also that of Theorem 4.1.

Proof of Proposition 4.1Assume first thatYhas geometric property(T).LetFbe a generating controlled set for the coarse structure onY,and letEbe any generating controlled set for the coarse structure onXthat containsF,the controlled setforX×X,and the composition of these two.Using Proposition 3.1,it will suffice to show that there exists some constantdepending only onFand the coversuch that for any unit vectorthere existssupported inEwith matrix coefficients bounded by a number depending only onandF,and with

Letc∈(0,1)be a constant,which will be chosen later in a way that depends only on the coverofXandF.Note that ifthen we are done,asThen assume that

Now,by part(4)of Lemma 4.5(and the fact thatApreserveswe may writewhereξ1is inis inandfor somec1>0 depending only on the cover{Uy}.Using geometric(T)forY(and the fact thatξ1is inthere exists a partial translationv∈Cu[Y]supported inFand a constantc2>0 depending only onFsuch that

Hence

Now,using Lemma 4.5(1)and the fact thatNcommutes withβ(vv∗),this implies that

wherec3>0 depends only onandFagain.Finally,this forces

wherecis as in(4.11).Noting that

and settingwe see that

Note thatis supported in

In summary,we have shown the desired conclusion withwe may takeT=A,and otherwise we may take

For the converse implication,assume thatXhas geometric property(T),and letFbe a controlled generating set forX.LetEbe any controlled set forXthat contains a generating set forYand such that

Using Proposition 3.1 and the isomorphismfrom Lemma 4.3(4),it will suffice to show that there exists some constantdepending only onFand the coversuch that for any unit vectorthere existssupported inEwith matrix coefficients bounded by some number depending only onFandand such that

Letξbe a unit vector inUsing Lemma 4.5(5)we may writewhereξ1is inis inandfor somec1>0 depending only on the cover

Using geometric property(T),there exists a partial translationvsupported inFanddepending only onFsuch that

We may splitvup as a finite sum of at mostpartial translations satisfying the support condition in(4.9),and thus assume that

whereandvsatis fies the support condition in(4.9).

Now,letCbe the “collapsing”operator defined as in(4.10)above for thisv.Using thatandwe see that

Asand N commutes with A,C and vv∗,this implies that

Hence by Lemma 4.5(2),we see that

so,asAcommutes withN,

Asadmits an upper boundc3depending only on the coverthis completes the proof:Take

The following corollary gives our first examples of spaces with geometric property(T);in some sense,these could be considered “trivial”examples.

Lemma 4.6Let X be a space which splits into coarse componentssuch thatis finite.Then X has geometric property(T).

ProofIf eachXnis a single point,then for any representationHof Cu[X],we haveH=Hc,so geometric property(T)is trivially satisfied.Any spaceXas in the statement is coarsely equivalent to such a space where eachXnis a single point,however.

5 Laplacians

In this section,we define Laplacian operators,and use them to give another characterisation of geometric property(T).This characterisation in terms of Laplacians was our original definition of geometric property(T)in[18,Section 7],and is more closely connected toK-theory.It also lets us relate geometric property(T)to expanding graphs.

Throughout this section,Xdenotes a space as in Definition 2.3.

Definition 5.1Let E be a controlled set for X.The Laplacian associated to E,denoted byis the element ofwith matrix coefficients defined by

Note thatΔEonly depends onAlso note that if E is empty,or is asubset of the diagonal,thenΔEis0.

Example 5.1Suppose thatXis the vertex set of an undirected graph,with the coarse structure generated by the subsetEofX×Xconsisting of all the edges as in Example 2.1.The(un-normalised)combinatorial Laplacian ofXin the sense of spectral graph theory(see[11,Section 4.2])is then the same as our ΔE.This is the motivating example.

The next two lemmas record some basic properties of Laplacians associated to antisymmetric elementary controlled sets(see Definition 2.4 for the terminology).

Lemma 5.1Let E be an antisymmetric elementary controlled set,andΔEbe the corresponding Laplacian.Let t:A→B be the partial translation8Recall that E being antisymmetric means that A∩B=∅.such that E=graph(t)and v be the partial translation operator corresponding to t.

(1)ΔEand v are related by the equation

(2)The image ofΔEin any representation is a positive operator.

(3)If H is any representation ofCu[X],then the kernel ofΔEconsists precisely of those vectors ξ∈H such that

ProofFor the first part,one checks directly that for both operators ΔEandvv∗+v∗v−v−v∗,the(x,y)thmatrix coefficient is equal to

The second and third parts both follow from the formula

Corollary 5.1Let E be an elementary controlled set such that

Let v be the corresponding partial translation operator,andΔEbe the corresponding Laplacian.Then in any representation H ofCu[X],the kernel ofΔEconsists precisely of those vectors ξ∈H such that

ProofLett:A→Bbe the partial translation underlyingE.Using Lemma 2.1,we may decompose

such that foriand so that the restriction ofttoA3is the identity.Writetifor the restriction ofttoAi,Eifor the graph ofti,andvifor the corresponding partial translation operator.The condition

onEimplies that we have a disjoint union

which implies by a direct computation of matrix coefficients that

Lemma 5.1(2)implies that all the operators ΔEiare positive,and combining this with Lemma 5.1(3),we have that in any∗-representation

On the other hand,the facts thatv0,v1,v2,v3have mutually orthogonal domains and mutually orthogonal ranges,and thatimply that

and moreover that the conditionfor allon vectors in H is equivalent toso we are done.

Lemma 5.2If E⊆F are controlled sets,then there exist antisymmetric elementary controlled sets E1,···,Ensuch that

In particular,in any∗-representation ofCu[X],we have the operator inequality

ProofLemma 2.2 implies that there exist antisymmetric elementary controlled setsE1,···,Ensuch that(F∪F−1)diag(F)can be written as the disjoint union

It follows by a direct computation of matrix coefficients that

The operator inequalitynow follows from positivity of each ΔEias in Lemma 5.1(2).The fact thatfor any controlled setEfollows from the special case inclusion

Proposition 5.1Let E be a controlled set and H be a representation ofTheconstant vectors Hcin H are contained in the kernel ofΔE.If moreover,E is generating,thenthe kernel ofΔEis precisely equal to Hc.

ProofAssume first thatEis a general controlled set.Lemma 5.2 implies that there are antisymmetric elementary controlled setsE1,···,En,such that

Lettingvibe the partial translation operator corresponding toEi,Lemma 5.1(3)implies that the kernel ofconsists precisely of thosesuch thatand thus containsOn the other hand,

whence Kernel

Now assume thatEis generating and thatLetvbe a partial translation operator;we must show thatSuppose thatvcorresponds to the partial translationt:A→B.AsEis generating,there existsnsuch thatE◦ncontains graph(t),whence Lemma 2.2 implies that there existsmand a decompositionsuch that ifthen there exist partial translationssuch that

(2)graphfor alli=1,···,mandj=1,···,n;

(3)for eachiand eachj=1,···,n−1 ,the range ofis equal to the domain of

(4)for eachiand eachj=1,···,n−1 ,eitheris the identity map,or the range ofis disjoint from its support.

Letbe the operator corresponding toandvibe the operator corresponding toti.For fixedi,j,let

Thenby Lemma 5.2 whenceCorollary 5.1 then implies that(and this is true for alli,j,as the choice of indices is arbitrary).

To complete the proof,assume inductively for someiandj=1,···,n−1 that ifu:=thenuu∗ξ=uξ.Then as the support ofis the range ofu,we have

whence by inductionfor eachi.Finally,note that

this,however,is equal tousing that the operatorsviall have orthogonal ranges,and we are done.

Definition 5.2Let T be an elementand H be a representationDefineσH(T)to be the spectrum of T considered as an operator on H via this representation.

Define the maximal spectrum ofto be the union of all the sets σH(T)as Hranges over all representations

We are now ready to relate geometric property(T)to Laplacians.Proposition 5.2The following are equivalent:

(1)X has geometric property(T).

(2)For any controlled set E,there exists c=c(E)>0 such that

(3)For some controlled set E,there exists c>0 such that

ProofWe will only prove that(1)and(2)are equivalent:One can show that(3)is equivalent to conditions(2)and(4)from Proposition 3.1 analogously.

Assume first thatXsatisfies condition(2).Noting that Φ(ΔE)=0 for any controlled setEand using Proposition 5.1,it is clear thatXthen satisfies condition(3)from Proposition 3.1 withT=ΔE.

Assume conversely thatXhas geometric property(T),and letEbe a controlled set,andc=c(E)>0 be as in the definition of geometric property(T).LetHbe a representation ofbe a unit vector.Letvbe a partial translation with support inEsuch thatwhich exists by geometric property(T).Lemma 2.1 implies that we may writewhere eachviis a partial translation corresponding to an antisymmetricelementary controlled set,and thevihave mutually orthogonal ranges.It follows that from the orthogonality of the ranges that

whence for somei,In particular,on altering the constantcand replacingvby one of thevi,we may assume thatvcomes from an antisymmetric elementary set.

Now by Lemmas 5.1–5.2,we may write

wherev1=v,and the otherviare partial translations with support inE.Using Lemma 5.1 again(and its proof),it follows that

asis an arbitrary element ofandis itself arbitrary,this shows thatis contained inso we are done.

Our work on Laplacians allows us to give an easy proof of the following consequence of geometric property(T)for sequences of graphs:It implies that the sequence of graphs is an expander in the sense of the following condition.

Definition 5.3Let(Xn)be a sequence of(vertex sets of)finite connected graphs.The sequence(Xn)is an expander if the following hold:

(i)the cardinalitiestend to infinity;

(ii)there is a uniform bound on the degrees of all vertices in each Xn;

(iii)there exists some c>0such that ifΔnis the graph Laplacian on l2(Xn)as in Example

5.1,then the spectrum ofΔnis contained in

Expanders have applications in several areas of pure mathematics,as well as computer science and information theory(see[11]for more information).

Corollary 5.2Let X be a space that decomposes into coarse components asand assume thatis finite and thattends to infinity.Let E be a symmetric generatingset for the coarse structure.Define a(connected)graph structure on each Xnby decreeing that E∩(Xn×Xn)is the edge set,and call the corresponding graph Gn.

Then the sequence(Xn)is an expander.

ProofLet Δnbe the graph Laplacian on eachXn.Then(Xn)is an expander if and only if the operator

has a spectrum contained in some set of the formThis follows,however,as Δ identifies with ΔEacting onl2(X),so the spectrum of Δ is equal towhich is contained inand is a subset ofby geometric property(T).

Remark 5.1The methods of Section 4 can be used to show that “being an expander”is a coarse invariant of a sequence of graphs in the obvious sense.Although known to some experts9It also admits a rather easier proof,as pointed out to us by Romain Tessera.,this does not seem to have been observed in the literature before.

6 Relationship with Amenability

In this section we discuss the relationship between geometric property(T)and amenability.Throughout the section,Xdenotes a space.

The main result is Proposition 6.1;phrased slightly dif f erently,it says that Cu[X]admits a representationHwhere the spaceHcof constant vectors is non-zero,if and only ifXis amenable.It follows(Corollary 6.1)that for coarsely connected spaces,geometric propertyTis equivalent to non-amenability.

We start with a lemma.Variants of this are very well-known,but we include a proof for the readers’convenience,as we could not find exactly what we needed in the literature.

Lemma 6.1The following are equivalent:

(1)There exists an invariant mean on X,a positive unital linear functional

such that if f∈l∞(X),and t:A→B is any partial translation such that B contains the support of f,then

(2)There exists a netof unit vectors in l2(X)such that for any partial translationv,

ProofAssume first condition(1).Fix a finite setof partial translations.It suffices to show that there exists a sequence of unit vectorssuch that

for alli=1,···,m.LetP(X)denote the space of finitely supported probability measures onX(a subset ofl1(X)),which identifies with a weak-∗dense subset of the space of positive unital linear functionals onl∞(X)via the standard pairing betweenl1andl∞.Letbe a net inP(X)that converges weak-∗toφ,and for eachi=1,···,m,letti:Ai→Bibe the partial bijection corresponding tovi.Then for eachiand anywe have

or in other words,

converges weakly to zero inl1(X),whence 0 is in the weak closure of the convex set

The Hahn-Banach theorem thus implies that it is in the norm closure,i.e.,there is a sequence(φn)of elements ofP(X)such that

fori=1,···,m.for eachnandx∈X,so eachξnis a unit vector inl2(X).For anyi=1,···,m,we have

which tends to zero asntends to infinity.

For the converse,let(ξi)be a net with the properties given.Then it is not difficult to check that any weak-∗limit point of the functionals

will have the desired properties.

Definition 6.1A space X is amenable ifit satisfies the conditions in Lemma6.1.

It is not difficult to see that this is equivalent to the definitions of amenability in[2,Section 3]or[13,Sections 3.3–3.6]

The equivalence of the first and third conditions in the proposition below is fairly well-known;the main point is that this equivalence still holds if one takes the “maximal spectrum”.

Proposition 6.1Let E be a generating controlled set for X.With notation as in Section 5,the following are equivalent:

ProofIt is clear that(1)implies(2).

To see that(2)implies(3),assume that 0 is an element ofThis is equivalent to 0 being an element of the spectrum of ΔEin theC∗-algebradefined as the completion of Cu[X]for the norm

(the arguments of[8,Section 3]show that this supremum is finite for eachAny point in the spectrum of a positive operator in aC∗-algebra can be realized as an eigenvalue in some representation,whence there exists a representationHof(equivalently,ofin which 0 is an eigenvalue of ΔE.Proposition 5.1 then implies that this representation contains non-zero constant vectors.Letbe any norm-one constant vector,and let

be the corresponding vector state.We will show that the restriction ofφtois an invariant mean.

Indeed,letfbe an element oflett:A→Bbe a partial translation such thatBcontains the support off,and letvbe the operator corresponding tot.Thenandso the fact thatis constant implies that

To see that(3)implies(1),let(ξi)be a net of functions inl2(X)with the properties in Lemma 6.1(2).Using Lemmas 5.1–5.2,we may write

for some partial translationsFor anywe then have that

which tends to zero in the limit overi.As ΔEis a positive operator onl2(X),this implies that its spectrum contains zero.

In particular,note that whether or not 0 is in the above variations of the spectrum of ΔEis a property not ofE,but of the coarse spaceX.

We now turn to the relationship between geometric property(T)and amenability.

Lemma 6.2Let X be a coarsely connected amenable space.Then for any generating con-trolled set E,0is a non-isolated point of the spectrum of

ProofUsing Proposition 6.1,0 is also init suffices to show that 0 is not an isolated point inIfit were,then 0 would be an eigenvector of ΔEfor its action onl2(X).Letξbe an eigenvector,and note that Proposition 5.1 implies thatξis a fixed vector.Letx0be a point in the support ofξ,and note that asXis infinite,coarsely connected and monogenic,there exists a sequence

of distinct points inXsuch thatis inEfor alli.Letvbe the partial translation operator corresponding to the partial translation defined by

Then asis a fixed vector,we have

for alln,and thus by induction all the valuesare non-zero and equal.This contradicts the fact thatξis in.

Finally,here is the characterization of coarsely connected spaces with geometric property(T).

Corollary 6.1Let X be an infinite coarsely connected space.Then X has geometric property(T),if and only ifit is not amenable.

The special case of this result whenXis a group was proved in[18,Lemma 7.2].

Proof of Corollary 6.1IfXis amenable,then by Lemma 6.2,0 is a non-isolated point infor any generating setEfor the coarse structure,and this contradicts geometric(T).Conversely,ifXis not amenable,then 0 is not in the spectrum offor any generating setEby Proposition 6.1.As the spectrum is closed and ΔEis positive,it is contained in a set of the form[c,∞)for somec>0,and this,in particular,implies geometric property(T).

This says that geometric property(T)is not very interesting for coarsely connected spaces!In the next section,we will finally look at a class of interesting examples of spaces with geometric property(T).

7 Relationship with Property(T)Groups

In this section,we give some non-trivial examples of spaces with geometric property(T).Up to trivial adjustments,these are the only examples we know.Most of this material is contained in[18,Section 7],but fairly sketchily;we provide more detail here for the readers’convenience.

It will be very convenient to use someC∗-algebraic machinery in this section,mainly as the followingC∗-algebras are useful to organize certain arguments.This material was already briefly used in the proof of Proposition 6.1.

Definition 7.1Let X be a space.The uniform Roe algebra of X,denoted byis thecompletionfor its natural∗-representation on l2(X).

The maximal uniform Roe algebra of X,denoted,is the completion offorthe norm

(see[8,Section3]for a proof that this norm is finite).

Using the notation from Definition 5.2,note that for anyis the spectrum ofTconsidered as an element ofandσmax(T)is the spectrum ofTconsidered as an element of

Definition 7.2LetΓbe an infinite finitely generated discrete group with a fixed finite generating set S.Assume that S=S−1.Let

be a nested sequence of finite index normal subgroups ofΓsuch that∩nΓn={e}.For each n,set Xn= Γ/Γn,and set

Set

and

Finally,equip X with the monogenic coarse structure generated by E.

The following theorem characterizes when a space built from a group as above has geometric property(T).

Theorem 7.1Let X be a space built from data(Γ,(Γn))as above.Then X has geometric property(T),if and only ifΓhas property(T).

In order to prove this,we need a lemma.Let C[Γ]denote the complex group algebra of Γ,and note that the right actions of Γ on the variousXngive rise to a∗-homomorphism

This∗-homomorphism is injective asMoreover,ifdenotes the completion of C[Γ]for the norm

thenιalso induces a∗-homomorphismι:by the universal property ofWe have the following injectivity result,which is stronger than the statement that the map in(7.1)is injective.

Lemma 7.1The∗-homomorphism ι:is injective.

ProofNote that the algebraic direct sumis an ideal inetIdenote its closure inIt follows from the argument of[12,Proposition 2.8]that

where the right-hand side denotes the maximal crossed product defined by using the action of Γ onl∞(X)/C0(X)induced by the right action onX.It suffices to prove that the composed map

is an injection.

Now,for eachn,letξnbe the normalised characteristic function ofXninl2(X),and let

be the corresponding vector state.Letφbe any cluster point of the sequence(φn)of vector states onl∞(X),and note thatφdescends to a state onl∞(X)/C0(X).It is Γ-invariant,as all theφnare.Finally,consider the maps

where the first map is the unit inclusion which is split by the ucp mapφ.As maximal crossed products are functorial for ucp maps(see[4,Exercise 4.1.4]),this gives rise to maps

whose composition is the identity;the first map is thus injective.

Remark 7.1The above proofis a disguised version of the following fact:We guess this is known,but do not know ifit appears in the literature.LetGbe a locally compact group acting on a compact Hausdorf ftopological spaceX.Then the canonical∗-homomorphism

is injective,if and only if there is an invariant measure onX.

Proof of Theorem 7.1Consider the element

of the group algebra of Γ,and letEbe the controlled set appearing in the definition of a box space(Definition 7.2).Then the image of ΔΓunderιis the Laplacian ΔEassociated toE.Asιis injective on the level of maximal completions(and injective maps ofC∗-algebras preserve spectra)it follows that the spectrum of ΔΓinis equal toHowever,it is well-known that Γ has property(T),if and only if the spectrum ofis contained in a set of the formfor somec>0(see[16,Theorem 3.2]);asEis generating(and using Proposition 3.1),this is equivalent to geometric property(T)forX.

Corollary 7.1For spaces built from sequences of quotients as above,having geometric property(T)is a strictly stronger property that being an expander.

ProofA space associated to the pair(Γ,(Γn))is an expander,if and only if the pair has property(τ)(see[11,Theorem 4.3.2]).The result now follows as there are many pairs(Γ,(Γn)),for example,with Γ a free group,which have property(τ),where Γ does not have property(T).

Also note that whether a space as above has property(T)depends only on the ambient group Γ,instead of the given sequence of subgroups;on the other hand,whether or not such a space is an expander in general does depend on the choice of sequence of subgroups.

8 Geometric Property(T)and Coarse a-T-Menability Properties

It follows from Lemma 4.6 and Corollary 6.1 that geometric property(T)is only really interesting when a spaceXadmits a decomposition

into non-empty finite coarse components such that|Xn|tends to infinity.We assume for simplicity10Slightly more general results,for example,allowing,index sets other than N,or only assuming that|Xn|is unbounded,are certainly possible,but we did not think that the extra messiness this would force on the statements is worthwhile.throughout this section that we are dealing with a space of this form.

Given such a space,we define its box spaceto be the setXequipped with the coarse structure generated by the original coarse structure onX,and all the singletons{(x,y)},asxandyvary acrossX(note that this coarse structure is never monogenic).

Our goal in this section is to show that geometric property(T)is incompatible with the following notions of“coarse a-T-menability” forX:Xadmits a coarse embedding into Hilbertspace(see[19]);Xadmits a fibered coarse embedding into Hilbert space(see[5]);the restriction of the coarse groupoid ofXto its boundary is a-T-menable(see[7]).

Theorem 8.1Assume thatsplits intofinite coarse components such that|Xn|tends to infinity as above,and X has geometric property(T).Then the following are impossible:

(1)admits a coarse embedding into Hilbert space;

(2)admits a fibered coarse embedding into Hilbert space;

(3)the restriction of the coarse groupoid of Xto its boundary is a-T-menable.

Natural examples satisfying conditions(2)and(3)are sequences of finite quotients of a-T-menable groups,and sequences of graphs(Xn)such that the girth11i.e.,length of shortest non-trivial cycle.ofXntends to infinity(see[5,Examples 2.4 and 2.5]).It follows from the theorem that such spaces can not have geometric property(T).On the other hand,note that there are many expander sequences with girth tending to infinity;this gives another dif f erence between geometric property(T)and general sequences of expanding graphs.

Special cases of this theorem follow from known results inK-theory(see[5–7,12,17–18]),but the proof we give here is more direct and a little more general.The theorem is definitely not true for coarsely connected spaces,this follows from Corollary 6.1.

The basic idea of the proofis to show that any of the coarse a-T-menability properties appearing in the statement allow one to construct representations of Cu[X]that contradict geometric property(T).Unfortunately,properties(2)and(3)from the above theorem are quite technical,and are not stated anywhere in the literature in a form that is particularly well-suited for our purposes;as a result,in order to keep the proof of Theorem 8.1 reasonably short and self-contained,we have had to be a little ad-hoc in some constructions below.

In order to cover part(3)of the above theorem,we must use the language of the Stone-ˇCech compactification ofX;we thus start by recalling the relevant facts.LetYbe a discrete topological space.The Stone-ˇCech compactification ofY,denoted byβY,is a compact Hausdorf f space containingYas a dense open subset.It is determined by the following universal property:For any compact Hausdorf fspaceKand any functionf:Y→K,there is a unique continuous extensionf:βY→K.We write∂Y:=βYYfor the associated Stone-ˇCech corona.Note that the universal property implies that for anyA⊆Y,the inclusion mapA→Yextends to an injectionβA→βY;in particular,the closure ofAinβYis canonically identified withβA.

Now letEdenote the coarse structure onX×X.For each controlled setdenote its closure inβX×βX.Define also∂Eto be the intersectionE∩(∂X×∂X).Define

We think of this space as a subset of(and correspondingly write elements as pairsbut equip it with the(weak)topology determined by the following condition:A subsetis open,if and only ifits intersection with eachis open in

equipped with the subspace topology fromβEX.

With this topology,βEXis a locally compact Hausdor ffspace,and∂EXis a closed subspace:βEXactually identifies as a topological space with the coarse groupoidG(X),and∂EXwith the “restriction to the boundary” ofG(X).We will not use this,but see[14]or[13,Chapter 10]for more information.

The following definition is easily seen to be equivalent to the property in part(3)of Theorem 8.1;we state it in this form to avoid having to introduce a lot of groupoid language.

Definition 8.1The space X is boundary a-T-menable if there exists a continuous functionsuch that the following hold:

(1)The function k is normalized:k(ω,ω)=0for all pairs

(2)The function k is symmetric:k(ω1,ω2)=k(ω2,ω1)for all

(3)The function k is negative type:for any finite subsetof∂X such that allthe pairs(ωi,ωj)belong to∂EX and any finite subsetofCsuch thatwehave

(4)The function k is proper:If

then the limit over the directed set of controlled sets(ordered by inclusion)is in finity.

The main result which we want to prove in this section is as follows.

Theorem 8.2Assume thatsplits intofinite coarse components such thattends to infinity,and X is boundary a-T-menable.Then X does not have geometric property(T).

Before we prove this,we show how it implies Theorem 8.1.

Proof of Theorem 8.1As already remarked,given the definition of the coarse groupoidG(X)(see[14]or[13,Chapter 10]),it is clear thatXis boundary a-T-menable in our sense if and only if the restriction ofG(X)to its boundary is a-T-menable in the sense of[15,Section 3](see also[14,Section 5]and[7]).The result of[6,Corollary 20]thus implies that ifadmits a fibered coarse embedding into Hilbert space in the sense of[5](and in particular ifadmits a coarse embedding into Hilbert space in the sense of[19]).ThenXis boundary a-T-menable.Theorem 8.1 follows.

The remainder of this section is devoted to the proof of Theorem 8.2.We assume from now on thatXis as in the statement of Theorem 8.2,and assume thatk:∂EX→C is as in the definition of boundary a-T-menability.

For eacht>0,define a functionkt:∂EX→[0,1]by

Lemma 8.1The functions ktfrom(8.1)have the following properties:

(1)They are normalized:kt(ω,ω)=1for all ω∈∂X.

(2)They are symmetric:for all

(3)They are positive type:For any finite subsetof∂X such that all the pairs(ωi,ωj)are in∂EX,and any finite subset

ProofParts(1)and(2)are obvious.Part(3)is essentially a version of a well-known theorem of Schoenberg(see[1,Theorem C.3.2]):It follows from the statement of[1,Theoremby applying that result separately to each finite subsetof∂Xsuch that

We will now use the functionsktto construct representations ofFirst,extend12The exact extensions we use will not af f ect the representations we build;however,we will make slightly refined choices of extension below in order to analyze properties of the representations.the functionsto continuous functionswe may assume thatfor allDefine a form

via the formula

asXhas bounded geometry,the sum contains uniformly finitely many terms for eachx,so this is well-defined.

In order to use these functions to build representations ofwe need some preliminaries.First,we have the following lemma about elementary controlled sets.

Lemma 8.2Let E be an elementary controlled set on X which is the graph of a partialtranslation tdenote the extension of t to the Stone-ech compactifica-tions.Then the closureis the set

which identifies homeomorphically with βE.

ProofDenote byg:A→X×Xthe “graph bijection”with imageE.Consider the maps

whereπis the projection onto the second factor.The composition of these maps is just the inclusion ofAintoβX.Now,the universal property of the Stone-ˇCech compactification gives rise to maps

where the image ofUniqueness of the extensionimplies that it must be equal to the mapwhich gives the characterization of.On the other hand,π◦is the identity inclusion(by uniqueness again),which implies thatis injective.Henceidentifies canonically withβA,and so withβE.

Now,letf:X×X→C be a bounded function with support in a controlled setE.Using Lemma 2.2,we may writewhere eachEiis elementary.SayEiis the graph of the partial translationFor eachdefinebyand extendThe extension ofmust be given by

by uniqueness.This formula then extendsfto all ofUsing subdivisions,it is not difficult to see that this extension does not depend on the choice of decomposition

IfTis any operator insupported in a controlled setE,andis an element of∂E,we defineby using the extension process above applied to the function fromEtodefined by

For a controlled setE,we also define

and for an elementTof Cu[X],define

Lemma 8.3For each t>0,the form

has the following properties:

(1)The formis linear in the second variable and conjugate linear in the first.

(2)For any S,T-norm ofis bounded by

(3)The restriction ofto∂X is given by the formula

(4)For any S inthe restriction ofto∂X only takes non-negative values.

ProofPart(1)is clear.Part(2)follows from the fact thatkttakes values in[0,1]and the triangle inequality.

For part(3),note that anyis a finite sum of operators with the property that

is elementary.Using part(1),it suffices to assume thatSandThave this property.Assuming this,lets:be the partial translation corresponding toS,and definef:Similarly,defineg:wheret:is thepartial translation corresponding toT.Then for any fixedwe have via the discussion preceding this lemma,that

and similarly forT.We thus have

On the other hand,for anyx∈X,

The claimed formula follows.

Part(4)follows from part(3)and Lemma 8.1.

Now,for eachn,define a state onby the formula

Letφbe any cluster point of the sequencein the state space ofNote thatφdescends to a state on the quotientwhich naturally identifies withC(∂X).

For eacht>0,we may thus define a form onby

Using Lemma 8.3(1)–(2),and thatφis a state,each formis linear in the second variable,conjugate linear in the first,and positive semi-definite.Separation and completion thus define a Hilbert spacefor eacht>0.An elementgives rise to an equivalence class[S]in this Hilbert space.Provisionally define a representationvia the formula

Lemma 8.3(3),and thatφhas norm one,imply that each operatorπt(T)extends to a bounded linear operator onHt,and thus we have a well-defined map

Lemma 8.4The map πtin(8.4)above is a∗-representation.It does not depend on the choice of extension of kt.

ProofLinearity and multiplicativity ofπtare clear,so to show thatπtis a∗-representation it suffices to check that it preserves adjoints.

Asφis cluster point of the functionalsφn,it suffices to show that

for allnand allComputing

The fact thatπtdoes not depend on the choice of extension ofktfollows from Lemma 8.3(3)and thatφonly depends on the restriction of a function inC(βX)to∂X.

Our eventual goal is to preclude geometric property(T)by showing that the representationsHt“come close”to containing constant vectors for smallt>0,although none of them actually does contain constant vectors.The following lemma is the next step.

Lemma 8.5For any t>0,the∗-representationcontains no constantvectors.

In order to prove this,we need a combinatorial lemma.

Lemma 8.6Let E be a symmetric generating set for the coarse structure on X that contains

the diagonal,and fix rThen there exist s,such that for all n≥N,there exists abijective partial translation tn:Xn→Xnsuch that

ProofLetrbe given,and letsbe so large thatwhereis as in(8.2).Astends to infinity,and using the bounded geometry assumption,there existsNsuch that for alland all pointsthere is a pointsuch thatFixn≥N,and letGbe the graph with vertex setXnwhere two verticesx,yare connected by an edge if and only if(x,y)is inIt suffices to show that there is a bijectionsuch thatis an edge inGfor allIt suffices by Tutte’s 2-matching theorem(see[10,Proposition 2])to show that ifCis a subset ofXn,no two vertices of which are connected by an edge inG,and if we set

then

Fix such a setC,and define a relation onCbyif and only if(x,y)This is an equivalence relation:It is symmetric and reflexive asEis symmetric and contains thediagonal.It is transitive,as ifandthen(x,z)is inasx,zare inC,they are not connected by an edge inG,and so this is impossible unless(x,z)is actually inFix a subsetcontaining one representative of each equivalence class,and for eachdefine

to be its equivalence class,which has at mostmembers.For eachdefine

The choice ofNimplies that there existssuch thatAsEis generating,it follows that there isand a sequence of distinct points

such thatis inEandHence in particular,are inDx,and thus

where the central inequality follows by choice ofs.Finally,note that ifxandyare distinct points inC0,then(x,y)is not an edge andIt follows by the definition ofGthatand in particular,Moreover,eachDxis contained ind(C)whence

thus completing the proof.

Proof of Lemma 8.5Fixt>0,and assume for contradiction thatis a constant vector of norm one.be an element of norm one coming fromsuch thatLetEbe a symmetric generating set for the coarse structure that contains the diagonal.Letsuch thatwheneverand de fine

Letr>2Kbe so large such that wheneverwe have that

whereN(T)is as in(8.3)(such anrexists by properness ofk).Adjusting the extensions ofkttoif necessary,we may assume that this estimate holds in the stronger form:

LetNandsbe as in Lemma 8.6 forras above,and letbe the bijective partial translation given by that lemma forand be the empty partial translation otherwise.Letvbe the partial translation that is de fined by usingtnon eachXn.As all but finitely many of thetnare bijections,is a unitary operator on

Now,for anyand

If the termis non-zero,then(x,y)andare inAsis not inhowever,this forcesand thus by line(8.5),Hence for all

for allx∈X,whencefor allnand so

On the other hand,the facts thatis unitary,ξis constant,andhave norm one,andtogether imply that

This contradicts(8.6),so we are done.

The following lemma completes the proof of Theorem 8.2.

Lemma 8.7LetΔEbe a Laplacian operator inCu[X],and letThen for all suitablysmall t>0,the spectrum of πt(ΔE)contains points from

ProofLetIbe the identity operator inThen for anywe have the formula

whence

Now,by boundedness ofkon∂E,there existst>0 such that

We may assume without loss of generality that the extension ofkttoX×Xsatisfies

Looking back at(8.7),we have

A simple computation shows that the first term on the right-hand side is zero,whence,however

Finally,asφis a cluster point of the sequenceit follows from this and the fact thatis a positive operator thatis inas the set of values

is contained in the convex hull of the spectrum ofthis completes the proof.

9 Questions and Comments

We conclude the paper with some open questions and comments.

Questions 9.1(1)Does a “generic” sequence of graphs have geometric property(T)?This is a strengthening of the well-known fact that a generic sequence of graphs is an expander(see[11,Proposition 1.2.1]).It is also possibly connected to the fact that a “generic”hyperbolic group has property(T)(see[20]).

(2)Are there useful necessary and/or sufficient conditions for geometric property(T)that can be stated purely in terms of graph theoretic properties? To answer question(1),it is probably necessary to answer this question first.

(3)Similarly,are there useful necessary and/or sufficient conditions for conditions(2)and/or(3)from Theorem 8.2 that can be stated purely in terms of graph theoretic properties,other than the known condition using girth?

(4)We suspect that the results of Section 8 are really part of a general result about groupoids.Precisely,treating Cu[X]as the convolution∗-algebraCc(G(X))of the coarse groupoidG(X),it is not too difficult to extrapolate the ideas of this paper to define a “topological property(T)”for this groupoid,and indeed any “reasonable”locally compact groupoid.We then suspect that the results of Section 8 say that this“topological property(T)” is incompatible with a-T-menability in the presence of an invariant measure on the unit space of the groupoid(the existence of an invariant measure corresponds to the amenability of the space in the assumptions of Theorem 8.1).It might be interesting to develop this further:For example,Theorem 4.1 naturally corresponds to a statement about Morita invariance of the general“topological property(T)”;but we have no idea if the corresponding general result would be true.

(5)Our version of geometric property(T)only really has good properties for disjoint unions of finite metric spaces;in the language of point(4)above,the issue is the presence of an invariant measure on the unit space of the groupoid.Is there a property that has more interesting consequences in the context of more general metric spaces,e.g.including Cayley graphs of infinite groups? Compare for example(see[13,Section 11.4.3]).

AcknowledgementsWe would like to thank Erik Guentner,J´anˇSpakula,and Romain Tessera for useful discussions on aspects of this work.We would also like to thank Jintao Dengfor pointing out some algebraic errors in an earlier version.The first author would like to thank the Shanghai Center for Mathematical Sciences for its hospitality during part of the work on this paper.

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