Qin WANGWenjing WANGXianjin WANG

1 Introduction

The metric sparsification property and finite decomposition complexity are notions of metric geometry,which were introduced very recently in studying the coarse Novikov conjecture(see[3–4])and the stable Borel conjecture(see[9]),respectively.Both properties were motivated by the notion of finite asymptotic dimension of a metric space introduced by Gromov[7].

Recall that a metric spaceXhas finite asymptotic dimension if there is an integern≥0,such that for any(large)numberr>0,the spaceXmay be written as a union ofn+1 subspacesXi,each of which may be further decomposed as anr-disjoint union:

in which the metric familyis bounded,i.e.,In general,we say that a countable family of metric spacesdecomposable over another metric familyY,if everyX∈Xadmits a decomposition as above,where each

Inspired by the feature of finite asymptotic dimension,Guentner,Tessera and Yu[9]introduced the notion of finite decomposition complexity as a measure of computational complexity of metric spaces.Roughly speaking,a metric familyXhas finite decomposition complexity,ifit can be decomposed,through a finite number of applications of the decomposability relation as above,into a bounded family.Guentner,Tessera and Yu proved that the stable Borel conjecture holds for an aspherical manifold whose fundamental group has finite decomposition complexity(see[9]).

On the other hand,the metric sparsification property was introduced by Chen,Tessera,Wang and Yu[3]to supply a more flexible geometric condition for operator norm localization,which can be applied to the coarse Novikov conjecture in operatorK-theory(see[6]).Roughly speaking,a metric space has the metric sparsification property,if there exists a constant 0

The motivation for studying the permanence property in coarse geometry is inspired by[5],in which it is proved that the operator norm localization is stable under some operations of coarse metric spaces.In[8],Guentner gave a survey on various permanence properties of coarse metric spaces.In this paper,we shall regard finite decomposition complexity as a type of operation of metric spaces to provide a permanence result for metric sparsification property.To do this,we introduce a notion of finite decomposition complexity with respect to metric sparsification property,and show that if a metric spaceXhas this property,thenXitself has the metric sparsification property.That is,the metric sparsification property is stable under large scale decompositions of finite complexity.Combined with a result of Guentner,Tessera and Yu[9]that all countable linear groups have finite decomposition complexity,this also implies another result of Guentner,Tessera and Yu[10]that all countable linear groups have the metric sparsification property,and hence the operator norm localization property.This fact can be used to prove the coarse Novikov conjecture for the box spaces associated to countable linear groups,including many interesting sequences of expander graphs(see[6,10]).

2 Preliminaries

Definition 2.1(see[3])Let X be a metric space.We say that X has metric sparsi fication property with constant0

When we need to be more explicit,we will say thatXhas MS(c)with functionf.Ifm,μare given,and if we want to say that a subset Ω satisfies the Definition 2.1,we will simply write Ω = Ω(μ,f,m,c).

Definition 2.2(see[3])We say that a family of metric spaces has uniform MS(c),if there is a common f that works for all the elements of the family.

The following property plays an important role in the next section.

Proposition 2.1(see[3])If X has metric sparsification property,then it has the property with constant c for all0

To recall the notion of finite decomposition complexity(see[9]),we shall useX,Y,etc.to denote the(countable)families of the metric spaces,and use Δ,Γ,etc.to denote the collections of the metric families.

A metric familyis bounded,if there is a uniform bound on the diameter of the individual spaceXj,namely,denote the collection of bounded families:

A metric familyXis(n,r)-decomposable over a metric familyY,if everyX∈Xadmits a decomposition

with eachXij∈Y.

Definition 2.3(see[9])LetFbe a collection of metric families.A family X is decomposable overF,if there exists an n≥0,such that for every r>0,there exists a Y∈F,such that X is(n,r)-decomposable over Y.The collectionFis closed under decomposability,if every family X decomposable overFactually belongs toF.

Note that a spaceX,always viewed as a singleton family,is decomposable over the collection D0of bounded families precisely when it has finite asymptotic dimension.A familyX={Xi}is decomposable over the collection of bounded families D0precisely when the metric spacesXicomprising it have uniformly finite asymptotic dimension in the sense of Bell and Dranishnikov[1–2].

Definition 2.4(see[9])The collection of metric familiesDhaving finite decomposition complexity is the smallest collection containing the bounded families and closed under decomposability.

LetFbe a Borel map from a metric spaceXto another metric spaceY.Recall thatFis said to be a coarse map if

(1)for everyR>0,there exists anS>0,such thatd(F(x),F(y))

(2)the inverse imageF−1(B)for every bounded subsetBofYis bounded.

We say thatXis coarsely equivalent toYif there exist coarse mapsF:X→YandG:Y→X,such that there exists a constantC>0 satisfyingd(G◦F(x),x)

In the following,let MSP denote the collection of metric families having uniform metric sparsification property.In order to express the idea of this paper,we introduce the following notion.

Definition 2.5The collectionof metric families,having finite decompositioncomplexity with respect to metric sparsificaton property,is the smallest metric collection containingMSP,which is closed under decomposability.A metric space X is said to have finite decomposition complexity with respect to metric sparsification property,if the singleton family{X}belongs to

3 Main Result

The main result of this paper is the following permanence property.

Theorem 3.1A metric space X has finite decomposition complexity with respect to metric sparsification property if and only if X itself has metric sparsification property.

It follows thator in other words,the collection MSP of metric families,having uniform metric sparsification property,is closed under decomposability.

Since the converse of the above theorem is obviously true,we only have to show the necessity of Theorem 3.1.To begin with,note that Definition 2.5 can be reformulated as follows.

Proposition 3.1A metric space X has finite decomposition complexity with respect tometric sparsi fication property if and only if,for any sequenceof positive numbers,there exists an integer m>0and m non-negative integerswhere n0depends onlyon X and each nkwith k>0depends only on r1,···,rk−1and n0,n1,···,nk−1,such that we have m levels of decomposition as follows:

(1)For X and r1>0,we have

(2)For all Xi1j1and r2>0,we have

···

(m)For alland rm>0,we have that

and the family of metric spaceshas uniform metric sparsificationproperty.

To prove Theorem 3.1,we first prove the following“quantitative version of the finite union theorem”for metric sparsification property of metric spaces.

Let

Lemma 3.1Let X be a metric space,expressed as a union of finite,i.e.,n+1,metric subspaces:

Let r>0be given,and fix a natural number k∈N.If there exist common constants S>0and0

wheredenote the least integer greater than or equal to x.

Then,for any finite positive measure μ on X,i.e.,there always exists a subsetwhere i runs through a countable index set,such that

ProofFirst,consider the case in whichXis a union of 2 subspaces,i.e.,n=1,

In this case,

LetFor eachm=0,1,2,···,let

ThenSinceμ(X)<∞,there is ani0,such thatandNote thatU0andU1are 2r-disjoint if none of them are empty.

Fix arbitrarily a measureμ∈M(X).Leti:U1→X1be the inclusion map.Take a Borel mapp:U0→X0,such that

and

LetonX0be the push-forward measure of the restriction measureThen for any Borel subsetA⊂X0,we have.Similarly,letbe the measure onX1.It is easy to check thatandμ1 ∈ M(X1)if none of them are empty.

By the conditions of this lemma,there are subsets

satisfying

for i=0,1.

Letandwhere j runs through a countable index set.

Then the subsetsatis fies>rdiam

The subsetsatisfies

Then we get cNow we rearrange the index set ofThen,we get

In the general case,suppose

Let r>0 and k ∈ N be given.and let N∈N be the least integer greater than or equal to1δ,i.e.,

wheredenotes the least integer greater than x.Fix arbitrarily a measureThe strategy of arguments in the above case of 2 subspaces implies that there exist 2r-disjoint subspaces U0and V1by cutting offa subset of X of measure at most(see Figure 1),such that

(1)U0is contained in the(2N −2)r-neighborhood of X0,i.e.,

(2)V1is a subspace of X1∪ X2∪ ···∪ Xn,such that

Figure 1 Incursive cutting of f

Similarly,there exist 2r-disjoint subspacesU1andV2ofX−X1by cutting of fa subset of measure at most,such that

(1)U1is contained in the(2N−2)r-neighborhood ofX1,i.e.,U1⊆N(2N−2)r(X1);

(2)V2is a subspace ofX2∪X3∪···∪Xn,such that

We get

By induction,we assume that at the(n−1)-th step,there are 2r-disjoint subspacesUn−2andVn−1,such that

(1)Un−2is contained in the(2N−2)r-neighborhood ofXn−2;

(2)Vn−1is a subspace ofXn−1,such that

Then at then-th step,finally there exist 2r-disjoint subspacesUn−1andUn=Vn,such that

(1)Un−1is contained in the(2N−2)r-neighborhood ofXn−1;

(2)Unis a subspace ofXn,such that

Further,we get that

i.e.,

Note that the family of subsetsare disjoint from each other,and eachUiis contained in(N−1 )·2rneighborhood ofXi.Hence,with the same technique as used to deal with the 2r-neighborhood in the above for the union of 2 subsets,associated to eachUi,i=0,1,···,n,there exists a subset,such that

LetThenif we rearrange the index of subsets.Hence,we have

The proofis complete.

Proof of Theorem 3.1Suppose that a metric spaceXhas finite decomposition complexity with respect to metric sparsification property.Letc0=0.4.We proceed to find anf:R+→R+forXto have the metric sparsification property.

Let

For anyr>0,by Proposition 2.1,there exists anm>0 andmnon-negative integerscorresponding to the sequence of positive numbers

r1=(4N(1,n0)−3)r,r2=(4N(2,n1)−3)r1,···,rm=(4N(m,nm−1)−3)rm−1,···,such that

(1)forXandr1,we have

(2)for allXi1j1andr2>0,we have

···

(m)for allandrm>0,we have

and the family of metric spaceshas uniform metric sparsification property.By proposition 2.1,we can take the common constantc=0.8 and the corresponding function:R+→R+as in Definition 2.2.

Letfor allm=0,1,2,···.By the step(m),for any positive finite measureμonXi1j1···imjm,there exists a subsetsuch that

LetSince distby rearranging the index of the subsetswe can write

Then we have

By Lemma 3.1,there exist nonempty subsets ofsuch that

Taking a measurewe have

By the above arguments and applying Lemma 3.1 formtimes,we have that,for any measurethere exists a subsetofX,such that

Note

Let

We have thatXhas metric sparsification property relative tofwith constantThe proofis complete.

Acknowledgements.The authors wish to thank the refrees for the kind comments.The third author is very grateful to the Laboratory of Mathematics for Nonlinear Science,Fudan University,for the hospitality and support during his visit.

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