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Metric properties of expandersPart 4: Towards characterization of spaces with

bounded geometry which are not coarselyembeddable into `2

Mikhail OstrovskiiSt. John’s University

e-mail: [email protected] page: http://facpub.stjohns.edu/ostrovsm/

In these notes OME means M. I. Ostrovskii, MetricEmbeddings, book in preparation

2013

Mikhail Ostrovskii, St. John’s University Metric properties of expandersPart 4: Towards characterization of spaces with bounded geometry which are not coarsely embeddable into `2
Definitions, Examples

I DefinitionLet ρ1, ρ2 : [0,∞)→ [0,∞) be two non-decreasing functions(important: ρ2 has finite values), and let F : (X , dX )→ (Y , dY )be a mapping between two metric spaces such that∀u, v ∈ X ρ1(dX (u, v)) ≤ dY (F (u),F (v)) ≤ ρ2(dX (u, v)).The mapping F is called a coarse embedding if ρ1 can be chosento satisfy limt→∞ ρ1(t) =∞.

I Example 1. The mapping F : R→ Z given by F (x) = bxc isa coarse embedding.

I Example 2. The vertex set V of an infinite dyadic tree Twith its graph distance can be coarsely embedded into `2 inthe following way: we consider a bijection between the set ofall edges of T and vectors of an orthonormal basis {ei} in `2,and map each vertex from V onto the sum of those vectorsfrom {ei} which correspond to a path from a root O of T tothe vertex, O is mapped to 0.

Applications of coarse embeddings

I The idea of Gromov was to approach some well-knownproblems in Topology using coarse embeddings of certainfinitely generated groups with their word metrics into “good”Banach spaces.

I This idea turned out to be very fruitful, see the survey of Yu[in: International Congress of Mathematicians. Vol. II,1623–1639, Eur. Math. Soc., Zürich, 2006].

I We need to recall that a discrete metric space A is said tohave a bounded geometry if for each r > 0 there exist apositive integer M(r) such that each ball in A of radius rcontains at most M(r) elements.

I The main references for this presentation: OME, Chapter 7and Section 11.2.

I DefinitionLet X be a space with bounded geometry and {Yn}∞n=1 be a familyof expanders. We say that X weakly contains {Yn} if there aremaps fn : Yn → X satisfying (with some abuse of notation we useYn to denote the vertex set of Yn)

I Lipschitz constants Lip(fn) are uniformly bounded

I limn→∞

maxy∈Yn

|f −1n (fn(y))||Yn|

= 0 (no dominating pre-images).

The images of Yn in X are called weak expanders.

I Main Open Problem. Suppose that a metric space M withbounded geometry is not coarsely embeddable into `2. Does itfollow that M weakly contains a family of expanders?

I Proposition. If X is a metric space with bounded geometryweakly containing a family {Yn} of expanders, then X is notcoarsely embeddable into L1.



I limn→∞

maxy∈Yn

|f −1n (fn(y))||Yn|







I limn→∞

maxy∈Yn

|f −1n (fn(y))||Yn|





L2 vs L1

I It turns out that coarse embeddability into L2 is equivalent tocoarse embeddability into L1. This statement follows from thefollowing well-known facts:

I L2 is linearly isometric to a subspace of L1 (can be provedusing independent Gaussian variables).

I The metric space (L1, || · ||1/21 ) is isometric to a subset of L2.I We define the embedding in the following way: we map each

function from L1(R) to the indicator function of the setbetween the graph of the function and the x-axis. Thisindicator function is considered as an element of L2(R2). Onecan check that this mapping has the desired properties. (Theobservation is due to Schoenberg, the presented proof wassuggested by Naor.)

I These results show that to prove coarseembeddability/non-embeddability results for a Hilbert space itsuffices to prove similar results for L1.

L2 vs L1






L2 vs L1



I The metric space (L1, || · ||1/21 ) is isometric to a subset of L2.

I We define the embedding in the following way: we map eachfunction from L1(R) to the indicator function of the setbetween the graph of the function and the x-axis. Thisindicator function is considered as an element of L2(R2). Onecan check that this mapping has the desired properties. (Theobservation is due to Schoenberg, the presented proof wassuggested by Naor.)


L2 vs L1






L2 vs L1






I This proof of the proposition is similar to the proof ofnon-embeddability of expanders into L1. Let fn : Yn → X bemaps showing that X weakly contains {Yn}.

I Assume that there is a coarse embedding f : X → L1. Thenthere are functions ρ1 and ρ2 such that

∀n ∈ N ∀u, v ∈ Yn ρ1(dX (fn(u), fn(v))) ≤ ||f (fn(u))− f (fn(v))||≤ ρ2(dX (fn(u), fn(v))).

(1)

and limt→∞ ρ1(t) =∞.I We apply the Poincaré inequality∑

u,v∈Vau,v ||ϕ(u)− ϕ(v)|| ≥

∑u,v∈V

h

|V |||ϕ(u)− ϕ(v)||. (2)

to ϕn(u) = f (fn(u)) and get∑u,v∈V (Yn)

au,v ||ϕn(u)−ϕn(v)|| ≥∑

u,v∈V (Yn)

h

|V (Yn)|||ϕn(u)−ϕn(v)||

(3)




(1)

and limt→∞ ρ1(t) =∞.

I We apply the Poincaré inequality∑u,v∈V

au,v ||ϕ(u)− ϕ(v)|| ≥∑

u,v∈V

h

|V |||ϕ(u)− ϕ(v)||. (2)


au,v ||ϕn(u)−ϕn(v)|| ≥∑

u,v∈V (Yn)

h

|V (Yn)|||ϕn(u)−ϕn(v)||

(3)




(1)

and limt→∞ ρ1(t) =∞.I We apply the Poincaré inequality∑

u,v∈Vau,v ||ϕ(u)− ϕ(v)|| ≥

∑u,v∈V

h

|V |||ϕ(u)− ϕ(v)||. (2)


au,v ||ϕn(u)−ϕn(v)|| ≥∑

u,v∈V (Yn)

h

|V (Yn)|||ϕn(u)−ϕn(v)||

(3)

I Since Yn are k-regular, we use (1) and get that the left-handside is ≤ k |V (Yn)| · ρ2(supn Lip(fn)).

I The right-hand side can be estimated from below by∑u,v∈V (Yn)

h

|V (Yn)|ρ1(dX (fn(u), fn(v))).

I We estimate the last sum from below using the followingargument. We claim that for each 0 < C


h




h


I In fact, the number of elements x ∈ X satisfyingdX (fn(u), x) ≤ C does not exceed MX (C ) (where MX (·) isthe function from the definition of bounded geometry).Therefore the number of elements v ∈ V (Yn) satisfyingdX (fn(u).fn(v)) ≤ C does not exceedMX (C ) ·maxx∈fn(Yn) |f −1n (x)|.

I The condition from Definition 2 (of weak containment ofexpanders) implies that

limn→∞

MX (C ) ·maxx∈fn(Yn) |f −1n (x)||V (Yn)|

= 0.

Therefore the desired n exists.

I In fact, the number of elements x ∈ X satisfyingdX (fn(u), x) ≤ C does not exceed MX (C ) (where MX (·) isthe function from the definition of bounded geometry).Therefore the number of elements v ∈ V (Yn) satisfyingdX (fn(u).fn(v)) ≤ C does not exceedMX (C ) ·maxx∈fn(Yn) |f −1n (x)|.

I The condition from Definition 2 (of weak containment ofexpanders) implies that

limn→∞

MX (C ) ·maxx∈fn(Yn) |f −1n (x)||V (Yn)|

= 0.

Therefore the desired n exists.

I For such C and n we get, by inequalities (1), (3), and theargument immediately after them, that:

|V (Yn)|2

2· h|V (Yn)|

· ρ1(C ) ≤ k |V (Yn)|ρ2(supn

Lip(fn)).

Thus

ρ1(C ) ≤2k

hρ2(sup

nLip(fn)).

Since C can be chosen to be arbitrarily large, this leads to acontradiction.

I The following more vague version of the Main Open Problemis also of interest: Find some expander-like structures in ametric space which is not coarsely embeddable into a Hilbertspace.

I The purpose of this lecture is to present some results on thisproblem.

I We are going to work with L1 instead of L2. (I alreadyexplained why.)

I The following result is the first attempt to find expander-likestructures in spaces which do not admit coarse embeddingsinto `2. Recall that a metric space is called locally finite is allballs in it have finitely many elements.

I Theorem (MO (2009), Tessera (2009))Let M be a locally finite metric space which is not coarselyembeddable into L1. Then there exists a constant D, depending onM only, such that for each n ∈ N there exists a finite setBn ⊂ M ×M and a probability measure µ on Bn such that

I dM(u, v) ≥ n for each (u, v) ∈ Bn.I For each Lipschitz function f : M → L1 we have∫

Bn

||f (u)− f (v)||L1dµ(u, v) ≤ DLip(f ). (4)

I The following result is the first attempt to find expander-likestructures in spaces which do not admit coarse embeddingsinto `2. Recall that a metric space is called locally finite is allballs in it have finitely many elements.

I Theorem (MO (2009), Tessera (2009))Let M be a locally finite metric space which is not coarselyembeddable into L1. Then there exists a constant D, depending onM only, such that for each n ∈ N there exists a finite setBn ⊂ M ×M and a probability measure µ on Bn such that

I dM(u, v) ≥ n for each (u, v) ∈ Bn.I For each Lipschitz function f : M → L1 we have∫

Bn

||f (u)− f (v)||L1dµ(u, v) ≤ DLip(f ). (4)

I LemmaLet M be a locally finite metric space which is not coarselyembeddable into L1. There exists a constant C depending on Monly such that for each Lipschitz function f : M → L1 there existsa subset Bf ⊂ M ×M such that sup

(x ,y)∈BfdM(x , y) =∞, but

sup(x ,y)∈Bf

||f (x)− f (y)||L1 ≤ CLip(f ).

I Proof. Assume the contrary. Then, for each n ∈ N, thenumber n3 cannot serve as C . This means, that for eachn ∈ N there exists a Lipschitz mapping fn : M → L1 such thatfor each subset U ⊂ M ×M with

sup(x ,y)∈U

dM(x , y) =∞,

we havesup

(x ,y)∈U||fn(x)− fn(y)|| > n3Lip(fn).

I LemmaLet M be a locally finite metric space which is not coarselyembeddable into L1. There exists a constant C depending on Monly such that for each Lipschitz function f : M → L1 there existsa subset Bf ⊂ M ×M such that sup

(x ,y)∈BfdM(x , y) =∞, but

sup(x ,y)∈Bf

||f (x)− f (y)||L1 ≤ CLip(f ).

I Proof. Assume the contrary. Then, for each n ∈ N, thenumber n3 cannot serve as C . This means, that for eachn ∈ N there exists a Lipschitz mapping fn : M → L1 such thatfor each subset U ⊂ M ×M with

sup(x ,y)∈U

dM(x , y) =∞,

we havesup

(x ,y)∈U||fn(x)− fn(y)|| > n3Lip(fn).

I We choose a point in M and denote it by O. Without loss ofgenerality we may assume that fn(O) = 0.

I Consider the mapping

f : M →

( ∞∑k=1

⊕L1

)1

⊂ L1

given by

f (x) =∞∑k=1

1

Kk2· fk(x)Lip(fk)

,

where K =∑∞

k=11k2

.

I It is clear that the series converges and Lip(f ) ≤ 1.



f : M →

( ∞∑k=1

⊕L1

)1

⊂ L1

given by

f (x) =∞∑k=1

1

Kk2· fk(x)Lip(fk)

,

where K =∑∞

k=11k2

.




f : M →

( ∞∑k=1

⊕L1

)1

⊂ L1

given by

f (x) =∞∑k=1

1

Kk2· fk(x)Lip(fk)

,

where K =∑∞

k=11k2

.


I Let us show that f is a coarse embedding. We need anestimate from below only (the estimate from above is satisfiedbecause f is Lipschitz).

I The assumption implies that for each n ∈ N there is N ∈ Nsuch that

dM(x , y) ≥ N ⇒ ||fn(x)− fn(y)|| > n3Lip(fn).

On the other hand

||fn(x)− fn(y)|| > n3Lip(fn)⇒

||f (x)− f (y)|| =∞∑k=1

1

Kk2· ||fk(x)− fk(y)||

Lip(fk)>

n

K.

Hence f : M → L1 is a coarse embedding and we get acontradiction.

I Let us show that f is a coarse embedding. We need anestimate from below only (the estimate from above is satisfiedbecause f is Lipschitz).

I The assumption implies that for each n ∈ N there is N ∈ Nsuch that

dM(x , y) ≥ N ⇒ ||fn(x)− fn(y)|| > n3Lip(fn).

On the other hand

||fn(x)− fn(y)|| > n3Lip(fn)⇒

||f (x)− f (y)|| =∞∑k=1

1

Kk2· ||fk(x)− fk(y)||

Lip(fk)>

n

K.

Hence f : M → L1 is a coarse embedding and we get acontradiction.

I LemmaLet C be the constant whose existence is proved in the previousLemma and let ε > be arbitrary. For each n ∈ N we can find afinite subset Mn ⊂ M such that for each Lipschitz mappingf : M → L1 there is a pair (uf ,n, vf ,n) ∈ Mn ×Mn such that

I dM(uf ,n, vf ,n) ≥ n.I ||f (uf ,n)− f (vf ,n)|| ≤ (C + ε)Lip(f ).

I Proof. The ball in M of radius R centered at O will bedenoted by B(R). It is clear that it suffices to prove the resultfor 1-Lipschitz mappings satisfying f (O) = 0.

I Assume the contrary. Since M is locally finite, this impliesthat for each R ∈ N there is a 1-Lipschitz mappingfR : M → L1 such that fR(O) = 0 and, for u, v ∈ B(R), theinequality dM(u, v) ≥ n implies ||fR(u)− fR(v)||L1 > C + ε.

I We form an ultraproduct of the mappings {fR}∞R=1, that is, amapping f : M → (L1)U , given by f (m) = {fR(m)}∞R=1, whereU is a non-trivial ultrafilter on N and (L1)U is thecorresponding ultrapower.

I It is well-known that each ultrapower of L1 is isometric to anL1 space on some measure space, and its separable subspacesare isometric to subspaces of L1(0, 1). Therefore we canconsider f as a mapping into L1(0, 1). It is easy to verify thatLip(f ) ≤ 1 and that f satisfies the condition

dM(u, v) ≥ n⇒ ||f (u)− f (v)||L1 ≥ (C + ε).

We get a contradiction with the definition of C .

I We form an ultraproduct of the mappings {fR}∞R=1, that is, amapping f : M → (L1)U , given by f (m) = {fR(m)}∞R=1, whereU is a non-trivial ultrafilter on N and (L1)U is thecorresponding ultrapower.

I It is well-known that each ultrapower of L1 is isometric to anL1 space on some measure space, and its separable subspacesare isometric to subspaces of L1(0, 1). Therefore we canconsider f as a mapping into L1(0, 1). It is easy to verify thatLip(f ) ≤ 1 and that f satisfies the condition

dM(u, v) ≥ n⇒ ||f (u)− f (v)||L1 ≥ (C + ε).

We get a contradiction with the definition of C .

I Proof of the Theorem. Let D be a number satisfyingD > C , and let B be a number satisfying C < B < D.

I According to the second Lemma, there is a finite subsetMn ⊂ M such that for each 1-Lipschitz function f on M thereis a pair (u, v) in Mn such that dM(u, v) ≥ n and||f (u)− f (v)|| ≤ B.

I Let αn be the cardinality of Mn, we choose a point in Mn anddenote it by O. Proving the theorem it is enough to consider1-Lipschitz functions f : Mn → L1 satisfying f (O) = 0. Eachαn-element subset of L1 is isometric to a subset in `

αn(αn−1)/21

(Witsenhausen (1986), Ball (1990)). Therefore it suffices to

prove the result for 1-Lipschitz embeddings into `αn(αn−1)/21 .




αn(αn−1)/21






αn(αn−1)/21



I It is clear that it suffices to prove the inequality∫Bn

||f (u)− f (v)||dµ(u, v) ≤ B

for a(D−B2

)-net in the set of all functions satisfying the

conditions mentioned above, endowed with the metric

τ(f , g) = maxm∈Mn

||f (m)− g(m)||

I By compactness there exists a finite net satisfying thecondition. Let N be such a net.

I We are going to use the minimax theorem.


||f (u)− f (v)||dµ(u, v) ≤ B

for a(D−B2




||f (m)− g(m)||




||f (u)− f (v)||dµ(u, v) ≤ B

for a(D−B2




||f (m)− g(m)||



I Let A be the matrix whose columns are labelled by functionsbelonging to N, whose rows are labelled by pairs (u, v) ofelements of Mn satisfying dM(u, v) ≥ n, and whose entry onthe intersection of the column corresponding to f , and therow corresponding to (u, v) is ||f (u)− f (v)||.

I Then, for each column vector x = {xf }f ∈N with xf ≥ 0 and∑f ∈N xf = 1, the entries of the product Ax are the differences

||F (u)− F (v)||, where F : M →

(∑f ∈N⊕`αn(αn−1)/21

)1

is

given by F (m) =∑f ∈N

xf f (m). The function F can be

considered as a function into L1. It satisfies Lip(F ) ≤ 1.Hence there is a pair (u, v) in Mn satisfying dM(u, v) ≥ n and||F (u)− F (v)|| ≤ B. Therefore we have maxx minµ µAx ≤ B,where the minimum is taken over all vectors µ = {µ(u, v)},indexed by u, v ∈ Mn, dM(u, v) ≥ n, and satisfying theconditions µ(u, v) ≥ 0 and

∑µ(u, v) = 1.

I Let A be the matrix whose columns are labelled by functionsbelonging to N, whose rows are labelled by pairs (u, v) ofelements of Mn satisfying dM(u, v) ≥ n, and whose entry onthe intersection of the column corresponding to f , and therow corresponding to (u, v) is ||f (u)− f (v)||.

I Then, for each column vector x = {xf }f ∈N with xf ≥ 0 and∑f ∈N xf = 1, the entries of the product Ax are the differences

||F (u)− F (v)||, where F : M →

(∑f ∈N⊕`αn(αn−1)/21

)1

is

given by F (m) =∑f ∈N

xf f (m). The function F can be

considered as a function into L1. It satisfies Lip(F ) ≤ 1.Hence there is a pair (u, v) in Mn satisfying dM(u, v) ≥ n and||F (u)− F (v)|| ≤ B. Therefore we have maxx minµ µAx ≤ B,where the minimum is taken over all vectors µ = {µ(u, v)},indexed by u, v ∈ Mn, dM(u, v) ≥ n, and satisfying theconditions µ(u, v) ≥ 0 and

∑µ(u, v) = 1.

I By the von Neumann minimax theorem we have

minµ

maxxµAx ≤ B,

which is exactly the inequality we need to prove because µcan be regarded as a probability measure on the set of pairsfrom Mn with distance ≥ n.

I The proof above can be summarized in the following way: wecan consider our situation as a kind of a two-person game:one person picks a 1-Lipschitz function and the other picks apair of points in Mn at distance ≥ n. The second person winsif ||f (u)− f (v)|| ≤ B.

I By the minimax theorem the second person has always awinning weighted strategy.


minµ

maxxµAx ≤ B,





minµ

maxxµAx ≤ B,




I This is still far from the desired result. In fact, one can provean analogue of the Poincaré inequality (introduced in theprevious lecture) for Lp-valued functions on expander graphs,and show that metric spaces containing families of expandersdo not embed coarsely into Lp for 1 ≤ p 2, which is not coarsely embeddable into `2, and thuscontains structures described above.

I Therefore properties of the structures whose existence weproved today are quite different from properties of realexpanders.

I The following result was proved with the purpose to get fromthe previous result some more satisfactory expander-likestructures.

Expansion properties of sets Mn.

I Let s be a positive integer. We consider graphsG (n, s) = (Mn,E (Mn, s)), where the edge set E (Mn, s) isobtained by joining those pairs of vertices of Mn which are atdistance ≤ s. The graphs G (n, s) have uniformly boundeddegrees if the metric space M has bounded geometry.

I Consider the following condition:

I (*) For some s ∈ N there is a number hs > 0 and subgraphsHn of G (n, s) of indefinitely growing sizes (as n→∞) suchthat the expansion constants of {Hn} are uniformly boundedfrom below by hs .

I If we would prove that in the bounded geometry case thecondition (*) is satisfied, it would solve the problemmentioned at the beginning of the talk: whether each metricspace with bounded geometry which does not embed coarselyinto a Hilbert space contains weak expanders?

I At this point we are able to prove only the following weakerexpansion property of the graphs G (n, s). We introduce themeasure νn on Mn by νn(A) = µn(A×Mn). Let F be aninduced subgraph of G (n, s). We denote the vertex boundaryof a set A of vertices in F by δFA. (The vertex boundary of Ais the set of vertices which are not in A but are adjacent tosome vertices of A.)

I Theorem (MO (2009))

Let s and n be such that 2n > s > 8D. Let ϕ(D, s) = s4D − 2.Then G (n, s) contains an induced subgraph F with dM -diameter≥ n − s2 , such that each subset A ⊂ F of dM -diameter < n −

s2

satisfies the condition: νn(δFA) > ϕ(D, s)νn(A).

I The proof is subject of the next presentation. It uses theexhaustion process similar to the one used by Linial-Saks(1993) and “random” signing of functions similar to the wayit was used by Rao (1999) in his work on Lipschitzembeddings of planar graphs into `2.




s2






s2

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