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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
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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.
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.
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.
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
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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.
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
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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.
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
-
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.
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
-
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.
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
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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.
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
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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.
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
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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.
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
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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.
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
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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.
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
-
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 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.)
I These results show that to prove coarseembeddability/non-embeddability results for a Hilbert space itsuffices to prove similar results for L1.
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
-
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.
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
-
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.
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
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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)
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
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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∈V
au,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)
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
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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)
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
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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
-
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
-
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
-
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.
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
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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.
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
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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.
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
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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.)
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
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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.)
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
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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.)
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
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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)
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
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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)
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
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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).
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
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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).
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
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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.
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
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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.
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
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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.
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
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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.
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
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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.
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
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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.
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
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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.
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
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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.
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
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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 .
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
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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 .
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
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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 .
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
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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 .
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
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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 .
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
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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.
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
-
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.
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
-
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.
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
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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.
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
-
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.
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
-
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.
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
-
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.
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
-
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.
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
-
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.
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
-
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.
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
-
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.
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
-
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.
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
-
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.
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
-
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?
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
-
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?
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
-
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?
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
-
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.
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
-
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.
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
-
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.
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