hilbert space compression of groups · definition of compression (guentner-kaminker)let x and y be...
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Hilbert space compression of groups
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The definition: compression
Definition of compression (Guentner-Kaminker)
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The definition: compression
Definition of compression (Guentner-Kaminker)Let X and Y
be two metric spaces and let φ : X → Y be a 1-Lipschitz map.
![Page 4: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/4.jpg)
The definition: compression
Definition of compression (Guentner-Kaminker)Let X and Y
be two metric spaces and let φ : X → Y be a 1-Lipschitz map.Thecompression of φ is the supremum over all α ≥ 0 such that
distY (f (u), f (v)) ≥ distX (u, v)α
for all u, v with large enough distX (u, v).
![Page 5: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/5.jpg)
The definition: compression
Definition of compression (Guentner-Kaminker)Let X and Y
be two metric spaces and let φ : X → Y be a 1-Lipschitz map.Thecompression of φ is the supremum over all α ≥ 0 such that
distY (f (u), f (v)) ≥ distX (u, v)α
for all u, v with large enough distX (u, v).
If E is a class of metric spaces, then the E–compression of X
![Page 6: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/6.jpg)
The definition: compression
Definition of compression (Guentner-Kaminker)Let X and Y
be two metric spaces and let φ : X → Y be a 1-Lipschitz map.Thecompression of φ is the supremum over all α ≥ 0 such that
distY (f (u), f (v)) ≥ distX (u, v)α
for all u, v with large enough distX (u, v).
If E is a class of metric spaces, then the E–compression of X is thesupremum over all compressions of 1-Lipschitz maps X → Y ,Y ∈ E .
![Page 7: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/7.jpg)
The definition: compression
Definition of compression (Guentner-Kaminker)Let X and Y
be two metric spaces and let φ : X → Y be a 1-Lipschitz map.Thecompression of φ is the supremum over all α ≥ 0 such that
distY (f (u), f (v)) ≥ distX (u, v)α
for all u, v with large enough distX (u, v).
If E is a class of metric spaces, then the E–compression of X is thesupremum over all compressions of 1-Lipschitz maps X → Y ,Y ∈ E .
In particular, if E is the class of Hilbert spaces, we get the Hilbert
space compression of X .
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The definition: compression
Definition of compression (Guentner-Kaminker)Let X and Y
be two metric spaces and let φ : X → Y be a 1-Lipschitz map.Thecompression of φ is the supremum over all α ≥ 0 such that
distY (f (u), f (v)) ≥ distX (u, v)α
for all u, v with large enough distX (u, v).
If E is a class of metric spaces, then the E–compression of X is thesupremum over all compressions of 1-Lipschitz maps X → Y ,Y ∈ E .
In particular, if E is the class of Hilbert spaces, we get the Hilbert
space compression of X .
The Hilbert space compression of a space is a q.i. invariant.
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Compression function of an embedding
Let (X , distX ) and (Y , distY ) be metric spaces.
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Compression function of an embedding
Let (X , distX ) and (Y , distY ) be metric spaces. Let ρ be anincreasing function ρ : R+ → R+, with limx→∞ ρ = ∞.
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Compression function of an embedding
Let (X , distX ) and (Y , distY ) be metric spaces. Let ρ be anincreasing function ρ : R+ → R+, with limx→∞ ρ = ∞. A mapφ : X → Y is called a ρ–embedding
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Compression function of an embedding
Let (X , distX ) and (Y , distY ) be metric spaces. Let ρ be anincreasing function ρ : R+ → R+, with limx→∞ ρ = ∞. A mapφ : X → Y is called a ρ–embedding if
ρ(distX (x1, x2)) ≤ distY (φ(x1), φ(x2)) ≤ distX (x1, x2) , (1)
for all x1, x2 with large enough dist(x1, x2).
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Compression function of an embedding
Let (X , distX ) and (Y , distY ) be metric spaces. Let ρ be anincreasing function ρ : R+ → R+, with limx→∞ ρ = ∞. A mapφ : X → Y is called a ρ–embedding if
ρ(distX (x1, x2)) ≤ distY (φ(x1), φ(x2)) ≤ distX (x1, x2) , (1)
for all x1, x2 with large enough dist(x1, x2). This ρ is called thecompression function of the embedding.
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An order on functions
Notation.
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An order on functions
Notation. Let f , g : R+ → R+.
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An order on functions
Notation. Let f , g : R+ → R+. We write f ≪ g if for some a, b, c ,
f (x) ≤ ag(bx) + c
for all x > 0.
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An order on functions
Notation. Let f , g : R+ → R+. We write f ≪ g if for some a, b, c ,
f (x) ≤ ag(bx) + c
for all x > 0.
Equivalence relation:
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An order on functions
Notation. Let f , g : R+ → R+. We write f ≪ g if for some a, b, c ,
f (x) ≤ ag(bx) + c
for all x > 0.
Equivalence relation: f ∼ g iff f ≪ g , g ≪ f .
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An order on functions
Notation. Let f , g : R+ → R+. We write f ≪ g if for some a, b, c ,
f (x) ≤ ag(bx) + c
for all x > 0.
Equivalence relation: f ∼ g iff f ≪ g , g ≪ f .
Not a linear order. So we cannot talk about the maximal
compression function.
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The result of Guentner-Kaminker
◮ (Guentner-Kaminker)
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The result of Guentner-Kaminker
◮ (Guentner-Kaminker) If the compression function of someembedding into a Hilbert space is ≫ √
x (say, if thecompression > 1
2)
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The result of Guentner-Kaminker
◮ (Guentner-Kaminker) If the compression function of someembedding into a Hilbert space is ≫ √
x (say, if thecompression > 1
2) then the group is exact,
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The result of Guentner-Kaminker
◮ (Guentner-Kaminker) If the compression function of someembedding into a Hilbert space is ≫ √
x (say, if thecompression > 1
2) then the group is exact, so the group
satisfies Yu’s property A.
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The result of Guentner-Kaminker
◮ (Guentner-Kaminker) If the compression function of someembedding into a Hilbert space is ≫ √
x (say, if thecompression > 1
2) then the group is exact, so the group
satisfies Yu’s property A.
(Amenability - for the equivariant compression.)
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Known results about compression
(Bourgain, Tessera) The free group has compression 1.
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Known results about compression
(Bourgain, Tessera) The free group has compression 1.
The standard embedding of a tree
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Known results about compression
(Bourgain, Tessera) The free group has compression 1.
The standard embedding of a tree Fn → l2(set of edges) :
w →∑
w [i ].
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Known results about compression
(Bourgain, Tessera) The free group has compression 1.
The standard embedding of a tree Fn → l2(set of edges) :
w →∑
w [i ].
Bourgain’s embedding: add coefficients to that sum:
![Page 29: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/29.jpg)
Known results about compression
(Bourgain, Tessera) The free group has compression 1.
The standard embedding of a tree Fn → l2(set of edges) :
w →∑
w [i ].
Bourgain’s embedding: add coefficients to that sum:
w →∑
(n − i)1
2−ǫw [i ].
![Page 30: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/30.jpg)
Known results about compression
(Bourgain, Tessera) The free group has compression 1.
The standard embedding of a tree Fn → l2(set of edges) :
w →∑
w [i ].
Bourgain’s embedding: add coefficients to that sum:
w →∑
(n − i)1
2−ǫw [i ].
That embedding has compression function x1−2ǫ.
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Trees continued
Theorem. (Tessera)
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Trees continued
Theorem. (Tessera)Every increasing function f : R+ → R+
satisfying∫
x>c
f (x)2
x3dx < ∞
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Trees continued
Theorem. (Tessera)Every increasing function f : R+ → R+
satisfying∫
x>c
f (x)2
x3dx < ∞
is ≪ the compression function of some embedding of Fn into aHilbert space.
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Trees continued
Theorem. (Tessera)Every increasing function f : R+ → R+
satisfying∫
x>c
f (x)2
x3dx < ∞
is ≪ the compression function of some embedding of Fn into aHilbert space.
Clearly for every sublinear g there exists a function f satisfyingthat condition such that f (n) > g(n) for infinitely many n.
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Trees continued
Theorem. (Tessera)Every increasing function f : R+ → R+
satisfying∫
x>c
f (x)2
x3dx < ∞
is ≪ the compression function of some embedding of Fn into aHilbert space.
Clearly for every sublinear g there exists a function f satisfyingthat condition such that f (n) > g(n) for infinitely many n.
g(x)
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Trees continued
Theorem. (Tessera)Every increasing function f : R+ → R+
satisfying∫
x>c
f (x)2
x3dx < ∞
is ≪ the compression function of some embedding of Fn into aHilbert space.
Clearly for every sublinear g there exists a function f satisfyingthat condition such that f (n) > g(n) for infinitely many n.
f (x)
g(x)
![Page 37: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/37.jpg)
The definition: compression gap and compression function
Definition (compression gap).
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces.
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces. Let f ≪ g : R+ → R+
be increasing functions with limx→∞ f (x) = limx→∞ g(x) = ∞.
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces. Let f ≪ g : R+ → R+
be increasing functions with limx→∞ f (x) = limx→∞ g(x) = ∞.
We say that (f , g) is a E–compression gap of (X , dist) if
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces. Let f ≪ g : R+ → R+
be increasing functions with limx→∞ f (x) = limx→∞ g(x) = ∞.
We say that (f , g) is a E–compression gap of (X , dist) if
(1) for some ρ ≫ f there exists a ρ-embedding of X into aE-space;
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces. Let f ≪ g : R+ → R+
be increasing functions with limx→∞ f (x) = limx→∞ g(x) = ∞.
We say that (f , g) is a E–compression gap of (X , dist) if
(1) for some ρ ≫ f there exists a ρ-embedding of X into aE-space;
(2) for every ρ-embedding of X into a E-space, ρ ≪ g .
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces. Let f ≪ g : R+ → R+
be increasing functions with limx→∞ f (x) = limx→∞ g(x) = ∞.
We say that (f , g) is a E–compression gap of (X , dist) if
(1) for some ρ ≫ f there exists a ρ-embedding of X into aE-space;
(2) for every ρ-embedding of X into a E-space, ρ ≪ g .
If f = g then we say that f is the E-compression of X .
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The definition: compression gap and compression function
Definition (compression gap). Let (X , dist) be a metric space,and let E be a collection of metric spaces. Let f ≪ g : R+ → R+
be increasing functions with limx→∞ f (x) = limx→∞ g(x) = ∞.
We say that (f , g) is a E–compression gap of (X , dist) if
(1) for some ρ ≫ f there exists a ρ-embedding of X into aE-space;
(2) for every ρ-embedding of X into a E-space, ρ ≪ g .
If f = g then we say that f is the E-compression of X . The
quotient gf
is called the size of the gap.
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Compression gaps of groups
The following groups have compression gaps
(
x
log x(log log x)1+ǫ, x
)
:
![Page 46: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/46.jpg)
Compression gaps of groups
The following groups have compression gaps
(
x
log x(log log x)1+ǫ, x
)
:
◮ (Bourgain, Tessera) Free groups;
![Page 47: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/47.jpg)
Compression gaps of groups
The following groups have compression gaps
(
x
log x(log log x)1+ǫ, x
)
:
◮ (Bourgain, Tessera) Free groups;
◮ (Bonk-Schramm, Dranishnikov-Schroeder) Hyperbolic groups;
![Page 48: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/48.jpg)
Compression gaps of groups
The following groups have compression gaps
(
x
log x(log log x)1+ǫ, x
)
:
◮ (Bourgain, Tessera) Free groups;
◮ (Bonk-Schramm, Dranishnikov-Schroeder) Hyperbolic groups;
◮ (Tessera) Lattices in semi-simple Lie groups;
![Page 49: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/49.jpg)
Compression gaps of groups
The following groups have compression gaps
(
x
log x(log log x)1+ǫ, x
)
:
◮ (Bourgain, Tessera) Free groups;
◮ (Bonk-Schramm, Dranishnikov-Schroeder) Hyperbolic groups;
◮ (Tessera) Lattices in semi-simple Lie groups;
◮ (Campbell-Niblo, Brodsky-Sonkin) Groups acting on finitedim. CAT(0)-cubings.
![Page 50: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/50.jpg)
Compression gaps of groups
The following groups have compression gaps
(
x
log x(log log x)1+ǫ, x
)
:
◮ (Bourgain, Tessera) Free groups;
◮ (Bonk-Schramm, Dranishnikov-Schroeder) Hyperbolic groups;
◮ (Tessera) Lattices in semi-simple Lie groups;
◮ (Campbell-Niblo, Brodsky-Sonkin) Groups acting on finitedim. CAT(0)-cubings.
Problem. Is there a non-virtually cyclic group with bettercompression gap than Fn?
![Page 51: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/51.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
![Page 52: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/52.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
![Page 53: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/53.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
![Page 54: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/54.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
![Page 55: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/55.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
Definition 2.
![Page 56: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/56.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
Definition 2. F is the group of all piecewise linearhomeomorphisms of [0, 1] with dyadic break-points and slopes 2n,n ∈ Z.
![Page 57: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/57.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
Definition 2. F is the group of all piecewise linearhomeomorphisms of [0, 1] with dyadic break-points and slopes 2n,n ∈ Z.
Problems.
![Page 58: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/58.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
Definition 2. F is the group of all piecewise linearhomeomorphisms of [0, 1] with dyadic break-points and slopes 2n,n ∈ Z.
Problems. Is F amenable?
![Page 59: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/59.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
Definition 2. F is the group of all piecewise linearhomeomorphisms of [0, 1] with dyadic break-points and slopes 2n,n ∈ Z.
Problems. Is F amenable? Does F satisfy G. Yu’s property A?
![Page 60: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/60.jpg)
The Thompson group
Definition 1. F is given by the following presentation:
F = 〈xi , i ≥ 0 | xixj = xj+1xi , i < j〉
= 〈x0, x1 | x20x1x
−2
0= (x1x0)x1(x1x0),
x30x1x
−3
0= (x2
1x0)x1(x21x0)
−1〉.
Definition 2. F is the group of all piecewise linearhomeomorphisms of [0, 1] with dyadic break-points and slopes 2n,n ∈ Z.
Problems. Is F amenable? Does F satisfy G. Yu’s property A?
(Farley) F is a-T-menable.
![Page 61: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/61.jpg)
Thompson group as a diagram group
Definition 3.
![Page 62: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/62.jpg)
Thompson group as a diagram group
Definition 3. F is generated by the following two pictures:
![Page 63: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/63.jpg)
Thompson group as a diagram group
Definition 3. F is generated by the following two pictures:
q q q q q q q q q
x0 x1
![Page 64: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/64.jpg)
Thompson group as a diagram group
Definition 3. F is generated by the following two pictures:
q q q q q q q q q
x0 x1
Elementary diagrams:
![Page 65: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/65.jpg)
Thompson group as a diagram group
Definition 3. F is generated by the following two pictures:
q q q q q q q q q
x0 x1
Elementary diagrams:
r
r r
r qi e
i e
![Page 66: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/66.jpg)
Thompson group as a diagram group
Definition 3. F is generated by the following two pictures:
q q q q q q q q q
x0 x1
Elementary diagrams:
r
r r
r qi e
i e
Operations:
![Page 67: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/67.jpg)
Thompson group as a diagram group
Definition 3. F is generated by the following two pictures:
q q q q q q q q q
x0 x1
Elementary diagrams:
r
r r
r qi e
i e
Operations:
+ =
× =
![Page 68: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/68.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995)
![Page 69: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/69.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F .
![Page 70: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/70.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
![Page 71: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/71.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
![Page 72: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/72.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
![Page 73: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/73.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
![Page 74: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/74.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
![Page 75: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/75.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
x
![Page 76: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/76.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
x
![Page 77: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/77.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
x
![Page 78: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/78.jpg)
Thompson group as a diagram group
Theorem. (Guba, S., 1995) The set of (1, 1)-diagrams forms agroup isomorphic to F . The length function of F is quasi-isometricto “the number of cells” function.
Representation by piecewise linear and other homeomorphisms ofthe interval:
q
q q
f : x → 2x
x
φ(x)
![Page 79: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/79.jpg)
Z ≀ Z as a diagram group
Theorem. (Guba, S., 1995)
![Page 80: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/80.jpg)
Z ≀ Z as a diagram group
Theorem. (Guba, S., 1995) The group Z ≀ Z is the(ac , ac)-diagram group corresponding to the following elementarydiagrams:
![Page 81: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/81.jpg)
Z ≀ Z as a diagram group
Theorem. (Guba, S., 1995) The group Z ≀ Z is the(ac , ac)-diagram group corresponding to the following elementarydiagrams:
q q
q qq q
p q
c a
b ac bb
b
a, b, c?
![Page 82: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/82.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
![Page 83: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/83.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
![Page 84: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/84.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
◮ The R. Thompson group F has compression 1
2
![Page 85: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/85.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
◮ The R. Thompson group F has compression 1
2
◮ and compression gap
(√x ,√
x log x)
.
![Page 86: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/86.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
◮ The R. Thompson group F has compression 1
2
◮ and compression gap
(√x ,√
x log x)
.
◮ Same for the equivariant compression.
![Page 87: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/87.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
◮ The R. Thompson group F has compression 1
2
◮ and compression gap
(√x ,√
x log x)
.
◮ Same for the equivariant compression.
Idea of the proof.
![Page 88: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/88.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
◮ The R. Thompson group F has compression 1
2
◮ and compression gap
(√x ,√
x log x)
.
◮ Same for the equivariant compression.
Idea of the proof. Free group acts on a tree,
![Page 89: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/89.jpg)
Compression gap of the Thompson group
Theorem. (Arzhantseva-Guba-S.)
◮ Any diagram group with the Burillo property has compressionfunction ≥ √
x .
◮ The R. Thompson group F has compression 1
2
◮ and compression gap
(√x ,√
x log x)
.
◮ Same for the equivariant compression.
Idea of the proof. Free group acts on a tree, Thompson group(and any other diagram group) acts of a 2-tree.
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2-treesHow to build the tree (Cayley graph of F3):
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2-treesHow to build the tree (Cayley graph of F3):
a
b
c
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2-treesHow to build the tree (Cayley graph of F3):
a
b
c
a
c
a
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2-treesHow to build the tree (Cayley graph of F3):
a
b
c
a
c
a
a b b
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2-treesHow to build the tree (Cayley graph of F3):
a
c
a
a b b
b
c
![Page 95: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/95.jpg)
2-treesHow to build the tree (Cayley graph of F3):
a
c
a
b
c
bb
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2-treesHow to build the tree (Cayley graph of F3):
a
c
a
b
c b
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The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
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The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
π
π
∆ × = ∆
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The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
π
π
∆ × = ∆
Let H = l2(set of 2-cells of the 2-tree).
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The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
π
π
∆ × = ∆
Let H = l2(set of 2-cells of the 2-tree). Every diagram ∆ hasunique image on the 2-tree.
![Page 101: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/101.jpg)
The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
π
π
∆ × = ∆
Let H = l2(set of 2-cells of the 2-tree). Every diagram ∆ hasunique image on the 2-tree. Map it to the sum of the cells of theimage.
![Page 102: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/102.jpg)
The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
π
π
∆ × = ∆
Let H = l2(set of 2-cells of the 2-tree). Every diagram ∆ hasunique image on the 2-tree. Map it to the sum of the cells of theimage. The compression function is
√x . The equivariant
compression of the embedding = 1
2
![Page 103: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/103.jpg)
The lower bound of the compression gap
F acts on the set of cells of the 2-tree T :
π
π
∆ × = ∆
Let H = l2(set of 2-cells of the 2-tree). Every diagram ∆ hasunique image on the 2-tree. Map it to the sum of the cells of theimage. The compression function is
√x . The equivariant
compression of the embedding = 1
2because of the Burillo property.
![Page 104: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/104.jpg)
The upper bound
Build the following collections of diagrams Ψn by induction.
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The upper bound
Build the following collections of diagrams Ψn by induction.Ψ0 = {x0}. Suppose that Ψn = {∆1, ...,∆2n} has beenconstructed.
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The upper bound
Build the following collections of diagrams Ψn by induction.Ψ0 = {x0}. Suppose that Ψn = {∆1, ...,∆2n} has beenconstructed. Let Ψn+1 be:
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The upper bound
Build the following collections of diagrams Ψn by induction.Ψ0 = {x0}. Suppose that Ψn = {∆1, ...,∆2n} has beenconstructed. Let Ψn+1 be:
∆i ∆i
i = 1, ..., 2n.
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The upper bound
Build the following collections of diagrams Ψn by induction.Ψ0 = {x0}. Suppose that Ψn = {∆1, ...,∆2n} has beenconstructed. Let Ψn+1 be:
∆i ∆i
i = 1, ..., 2n.
The diagrams of Ψn pairwise commute and have 2n + 4 cells.
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The upper bound
Build the following collections of diagrams Ψn by induction.Ψ0 = {x0}. Suppose that Ψn = {∆1, ...,∆2n} has beenconstructed. Let Ψn+1 be:
∆i ∆i
i = 1, ..., 2n.
The diagrams of Ψn pairwise commute and have 2n + 4 cells.Thus the Cayley graph of F contains cubes of dimension 2n withsides of edges ∼ n.
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Skew cubes
Use the “skew cube” inequality:
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Skew cubes
Use the “skew cube” inequality:
For every skew cube in a Hilbert space, the sum of squares of its
diagonals does not exceed the sum of squares of its edges.
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Skew cubes
Use the “skew cube” inequality:
For every skew cube in a Hilbert space, the sum of squares of its
diagonals does not exceed the sum of squares of its edges.
This gives the upper bound for compression√
x log x
![Page 113: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/113.jpg)
Skew cubes
Use the “skew cube” inequality:
For every skew cube in a Hilbert space, the sum of squares of its
diagonals does not exceed the sum of squares of its edges.
This gives the upper bound for compression√
x log x and acompression gap
(√x ,√
x log x)
of logarithmic size.
![Page 114: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/114.jpg)
The problem
Problem. Is it true that a compression function of someembedding of F into a Hilbert space is ≫ √
x?
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Wreath products
Theorem (Arzhantseva-Guba-Sapir)
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Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4.
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Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4. The Hilbert space
compression of Z ≀ B where B has exponential growth is between 0and 1
2.
![Page 118: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/118.jpg)
Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4. The Hilbert space
compression of Z ≀ B where B has exponential growth is between 0and 1
2. If the polynomial growth rate is k then the compression
does not exceed 1+k/2
1+k.
![Page 119: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/119.jpg)
Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4. The Hilbert space
compression of Z ≀ B where B has exponential growth is between 0and 1
2. If the polynomial growth rate is k then the compression
does not exceed 1+k/2
1+k.
Problem. What is the compression of Z ≀ Z?
![Page 120: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/120.jpg)
Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4. The Hilbert space
compression of Z ≀ B where B has exponential growth is between 0and 1
2. If the polynomial growth rate is k then the compression
does not exceed 1+k/2
1+k.
Problem. What is the compression of Z ≀ Z? Tessera: ≥ 2
3.
![Page 121: Hilbert space compression of groups · Definition of compression (Guentner-Kaminker)Let X and Y be two metric spaces and let φ: X → Y be a 1-Lipschitz map.The compression of φ](https://reader033.vdocuments.mx/reader033/viewer/2022053023/60525ae6b0b1ff512023e4bc/html5/thumbnails/121.jpg)
Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4. The Hilbert space
compression of Z ≀ B where B has exponential growth is between 0and 1
2. If the polynomial growth rate is k then the compression
does not exceed 1+k/2
1+k.
Problem. What is the compression of Z ≀ Z? Tessera: ≥ 2
3.
Problem. What is the compression of Grigorchuk’s group ofsubexponential growth?
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Wreath products
Theorem (Arzhantseva-Guba-Sapir)The Hilbert spacecompression of Z ≀ Z is between 1
2and 3
4. The Hilbert space
compression of Z ≀ B where B has exponential growth is between 0and 1
2. If the polynomial growth rate is k then the compression
does not exceed 1+k/2
1+k.
Problem. What is the compression of Z ≀ Z? Tessera: ≥ 2
3.
Problem. What is the compression of Grigorchuk’s group ofsubexponential growth?
Problem. Is there an amenable group with compression 0?