Large scale geometry of metrisable groups Christian Rosendal, University of Illinois at Chicago Mostowski Centennial, Warsaw 2013 Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
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Large scale geometry of metrisable groups
Christian Rosendal,University of Illinois at Chicago
Mostowski Centennial, Warsaw 2013
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of discrete groups
One of the principal tenets of current day infinite group theory isthat countable discrete groups should be viewed as geometricobjects.
For a finitely generated group Γ there is an almost canonicalmanner of doing this.
Fix a finite symmetric generating set Σ for Γ and define thecorresponding Cayley graph on Γ by letting the edges be all
(g , gs)
where g ∈ Γ and s ∈ Σ \ {1}.
The resulting graph Cayley(Γ,Σ) is connected and hence Γ is ametric space, when given the shortest-path metric ρΣ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of discrete groups
One of the principal tenets of current day infinite group theory isthat countable discrete groups should be viewed as geometricobjects.
For a finitely generated group Γ there is an almost canonicalmanner of doing this.
Fix a finite symmetric generating set Σ for Γ and define thecorresponding Cayley graph on Γ by letting the edges be all
(g , gs)
where g ∈ Γ and s ∈ Σ \ {1}.
The resulting graph Cayley(Γ,Σ) is connected and hence Γ is ametric space, when given the shortest-path metric ρΣ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of discrete groups
One of the principal tenets of current day infinite group theory isthat countable discrete groups should be viewed as geometricobjects.
For a finitely generated group Γ there is an almost canonicalmanner of doing this.
Fix a finite symmetric generating set Σ for Γ and define thecorresponding Cayley graph on Γ by letting the edges be all
(g , gs)
where g ∈ Γ and s ∈ Σ \ {1}.
The resulting graph Cayley(Γ,Σ) is connected and hence Γ is ametric space, when given the shortest-path metric ρΣ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of discrete groups
One of the principal tenets of current day infinite group theory isthat countable discrete groups should be viewed as geometricobjects.
For a finitely generated group Γ there is an almost canonicalmanner of doing this.
Fix a finite symmetric generating set Σ for Γ and define thecorresponding Cayley graph on Γ by letting the edges be all
(g , gs)
where g ∈ Γ and s ∈ Σ \ {1}.
The resulting graph Cayley(Γ,Σ) is connected and hence Γ is ametric space, when given the shortest-path metric ρΣ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Consider first (Z,+) with generating set Σ = {−1, 1}.
• • • • • • • • • •−2 −1 0 1 2 3
Similarly, let F2 be the free non-abelian group on generators a, band set Σ = {a, b, a−1, b−1}.
• •
•
•
•
•
•a−1
bba−1 ba
1
b−1
a
aba−1b
a−1b−1 ab−1
b−1a−1 b−1a
b−2
b2
a2a−2• •
• •
• •
•
••
•
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Consider first (Z,+) with generating set Σ = {−1, 1}.
• • • • • • • • • •−2 −1 0 1 2 3
Similarly, let F2 be the free non-abelian group on generators a, band set Σ = {a, b, a−1, b−1}.
• •
•
•
•
•
•a−1
bba−1 ba
1
b−1
a
aba−1b
a−1b−1 ab−1
b−1a−1 b−1a
b−2
b2
a2a−2• •
• •
• •
•
••
•
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
SoρΣ(g , f ) = min(k
∣∣ ∃s1, . . . , sk ∈ Σ: f = gs1 · · · sk).
We note that, since
f = gs1 · · · sk ⇔ hf = hgs1 · · · sk ,
the left-multiplication action by Γ on (Γ, ρΣ) is an isometric action.
Therefore, ρΣ is left-invariant, i.e.,
ρΣ(hg , hf ) = ρΣ(g , f )
for all f , g , h.
ρΣ is called the word metric induced by the generating set Σ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
SoρΣ(g , f ) = min(k
∣∣ ∃s1, . . . , sk ∈ Σ: f = gs1 · · · sk).
We note that, since
f = gs1 · · · sk ⇔ hf = hgs1 · · · sk ,
the left-multiplication action by Γ on (Γ, ρΣ) is an isometric action.
Therefore, ρΣ is left-invariant, i.e.,
ρΣ(hg , hf ) = ρΣ(g , f )
for all f , g , h.
ρΣ is called the word metric induced by the generating set Σ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
SoρΣ(g , f ) = min(k
∣∣ ∃s1, . . . , sk ∈ Σ: f = gs1 · · · sk).
We note that, since
f = gs1 · · · sk ⇔ hf = hgs1 · · · sk ,
the left-multiplication action by Γ on (Γ, ρΣ) is an isometric action.
Therefore, ρΣ is left-invariant, i.e.,
ρΣ(hg , hf ) = ρΣ(g , f )
for all f , g , h.
ρΣ is called the word metric induced by the generating set Σ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
SoρΣ(g , f ) = min(k
∣∣ ∃s1, . . . , sk ∈ Σ: f = gs1 · · · sk).
We note that, since
f = gs1 · · · sk ⇔ hf = hgs1 · · · sk ,
the left-multiplication action by Γ on (Γ, ρΣ) is an isometric action.
Therefore, ρΣ is left-invariant, i.e.,
ρΣ(hg , hf ) = ρΣ(g , f )
for all f , g , h.
ρΣ is called the word metric induced by the generating set Σ.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ)
and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ,
which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g
gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1
gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2
. . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
As mentioned, Cayley(Γ,Σ) is almost canonical for Γ, except thatit involves the choice of a finite generating set Σ.
Now, if Σ′ was a larger finite generating set, then Cayley(Γ,Σ′)would have more edges than Cayley(Γ,Σ) and thus the shortestpaths could become shorter meaning that
ρΣ′ 6 ρΣ.
On the other hand, every s ′ ∈ Σ′ can be written as s ′ = s1 · · · snfor some si ∈ Σ, which means that, for every g ∈ Γ,
g gs1 gs1s2 . . . gs1s2 · · · sn
is a path of length n in Cayley(Γ,Σ) from g to gs ′ = gs1s2 · · · sn.
Taking N to be the largest n needed for the finitely many s ′ ∈ Σ′,one sees that
ρΣ′ 6 ρΣ 6 N · ρΣ′ .
So the two metrics are bi-Lipschitz equivalent.Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We can compare any two generating sets Σ and Σ′ with thegenerating common superset Σ ∪ Σ′.
So, it follows that, for afinitely generated group Γ,
the word metric ρΣ is canonical up to bi-Lipschitzequivalence.
In particular, Γ always carries a well-defined (up to bi-Lipschitzequivalence) large scale geometry.
Take again the example of (Z,+) with generating set Σ = {−1, 1}.• • • • • • • • • •
−2 −1 0 1 2 3
Whereas, with generating set Σ = {−2,−1, 1, 2}, we have
•
•
•
•
•
•
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−3
−2
−1
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1
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3
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We can compare any two generating sets Σ and Σ′ with thegenerating common superset Σ ∪ Σ′. So, it follows that, for afinitely generated group Γ,
the word metric ρΣ is canonical up to bi-Lipschitzequivalence.
In particular, Γ always carries a well-defined (up to bi-Lipschitzequivalence) large scale geometry.
Take again the example of (Z,+) with generating set Σ = {−1, 1}.• • • • • • • • • •
−2 −1 0 1 2 3
Whereas, with generating set Σ = {−2,−1, 1, 2}, we have
•
•
•
•
•
•
•
•
•
•
−4
−3
−2
−1
0
1
2
3
4
5
���
���
���
���
���
@@
@@@
@@@
@@@
@@@
@@
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We can compare any two generating sets Σ and Σ′ with thegenerating common superset Σ ∪ Σ′. So, it follows that, for afinitely generated group Γ,
the word metric ρΣ is canonical up to bi-Lipschitzequivalence.
In particular, Γ always carries a well-defined (up to bi-Lipschitzequivalence) large scale geometry.
Take again the example of (Z,+) with generating set Σ = {−1, 1}.• • • • • • • • • •
−2 −1 0 1 2 3
Whereas, with generating set Σ = {−2,−1, 1, 2}, we have
•
•
•
•
•
•
•
•
•
•
−4
−3
−2
−1
0
1
2
3
4
5
���
���
���
���
���
@@
@@@
@@@
@@@
@@@
@@
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We can compare any two generating sets Σ and Σ′ with thegenerating common superset Σ ∪ Σ′. So, it follows that, for afinitely generated group Γ,
the word metric ρΣ is canonical up to bi-Lipschitzequivalence.
In particular, Γ always carries a well-defined (up to bi-Lipschitzequivalence) large scale geometry.
Take again the example of (Z,+) with generating set Σ = {−1, 1}.• • • • • • • • • •
−2 −1 0 1 2 3
Whereas, with generating set Σ = {−2,−1, 1, 2}, we have
•
•
•
•
•
•
•
•
•
•
−4
−3
−2
−1
0
1
2
3
4
5
���
���
���
���
���
@@
@@@
@@@
@@@
@@@
@@
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We can compare any two generating sets Σ and Σ′ with thegenerating common superset Σ ∪ Σ′. So, it follows that, for afinitely generated group Γ,
the word metric ρΣ is canonical up to bi-Lipschitzequivalence.
In particular, Γ always carries a well-defined (up to bi-Lipschitzequivalence) large scale geometry.
Take again the example of (Z,+) with generating set Σ = {−1, 1}.• • • • • • • • • •
−2 −1 0 1 2 3
Whereas, with generating set Σ = {−2,−1, 1, 2}, we have
•
•
•
•
•
•
•
•
•
•
−4
−3
−2
−1
0
1
2
3
4
5
���
���
���
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���
@@
@@@
@@@
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of locally compact groups
Locally compact groups may be studied from the same perspectiveassuming now that they are metrisable, that is, second countableor equivalently Polish (i.e., separable and completely metrisable).
Instead of finite generation, one should now consider compactgeneration.
Moreover, we should relax the notion of bi-Lipschitz equivalence inorder to get canonical metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of locally compact groups
Locally compact groups may be studied from the same perspectiveassuming now that they are metrisable, that is, second countableor equivalently Polish (i.e., separable and completely metrisable).
Instead of finite generation, one should now consider compactgeneration.
Moreover, we should relax the notion of bi-Lipschitz equivalence inorder to get canonical metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of locally compact groups
Locally compact groups may be studied from the same perspectiveassuming now that they are metrisable, that is, second countableor equivalently Polish (i.e., separable and completely metrisable).
Instead of finite generation, one should now consider compactgeneration.
Moreover, we should relax the notion of bi-Lipschitz equivalence inorder to get canonical metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be aquasi-isometric embedding if there are constants K ,C so that
1
K· d(x , y)− C 6 ∂(Fx ,Fy) 6 K · d(x , y) + C .
Moreover, F is a quasi-isometry if, in addition, its image iscobounded, meaning that
supy∈Y
∂(y ,F [X ]) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be aquasi-isometric embedding if there are constants K ,C so that
1
K· d(x , y)− C 6 ∂(Fx ,Fy) 6 K · d(x , y) + C .
Moreover, F is a quasi-isometry if, in addition, its image iscobounded, meaning that
supy∈Y
∂(y ,F [X ]) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, a result of Struble tells us that
if G is locally compact metrisable group generated by acompact symmetric set Σ, then G admits a compatibleleft-invariant metric d quasi-isometric with ρΣ.
Moreover, as for the case of finite generating sets, one can verifythat all word metrics ρΣ with Σ compact are quasi-isometric.
Thus, the metric d is canonical up to quasi-isometric equivalence.
We take this d to define the large scale geometry of G .
*********************
As an aside, we should mention that by work of Gleason, G carriescanonical small scale geometry if and only if G is a Lie group.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, a result of Struble tells us that
if G is locally compact metrisable group generated by acompact symmetric set Σ, then G admits a compatibleleft-invariant metric d quasi-isometric with ρΣ.
Moreover, as for the case of finite generating sets, one can verifythat all word metrics ρΣ with Σ compact are quasi-isometric.
Thus, the metric d is canonical up to quasi-isometric equivalence.
We take this d to define the large scale geometry of G .
*********************
As an aside, we should mention that by work of Gleason, G carriescanonical small scale geometry if and only if G is a Lie group.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, a result of Struble tells us that
if G is locally compact metrisable group generated by acompact symmetric set Σ, then G admits a compatibleleft-invariant metric d quasi-isometric with ρΣ.
Moreover, as for the case of finite generating sets, one can verifythat all word metrics ρΣ with Σ compact are quasi-isometric.
Thus, the metric d is canonical up to quasi-isometric equivalence.
We take this d to define the large scale geometry of G .
*********************
As an aside, we should mention that by work of Gleason, G carriescanonical small scale geometry if and only if G is a Lie group.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, a result of Struble tells us that
if G is locally compact metrisable group generated by acompact symmetric set Σ, then G admits a compatibleleft-invariant metric d quasi-isometric with ρΣ.
Moreover, as for the case of finite generating sets, one can verifythat all word metrics ρΣ with Σ compact are quasi-isometric.
Thus, the metric d is canonical up to quasi-isometric equivalence.
We take this d to define the large scale geometry of G .
*********************
As an aside, we should mention that by work of Gleason, G carriescanonical small scale geometry if and only if G is a Lie group.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, a result of Struble tells us that
if G is locally compact metrisable group generated by acompact symmetric set Σ, then G admits a compatibleleft-invariant metric d quasi-isometric with ρΣ.
Moreover, as for the case of finite generating sets, one can verifythat all word metrics ρΣ with Σ compact are quasi-isometric.
Thus, the metric d is canonical up to quasi-isometric equivalence.
We take this d to define the large scale geometry of G .
*********************
As an aside, we should mention that by work of Gleason, G carriescanonical small scale geometry if and only if G is a Lie group.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
If G is no longer compactly generated, there are still left-invariantmetrics on G , but they are only canonical to a lesser extent.
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be acoarse embedding if,
1 for all R > 0 there is S > 0 so that
d(x , y) 6 R ⇒ ∂(Fx ,Fy) 6 S ,
2 for all S > 0 there is R > 0 so that
d(x , y) > R ⇒ ∂(Fx ,Fy) > S .
Moreover, F is a coarse equivalence if, in addition, its image iscobounded.
So, as opposed to quasi-isometry, the interdependence of R and Sis no longer affine.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
If G is no longer compactly generated, there are still left-invariantmetrics on G , but they are only canonical to a lesser extent.
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be acoarse embedding if,
1 for all R > 0 there is S > 0 so that
d(x , y) 6 R ⇒ ∂(Fx ,Fy) 6 S ,
2 for all S > 0 there is R > 0 so that
d(x , y) > R ⇒ ∂(Fx ,Fy) > S .
Moreover, F is a coarse equivalence if, in addition, its image iscobounded.
So, as opposed to quasi-isometry, the interdependence of R and Sis no longer affine.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
If G is no longer compactly generated, there are still left-invariantmetrics on G , but they are only canonical to a lesser extent.
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be acoarse embedding if,
1 for all R > 0 there is S > 0 so that
d(x , y) 6 R ⇒ ∂(Fx ,Fy) 6 S ,
2 for all S > 0 there is R > 0 so that
d(x , y) > R ⇒ ∂(Fx ,Fy) > S .
Moreover, F is a coarse equivalence if, in addition, its image iscobounded.
So, as opposed to quasi-isometry, the interdependence of R and Sis no longer affine.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
If G is no longer compactly generated, there are still left-invariantmetrics on G , but they are only canonical to a lesser extent.
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be acoarse embedding if,
1 for all R > 0 there is S > 0 so that
d(x , y) 6 R ⇒ ∂(Fx ,Fy) 6 S ,
2 for all S > 0 there is R > 0 so that
d(x , y) > R ⇒ ∂(Fx ,Fy) > S .
Moreover, F is a coarse equivalence if, in addition, its image iscobounded.
So, as opposed to quasi-isometry, the interdependence of R and Sis no longer affine.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
If G is no longer compactly generated, there are still left-invariantmetrics on G , but they are only canonical to a lesser extent.
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be acoarse embedding if,
1 for all R > 0 there is S > 0 so that
d(x , y) 6 R ⇒ ∂(Fx ,Fy) 6 S ,
2 for all S > 0 there is R > 0 so that
d(x , y) > R ⇒ ∂(Fx ,Fy) > S .
Moreover, F is a coarse equivalence if, in addition, its image iscobounded.
So, as opposed to quasi-isometry, the interdependence of R and Sis no longer affine.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
If G is no longer compactly generated, there are still left-invariantmetrics on G , but they are only canonical to a lesser extent.
Definition
A map F : (X , d)→ (Y , ∂) between metric spaces is said to be acoarse embedding if,
1 for all R > 0 there is S > 0 so that
d(x , y) 6 R ⇒ ∂(Fx ,Fy) 6 S ,
2 for all S > 0 there is R > 0 so that
d(x , y) > R ⇒ ∂(Fx ,Fy) > S .
Moreover, F is a coarse equivalence if, in addition, its image iscobounded.
So, as opposed to quasi-isometry, the interdependence of R and Sis no longer affine.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, the full result of Struble gives us that
if G is locally compact second countable (not necessarilycompactly generated), then G admits a compatibleleft-invariant proper metric d ,
i.e., so that finite-diametersets are relatively compact.
Moreover, any two proper left-invariant compatible metrics on Gare coarsely equivalent.
We take this d to define the coarse geometry of G .
Observation
Let d be a proper left-invariant metric on a locally compact groupG and suppose gn ∈ G .Then
d(gn, 1)→∞ ⇔ gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, the full result of Struble gives us that
if G is locally compact second countable (not necessarilycompactly generated), then G admits a compatibleleft-invariant proper metric d , i.e., so that finite-diametersets are relatively compact.
Moreover, any two proper left-invariant compatible metrics on Gare coarsely equivalent.
We take this d to define the coarse geometry of G .
Observation
Let d be a proper left-invariant metric on a locally compact groupG and suppose gn ∈ G .Then
d(gn, 1)→∞ ⇔ gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, the full result of Struble gives us that
if G is locally compact second countable (not necessarilycompactly generated), then G admits a compatibleleft-invariant proper metric d , i.e., so that finite-diametersets are relatively compact.
Moreover, any two proper left-invariant compatible metrics on Gare coarsely equivalent.
We take this d to define the coarse geometry of G .
Observation
Let d be a proper left-invariant metric on a locally compact groupG and suppose gn ∈ G .Then
d(gn, 1)→∞ ⇔ gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, the full result of Struble gives us that
if G is locally compact second countable (not necessarilycompactly generated), then G admits a compatibleleft-invariant proper metric d , i.e., so that finite-diametersets are relatively compact.
Moreover, any two proper left-invariant compatible metrics on Gare coarsely equivalent.
We take this d to define the coarse geometry of G .
Observation
Let d be a proper left-invariant metric on a locally compact groupG and suppose gn ∈ G .Then
d(gn, 1)→∞ ⇔ gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, the full result of Struble gives us that
if G is locally compact second countable (not necessarilycompactly generated), then G admits a compatibleleft-invariant proper metric d , i.e., so that finite-diametersets are relatively compact.
Moreover, any two proper left-invariant compatible metrics on Gare coarsely equivalent.
We take this d to define the coarse geometry of G .
Observation
Let d be a proper left-invariant metric on a locally compact groupG and suppose gn ∈ G .
Thend(gn, 1)→∞ ⇔ gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, the full result of Struble gives us that
if G is locally compact second countable (not necessarilycompactly generated), then G admits a compatibleleft-invariant proper metric d , i.e., so that finite-diametersets are relatively compact.
Moreover, any two proper left-invariant compatible metrics on Gare coarsely equivalent.
We take this d to define the coarse geometry of G .
Observation
Let d be a proper left-invariant metric on a locally compact groupG and suppose gn ∈ G .Then
d(gn, 1)→∞ ⇔ gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Problems for general metrisable groups
In a general metrisable group, there is a priori no canonical way ofpicking out a generating subset as in finitely or compactlygenerated groups.
So there is no known canonical word metric or large scale structure.
What is worse is that it is not clear that there should be anywell-defined coarse geometric structure either.
(a) How should we define gn →∞ in a general metrisable group?
(b) Is there a reasonable definition of proper metrics on G?
(c) Is there a canonical large scale or coarse geometry on such G?
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Problems for general metrisable groups
In a general metrisable group, there is a priori no canonical way ofpicking out a generating subset as in finitely or compactlygenerated groups.
So there is no known canonical word metric or large scale structure.
What is worse is that it is not clear that there should be anywell-defined coarse geometric structure either.
(a) How should we define gn →∞ in a general metrisable group?
(b) Is there a reasonable definition of proper metrics on G?
(c) Is there a canonical large scale or coarse geometry on such G?
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Problems for general metrisable groups
In a general metrisable group, there is a priori no canonical way ofpicking out a generating subset as in finitely or compactlygenerated groups.
So there is no known canonical word metric or large scale structure.
What is worse is that it is not clear that there should be anywell-defined coarse geometric structure either.
(a) How should we define gn →∞ in a general metrisable group?
(b) Is there a reasonable definition of proper metrics on G?
(c) Is there a canonical large scale or coarse geometry on such G?
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Problems for general metrisable groups
In a general metrisable group, there is a priori no canonical way ofpicking out a generating subset as in finitely or compactlygenerated groups.
So there is no known canonical word metric or large scale structure.
What is worse is that it is not clear that there should be anywell-defined coarse geometric structure either.
(a) How should we define gn →∞ in a general metrisable group?
(b) Is there a reasonable definition of proper metrics on G?
(c) Is there a canonical large scale or coarse geometry on such G?
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Problems for general metrisable groups
In a general metrisable group, there is a priori no canonical way ofpicking out a generating subset as in finitely or compactlygenerated groups.
So there is no known canonical word metric or large scale structure.
What is worse is that it is not clear that there should be anywell-defined coarse geometric structure either.
(a) How should we define gn →∞ in a general metrisable group?
(b) Is there a reasonable definition of proper metrics on G?
(c) Is there a canonical large scale or coarse geometry on such G?
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Problems for general metrisable groups
In a general metrisable group, there is a priori no canonical way ofpicking out a generating subset as in finitely or compactlygenerated groups.
So there is no known canonical word metric or large scale structure.
What is worse is that it is not clear that there should be anywell-defined coarse geometric structure either.
(a) How should we define gn →∞ in a general metrisable group?
(b) Is there a reasonable definition of proper metrics on G?
(c) Is there a canonical large scale or coarse geometry on such G?
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
A first observation is that, by a classical result due to Birkhoff(fils) and Kakutani, the following are equivalent for a topologicalgroup G ,
G is metrisable,
G is first-countable,
G admits a compatible left-invariant metric.
This suggests the following definition.
Definition
A sequence gn in a metrisable group G is said to tend to infinity,gn →∞, if and only if there exists a compatible left-invariantmetric d on G so that
d(gn, 1)→∞.
This agrees with the usual definition in locally compact groups.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Bad or surprising news
In the infinite symmetric group S∞ there are no sequencesgn →∞.
In fact, S∞ has finite diameter with respect to every left-invariantmetric (compatible with the topology or not) [G. M. Bergman].
If Gn is a sequence of non-compact locally compact metrisablegroups, then there is no single compatible left-invariant metricd on G1×G2×G3× . . . characterising the sequences gn →∞.
Let us give a name to those metrics that do define the coarsegeometry.
Definition
A compatible left-invariant metric d on a metrisable group G issaid to be metrically proper if d(gn, 1)→∞ whenever gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Bad or surprising news
In the infinite symmetric group S∞ there are no sequencesgn →∞.
In fact, S∞ has finite diameter with respect to every left-invariantmetric (compatible with the topology or not) [G. M. Bergman].
If Gn is a sequence of non-compact locally compact metrisablegroups, then there is no single compatible left-invariant metricd on G1×G2×G3× . . . characterising the sequences gn →∞.
Let us give a name to those metrics that do define the coarsegeometry.
Definition
A compatible left-invariant metric d on a metrisable group G issaid to be metrically proper if d(gn, 1)→∞ whenever gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Bad or surprising news
In the infinite symmetric group S∞ there are no sequencesgn →∞.
In fact, S∞ has finite diameter with respect to every left-invariantmetric (compatible with the topology or not) [G. M. Bergman].
If Gn is a sequence of non-compact locally compact metrisablegroups, then there is no single compatible left-invariant metricd on G1×G2×G3× . . . characterising the sequences gn →∞.
Let us give a name to those metrics that do define the coarsegeometry.
Definition
A compatible left-invariant metric d on a metrisable group G issaid to be metrically proper if d(gn, 1)→∞ whenever gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Bad or surprising news
In the infinite symmetric group S∞ there are no sequencesgn →∞.
In fact, S∞ has finite diameter with respect to every left-invariantmetric (compatible with the topology or not) [G. M. Bergman].
If Gn is a sequence of non-compact locally compact metrisablegroups, then there is no single compatible left-invariant metricd on G1×G2×G3× . . . characterising the sequences gn →∞.
Let us give a name to those metrics that do define the coarsegeometry.
Definition
A compatible left-invariant metric d on a metrisable group G issaid to be metrically proper if d(gn, 1)→∞ whenever gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Bad or surprising news
In the infinite symmetric group S∞ there are no sequencesgn →∞.
In fact, S∞ has finite diameter with respect to every left-invariantmetric (compatible with the topology or not) [G. M. Bergman].
If Gn is a sequence of non-compact locally compact metrisablegroups, then there is no single compatible left-invariant metricd on G1×G2×G3× . . . characterising the sequences gn →∞.
Let us give a name to those metrics that do define the coarsegeometry.
Definition
A compatible left-invariant metric d on a metrisable group G issaid to be metrically proper if d(gn, 1)→∞ whenever gn →∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Relative property (OB)
Definition
A subset A of a metrisable group G is said to have property (OB)relative to G if, for every compatible left-invariant metric d on G ,one has
diamd(A) <∞.
Also, the group G is said to have property (OB) if it has property(OB) relative to itself.
It is not hard to show that the class of sets with property (OB)relative to G is an ideal closed under multiplication (A,B) 7→ ABand inversion A 7→ A−1.
Also, by the existence of proper metrics, in locally compact groups,the relative property (OB) coincides with relative compactness.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Relative property (OB)
Definition
A subset A of a metrisable group G is said to have property (OB)relative to G if, for every compatible left-invariant metric d on G ,one has
diamd(A) <∞.
Also, the group G is said to have property (OB) if it has property(OB) relative to itself.
It is not hard to show that the class of sets with property (OB)relative to G is an ideal closed under multiplication (A,B) 7→ ABand inversion A 7→ A−1.
Also, by the existence of proper metrics, in locally compact groups,the relative property (OB) coincides with relative compactness.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Relative property (OB)
Definition
A subset A of a metrisable group G is said to have property (OB)relative to G if, for every compatible left-invariant metric d on G ,one has
diamd(A) <∞.
Also, the group G is said to have property (OB) if it has property(OB) relative to itself.
It is not hard to show that the class of sets with property (OB)relative to G is an ideal closed under multiplication (A,B) 7→ ABand inversion A 7→ A−1.
Also, by the existence of proper metrics, in locally compact groups,the relative property (OB) coincides with relative compactness.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Relative property (OB)
Definition
A subset A of a metrisable group G is said to have property (OB)relative to G if, for every compatible left-invariant metric d on G ,one has
diamd(A) <∞.
Also, the group G is said to have property (OB) if it has property(OB) relative to itself.
It is not hard to show that the class of sets with property (OB)relative to G is an ideal closed under multiplication (A,B) 7→ ABand inversion A 7→ A−1.
Also, by the existence of proper metrics, in locally compact groups,the relative property (OB) coincides with relative compactness.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Examples of Polish groups with property (OB)
1 Automorphism groups of countable ℵ0-categorical structures(Cameron),
2 Automorphism groups of separably categorical bounded metricstructures (R.), e.g., U(`2), Aut([0, 1], λ), Iso(U1),
3 Homeo(Sn), Homeo([0, 1]N) (R., also Calegari, Freedman, deCornulier),
4 Isom(QU1) (R.).
Note that Isom(U) acts transitively on the unbounded metricspace U and therefore cannot have property (OB) as the metric onU pulls back to an unbounded left-invariant metric on Isom(U).
Also, QU1 is not ℵ0-categorical.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Examples of Polish groups with property (OB)
1 Automorphism groups of countable ℵ0-categorical structures(Cameron),
2 Automorphism groups of separably categorical bounded metricstructures (R.), e.g., U(`2), Aut([0, 1], λ), Iso(U1),
3 Homeo(Sn), Homeo([0, 1]N) (R., also Calegari, Freedman, deCornulier),
4 Isom(QU1) (R.).
Note that Isom(U) acts transitively on the unbounded metricspace U and therefore cannot have property (OB) as the metric onU pulls back to an unbounded left-invariant metric on Isom(U).
Also, QU1 is not ℵ0-categorical.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Examples of Polish groups with property (OB)
1 Automorphism groups of countable ℵ0-categorical structures(Cameron),
2 Automorphism groups of separably categorical bounded metricstructures (R.), e.g., U(`2), Aut([0, 1], λ), Iso(U1),
3 Homeo(Sn), Homeo([0, 1]N) (R., also Calegari, Freedman, deCornulier),
4 Isom(QU1) (R.).
Note that Isom(U) acts transitively on the unbounded metricspace U and therefore cannot have property (OB) as the metric onU pulls back to an unbounded left-invariant metric on Isom(U).
Also, QU1 is not ℵ0-categorical.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Examples of Polish groups with property (OB)
1 Automorphism groups of countable ℵ0-categorical structures(Cameron),
2 Automorphism groups of separably categorical bounded metricstructures (R.), e.g., U(`2), Aut([0, 1], λ), Iso(U1),
3 Homeo(Sn), Homeo([0, 1]N) (R., also Calegari, Freedman, deCornulier),
4 Isom(QU1) (R.).
Note that Isom(U) acts transitively on the unbounded metricspace U and therefore cannot have property (OB) as the metric onU pulls back to an unbounded left-invariant metric on Isom(U).
Also, QU1 is not ℵ0-categorical.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Examples of Polish groups with property (OB)
1 Automorphism groups of countable ℵ0-categorical structures(Cameron),
2 Automorphism groups of separably categorical bounded metricstructures (R.), e.g., U(`2), Aut([0, 1], λ), Iso(U1),
3 Homeo(Sn), Homeo([0, 1]N) (R., also Calegari, Freedman, deCornulier),
4 Isom(QU1) (R.).
Note that Isom(U) acts transitively on the unbounded metricspace U and therefore cannot have property (OB) as the metric onU pulls back to an unbounded left-invariant metric on Isom(U).
Also, QU1 is not ℵ0-categorical.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Examples of Polish groups with property (OB)
1 Automorphism groups of countable ℵ0-categorical structures(Cameron),
2 Automorphism groups of separably categorical bounded metricstructures (R.), e.g., U(`2), Aut([0, 1], λ), Iso(U1),
3 Homeo(Sn), Homeo([0, 1]N) (R., also Calegari, Freedman, deCornulier),
4 Isom(QU1) (R.).
Note that Isom(U) acts transitively on the unbounded metricspace U and therefore cannot have property (OB) as the metric onU pulls back to an unbounded left-invariant metric on Isom(U).
Also, QU1 is not ℵ0-categorical.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using the mechanics of the Birkhoff–Kakutani theorem andArens–Eells spaces, we have the following characterisation.
Lemma
TFAE for a subset A of a separable metrisable group G ,
1 A has property (OB) relative to G ,
2 for every open V 3 1 there are a finite subset F ⊆ G andsome k > 1 so that A ⊆ (FV )k ,
3 for every continuous isometric action G y (X , d) on a metricspace and every x ∈ X ,
diamd(A · x) <∞,
4 for every continuous affine isometric action G y V on aBanach space and every v ∈ V ,
diamd(A · v) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using the mechanics of the Birkhoff–Kakutani theorem andArens–Eells spaces, we have the following characterisation.
Lemma
TFAE for a subset A of a separable metrisable group G ,
1 A has property (OB) relative to G ,
2 for every open V 3 1 there are a finite subset F ⊆ G andsome k > 1 so that A ⊆ (FV )k ,
3 for every continuous isometric action G y (X , d) on a metricspace and every x ∈ X ,
diamd(A · x) <∞,
4 for every continuous affine isometric action G y V on aBanach space and every v ∈ V ,
diamd(A · v) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using the mechanics of the Birkhoff–Kakutani theorem andArens–Eells spaces, we have the following characterisation.
Lemma
TFAE for a subset A of a separable metrisable group G ,
1 A has property (OB) relative to G ,
2 for every open V 3 1 there are a finite subset F ⊆ G andsome k > 1 so that A ⊆ (FV )k ,
3 for every continuous isometric action G y (X , d) on a metricspace and every x ∈ X ,
diamd(A · x) <∞,
4 for every continuous affine isometric action G y V on aBanach space and every v ∈ V ,
diamd(A · v) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using the mechanics of the Birkhoff–Kakutani theorem andArens–Eells spaces, we have the following characterisation.
Lemma
TFAE for a subset A of a separable metrisable group G ,
1 A has property (OB) relative to G ,
2 for every open V 3 1 there are a finite subset F ⊆ G andsome k > 1 so that A ⊆ (FV )k ,
3 for every continuous isometric action G y (X , d) on a metricspace and every x ∈ X ,
diamd(A · x) <∞,
4 for every continuous affine isometric action G y V on aBanach space and every v ∈ V ,
diamd(A · v) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using the mechanics of the Birkhoff–Kakutani theorem andArens–Eells spaces, we have the following characterisation.
Lemma
TFAE for a subset A of a separable metrisable group G ,
1 A has property (OB) relative to G ,
2 for every open V 3 1 there are a finite subset F ⊆ G andsome k > 1 so that A ⊆ (FV )k ,
3 for every continuous isometric action G y (X , d) on a metricspace and every x ∈ X ,
diamd(A · x) <∞,
4 for every continuous affine isometric action G y V on aBanach space and every v ∈ V ,
diamd(A · v) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using the mechanics of the Birkhoff–Kakutani theorem andArens–Eells spaces, we have the following characterisation.
Lemma
TFAE for a subset A of a separable metrisable group G ,
1 A has property (OB) relative to G ,
2 for every open V 3 1 there are a finite subset F ⊆ G andsome k > 1 so that A ⊆ (FV )k ,
3 for every continuous isometric action G y (X , d) on a metricspace and every x ∈ X ,
diamd(A · x) <∞,
4 for every continuous affine isometric action G y V on aBanach space and every v ∈ V ,
diamd(A · v) <∞.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Lemma
TFAE for a compatible left-invariant metric d on G ,
1 d is metrically proper,
2 for all A ⊆ G , we have diamd(A) <∞ if and only if A hasproperty (OB) relative to G ,
3 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , ∂)→ (G , d)
is metrically proper, i.e., every set of infinite ∂-diameter hasinfinite d-diameter,
4 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , d)→ (G , ∂)
is bornologous, i.e., for all R > 0 there is S > 0 so that
d(g , f ) < R ⇒ ∂(g , f ) < S .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Lemma
TFAE for a compatible left-invariant metric d on G ,
1 d is metrically proper,
2 for all A ⊆ G , we have diamd(A) <∞ if and only if A hasproperty (OB) relative to G ,
3 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , ∂)→ (G , d)
is metrically proper, i.e., every set of infinite ∂-diameter hasinfinite d-diameter,
4 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , d)→ (G , ∂)
is bornologous, i.e., for all R > 0 there is S > 0 so that
d(g , f ) < R ⇒ ∂(g , f ) < S .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Lemma
TFAE for a compatible left-invariant metric d on G ,
1 d is metrically proper,
2 for all A ⊆ G , we have diamd(A) <∞ if and only if A hasproperty (OB) relative to G ,
3 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , ∂)→ (G , d)
is metrically proper, i.e., every set of infinite ∂-diameter hasinfinite d-diameter,
4 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , d)→ (G , ∂)
is bornologous, i.e., for all R > 0 there is S > 0 so that
d(g , f ) < R ⇒ ∂(g , f ) < S .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Lemma
TFAE for a compatible left-invariant metric d on G ,
1 d is metrically proper,
2 for all A ⊆ G , we have diamd(A) <∞ if and only if A hasproperty (OB) relative to G ,
3 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , ∂)→ (G , d)
is metrically proper, i.e., every set of infinite ∂-diameter hasinfinite d-diameter,
4 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , d)→ (G , ∂)
is bornologous, i.e., for all R > 0 there is S > 0 so that
d(g , f ) < R ⇒ ∂(g , f ) < S .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Lemma
TFAE for a compatible left-invariant metric d on G ,
1 d is metrically proper,
2 for all A ⊆ G , we have diamd(A) <∞ if and only if A hasproperty (OB) relative to G ,
3 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , ∂)→ (G , d)
is metrically proper, i.e., every set of infinite ∂-diameter hasinfinite d-diameter,
4 for every compatible left-invariant metric ∂ on G , the mapping
id : (G , d)→ (G , ∂)
is bornologous, i.e., for all R > 0 there is S > 0 so that
d(g , f ) < R ⇒ ∂(g , f ) < S .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Coarse geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 G admits a metrically proper compatible left-invariant metric,
2 G has the local property (OB), i.e., there is a neighbourhoodV 3 1 with property (OB) relative to G .
We should mention that this fails without the assumption ofseparability.
Also, the metrically proper compatible left-invariant metric d isunique up to coarse equivalence and thus defines the coarsegeometry of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Coarse geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 G admits a metrically proper compatible left-invariant metric,
2 G has the local property (OB), i.e., there is a neighbourhoodV 3 1 with property (OB) relative to G .
We should mention that this fails without the assumption ofseparability.
Also, the metrically proper compatible left-invariant metric d isunique up to coarse equivalence and thus defines the coarsegeometry of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Coarse geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 G admits a metrically proper compatible left-invariant metric,
2 G has the local property (OB), i.e., there is a neighbourhoodV 3 1 with property (OB) relative to G .
We should mention that this fails without the assumption ofseparability.
Also, the metrically proper compatible left-invariant metric d isunique up to coarse equivalence and thus defines the coarsegeometry of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Coarse geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 G admits a metrically proper compatible left-invariant metric,
2 G has the local property (OB), i.e., there is a neighbourhoodV 3 1 with property (OB) relative to G .
We should mention that this fails without the assumption ofseparability.
Also, the metrically proper compatible left-invariant metric d isunique up to coarse equivalence and thus defines the coarsegeometry of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Coarse geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 G admits a metrically proper compatible left-invariant metric,
2 G has the local property (OB), i.e., there is a neighbourhoodV 3 1 with property (OB) relative to G .
We should mention that this fails without the assumption ofseparability.
Also, the metrically proper compatible left-invariant metric d isunique up to coarse equivalence and thus defines the coarsegeometry of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Coarse geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 G admits a metrically proper compatible left-invariant metric,
2 G has the local property (OB), i.e., there is a neighbourhoodV 3 1 with property (OB) relative to G .
We should mention that this fails without the assumption ofseparability.
Also, the metrically proper compatible left-invariant metric d isunique up to coarse equivalence and thus defines the coarsegeometry of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Maximal (or word) metrics
Having characterised the separable metrisable groups admittingcanonical metrics up to coarse equivalence, we turn to the finerissue of characterising those admitting canonical metrics up toquasi-isometry.
First we need to identify the right class of metrics.
Definition
A compatible left-invariant metric d on G is said to be maximal forlarge distances if, for every other compatible left-invariant metric ∂on G , the map
id : (G , d)→ (G , ∂)
is Lipschitz for large distances, i.e., there are constants K ,C sothat
∂(g , f ) 6 K · d(g , f ) + C .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Maximal (or word) metrics
Having characterised the separable metrisable groups admittingcanonical metrics up to coarse equivalence, we turn to the finerissue of characterising those admitting canonical metrics up toquasi-isometry.
First we need to identify the right class of metrics.
Definition
A compatible left-invariant metric d on G is said to be maximal forlarge distances if, for every other compatible left-invariant metric ∂on G , the map
id : (G , d)→ (G , ∂)
is Lipschitz for large distances, i.e., there are constants K ,C sothat
∂(g , f ) 6 K · d(g , f ) + C .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Maximal (or word) metrics
Having characterised the separable metrisable groups admittingcanonical metrics up to coarse equivalence, we turn to the finerissue of characterising those admitting canonical metrics up toquasi-isometry.
First we need to identify the right class of metrics.
Definition
A compatible left-invariant metric d on G is said to be maximal forlarge distances if, for every other compatible left-invariant metric ∂on G , the map
id : (G , d)→ (G , ∂)
is Lipschitz for large distances
Lipschitz for large distances, i.e.,there are constants K ,C so that
∂(g , f ) 6 K · d(g , f ) + C .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Maximal (or word) metrics
Having characterised the separable metrisable groups admittingcanonical metrics up to coarse equivalence, we turn to the finerissue of characterising those admitting canonical metrics up toquasi-isometry.
First we need to identify the right class of metrics.
Definition
A compatible left-invariant metric d on G is said to be maximal forlarge distances if, for every other compatible left-invariant metric ∂on G , the map
id : (G , d)→ (G , ∂)
is Lipschitz for large distances, i.e., there are constants K ,C sothat
∂(g , f ) 6 K · d(g , f ) + C .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Note that, up to quasi-isometry, there is at most one compatibleleft-invariant metric on G maximal for large distances.
The problem is their existence.
If it exists, we can therefore talk unambigously of thequasi-isometric structure of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Note that, up to quasi-isometry, there is at most one compatibleleft-invariant metric on G maximal for large distances.
The problem is their existence.
If it exists, we can therefore talk unambigously of thequasi-isometric structure of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Note that, up to quasi-isometry, there is at most one compatibleleft-invariant metric on G maximal for large distances.
The problem is their existence.
If it exists, we can therefore talk unambigously of thequasi-isometric structure of G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Definition
A metric space (X , d) is said to be large scale geodesic if there isK > 1 so that, for all x , y ∈ X , there are z0 = x , z1, z2, . . . , zn = ysatisfying
1 d(zi , zi+1) 6 K ,
2∑n−1
i=0 d(zi , zi+1) 6 K · d(x , y).
•
••
•
•
x = z0
z1
z3
z2
y = z4
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Definition
A metric space (X , d) is said to be large scale geodesic if there isK > 1 so that, for all x , y ∈ X , there are z0 = x , z1, z2, . . . , zn = ysatisfying
1 d(zi , zi+1) 6 K ,
2∑n−1
i=0 d(zi , zi+1) 6 K · d(x , y).
•
• y
x
•
••
•
•
x = z0
z1
z3
z2
y = z4
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Definition
A metric space (X , d) is said to be large scale geodesic if there isK > 1 so that, for all x , y ∈ X , there are z0 = x , z1, z2, . . . , zn = ysatisfying
1 d(zi , zi+1) 6 K ,
2∑n−1
i=0 d(zi , zi+1) 6 K · d(x , y).
•
••
•
•
x = z0
z1
z3
z2
y = z4
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Proposition
The following conditions are equivalent for a metrically propercompatible left-invariant metric d on a metrisable group G ,
1 d is maximal for large distances,
2 (G , d) is large scale geodesic,
3 d is quasi-isometric to a word metric ρΣ, where Σ is asymmetric generating set with property (OB) relative to G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Proposition
The following conditions are equivalent for a metrically propercompatible left-invariant metric d on a metrisable group G ,
1 d is maximal for large distances,
2 (G , d) is large scale geodesic,
3 d is quasi-isometric to a word metric ρΣ, where Σ is asymmetric generating set with property (OB) relative to G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Proposition
The following conditions are equivalent for a metrically propercompatible left-invariant metric d on a metrisable group G ,
1 d is maximal for large distances,
2 (G , d) is large scale geodesic,
3 d is quasi-isometric to a word metric ρΣ, where Σ is asymmetric generating set with property (OB) relative to G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Proposition
The following conditions are equivalent for a metrically propercompatible left-invariant metric d on a metrisable group G ,
1 d is maximal for large distances,
2 (G , d) is large scale geodesic,
3 d is quasi-isometric to a word metric ρΣ, where Σ is asymmetric generating set with property (OB) relative to G .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 there is a compatible left-invariant metric d on G maximal forlarge distances,
2 G is generated by an open set with the relative property (OB).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 there is a compatible left-invariant metric d on G maximal forlarge distances,
2 G is generated by an open set with the relative property (OB).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem
The following are equivalent for a separable metrisable group G ,
1 there is a compatible left-invariant metric d on G maximal forlarge distances,
2 G is generated by an open set with the relative property (OB).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Svarc–Milnor lemmas
Definition
An continuous isometric action G y (X , d) by a metrisable groupon a metric space is said to be metrically proper if, for all x ∈ X ,
d(gnx , x)→∞ whenever gn →∞.
Moreover, the action is cobounded if there is a set A ⊆ X of finitediameter so that X = G · A.
Theorem (Existence)
Suppose G is a separable metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on aconnected metric space.
Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Svarc–Milnor lemmas
Definition
An continuous isometric action G y (X , d) by a metrisable groupon a metric space is said to be metrically proper if, for all x ∈ X ,
d(gnx , x)→∞ whenever gn →∞.
Moreover, the action is cobounded if there is a set A ⊆ X of finitediameter so that X = G · A.
Theorem (Existence)
Suppose G is a separable metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on aconnected metric space.
Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Svarc–Milnor lemmas
Definition
An continuous isometric action G y (X , d) by a metrisable groupon a metric space is said to be metrically proper if, for all x ∈ X ,
d(gnx , x)→∞ whenever gn →∞.
Moreover, the action is cobounded if there is a set A ⊆ X of finitediameter so that X = G · A.
Theorem (Existence)
Suppose G is a separable metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on aconnected metric space.
Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Svarc–Milnor lemmas
Definition
An continuous isometric action G y (X , d) by a metrisable groupon a metric space is said to be metrically proper if, for all x ∈ X ,
d(gnx , x)→∞ whenever gn →∞.
Moreover, the action is cobounded if there is a set A ⊆ X of finitediameter so that X = G · A.
Theorem (Existence)
Suppose G is a separable metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on aconnected metric space.
Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem (Existence and characterisation)
Suppose G is a metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on a largescale geodesic metric space.
(a) Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
(b) Moreover, for every x ∈ X , the map
g ∈ G 7→ gx ∈ X
is a quasi-isometry between (G , ∂) and (X , d).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem (Existence and characterisation)
Suppose G is a metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on a largescale geodesic metric space.
(a) Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
(b) Moreover, for every x ∈ X , the map
g ∈ G 7→ gx ∈ X
is a quasi-isometry between (G , ∂) and (X , d).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem (Existence and characterisation)
Suppose G is a metrisable group with a metrically propercobounded continuous isometric action G y (X , d) on a largescale geodesic metric space.
(a) Then G admits a compatible left-invariant metric ∂ that ismaximal for large distances.
(b) Moreover, for every x ∈ X , the map
g ∈ G 7→ gx ∈ X
is a quasi-isometry between (G , ∂) and (X , d).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of first order structures and theirautomorphism groups
As mentioned earlier, by a lemma due to Cameron, theautomorphism group Aut(K) of a countable ℵ0-categoricalstructure K has property (OB) and therefore admits a compatibleleft-invariant metric d maximal for large distances.
However, this d is bounded and therefore Aut(K) isquasi-isometric to a point.
Nevertheless, there are plenty of more interesting examples.
But to identify these, it will be useful to have some verifiablecriteria for admitting metrically proper or maximal metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of first order structures and theirautomorphism groups
As mentioned earlier, by a lemma due to Cameron, theautomorphism group Aut(K) of a countable ℵ0-categoricalstructure K has property (OB) and therefore admits a compatibleleft-invariant metric d maximal for large distances.
However, this d is bounded and therefore Aut(K) isquasi-isometric to a point.
Nevertheless, there are plenty of more interesting examples.
But to identify these, it will be useful to have some verifiablecriteria for admitting metrically proper or maximal metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of first order structures and theirautomorphism groups
As mentioned earlier, by a lemma due to Cameron, theautomorphism group Aut(K) of a countable ℵ0-categoricalstructure K has property (OB) and therefore admits a compatibleleft-invariant metric d maximal for large distances.
However, this d is bounded and therefore Aut(K) isquasi-isometric to a point.
Nevertheless, there are plenty of more interesting examples.
But to identify these, it will be useful to have some verifiablecriteria for admitting metrically proper or maximal metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Large scale geometry of first order structures and theirautomorphism groups
As mentioned earlier, by a lemma due to Cameron, theautomorphism group Aut(K) of a countable ℵ0-categoricalstructure K has property (OB) and therefore admits a compatibleleft-invariant metric d maximal for large distances.
However, this d is bounded and therefore Aut(K) isquasi-isometric to a point.
Nevertheless, there are plenty of more interesting examples.
But to identify these, it will be useful to have some verifiablecriteria for admitting metrically proper or maximal metrics.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Non-Archimedean Polish groups
Recall that a Polish group is called non-Archimedean if it admits aneighbourhood basis at 1 consisting of open subgroups or,equivalently, if it is isomorphic to a closed subgroup of S∞.
Also, for a closed subgroup G 6 S∞ = Sym(N), there is acountable relational language L and an ultrahomogeneousL-structure K with universe N so that
G = Aut(K).
So, when studying non-Archimedean Polish groups, we may alwaysrepresent them as automorphism groups of ultrahomogeneousrelational structures.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Non-Archimedean Polish groups
Recall that a Polish group is called non-Archimedean if it admits aneighbourhood basis at 1 consisting of open subgroups or,equivalently, if it is isomorphic to a closed subgroup of S∞.
Also, for a closed subgroup G 6 S∞ = Sym(N), there is acountable relational language L and an ultrahomogeneousL-structure K with universe N so that
G = Aut(K).
So, when studying non-Archimedean Polish groups, we may alwaysrepresent them as automorphism groups of ultrahomogeneousrelational structures.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Non-Archimedean Polish groups
Recall that a Polish group is called non-Archimedean if it admits aneighbourhood basis at 1 consisting of open subgroups or,equivalently, if it is isomorphic to a closed subgroup of S∞.
Also, for a closed subgroup G 6 S∞ = Sym(N), there is acountable relational language L and an ultrahomogeneousL-structure K with universe N so that
G = Aut(K).
So, when studying non-Archimedean Polish groups, we may alwaysrepresent them as automorphism groups of ultrahomogeneousrelational structures.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Fraısse theory
A Fraısse class K is a countable class of isomorphism types offinitely generated L-structures satisfying the hereditary, jointembedding and amalgamation properties.
By a theorem of Fraısse, there is a correspondence betweencountable ultrahomogeneous L-structures K and Fraısse classes Kgiven by
K 7→ Age(K),
where K = Age(K) is the collection of isomorphism classes of allfinitely generated substructures of K.
Here, K is uniquely determined up to isomorphism by K = Age(K)and we say that K is the Fraısse limit of K.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Fraısse theory
A Fraısse class K is a countable class of isomorphism types offinitely generated L-structures satisfying the hereditary, jointembedding and amalgamation properties.
By a theorem of Fraısse, there is a correspondence betweencountable ultrahomogeneous L-structures K and Fraısse classes Kgiven by
K 7→ Age(K),
where K = Age(K) is the collection of isomorphism classes of allfinitely generated substructures of K.
Here, K is uniquely determined up to isomorphism by K = Age(K)and we say that K is the Fraısse limit of K.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Fraısse theory
A Fraısse class K is a countable class of isomorphism types offinitely generated L-structures satisfying the hereditary, jointembedding and amalgamation properties.
By a theorem of Fraısse, there is a correspondence betweencountable ultrahomogeneous L-structures K and Fraısse classes Kgiven by
K 7→ Age(K),
where K = Age(K) is the collection of isomorphism classes of allfinitely generated substructures of K.
Here, K is uniquely determined up to isomorphism by K = Age(K)and we say that K is the Fraısse limit of K.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We now wish to reformulate the existence of metrically propermetrics on Aut(K) in terms of combinatorial conditions on theFraısse class K.
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible metrically proper left-invariantmetric,
2 there is A ∈ K satisfying the following condition
(∗) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We now wish to reformulate the existence of metrically propermetrics on Aut(K) in terms of combinatorial conditions on theFraısse class K.
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible metrically proper left-invariantmetric,
2 there is A ∈ K satisfying the following condition
(∗) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We now wish to reformulate the existence of metrically propermetrics on Aut(K) in terms of combinatorial conditions on theFraısse class K.
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible metrically proper left-invariantmetric,
2 there is A ∈ K satisfying the following condition
(∗) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We now wish to reformulate the existence of metrically propermetrics on Aut(K) in terms of combinatorial conditions on theFraısse class K.
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible metrically proper left-invariantmetric,
2 there is A ∈ K satisfying the following condition
(∗) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
We now wish to reformulate the existence of metrically propermetrics on Aut(K) in terms of combinatorial conditions on theFraısse class K.
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible metrically proper left-invariantmetric,
2 there is A ∈ K satisfying the following condition
(∗) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.
We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
�
�
�
��� ��nA
B′
B′′C
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
To visualise (2)(∗), let C be an amalgam of two copies B′ and B′′
of B over A.We must find a path B0,B1, . . . ,Bn of isomorphic copies of Bbeginning at B0 = B′ and ending at Bn = B′′, so that the〈Bi ∪ Bi+1〉 have one of finitely many isomorphism types S ⊆ K.
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Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible left-invariant metric maximal forlarge distances,
2 there is A ∈ K satisfying the following two conditions
(i) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
(ii) there is a finite family R ⊆ K so that, for all B ∈ K andisomorphic copies A′,A′′ ⊆ B of A, there is some C ∈ Kcontaining B and isomorphic copiesA0 = A′,A1, . . . ,Am = A′′ ⊆ C of A so that 〈Ai ∪Ai+1〉 ∈ R.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Theorem (Existence)
Suppose K is a Fraısse class of finite structures with Fraısse limitK. Then the following are equivalent.
1 Aut(K) admits a compatible left-invariant metric maximal forlarge distances,
2 there is A ∈ K satisfying the following two conditions
(i) for every B ∈ K containing A, there are n > 1 and a finitefamily S ⊆ K so that, for all C ∈ K and embeddingsη1, η2 : B ↪→ C with η1|A = η2|A, one can find some D ∈ Kcontaining C and a path B0 = η1B,B1, . . . ,Bn = η2B ofisomorphic copies of B inside D with 〈Bi ∪ Bi+1〉 ∈ S,
(ii) there is a finite family R ⊆ K so that, for all B ∈ K andisomorphic copies A′,A′′ ⊆ B of A, there is some C ∈ Kcontaining B and isomorphic copiesA0 = A′,A1, . . . ,Am = A′′ ⊆ C of A so that 〈Ai ∪Ai+1〉 ∈ R.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using our Svarc–Milnor lemma, we also have a concretecomputation of the large scale geometry of non-Archimedeangroups.
Theorem (Characterisation)
Moreover, if A and R are as in (2), let X denote the set ofisomorphic copies of A in K and put
(A′,A′′) ∈ E ⇔ 〈A′ ∪ A′′〉 ∈ R.
Then (X ,E ) is a connected graph and the mapping
g ∈ Aut(K) 7→ g · A ∈ X
is a quasi-isometry between Aut(K) and X equipped with its pathmetric.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using our Svarc–Milnor lemma, we also have a concretecomputation of the large scale geometry of non-Archimedeangroups.
Theorem (Characterisation)
Moreover, if A and R are as in (2), let X denote the set ofisomorphic copies of A in K and put
(A′,A′′) ∈ E ⇔ 〈A′ ∪ A′′〉 ∈ R.
Then (X ,E ) is a connected graph and the mapping
g ∈ Aut(K) 7→ g · A ∈ X
is a quasi-isometry between Aut(K) and X equipped with its pathmetric.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Using our Svarc–Milnor lemma, we also have a concretecomputation of the large scale geometry of non-Archimedeangroups.
Theorem (Characterisation)
Moreover, if A and R are as in (2), let X denote the set ofisomorphic copies of A in K and put
(A′,A′′) ∈ E ⇔ 〈A′ ∪ A′′〉 ∈ R.
Then (X ,E ) is a connected graph and the mapping
g ∈ Aut(K) 7→ g · A ∈ X
is a quasi-isometry between Aut(K) and X equipped with its pathmetric.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Example: The rational Urysohn space
As is well-known, the rational Urysohn space QU is the Fraısselimit of the class U of isometry types of finite metric spaces withrational distances.
We let A = {p} be the one point metric space and claim that Asatisfies condition (2)(i) of the preceding theorem.
To see this, let B ∈ U contain the point p and set δ = diam(B).
We letB⊕2δ B
be the disjoint union of two copies of B so that every point in thefirst copy is of distance 2δ from every point in the second copy.
Let S ⊆ U consist of the isometry class of B⊕2δ B and put n = 2.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Example: The rational Urysohn space
As is well-known, the rational Urysohn space QU is the Fraısselimit of the class U of isometry types of finite metric spaces withrational distances.
We let A = {p} be the one point metric space and claim that Asatisfies condition (2)(i) of the preceding theorem.
To see this, let B ∈ U contain the point p and set δ = diam(B).
We letB⊕2δ B
be the disjoint union of two copies of B so that every point in thefirst copy is of distance 2δ from every point in the second copy.
Let S ⊆ U consist of the isometry class of B⊕2δ B and put n = 2.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Example: The rational Urysohn space
As is well-known, the rational Urysohn space QU is the Fraısselimit of the class U of isometry types of finite metric spaces withrational distances.
We let A = {p} be the one point metric space and claim that Asatisfies condition (2)(i) of the preceding theorem.
To see this, let B ∈ U contain the point p and set δ = diam(B).
We letB⊕2δ B
be the disjoint union of two copies of B so that every point in thefirst copy is of distance 2δ from every point in the second copy.
Let S ⊆ U consist of the isometry class of B⊕2δ B and put n = 2.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Example: The rational Urysohn space
As is well-known, the rational Urysohn space QU is the Fraısselimit of the class U of isometry types of finite metric spaces withrational distances.
We let A = {p} be the one point metric space and claim that Asatisfies condition (2)(i) of the preceding theorem.
To see this, let B ∈ U contain the point p and set δ = diam(B).
We letB⊕2δ B
be the disjoint union of two copies of B so that every point in thefirst copy is of distance 2δ from every point in the second copy.
Let S ⊆ U consist of the isometry class of B⊕2δ B and put n = 2.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Example: The rational Urysohn space
As is well-known, the rational Urysohn space QU is the Fraısselimit of the class U of isometry types of finite metric spaces withrational distances.
We let A = {p} be the one point metric space and claim that Asatisfies condition (2)(i) of the preceding theorem.
To see this, let B ∈ U contain the point p and set δ = diam(B).
We letB⊕2δ B
be the disjoint union of two copies of B so that every point in thefirst copy is of distance 2δ from every point in the second copy.
Let S ⊆ U consist of the isometry class of B⊕2δ B and put n = 2.
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, suppose C is an amalgam of two copies B′ and B′′ of B overthe point p.
Then we can adjoin a further copy B of B to C so that every pointof B is of distance 2δ from every point point in C.
�
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B′
B′′C
B
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�Thus, 〈B′ ∪ B〉 ∼= B⊕2δ B ∈ S and 〈B ∪ B′′〉 ∼= B⊕2δ B ∈ S.
We remark that n = 2 is minimal, since there are infinitely manydistinct ways of amalgamating two copies of B over the singlepoint p (provided, of course, that B is non-trivial).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, suppose C is an amalgam of two copies B′ and B′′ of B overthe point p.
Then we can adjoin a further copy B of B to C so that every pointof B is of distance 2δ from every point point in C.
�
�
�
��� ��•p
B′
B′′C
B
�
�
�
�Thus, 〈B′ ∪ B〉 ∼= B⊕2δ B ∈ S and 〈B ∪ B′′〉 ∼= B⊕2δ B ∈ S.
We remark that n = 2 is minimal, since there are infinitely manydistinct ways of amalgamating two copies of B over the singlepoint p (provided, of course, that B is non-trivial).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, suppose C is an amalgam of two copies B′ and B′′ of B overthe point p.
Then we can adjoin a further copy B of B to C so that every pointof B is of distance 2δ from every point point in C.
�
�
�
��� ��•p
B′
B′′C
B
�
�
�
�Thus, 〈B′ ∪ B〉 ∼= B⊕2δ B ∈ S and 〈B ∪ B′′〉 ∼= B⊕2δ B ∈ S.
We remark that n = 2 is minimal, since there are infinitely manydistinct ways of amalgamating two copies of B over the singlepoint p (provided, of course, that B is non-trivial).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, suppose C is an amalgam of two copies B′ and B′′ of B overthe point p.
Then we can adjoin a further copy B of B to C so that every pointof B is of distance 2δ from every point point in C.
�
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��� ��•p
B′
B′′C
B
�
�
�
�
Thus, 〈B′ ∪ B〉 ∼= B⊕2δ B ∈ S and 〈B ∪ B′′〉 ∼= B⊕2δ B ∈ S.
We remark that n = 2 is minimal, since there are infinitely manydistinct ways of amalgamating two copies of B over the singlepoint p (provided, of course, that B is non-trivial).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, suppose C is an amalgam of two copies B′ and B′′ of B overthe point p.
Then we can adjoin a further copy B of B to C so that every pointof B is of distance 2δ from every point point in C.
�
�
�
��� ��•p
B′
B′′C
B
�
�
�
�Thus, 〈B′ ∪ B〉 ∼= B⊕2δ B ∈ S and 〈B ∪ B′′〉 ∼= B⊕2δ B ∈ S.
We remark that n = 2 is minimal, since there are infinitely manydistinct ways of amalgamating two copies of B over the singlepoint p (provided, of course, that B is non-trivial).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Now, suppose C is an amalgam of two copies B′ and B′′ of B overthe point p.
Then we can adjoin a further copy B of B to C so that every pointof B is of distance 2δ from every point point in C.
�
�
�
��� ��•p
B′
B′′C
B
�
�
�
�Thus, 〈B′ ∪ B〉 ∼= B⊕2δ B ∈ S and 〈B ∪ B′′〉 ∼= B⊕2δ B ∈ S.
We remark that n = 2 is minimal, since there are infinitely manydistinct ways of amalgamating two copies of B over the singlepoint p (provided, of course, that B is non-trivial).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Similarly, if R consists of the isometry class of 2 points withdistance 1 between them, then A = {p} and R also satisfycondition (2)(ii).
Thus, if we let X denote the set of isometric copies of A in QU,we simply have X = QU.
On the other hand, if ρ is the path metric induced by the graphstructure on QU given by R, then
ρ(x , y) 6 k ⇔ ∃z0 = x , z1, z2, . . . , zk = y : d(zi , zi+1) = 1.
By a bit of Urysohn geometry, it is not hard to see that (QU, ρ) isquasi-isometric to QU with the usual metric d .
Thus, by our theorem, Isom(QU) is quasi-isometric to QU.
Similarly, e.g., Aut(Tℵ0) is quasi-isometric to Tℵ0 .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Similarly, if R consists of the isometry class of 2 points withdistance 1 between them, then A = {p} and R also satisfycondition (2)(ii).
Thus, if we let X denote the set of isometric copies of A in QU,we simply have X = QU.
On the other hand, if ρ is the path metric induced by the graphstructure on QU given by R, then
ρ(x , y) 6 k ⇔ ∃z0 = x , z1, z2, . . . , zk = y : d(zi , zi+1) = 1.
By a bit of Urysohn geometry, it is not hard to see that (QU, ρ) isquasi-isometric to QU with the usual metric d .
Thus, by our theorem, Isom(QU) is quasi-isometric to QU.
Similarly, e.g., Aut(Tℵ0) is quasi-isometric to Tℵ0 .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Similarly, if R consists of the isometry class of 2 points withdistance 1 between them, then A = {p} and R also satisfycondition (2)(ii).
Thus, if we let X denote the set of isometric copies of A in QU,we simply have X = QU.
On the other hand, if ρ is the path metric induced by the graphstructure on QU given by R, then
ρ(x , y) 6 k ⇔ ∃z0 = x , z1, z2, . . . , zk = y : d(zi , zi+1) = 1.
By a bit of Urysohn geometry, it is not hard to see that (QU, ρ) isquasi-isometric to QU with the usual metric d .
Thus, by our theorem, Isom(QU) is quasi-isometric to QU.
Similarly, e.g., Aut(Tℵ0) is quasi-isometric to Tℵ0 .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Similarly, if R consists of the isometry class of 2 points withdistance 1 between them, then A = {p} and R also satisfycondition (2)(ii).
Thus, if we let X denote the set of isometric copies of A in QU,we simply have X = QU.
On the other hand, if ρ is the path metric induced by the graphstructure on QU given by R, then
ρ(x , y) 6 k ⇔ ∃z0 = x , z1, z2, . . . , zk = y : d(zi , zi+1) = 1.
By a bit of Urysohn geometry, it is not hard to see that (QU, ρ) isquasi-isometric to QU with the usual metric d .
Thus, by our theorem, Isom(QU) is quasi-isometric to QU.
Similarly, e.g., Aut(Tℵ0) is quasi-isometric to Tℵ0 .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Similarly, if R consists of the isometry class of 2 points withdistance 1 between them, then A = {p} and R also satisfycondition (2)(ii).
Thus, if we let X denote the set of isometric copies of A in QU,we simply have X = QU.
On the other hand, if ρ is the path metric induced by the graphstructure on QU given by R, then
ρ(x , y) 6 k ⇔ ∃z0 = x , z1, z2, . . . , zk = y : d(zi , zi+1) = 1.
By a bit of Urysohn geometry, it is not hard to see that (QU, ρ) isquasi-isometric to QU with the usual metric d .
Thus, by our theorem, Isom(QU) is quasi-isometric to QU.
Similarly, e.g., Aut(Tℵ0) is quasi-isometric to Tℵ0 .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Similarly, if R consists of the isometry class of 2 points withdistance 1 between them, then A = {p} and R also satisfycondition (2)(ii).
Thus, if we let X denote the set of isometric copies of A in QU,we simply have X = QU.
On the other hand, if ρ is the path metric induced by the graphstructure on QU given by R, then
ρ(x , y) 6 k ⇔ ∃z0 = x , z1, z2, . . . , zk = y : d(zi , zi+1) = 1.
By a bit of Urysohn geometry, it is not hard to see that (QU, ρ) isquasi-isometric to QU with the usual metric d .
Thus, by our theorem, Isom(QU) is quasi-isometric to QU.
Similarly, e.g., Aut(Tℵ0) is quasi-isometric to Tℵ0 .
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Isom(QU) is quasi-isometric to QU.
Aut(Tℵ0) is quasi-isometric to Tℵ0 .
However, since QU is not quasi-isometric to Tℵ0 , it follows that
Isom(QU) 6∼= Aut(Tℵ0).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
Isom(QU) is quasi-isometric to QU.
Aut(Tℵ0) is quasi-isometric to Tℵ0 .
However, since QU is not quasi-isometric to Tℵ0 , it follows that
Isom(QU) 6∼= Aut(Tℵ0).
Christian Rosendal, University of Illinois at Chicago Large scale geometry of metrisable groups
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