Coarse rays
We give some characterizations of geodesic metric spaces coarsely equivalent to the ray R⁺.
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irk-123456789-1243352017-09-24T03:03:37Z Coarse rays Kuchaiev, O. Protasov, I.V. We give some characterizations of geodesic metric spaces coarsely equivalent to the ray R⁺. 2008 Article Coarse rays / O. Kuchaiev, I.V. Protasov // Український математичний вісник. — 2008. — Т. 5, № 2. — С. 185-192. — Бібліогр.: 5 назв. — англ. 1810-3200 2000 MSC. 54C99, 54F15, 05C10 http://dspace.nbuv.gov.ua/handle/123456789/124335 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We give some characterizations of geodesic metric spaces coarsely equivalent to the ray R⁺. |
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Kuchaiev, O. Protasov, I.V. |
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Kuchaiev, O. Protasov, I.V. Coarse rays Український математичний вісник |
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Kuchaiev, O. Protasov, I.V. |
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Coarse rays |
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Coarse rays |
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Coarse rays |
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Coarse rays |
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Coarse rays |
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coarse rays |
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Інститут прикладної математики і механіки НАН України |
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Coarse rays / O. Kuchaiev, I.V. Protasov // Український математичний вісник. — 2008. — Т. 5, № 2. — С. 185-192. — Бібліогр.: 5 назв. — англ. |
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Український математичний вісник |
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AT kuchaievo coarserays AT protasoviv coarserays |
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2025-07-09T01:16:49Z |
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1837130122032840704 |
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Український математичний вiсник
Том 5 (2008), № 2, 185 – 192
Coarse rays
O. Kuchaiev, Igor V. Protasov
Abstract. We give some characterizations of geodesic metric spaces
coarsely equivalent to the ray R
+.
2000 MSC. 54C99, 54F15, 05C10.
Key words and phrases. Coarse equivalence, coarse ray, geodesic ray,
almost geodesic ray.
Let (X, d), (Y, ρ) be metric spaces. A mapping f : X → Y is called a
coarse embedding if, for every r > 0, there exists s > 0 such that, for all
x1, x2 ∈ X,
d(x1, x2) ≤ r =⇒ ρ(f(x1), f(x2)) ≤ s,
ρ(f(x1), f(x2)) ≤ r =⇒ d(x1, x2) ≤ s.
The metric spaces (X, d), (Y, ρ) are called coarsely equivalent if there
exists a coarse embedding f : X → Y such that f(X) is large in Y , i.e.
there exists t > 0 such that, for every y ∈ Y , there is z ∈ f(X) such that
ρ(y, z) ≤ t.
The space R
+ = {r ∈ R : r ≥ 0} endowed with the Euclidian metric
is called the ray. By a coarse ray we mean any metric space coarsely
equivalent to ray.
For motivation of study metric spaces from the “coarse” point of view
see [3, 5, 9, 10]. As is the segment [0, 1] in topology, the ray is one of the
simplest non-trivial objects in coarse geometry, so it is natural to ask for
its characterization up to coarse equivalence. There is a simple test [5,
Proposition 2.57] to recognize whether a given metric space is coarsely
equivalent to some geodesic metric space. Thus, to answer this question
we can work with only geodesic metric spaces. Theorem 1 and 3 provide
several characterizations; Theorem 2 gives a better characterization in
Received 28.12.2008
ISSN 1810 – 3200. c© Iнститут математики НАН України
186 Coarse rays
case of proper metric spaces; Example 2 shows that Theorem 2 does
not hold for non-proper metric spaces. By Theorem 1, every geodesic
metric space coarsely emenddable in the ray is a coarse ray. Clearly, it
is not true outside of the geodesic case. Using the anti-Cantor set and
Theorem 3.11 from [2], in Theorem 4 we describe the ultrametric spaces
coarsely embeddable to the ray.
Let (X, d) be a metric space. We fix some point x0 ∈ X and define a
preordering ≤ on X by the rule: x ≤ y if and only if d(x0, x) ≤ d(x0, y).
For ε ≥ 0, a space (X, d) is said to be ε-directed (with respect to the base
point x0) if, for any x, y ∈ X, x ≤ y, we have
d(x0, x) + d(x, y) ≤ d(x0, y) + ε.
If (X, d) is ε-directed then, for every x′ ∈ X there exists ε′ ≥ 0 such that
(X, d) is ε′-directed with respect to x′.
Lemma 1. Every ε-directed space is coarsely emenddable in the ray.
Proof. Let (X, d) be ε-directed with respect to x0. We define a mapping
f : X → R
+ by the rule f(x) = d(x0, x), and note that, for any x, y ∈ X
with x ≤ y, we have
d(x, y) − ε ≤ f(y) − f(x) ≤ d(x, y),
so f is a coarse embedding.
By Theorem 1, the converse statement is true for every geodesic metric
space (X, d), but in general case it does not hold.
Example 1. Let (X, d) be a half-parabola {(x, y) ∈ R
+ × R
+ : y = x2}
with the metric d inherited from the plane. It is easy to see that the
mapping f : (X, d) → R
+, f(x, y) = y is a coarse embedding. On the
other hand, (X, d) is not ε-directed for every ε ≥ 0.
A subset Y of a metric space (X, d) is called bounded if there exists
C > 0 such that diam Y ≤ C where diam Y = sup{d(x, y) : x, y ∈ X}.
A family ℑ of subsets of a metric space (X, d) is called uniformly bounded
if there exists C > 0 such that diam F ≤ C for every F ∈ ℑ.
Given a metric space (X, d) and any x ∈ X, r ∈ R
+, we put
B(x, r) = {y ∈ X : d(x, y) ≤ r}, S(x, r) = {y ∈ X : d(x, y) = r}
O. Kuchaiev, I. V. Protasov 187
Lemma 2. If (X, d) is an ε-directed space with the base point x0 then
the family {S(x0, r) : r ∈ R
+} is uniformly bounded.
Proof. Let x, y ∈ X, d(x0, x) = d(x0, y) and x ≤ y. Since d(x0, x) +
d(x, y) ≤ d(x0, y)+ε, we have d(x, y) ≤ ε, so diam S(x0, r) ≤ ε for every
r ∈ R
+.
By Theorem 1 the converse statement is true for every geodesic metric
space. On the other hand, Example 1 shows that in general case it does
not hold.
Let (X, d), (Y, ρ) be metric spaces. Given λ > 0,c ≥ 0, a mapping
f : X → Y is called a (λ, c)-isometric embedding if, for all x1, x2 ∈ X,
λ−1d(x1, x2) − c ≤ ρ(f(x1), f(x2)) ≤ λd(x1, x2) + c.
If in addition f(X) is large in Y , we say that f is a (λ, c)-isometry. The
metric spaces (X, d), (Y, ρ) are called quasi-isometric if there exists a
(λ, c)-isometry f : X → Y .
Let (X, d) be a metric space, r ≥ 0, f : [0, r] → X be an isometric
embedding. We say that f([0, r]) is a geodesic segment with the endpoints
f(0), f(r). A metric space (X, d) is called geodesic if any two points of
X can be joined by a geodesic segment.
Lemma 3. If the geodesic metric spaces (X, d), (Y, ρ) are coarsely equiv-
alent then (X, d), (Y, ρ) are quasi-isometric.
Proof. Let f : X → Y be a coarse embedding such that f(X) is large in
Y . By [1, Proposition 1.4] or [5, Lemma 1.10], there exist λ, c such that
ρ(f(x), f(x
′
)) ≤ λd(x, x
′
) + c for all x, x
′
∈ X.
Since f(X) is large in Y , there exists t > 0 such that, for every
y ∈ Y , we can find y
′
∈ f(X) with ρ(y, y
′
) < t. We fix some points x, x
′
and some geodesic segment [f(x), f(x
′
)]. On this segment we choose the
points y1, . . . , yn such that
y1 = f(x), d(y1, y2) = · · · = d(yn−1, yn) = 1, ρ(yn, f(x
′
)) = ε, ε < 1,
so ρ(f(x), f(x
′
)) = n+ε. Then we pick the points x2, . . . , xn ∈ X so that
ρ(f(x2), y2) < t, . . . , ρ(f(xn), yn) < t. We put s = 2t + 1, and choose
r > 0 such that ρ(f(a), f(b)) ≤ s implies d(a, b) ≤ r for all a, b ∈ X.
Then
d(x, x1) ≤ d(x, x2) + d(x2, x3) + . . .
+ d(xn−1, xn) + d(xn, x
′
)
≤ (n + 1)r = (n + ε)r + r(1 − ε)
= rρ(f(x), f(x
′
)) + r(1 − ε).
188 Coarse rays
Since the choice of r does not depend on x, x
′
, in view of the first para-
graph, we conclude that f is a quasi-isometry.
Theorem 1. For an unbounded geodesic metric space (X, d), x0 ∈ X,
the following statements are equivalent:
• (i) (X, d) is a coarse ray;
• (ii) (X, d) is coarsely emenddable in the ray;
• (iii) (X, d) is ε-directed;
• (iv) the family {S(x0, r) : r ∈ R
+} is uniformly bounded.
Proof. (i) ⇔ (ii). The implication (i) ⇒ (ii) is trivial. To check (ii) ⇒
(i), we fix some coarse embedding f : X → R
+. Then we pick λ > 0
such that d(a, b) ≤ 1 implies |f(a)−f(b)| < λ. Given an arbitrary points
x, x
′
∈ X, we choose the points x1, . . . , xn on the geodesic segment [x, x
′
]
such that x = x1, d(x1, x2) = . . . d(xn−1, xn) = 1, d(xn, x
′
) < 1. Then
every segment of length λ on [f(x), f(x
′
)] contains at least one point
f(x1), . . . , f(xn). Since f(X) is unbounded in R
+, it follows that f(X)
is large, so we get (i).
(i) ⇒ (iii) Let f be a coarse embedding of (X, d) into R
+ such that
f(X) is large in R
+. By Lemma 3, f is a (λ, C)-isometric embedding.
Changing the value of f in x0 we get a (λ′, C ′)-isometric embedding
for some parameters λ′, C ′, so we may suppose that f(x0) = 0 and
f(x0) ≤ f(x) for every x ∈ X. We put g = 1
λ
f , C1 = C
λ
. Then g is
(1, C1) isometric embedding of (X, d) into R
+ and, for all x, y ∈ X, we
have
d(x, y) − C1 ≤ |g(x) − g(y)| ≤ d(x, y) + C1
|g(x0) − g(x)| − C1 ≤ d(x0, x) ≤ |g(x0) − g(y)| + C1.
Now let d(x0, x) ≤ d(x0, y). Then
g(x) − g(x0) − C1 ≤ g(y) − g(x0) + C1, g(x) − g(y) ≤ 2C1.
If g(y) ≥ g(x) we have
d(x0, x) + d(x, y) ≤ |g(x0) − g(x)| + |g(x) − g(y)| + 2C1
= g(x) − g(x0) + g(y) − g(x) + 2C1
= |g(y) − g(x0)| + 2C1 ≤ d(x0, y) + 3C1.
O. Kuchaiev, I. V. Protasov 189
If g(y) ≤ g(x) then g(x) − g(y) ≤ 2C1, and we have
d(x0, x) + d(x, y) ≤ |g(x0) − g(x)| + |g(x) − g(y)| + 2C1
≤ |g(x0) − g(x)| + 4C1 ≤ d(x0, x) + 5C1 ≤ d(x0, y) + 5C1.
In both cases we see that (X, d) is a 5C1-ray.
(iii) ⇒ (iv) follows from Lemma 2.
(iv) ⇒ (iii). We choose ε > 0 such that diam S(x0, r) ≤ ε for
every r ≥ 0. Let x, y ∈ X and d(x0, x) ≤ d(x0, y). Since (X, d) is
geodesic, there exists a point x′ on the geodesic segment [x0, y] such that
d(x0, x) = d(x0, x
′). Then
d(x0, x) + d(x, y) ≤ d(x0, x
′) + d(x, x′) + d(x′, y)
= d(x, y) + d(x, x′) ≤ d(x0, y) + ε.
(iii) ⇒ (ii) follows from Lemma 1.
A subspace L of a metric space is called a geodesic ray if L is an
isometric copy of R
+.
An unbounded metric space (X, d) is called proper if every closed ball
B(x, r) in (X, d) is compact.
The next lemma is a geodesic version of Kønig Lemma stating that
every infinite locally finite graph has an infinite chain.
Lemma 4. Every proper geodesic metric space (X, d) has a geodesic ray.
Proof. We use the Hausdorff distance dH defined on the set C(X) of all
compact subsets of (X, d) by the rule
dH(C, C ′) = inf{ε > 0 : C ⊆ B(C ′, ε), C ′ ⊆ B(C, ε)},
where B(C, ε) =
⋃
c∈C
B(c, ε). By [5, Proposition 7.2], C(Y ) is compact
for every compact metric space Y . We fix an arbitrary point x0 ∈ X and,
for every n ∈ ω, pick xn ∈ X such that d(x0, xn) = n. For every n ∈ ω, we
choose a geodesic segment [x0, xn]. Since every space C(B(x0, m)), m ∈ ω
is compact, there exists a subsequence (nk)k∈ω of ω such, that for every
m ∈ ω the sequence (B(x0, m) ∩ [x0, xnk
])k∈ω converges to some subset
Lm. It is a standard verification that L =
⋃
m∈ω
Lm is a geodesic ray.
190 Coarse rays
Theorem 2. A proper geodesic metric space (X, d) is a coarse ray if and
only if (X, d) has a large geodesic ray.
Proof. Let (X, d) be a coarse ray. By Lemma 4, (X, d) has a geodesic L.
By the equivalence (i) ⇔ (iv) of Theorem 1, L is large in (X, d). On the
other hand, if (X, d) has a large geodesic ray then (X, d) is a coarse ray
by definition.
Example 2. We construct a geodesic (non-proper) coarse ray (X, d)
which has no geodesic rays. To this end we take a disjoint family
{[an, bn] : n ∈ ω} of segments of length n, and stick together all the
points an, n ∈ ω. Denote by a the resulting point. Then, for every
m ∈ ω, we choose the points xn, n ∈ ω, n ≥ m such that xn ∈ [a, bn],
|[a, xn]| = m, and connect any two points xn, xk, m ≤ n < k by the
segment of length 1. Let X be the resulting set endowed with the path
metric d. By the equivalence (i) ⇔ (iv) of Theorem 1, (X, d) is a coarse
ray. To see that (X, d) has no geodesic rays it suffices to observe that
every geodesic segment connecting the points a, x, where x ∈ [a, bn], lies
on the segment [a, bn].
We say that a subspace L of a metric space (X, d) is an almost geodesic
ray if there exist c ≥ 0 and a bijection f : R
+ → L such that, for all
t1, t2 ∈ R
+, we have
|t2 − t1| ≤ d(f(t1), f(t2)) ≤ |t2 − t1| + c.
Clearly, every almost geodesic ray is a coarse ray.
Theorem 3. A geodesic metric space (X, d) is a coarse ray if and only
if (X, d) has a large almost geodesic ray.
Proof. Let (X, d) be a coarse ray, x0 ∈ X. By Theorem 1, there exists
C ≥ 0 such that diamS(x0, t) ≤ C for every t ∈ R
+. Since (X, d) is
geodesic, for every t ≥ 0, the set S(x0, t) is non-empty, so we can take
some point f(t) ∈ S(x0, t) and get the mapping f : R
+ → X. We put
L = f(R+), note that L is large in (X, d) and show that L is an almost
geodesic ray. Let t1, t2 ∈ R
+ and t1 ≤ t2. Since (X, d) is geodesic, there
exists y ∈ S(x0, t1) such that d(f(t2), z) = t2 − t1. Then
t2 − t1 ≤ d(f(t1), f(t2)) ≤ d(f(t1), z) + d(z, f(t2)) ≤ (t2 − t1) + C.
O. Kuchaiev, I. V. Protasov 191
On the other hand, if L is a large almost geodesic ray in (X, d), then L
is a coarse ray, so (X, d) is also a coarse ray.
For r > 0, a subset Y of a metric space (X, d) is called r-discrete if
d(a, b) ≥ r for any a, b ∈ Y, a 6= b. The r-capacity of Y is the cardinal
sup{|Z| : Z is r-discrete subset of Y }. A metric space (X, d) is of bounded
geometry if there exists a number r > 0 and a function c : R
+ → R
+ such
that the r-capacity of every ball B(x, t) does not exceed c(t).
A metric d on a set X is called ultrametric if, for all x, y, z,
d(x, y) ≤ max{d(x, y), d(z, y)}
Theorem 4. An ultrametric space (X, d) is coarsely emenddable in the
ray if and only if (X, d) is of bounded geometry.
Proof. Let (X, d) be a space of bounded geometry. We fix the corre-
sponding r > 0, c : R
+ → R
+ and choose a maximal r-discrete subspace
Y of X. Given any x ∈ X, there exists y ∈ Y such that d(x, y) ≤ r. It
follows that Y is large in (X, d), so Y is coarsely equivalent to (X, d).
Every ball of radius t in Y has at most c(t) points, in particular, Y is
a proper metric space. By [2, Theorem 3.11], Y is coarsely emenddable
into the subspace M of R
+ consisting of all integers whose tercimal de-
composition does not contain 1. Since X is coarsely equivalent to Y ,
there exists a coarse embedding of X into R
+.
On the other hand, let (X, d) be coarsely equivalent to some subspace
Z of R
+. Since Z is of bounded geometry, it is easy to check that (X, d)
is also of bounded geometry.
Problem. Detect all metric spaces coarsely emenddable in the ray R
+.
References
[1] A. Dranishnikov, Asymptotic topology // Russian Math. Surveys, 55 (2000), N 6,
71–116.
[2] A. Dranishnikov, M. Zarichnyi, Universal spaces for asymptotic dimension //
Topol. Appl. 140 (2004), N 2–3, 203–225.
[3] M. Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture
Note Ser. 182, 1993.
[4] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Sec., 13,
2006.
[5] J. Roe, Lectures on Coarse Geometry, AMS University Lecture Series, 31, 2003.
192 Coarse rays
Contact information
O. Kuchaiev,
Igor V. Protasov
Department of Cybernetics,
Kyiv National University
Volodimirska 64,
Kyiv, 01033,
Ukraine
E-Mail: akuchaev@inbox.ru
protasov@unicyb.kiev.ua
|