Balleans of bounded geometry and G-spaces
A ballean (or a coarse structure) is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that a ballean can be considered as an asymptotical counterpart of a uniform topological space. We prove that every ballean...
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irk-123456789-1533612019-06-15T01:25:42Z Balleans of bounded geometry and G-spaces Protasov, I.V. A ballean (or a coarse structure) is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that a ballean can be considered as an asymptotical counterpart of a uniform topological space. We prove that every ballean of bounded geometry is coarsely equivalent to a ballean on some set X determined by some group of permutations of X. 2008 Article Balleans of bounded geometry and G-spaces / I.V. Protasov // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 2. — С. 101–108. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 37B05, 54E15. http://dspace.nbuv.gov.ua/handle/123456789/153361 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A ballean (or a coarse structure) is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that a ballean can be considered as an asymptotical counterpart of a uniform topological space.
We prove that every ballean of bounded geometry is coarsely equivalent to a ballean on some set X
determined by some group of permutations of X. |
| format |
Article |
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Protasov, I.V. |
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Protasov, I.V. Balleans of bounded geometry and G-spaces Algebra and Discrete Mathematics |
| author_facet |
Protasov, I.V. |
| author_sort |
Protasov, I.V. |
| title |
Balleans of bounded geometry and G-spaces |
| title_short |
Balleans of bounded geometry and G-spaces |
| title_full |
Balleans of bounded geometry and G-spaces |
| title_fullStr |
Balleans of bounded geometry and G-spaces |
| title_full_unstemmed |
Balleans of bounded geometry and G-spaces |
| title_sort |
balleans of bounded geometry and g-spaces |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/153361 |
| citation_txt |
Balleans of bounded geometry and G-spaces / I.V. Protasov // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 2. — С. 101–108. — Бібліогр.: 8 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT protasoviv balleansofboundedgeometryandgspaces |
| first_indexed |
2025-07-14T04:35:22Z |
| last_indexed |
2025-07-14T04:35:22Z |
| _version_ |
1837595594668900352 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2008). pp. 101 – 108
c© Journal “Algebra and Discrete Mathematics”
Balleans of bounded geometry and G-spaces
Igor V. Protasov
Communicated by V. I. Sushchansky
Abstract. A ballean (or a coarse structure) is a set en-
dowed with some family of subsets which are called the balls. The
properties of the family of balls are postulated in such a way that
a ballean can be considered as an asymptotical counterpart of a
uniform topological space.
We prove that every ballean of bounded geometry is coarsely
equivalent to a ballean on some set X determined by some group
of permutations of X.
1. Ball structures and balleans
A ball structure is a triple B = (X, P, B), where X, P are nonempty
sets and, for any x ∈ X and α ∈ P , B(x, α) is a subset of X which is
called a ball of radius α around x. It is supposed that x ∈ B(x, α) for all
x ∈ X, α ∈ P . The set X is called the support of B, P is called the set of
radii. Given any x ∈ X, A ⊆ X, α ∈ P we put
B∗(x, α) = {y ∈ X : x ∈ B(y, α)}, B(A, α) =
⋃
a∈A
B(a, α)
A ball structure is called
• lower symmetric if, for any α, β ∈ P , there exist α′, β′ such that,
for every x ∈ X,
B∗(x, α′) ⊆ B(x, α), B(x, β′) ⊆ B∗(x, β);
Thanks to my daughters.
2000 Mathematics Subject Classification: 37B05, 54E15.
Key words and phrases: ballean, coarse equivalence, G-space.
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.102 Balleans of bounded geometry
• upper symmetric if, for any α, β ∈ P , there exist α′, β′ such that,
for every x ∈ X,
B(x, α) ⊆ B∗(x, α′), B∗(x, β) ⊆ B(x, β′);
• lower multiplicative if, for any α, β ∈ P , there exists γ ∈ P such
that, for every x ∈ X,
B(B(x, γ), γ) ⊆ B(x, α) ∩ B(x, β);
• upper multiplicative if, for any α, β ∈ P , there exists γ ∈ P such
that, for every x ∈ X,
B(B(x, α), β) ⊆ B(x, γ).
Let B = (X, P, B) be a lower symmetric and lower multiplicative ball
structure. Then the family
{
⋃
x∈X
B(x, α) × B(x, α) : α ∈ P
}
is a base of entourages for some (uniquely determined) uniformity on X.
On the other hand, if U ⊆ X × X is a uniformity on X, then the ball
structure (X,U , B) is lower symmetric and lower multiplicative, where
B(x, U) = {y ∈ X : (x, y) ∈ U}. Thus, the lower symmetric and lower
multiplicative ball structures can be identified with the uniform topolog-
ical spaces.
We say that a ball structure B is a ballean if B is upper symmetric and
upper multiplicative. In this paper we follow terminology from [6, 7]. A
structure on X, equivalent to a ballean, can also be defined in terminology
of entourages. In this case it is called a coarse structure [8] or a uniformly
bounded space [5]. For motivations to study balleans see also [1, 2, 4].
2. Morphisms
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans. A mapping f :
X1 → X2 is called a ≺-mapping if, for every α ∈ P1, there exists β ∈ P2
such that, for every x ∈ X1,
f(B1(x, α)) ⊆ B2(f(x), β)
A bijection f : X1 → X2 is called an asymorphism between B1 and B2 if
f and f−1 are ≺-mappings.
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.I. Protasov 103
Let B = (X, P, B) be a ballean, S be a set. Two mappings f, f ′ : S →
X are called close if there exists α ∈ P such that f ′(s) ∈ B(f(s), α) for
every s ∈ S.
Two balleans B1 = (X1, P1, B1) and B2 = (X2, P2, B2) are called
coarsely equivalent if there exist the ≺-mappings f1 : X1 → X2, f2 :
X2 → X1 such that f1 ◦ f2, f2 ◦ f1 are close to the identity mappings
idX1
, idX2
.
Let B = (X, P, B) be a ballean. Every non-empty subset Y ⊆ X
determines the subballean BY = (Y, P, BY ), where BY (y, α) = B(Y, α)∩
Y , y ∈ Y , α ∈ P . A subset Y is called large if there exists γ ∈ P such that
B(Y, γ) = X. If Y is large, then BY and B are coarsely equivalent. We
shall use also the following observations. Two balleans B1 = (X1, P1, B1)
and B2 = (X2, P2, B2) are coarsely equivalent if and only if there exist
the large subsets Y1 ⊆ X1,Y2 ⊆ X2 such that the subballeans BY1
and
BY2
are asymorphic.
3. Density and capacity
Let B = (X, P, B) be a ballean, Y ⊆ X, S ⊆ Y , α ∈ P . We say that
a subset S is α-dense in Y if Y ⊆ B(S, α). An α-density of Y is the
cardinal
denα(Y ) = min{|S| : S is an α − dense subset of Y }.
A subset S of X is called α-separated if B(x, α) ∩ B(y, α) = ∅ for all
distinct x, y ∈ S. An α-capacity of Y is the cardinal
capα(Y ) = sup{|S| : S is an α − separated subset of Y }.
Let B = (X, P, B) be an arbitrary ballean. Replacing every ball
B(x, α) to B′(x, α) = B(x, α) ∩ B∗(x, α), we get the asymorphic ballean
B′ = (X, P, B′) with (B′)∗ = B′. Thus, in what follows we may suppose
that B∗(x, α) = B(x, α) for all x ∈ X, α ∈ P .
Lemma 1. Let B = (X, P, B) be a ballean, Y ⊆ X, α, β ∈ P and
B(B(x, α)) ⊆ B(x, β) for every x ∈ X. Then the following statements
hold
(i) denβ(Y ) 6 capα(Y ) 6 denα(Y );
(ii) if Z ⊆ X and Y ⊆ B(Z, α), then denβ(Y ) 6 |Z|.
Proof. (i) Let S be an α-separated subset of Y , D be an α-dense subset
of Y . Then every ball B(x, α), x ∈ D has at most one point of S. Since
S ⊆ Y ⊆
⋃
x∈D
B(x, α), we have |S| 6 |D|, so capα(Y ) 6 denα(Y ).
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.104 Balleans of bounded geometry
Let S be a maximal by inclusion α-separated subset of Y . Then every
ball B(x, α), x ∈ Y meets at least one ball B(y, α), y ∈ S. It follows that
Y ⊆
⋃
x∈S
B(x, β), so S is β-dense in Y and denβ(Y ) 6 capα(Y ).
(ii) We put Z ′ = {z ∈ Z : B(z, α) ∩ Y 6= ∅} and, for every z ∈ Z ′,
pick some point yz ∈ B(z, α) ∩ Y . Then the subset {yz : z ∈ Z ′} of Y is
β-dense in Y , so denβ(Y ) 6 |Z ′| 6 |Z|.
4. Locally finite balleans
A ballean B = (X, P, B) is called locally finite if every ball B(x, α), x ∈ X,
α ∈ P is finite.
Let B = (X, P, B), B′ = (X ′, P ′, B′) be balleans, f : X → X ′ be
an injective ≺-mapping. If B′ is locally finite then B is locally finite. In
particular, every ballean asymorphic to a locally finite ballean is locally
finite.
We say that a ballean B is coarsely locally finite if B is coarsely equiv-
alent to some locally finite ballean.
Proposition 1. A ballean B = (X, P, B) is coarsely locally finite if and
only if there exists β ∈ P such that β-capacity of every ball B(x, γ),x ∈ X,
γ ∈ P is finite.
Proof. Let B′ = (X ′, P ′, B′) be a locally finite ballean coarsely equivalent
to B. Then there exist the large subsets Y ⊆ X, Y ′ ⊆ X ′ such that
the subballeans BY and BY ′ are asymorphic. We choose α ∈ P such
that B(Y, α) = X and take an arbitrary x ∈ X, γ ∈ P . Since BY
is locally finite then the subset Z = B(B(x, γ), α) ∩ Y is finite. Since
B(x, γ) ⊆ B(Z, α), by Lemma 1 (ii), denβ(B(x, γ)) 6 |Z|. Since Z is
finite, by Lemma 1(i), β-capacity of B(x, γ) is finite.
On the other hand, let β-capacity of every ball B(x, γ) is finite. We
choose a maximal by inclusion β-separated subset Y of X. Clearly, Y is
large in X, so BY is coarsely equivalent to B. Since capβB(x, γ) is finite,
then B(x, γ) ∩ Y is finite. Hence, BY is locally finite.
Every metric space (X, d) determines the metric ballean B(X, d) =
(X, R+, Bd), where Bd(x, r) = {y ∈ X : d(x, y) 6 r}. For criterion of
metrizability of balleans see [7, Theorem 2.1.1]. A metric space is called
proper if every ball Bd(x, r) is compact.
Corollary 1. Let (X, d) be a proper metric space. Then the metric bal-
lean B(X, d) is coarsely locally finite.
Proof. It suffices to note that an 1-capacity of every ball in (X, d) is finite,
and apply Proposition 1.
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.I. Protasov 105
5. Uniformly locally finite balleans
A ballean B = (X, P, B) is called uniformly locally finite if there exists a
function h : P → ω such that |B(x, α)| 6 h(α) for all x ∈ X, α ∈ P .
Let B = (X, P, B), B′ = (X ′, P ′, B′) be balleans, f : X → X ′ be
an injective ≺-mapping. If B′ is uniformly locally finite then so is B. In
particular, every ballean asymorphic to an uniformly locally finite ballean
is uniformly locally finite.
We say that a ballean B = (X, P, B) has bounded geometry if there
exist β ∈ P and a function h : P → ω such that capβB(x, α) 6 h(α) for
all x ∈ X, α ∈ P .
Repeating the arguments proving Proposition 1 we get the following
statements.
Proposition 2. A ballean B = (X, P, B) has bounded geometry if and
only if B is coarsely equivalent to some uniformly locally finite ballean.
Example 1. Let Γ(V, E) be a connected graph with the set of vertices
V and the set of edges E. Given any u, v ∈ V , we denote by d(u, v) the
length of a shortest path between u and v. Then we get the metric space
(V, d) associated with Γ(V, E) and the metric ballean B(V, d). Clearly,
B(V, d) is uniformly locally finite if and only if there exists a natural
number r such that |Bd(v, 1)| 6 r for every v ∈ V .
Example 2. Let G be a finitely generated group with the identity e, F
be a symmetric (F = F−1) set of generators of G such that e /∈ F . The
Cayley graph Cay(G, F ) is a graph with the set of vertices G and set of
edges {{u, v} : uv−1 ∈ F}. Let dF be a path metric on Cay(G, F ). Then
the metric ballean B(G, dF ) is uniformly locally finite.
Example 3. Let G be an arbitrary group, Fe the family of all symmetric
subsets of G containing e. Then we get a ballean B(G) = (G,Fe, B),
where B(g, F ) = Fg. Clearly, B(G) is uniformly locally finite and in the
case G is finitely generated, B(G) is asymorphic to the ballean B(G, dF )
determined in Example 2.
Example 4. Let G be a group and X be a G-space with the action
of G on X defined by (g, x) 7→ g(x). We denote by Fe the family of
all finite symmetric subsets of G containing e. Then we get the ballean
B(G, X) = (X,Fe, B), where B(x, F ) = {g(x) : g ∈ F}, x ∈ X, F ∈ Fe.
Clearly, B(G, X) is uniformly locally finite.
Example 5. Let G be a gruppoid (=inverse semigroup) of partial bi-
jections of a set X, F be a family of all finite subsets of G such that
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.106 Balleans of bounded geometry
F = F−1 for every F ∈ F . Given any x ∈ X and F ∈ F , we put
B(x, F ) = {x} ∪ {g(x) : g ∈ F} and get the uniformly locally finite
ballean B(G, X).
Example 6. Let G be a locally compact topological group, C be the
family of all compact symmetric subsets of G containing e. Then, by
Proposition 5.1, the ballean B(G) = (G, C, B), where B(x, C) = Cx, is
of bounded geometry.
Remark 1. Let G be a locally compact group. Does there exist a discrete
group D such that the balleans B(G) and B(D) are coarsely equivalent?
This is so if G is Abelian or a connected Lie group.
6. G-space realization
Let B, B′ be balleans with the same support X. We write B ≺ B′ if the
identity mapping id : X → X is a ≺-mapping from B to B′. If B ≺ B′
and B′ ≺ B, we identify B and B′ and write B = B′.
Let B be a uniformly locally finite ballean with the support X. Ap-
plying Lemma 4.10 from [8], one can show that there exists a gruppoid
G of partial bijections of X such that B = B(G, X) where B(G, X) is a
ballean determined in Example 5. Our next result states that instead of
the gruppoid G we can take some group of permutations of X.
Theorem 1. For every uniformly locally finite ballean B = (X, P, B),
there exists a group G of permutations of X such that B = B(G, X).
Proof. We fix an arbitrary α ∈ P and choose β ∈ P such that
B(B(x, α), α) ⊆ B(x, β)
for each x ∈ X. Then we define the graph Γβ with the set of vertices
X and the set of edges Eβ defined by the rule: {x, y} ∈ Eβ if and only
if x ∈ B(y, β). Since B is uniformly locally finite, there exists a natural
number n(α) such that the local degree of every vertex of Γβ does not
exceed n(α). By [3, Corollary 12.2], the chromatic number of Γβ does not
exceed n(α) + 1. It follows that we can partition X = X1 ∪ . . .∪Xn(α)+1
so that any two vertices from Xj are non-adjacent, in particular, every
subset Xi is α-separated.
Now we fix i ∈ {1, . . . , n(α) + 1} and, for every vertex x ∈ Xi, enu-
merate the set B(x, α)\{x} = {x(1), . . . , x(nx)}, where nx 6 n(α). Then
we define the set Si(α) of n(α) permutations of X as follows. For each
j ∈ {1, . . . , n(α)} and x ∈ Xi, we put πj(x) = x(j), πj(x(j)) = x if
j 6 nx, and πj(x) = x otherwise. Then we extend π to X putting
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.I. Protasov 107
πj(y) = y for all y ∈ X \
⋃
x∈Xi
{x, x(j)}. Since Xi is α-separated, this
definition is correct. Thus, we get the set Si(α) = {π1, . . . , πn(α)} of per-
mutations of X. We put S(α) = S1(α) ∪ . . . ∪ Sn(α)+1(α) and denote by
G the group of permutations of X generated by
⋃
α∈P
S(α).
At last we show that the identity mapping id : X → X is an asymor-
phism between B and the ballean B(G, X) = (X,Fe, B
′) determined in
Example 5.4. Given any α ∈ P and x ∈ X, we have B(x, α) ⊆ B′(x, Sα).
On the other hand, let F be a finite subset of G, g ∈ F . Then there
exists α1, . . . , αm ∈ P and s(α1) ∈ S(α1), . . . , s(αm) ∈ S(αm) such that
g = s(αm) . . . s(α1). We choose γg ∈ P such that
B(. . . (B(B(x, α1), α2), . . .), αm) ⊆ B(x, γg)
for every x ∈ X. Then B′(x, {g}) ⊆ B(x, γg) for every x ∈ X. Since
F is finite, there exists γ ∈ P such that, for each x ∈ X, we have
B′(x, F ) ⊆ B(x, γ).
Sticking together Proposition 1 and Theorem 1 we get the following
statement.
Theorem 2. Every ballean of bounded geometry is coarsely equivalent to
some ballean B(G, X) of G-space X.
We conclude our paper with two applications of Theorem 1.
Theorem 3. Let X be a set, SX be a group of all permutations of X.
Then B(SX , X) is the strongest uniformly locally finite ballean on X.
Proof. Let B′ be a uniformly locally finite ballean on X. Using Theorem
1, we choose a group G of permutations of X such that B′ = B(G, X).
Since G is a subgroup of SX , we have B′ ≺ B(SX , X).
A ballean B = (X, P, B) is called connected if, for any x, y ∈ X, there
exists α ∈ P such that y ∈ B(x, α). Clearly, a ballean B(G, X) of a
G-space is connected if and only if G acts transitively on X.
Let B1 = (X1, P1, B1), B2 = (X2, P2, B2) be balleans. A mapping
f : X1 → X2 is called a ≻-mapping if, for every β ∈ P2, there exists
α ∈ P1 such that B2(f(x), β)) ⊆ f(B1(x, α)) for each x ∈ X1. A bijection
f : X1 → X2 is a ≻-mapping if and only if f−1 is a ≺-mapping. Thus,
B1 and B2 are asymorphic if and only if there is a bijection f : X1 → X2
which is a ≺-mapping and a ≻-mapping.
Theorem 4. For every connected uniformly locally finite ballean B on
a set X, there exist a group G of permutations of X and a surjective
mapping f : G → X which is a ≺-mapping and a ≻-mapping from B(G)
to B.
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.108 Balleans of bounded geometry
Proof. Applying Theorem 1, we identify B with B(G, X) for some group
G of permutations of X. Then we fix x0 ∈ X and, for every g ∈ G, put
f(g) = g(x0). Since B is connected, (G, X) is a transitive G-space, so f
is surjective. For any finite subset F of G, we have f(Fg) = Fg(x0) =
F (g(x0)) = F (f(g)). It follows that f is a ≺-mapping and a ≻-mapping.
Let (G, X) be a transitive G-space, x0 ∈ X. If St(x0) = {g ∈ G :
g(x0) = x0} is finite, applying Theorem 4, it is easy to show that the
balleans B(G) and B(G, X) are coarsely equivalent.
Remark 2. Let (G, X) be a transitive G-space. How to detect whether
the ballean B(G, X) is asymorphic (coarsely equivalent) to the ballean
B(H) of some group H?
References
[1] A. Dranishnikov, Asymptotic topology, Russian Math. Surveys, 55(2000), 1085-
1129.
[2] M. Gromov, Asymptotic invariants for infinite groups, in Geometric Group The-
ory, vol.2, 1-295, Cambridge University Press, 1993.
[3] F. Harary, Graph Theory, Addison-Wesley Publ. Comp., 1969.
[4] P. Harpe, Topics in Geometrical Group Theory, University Chicago Press, 2000.
[5] V. Nekrashevych, Uniformly bounded spaces, Problems in Algebra, 14, 47-67,
Gomel University Press, 1999.
[6] I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs,
Math. Stud. Monogr. Ser., vol.11, VNTL, Lviv, 2003.
[7] I. Protasov, M. Zarichniy, General Asymptology, Math. Stud. Monogr. Ser.,
vol.12, VNTL, Lviv, 2007.
[8] J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol.31, American
Mathematical Society, Providence, RI, 2003.
Contact information
I. Protasov Kyiv Taras Shevchenko Univ. (for Depart-
ment of Cybernetics), Volodimirska str., 64,
01033 Kyiv, Ukraine
E-Mail: protasov@unicyb.kiev.ua
URL: http://do.unicyb.kiev.ua/
Received by the editors: 23.03.2008
and in final form 23.03.2008.
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