A metric space with its transfinite asymptotic dimension omega + 1
Yan Wu, Jingming Zhu

TL;DR
This paper constructs a specific metric space demonstrating that the transfinite asymptotic dimension can reach omega+1, providing a counterexample to the omega conjecture in geometric group theory.
Contribution
It presents the first explicit example of a metric space with transfinite asymptotic dimension omega+1, disproving the omega conjecture.
Findings
Transfinite asymptotic dimension can be omega+1.
Counterexample to the omega conjecture.
Both dimensions are omega+1 in the constructed space.
Abstract
We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both omega+1, where omega is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Topology and Set Theory · Advanced Operator Algebra Research
A metric space with its transfinite asymptotic dimension
Yan Wu∗ Jingming Zhu*∗∗* 111 College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing , 314001, P.R.China. ∗ E-mail: [email protected] E-mail: [email protected]
Abstract. We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both , where is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.
**Keywords ** Asymptotic dimension, Transfinite asymptotic dimension, Complementary-finite asymptotic dimension;
222 This research was supported by the National Natural Science Foundation of China under Grant (No.11871342,11301224,11326104,11401256,11501249)
1 Introduction
In coarse geometry, asymptotic dimension of a metric space is an important concept which was defined by Gromov for studying asymptotic invariants of discrete groups [1]. This dimension can be considered as an asymptotic analogue of the Lebesgue covering dimension. T. Radul defined the trasfinite asymptotic dimension (trasdim) which can be viewed as a transfinite extension for asymptotic dimension and gave examples of metric spaces with trasdim and with trasdim, where is the smallest infinite ordinal number (see [2]). But whether there is a metric space with trasdim is still unknown so far. M. Satkiewicz stated ”omega conjecture”(see [3]). That is, trasdim implies trasdim. In this paper, we prove that the ”omega conjecture” is not true by constructing a metric space with trasdim.
In the paper [4], we introduced another approach to classify the metric spaces with infinite asymptotic dimension, which is called complementary-finite asymptotic dimension (coasdim), and gave metric spaces with coasdim and trasdim, where . The metric space constructed in this paper is the first metric space with trasdim and trasdimcoasdim.
The paper is organized as follows: In Section 2, we recall some definitions and properties of transfinite asymptotic dimension. In Section 3, we introduce a concrete metric space and prove that transfinite asymptotic dimension and complementary-finite asymptotic dimension of are both .
2 Preliminaries
Our terminology concerning the asymptotic dimension follows from [5] and for undefined terminology we refer to [2]. Let be a metric space and , let
[TABLE]
Let and be a family of subsets of . is said to be -bounded if
[TABLE]
In this case, is said to be uniformly bounded. Let , a family is said to be* -disjoint* if
[TABLE]
In this paper, we denote by and denote by .
**Definition 2.1. ** A metric space is said to have finite asymptotic dimension if there is an , such that for every , there exists a sequence of uniformly bounded families of subsets of such that the family covers and each is -disjoint for . In this case, we say that the asymptotic dimension of less than or equal to , which is denoted by asdim. We say that asdim if asdim and asdim is not true.
T. Radul generalized asymptotic dimension of a metric space to transfinite asymptotic dimension which is denoted by trasdim (see [2]). Let denote the collection of all finite, nonempty subsets of and let . For , let
[TABLE]
Let abbreviate for . Define the ordinal number Ord inductively as follows:
[TABLE]
Given a metric space , define the following collection:
[TABLE]
The transfinite asymptotic dimension of is defined as trasdim=Ord.
3 Main result
Let be a subset of a metric space and , we denote by . For the unit cube , facets is the subset with the -th coordinate or for some integer .
To prove the main result, we will use another version of Lebesgue theorem:
Theorem 3.1**.**
(see [8], Theorem 4.3)
Let the unit cube be covered by a family of closed sets . Then some connected component of intersects both the corresponding opposite facets and .
Lemma 3.1**.**
(see [4], Proposition 2.1)
Given a metric space , let , then the following conditions are equivalent:
- •
(1) trasdim;
- •
(2) For every , there exists such that for every , there are uniformly bounded families satisfying is -disjoint for , is -disjoint for and covers .
Let , , ,… For and with , let , and put if and if . Define a metric on by
[TABLE]
where is the maximum metric in Let .
Proposition 3.1**.**
trasdim is not true.
Proof.
Suppose that trasdim. Then by Lemma 3.1, for every , there exists such that for every , there exist and -bounded families such that is -disjoint, is -disjoint for and covers . Let for .
Without loss of generality, we assume that , and . Then contains -dimensional closed cubes , where , whose edges are contained in for .
Let and , be the pair of two opposite facets of for .
Let for .
Then, by the definition, is -disjoint and -bounded subset family in for .
Let
[TABLE]
Then is a closed cover of with covering multiplicity at most . By the Theorem 3.1, at least one which intersects both the corresponding opposite facets and . But every connected component in for is -bounded, so and there is a path such that and belong to opposite facets and respectively.
Let . Then is a connected subset in which intersects both the corresponding opposite facets and . But implies that any can not intersect both and for any . Then for . So a path such that is a subset of the 1-dimensional skeleton of the ’s in (and hence a subset of ) and intersects both the corresponding opposite facets and . Since for and , for . So which is a contradiction with the fact that is -bounded and -disjoint.
∎
**Definition 3.2. ** Every ordinal number can be represented as , where is the limit ordinal or [math] and . Let be a metric space, we define complementary-finite asymptotic dimension coasdim inductively as follows:
- •
coasdim iff ,
- •
coasdim iff for every there exist -disjoint uniformly bounded families of subsets of such that coasdim,
- •
coasdim iff coasdim and for every , coasdim is not true.
- •
coasdim iff for every ordinal number , coasdim is not true.
Remark 3.1.
- •
It is easy to see that for every , coasdim if and only if asdim.
- •
To simplify, we will abuse the notation a little bit. In the following, asdim means that asdim for some .
Lemma 3.2**.**
(see [4], Theorem 3.2)
Let be a metric space, if coasdim for some , then trasdim. Especially, coasdim implies trasdim.
Proposition 3.2**.**
trasdim.
Proof.
By the Lemma 3.2, it suffices to show that coasdim. i.e., for every there exist -disjoint uniformly bounded families of subsets of such that coasdim. Let and , let
- •
If , then let
[TABLE]
[TABLE]
- •
If , then let
[TABLE]
[TABLE]
Note that , are -disjoint, -bounded and covers . Let
[TABLE]
[TABLE]
[TABLE]
It is easy to see that are -disjoint and -bounded for . Let
[TABLE]
Since is -disjoint and -bounded, is -disjoint and -bounded. For every , without loss of generality, we assume that . Then is in one of the following cases.
- •
and for some , it is easy to see that .
- •
and for some , it is easy to see that .
- •
and for some , it is easy to see that .
So covers . Let
[TABLE]
Since when and , then are -disjoint , -bounded and covers . It follows that
[TABLE]
which implies coasdim.
∎
Remark 3.2.
- •
By the Proposition 3.1 and the Proposition 3.2, we can obtain that trasdim.
- •
By the Proposition 3.1 and the Lemma 3.2, we can obtain that coasdim is not true. Moreover, we obtain that coasdim by the Proposition 3.2.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] T. Radul, On transfinite extension of asymptotic dimension. Topol. Appl. 157 (2010), 2292–2296.
- 3[3] M. Satkiewicz, Transfinite Asymptotic Dimension. ar Xiv:1310.1258 v 1, 2013.
- 4[4] Yan Wu, Jingming Zhu, Classification of metric spaces with infinite asymptotic dimension. Topology and its Applications 238 (2018) 90–101.
- 5[5] G. Bell, A. Dranishnikov, Asymptotic dimension in Bedlewo. Topol. Proc. 38 (2011), 209–236.
- 6[6] P. Borst, Classification of weakly infinite-dimensional spaces. Fund. Math. 130(1988), 1–25.
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