Asymptotic dimension and small subsets in locally compact topological groups
Abstract
We prove that for a coarse space \(X\) the ideal \(\mathcal{S }(X)\) of small subsets of \(X\) coincides with the ideal \(\mathcal{D }_<(X)=\{A\subset X:\mathrm{asdim}(A)<\mathrm{asdim}(X)\}\) provided that \(X\) is coarsely equivalent to a Euclidean space \(\mathbb{R }^n\). Also we prove that for a locally compact Abelian group \(X\), the equality \(\mathcal{S }(X)=\mathcal{D }_<(X)\) holds if and only if the group \(X\) is compactly generated.
Keywords
Asymptotic dimension Locally compact group Coarse structure Small setMathematics Subject Classification (1991)
20F65 20F69 54F45 55M101 Introduction
In this paper we study the interplay between the ideal \(\mathcal{S }(X)\) of small subsets of a coarse space \(X\) and the ideal \(\mathcal{D }_{<}(X)\) of subsets of asymptotic dimension less than \(\mathrm{asdim}(X)\) in \(X\). We show that these two ideals coincide in spaces that are coarsely equivalent to \(\mathbb{R }^n\), in particular, they coincide in each compactly generated locally compact abelian group.
 (A)
each \(\varepsilon \in \mathcal{E }\) contains the diagonal \(\Delta _X=\{(x,y)\in X^2:x=y\}\) and is symmetric in the sense that \({\varepsilon }={\varepsilon }^{1}\) where \({\varepsilon }^{1}=\{(y,x):(x,y)\in {\varepsilon }\}\);
 (B)
for any entourages \({\varepsilon },\delta \in \mathcal{E }\) there is an entourage \(\eta \in \mathcal{E }\) that contains the composition\(\delta \circ {\varepsilon }=\{(x,z)\in X^2:\exists y\in X\) with \((x,y)\in {\varepsilon }\) and \((y,z)\in \delta \}\);
 (C)
a subset \(\delta \subset X^2\) belongs to \(\mathcal{E }\) if \(\Delta _X\subset \delta =\delta ^{1}\subset {\varepsilon }\) for some \({\varepsilon }\in \mathcal{E }\).
Each subset \(A\) of a coarse space \((X,\mathcal{E })\) carries the induced coarse structure \(\mathcal{E }_A=\{{\varepsilon }\cap A^2:{\varepsilon }\in \mathcal{E }\}\). Endowed with this structure, the space \((A,\mathcal{E }_A)\) is called a subspace of \((X,\mathcal{E })\).
For an entourage \({\varepsilon }\subset X^2\), a point \(x\in X\), and a subset \(A\subset X\) let \(B(x,{\varepsilon })=\{y\in X:(x,y)\in {\varepsilon }\}\) be the \({\varepsilon }\) ball centered at \(x\), \(B(A,{\varepsilon })=\bigcup _{a\in A}B(a,{\varepsilon })\) be the \({\varepsilon }\)neighborhood of \(A\) in \(X\), and \(\mathrm{diam}(A)=A\times A\) be the diameter of \(A\). For a family \(\mathcal{U }\) of subsets of \(X\) we put \(\mathrm{mesh}(\mathcal{U })=\bigcup _{U\in \mathcal{U }}\mathrm{diam}(U)\).
Now we consider two basic examples of coarse spaces. The first of them is any metric space \((X,d)\) carrying the metric coarse structure whose base consists of the entourages \(\{(x,y)\in X^2:d(x,y)<{\varepsilon }\}\) where \(0\le {\varepsilon }<\infty \). A coarse space is metrizable if its coarse structure is generated by some metric.
The second basic example is a topological group \(G\) endowed with the left coarse structure whose base consists of the entourages \(\{(x,y)\in G^2:x\in yK\}\) where \(K=K^{1}\) runs over compact symmetric subsets of \(G\) that contain the identity element \(1_G\) of \(G\). Let us observe that the left coarse structure on \(G\) coincides with the metric coarse structure generated by any leftinvariant continuous metric \(d\) on \(G\) which is proper in the sense that each closed ball \(B(e,R)=\{x\in G:d(x,e)\le R\}\) is compact. In particular, the coarse structure on \(\mathbb{R }^n\), generated by the Euclidean metric coincides with the left coarse structure of the Abelian topological group \(\mathbb{R }^n\).

large if \(B(A,{\varepsilon })=X\) for some \({\varepsilon }\in \mathcal{E }\);

small if for each large set \(L\subset X\) the set \(L\setminus A\) remains large in \(X\).
Small sets can be considered as coarse counterparts of nowhere dense subsets in topological spaces, see [2]. It is wellknown [8, 7.4.18] that the ideal of nowhere dense subsets in a Euclidean space \(\mathbb{R }^n\) coincides with the ideal generated by closed subsets of topological dimension \(<n\) in \(\mathbb{R }^n\). The aim of this paper is to prove a coarse counterpart of this fundamental fact.
For this we need to recall [14, 9.4] the definition of the asymptotic dimension \(\mathrm{asdim}(X)\) of a coarse space \(X\).
Definition 1.1
The asymptotic dimension \(\mathrm{asdim}(X)\) of a coarse space \((X,\mathcal{E })\) is the smallest number \(n\in {\omega }\) such that for each entourage \({\varepsilon }\in \mathcal{E }\) there is a cover \(\mathcal{U }\) of \(X\) such that \(\mathrm{mesh}(\mathcal{U })\subset \delta \) for some \(\delta \in \mathcal{E }\) and each \({\varepsilon }\)ball \(B(x,{\varepsilon })\), \(x\in X\), meets at most \(n+1\) sets \(U\in \mathcal{U }\). If such a number \(n\in {\omega }\) does not exist, then we put \(\mathrm{asdim}(X)=\infty \).
The main result of this paper is:
Theorem 1.2
For every \(n\in \mathbb{N }\) the ideal \(\mathcal{S }(X)\) of small subsets in the space \(X=\mathbb{R }^n\) coincides with the ideal \(\mathcal{D }_{<}(X)\).
Theorem 1.2 will be proved in Sect. 5 with help of some tools of Combinatorial Topology. In light of this theorem the following problem arises naturally:
Problem 1.3
Detect coarse spaces \(X\) for which \(\mathcal{S }(X)=\mathcal{D }_<(X)\).
It should be mentioned that the class of coarse spaces \(X\) with \(\mathcal{S }(X)=\mathcal{D }_<(X)\) is closed under coarse equivalences.

coarse if for each \(\delta _X\in \mathcal{E }_X\) there is \({\varepsilon }_Y\in \mathcal{E }_Y\) such that for any pair \((x,y)\in \delta _X\) we get \((f(x),f(y))\in {\varepsilon }_Y\);

a coarse equivalence if \(f\) is coarse and there is a coarse map \(g:Y\rightarrow X\) such that \(\{(x,g\circ f(x)):x\in X\}\subset {\varepsilon }_X\) and \(\{(y,f\circ g(y)):y\in Y\}\subset {\varepsilon }_Y\) for some entourages \({\varepsilon }_X\in \mathcal{E }_X\) and \({\varepsilon }_Y\in \mathcal{E }_Y\).
Proposition 1.4
 (1)
\(\mathrm{asdim}(X)=\mathrm{asdim}(Y)\);
 (2)
\(\mathcal{D }_<(X)=\mathcal{S }(X)\) if and only if \(\mathcal{D }_{<}(Y)=\mathcal{S }(Y)\).
This proposition will be proved in Sect. 3. Combined with Theorem 1.2 it implies:
Corollary 1.5
If a coarse space \(X\) is coarsely equivalent to a Euclidean space \(\mathbb{R }^n\), then \(\mathcal{D }_{<}(X)=\mathcal{S }(X)\).
Problem 1.3 can be completely resolved for locally compact Abelian topological groups \(G\), endowed with their left coarse structure. First we establish the following general fact, which will be proved in Sect. 4.
Theorem 1.6
For each topological group \(X\) endowed with its left coarse structure we get \(\mathcal{D }_<(X)\subset \mathcal{S }(X)\).
We recall that a topological group \(G\) is compactly generated if \(G\) is algebraically generated by some compact subset \(K\subset G\).
Theorem 1.7
 (1)
\(\mathcal{S }(X)=\mathcal{D }_<(X)\);
 (2)
\(X\) is compactly generated;
 (3)
\(X\) is coarsely equivalent to the Euclidean space \(\mathbb{R }^n\) for some \(n\in {\omega }\).
This theorem will be proved in Sect. 6.
Remark 1.8
The equivalence (1)\(\Leftrightarrow \)(2) in Theorem 1.7 does not hold beyond the class of Abelian groups. The simplest counterexample is the free group \(F_2\) with two generators, endowed with the discrete topology. Any infinite cyclic subgroup \(Z\subset F_2\) has infinite index in \(F_2\) and hence is small, yet \(\mathrm{asdim}(Z)=\mathrm{asdim}(F_2)=1\).
Problem 1.9
Is \(\mathcal{S }(X)=\mathcal{D }_<(X)\) for each connected Lie group \(X\)? For each discrete polycyclic group \(X\)?
2 The asymptotic dimension of coarse spaces
In this section we present various characterizations of the asymptotic dimension of coarse spaces. First we fix some notation. Let \((X,\mathcal{E })\) be a coarse space, \({\varepsilon }\in \mathcal{E }\) and \(A\subset X\). We shall say that \(A\) has diameter less than \({\varepsilon }\) if \(\mathrm{diam}(A)\subset {\varepsilon }\) where \(\mathrm{diam}(A)=A\times A\). A sequence \(x_0,\dots ,x_m\in X\) is called an \({\varepsilon }\) chain if \((x_i,x_{i+1})\in {\varepsilon }\) for all \(i<m\). In this case the finite set \(C=\{x_0,\dots ,x_m\}\) also will be called an \({\varepsilon }\) chain. A set \(C\subset X\) is called \({\varepsilon }\)connected if any two points \(x,y\in C\) can be linked by an \({\varepsilon }\)chain \(x=x_0,\dots ,x_m=y\). The maximal \({\varepsilon }\)connected subset \(C(x,{\varepsilon })\subset X\) containing a given point \(x\in X\) is called the \({\varepsilon }\) connected component of \(x\). It consists of all points \(y\in X\) that can be linked with \(x\) by an \({\varepsilon }\)chain \(x=x_0,\dots ,x_m=y\).
A family \(\mathcal{U }\) of subsets of \(X\) is called \({\varepsilon }\) disjoint if \((U\times V)\cap {\varepsilon }=\emptyset \) for any distinct sets \(U,V\in \mathcal{U }\). Each natural number \(n\) is identified with the set \(\{0,\dots ,n1\}\).
We shall study the interplay between the asymptotic dimension introduced in Definition 1.1 and the following modification:
Definition 2.1
The colored asymptotic dimension \(\mathrm{asdim}_{col}(X)\) of a coarse space \((X,\mathcal{E })\) is the smallest number \(n\in {\omega }\) such that for every entourage \(\varepsilon \in \mathcal{E }\) there is a cover \(\mathcal{U }\) of \(X\) such that \(\mathrm{mesh}(\mathcal{U })\subset \delta \) for some \(\delta \in \mathcal{E }\) and \(\mathcal{U }\) can be written as the union \(\mathcal{U }=\bigcup _{i\in n+1}\mathcal{U }_i\) of \(n+1\) many \({\varepsilon }\)disjoint subfamilies \(\mathcal{U }_i\). If such a number \(n\in {\omega }\) does not exist, then we put \(\mathrm{asdim}_{col}(X)=\infty \).
Without lost of generality we can assume that the cover \(\mathcal{U }=\bigcup _{i\in n+1}\mathcal{U }_i\) in the above definition consists of pairwise disjoint sets. In this case we can consider the coloring \(\chi : X\rightarrow n+1=\{0,\dots ,n\}\) such that \(\chi ^{1}(i)=\bigcup \mathcal{U }_i\) for every \(i\in n+1\). For this coloring every \(\chi \)monochrome \({\varepsilon }\)connected subset \(C\subset X\) lies in some \(U\in \mathcal{U }\) and hence has diameter \(\mathrm{diam}(C)\subset \mathrm{diam}(U)\subset \mathrm{mesh}(\mathcal{U })\subset \delta \). A subset \(A\subset X\) is \(\chi \) monochrome if \(\chi (A)\) is a singleton. Thus we arrive to the following useful characterization of the colored asymptotic dimension.
Proposition 2.2
A coarse space \((X,\mathcal{E })\) has \(\mathrm{asdim}_{col}(X)\le n\) for some number \(n\in {\omega }\) if and only if for any \(\varepsilon \in \mathcal{E }\) there are a coloring \(\chi : X\rightarrow n+1\) and an entourage \(\delta \in \mathcal{E }\) such that each \(\chi \)monochrome \(\varepsilon \)chain \(C\subset X\) has \(\mathrm{diam}(C)\subset \delta \).
Proof
The “only if” part follows from the above discussion. To prove the “if” part, for every \({\varepsilon }\in \mathcal{E }\) we need to construct a cover \(\mathcal{U }=\bigcup _{i\in n+1}\mathcal{U }_i\) such that \(\mathrm{mesh}(\mathcal{U })\in \mathcal{E }\) and each family \(\mathcal{U }_i\) is \({\varepsilon }\)disjoint. By our assumption, there is a coloring \(\chi :X\rightarrow n+1\) and an entourage \(\delta \in \mathcal{E }\) such that each \(\chi \)monochrome \({\varepsilon }\)chain \(C\subset X\) has \(\mathrm{diam}(C)\subset \delta \).
For each \(x\in X\) let \(C_\chi (x,{\varepsilon })\) be the set of all points \(y\in X\) that can be linked with \(x\) by a \(\chi \)monochrome \({\varepsilon }\)chain \(x=x_0,x_1,\dots ,x_m=y\). It follows that \(\mathrm{diam}(C_\chi (x,{\varepsilon }))\subset \delta \). For every \(i\in n+1\) consider the \({\varepsilon }\)disjoint family \(\mathcal{U }_i=\{C_\chi (x,{\varepsilon }): x\in \chi ^{1}(i)\}\). It is clear that \(\mathcal{U }=\bigcup _{i\in n+1}\mathcal{U }_i\) is a cover with \(\mathrm{mesh}(\mathcal{U })\subset \delta \in \mathcal{E }\), witnessing that \(\mathrm{asdim}_{col}(X)\le n\). \(\square \)
Now we are ready to prove the equivalence of two definitions of asymptotic dimension. For metrizable coarse spaces this equivalence was proved in [5, 9.3.7].
Proposition 2.3
Each coarse space \((X,\mathcal{E })\) has \(\mathrm{asdim}(X)=\mathrm{asdim}_{col}(X)\).
Proof
To prove that \(\mathrm{asdim}(X)\le \mathrm{asdim}_{col}(X)\), put \(n=\mathrm{asdim}_{col}(X)\) and take any entourage \({\varepsilon }\in \mathcal{E }\). By Definition 2.1, for the entourage \({\varepsilon }\circ {\varepsilon }\in \mathcal{E }\) we can find a cover \(\mathcal{U }=\bigcup _{i\in n+1}\mathcal{U }_i\) with \(\mathrm{mesh}(\mathcal{U })\in \mathcal{E }\) such that each family \(\mathcal{U }_i\) is \({\varepsilon }\circ {\varepsilon }\)disjoint. We claim that each \({\varepsilon }\)ball \(B(x,{\varepsilon })\), \(x\in X\), meets at most one set of each family \(\mathcal{U }_i\). Assuming that \(B(x,{\varepsilon })\) meets two distinct sets \(U,V\in \mathcal{U }_i\), we can find points \(u\in U\) and \(v\in V\) with \((x,u),(x,v)\in {\varepsilon }\) and conclude that \((u,v)\in {\varepsilon }\circ {\varepsilon }\), which is not possible as \(\mathcal{U }_i\) is \({\varepsilon }\circ {\varepsilon }\)disjoint. Now we see that the ball \(B(x,{\varepsilon })\) meets at most \(n+1\) element of the cover \(\mathcal{U }\) and hence \(\mathrm{asdim}(X)\le n\).
The proof of the inequality \(\mathrm{asdim}_{col}(X)\le \mathrm{asdim}(X)\) is a bit longer. If the dimension \(n=\mathrm{asdim}(X)\) is infinite, then there is nothing to prove. So, we assume that \(n\in {\omega }\). To prove that \(\mathrm{asdim}_{col}(X)\le n\), fix an entourage \({\varepsilon }\in \mathcal{E }\). Let \({\varepsilon }^0=\Delta _X\) and \({\varepsilon }^{k+1}={\varepsilon }^k\circ {\varepsilon }\) for \(k\in {\omega }\). Since \(\mathrm{asdim}(X)\le n\), for the entourage \({\varepsilon }^{n+1}\in \mathcal{E }\) we can find a cover \(\mathcal{U }\) of \(X\) such that \(\delta =\mathrm{mesh}(\mathcal{U })\in \mathcal{E }\) and each \({\varepsilon }^{n+1}\)ball \(B(x,{\varepsilon }^{n+1})\) meets at most \(n+1\) many sets \(U\in \mathcal{U }\). For every \(i\le n+1\) and \(x\in X\) consider the subfamily \(\mathcal{U }(x,{\varepsilon }^i)=\{U\in \mathcal{U }:B(x,{\varepsilon }^i)\cap U\ne \emptyset \}\) of \(\mathcal{U }\). It follows that \(1\le \mathcal{U }(x,{\varepsilon }^i)\le \mathcal{U }(x,{\varepsilon }^{i+1})\le n+1\) for every \(0\le i\le n\). Consequently, \(\mathcal{U }(x,{\varepsilon }^i)=i\) for some \(i\le n+1\). Let \(\chi (x)\) be the maximal number \(k\le n\) such that \(\mathcal{U }(x,{\varepsilon }^{k+1})=k+1\). In such a way we have defined a coloring \(\chi :X\rightarrow n+1=\{0,\dots ,n\}\).
To finish the proof it suffices to show that any \(\chi \)monochrome \(\varepsilon \)chain \(C=\{x_0,\dots ,x_m\}\subset X\) has \(\mathrm{diam}(C)\subset \delta \circ \varepsilon ^{n+1}\). Let \(k=\chi (x_0)\) be the color of the chain \(C\). It follows that \(\mathcal{U }(x_i,{\varepsilon }^{k+1})=k+1\) for all \(x_i\in C\). We claim that \(\mathcal{U }(x_i,{\varepsilon }^{k+1})=\mathcal{U }(x_{i+1},{\varepsilon }^{k+1})\) for all \(i<m\). Assuming the converse, we would get that \(\mathcal{U }(x_i,{\varepsilon }^{k+1})\cup \mathcal{U }(x_{i+1},{\varepsilon }^{k+1})\ge k+3\) and then the family \(\mathcal{U }(x_i,{\varepsilon }^{k+2})\supset \mathcal{U }(x_i,{\varepsilon }^{k+1})\cup \mathcal{U }(x_{i+1},{\varepsilon }^{k+1})\) has cardinality \(\mathcal{U }(x_i,{\varepsilon }^{k+2})\ge k+3\), which implies that \(\mathcal{U }(x_i,{\varepsilon }^i)=i\) for some \(k+3\le i\le n+1\). But this contradicts the definition of \(k=\chi (x_i)\). Hence \(\mathcal{U }(x_i,{\varepsilon }^{k+1})=\mathcal{U }(x_{0},{\varepsilon }^{k+1})\) for all \(i\le m\) and then \(C\subset B(U,{\varepsilon }^{k+1})\) for every \(U\in \mathcal{U }(x_0,{\varepsilon }^{k+1})\). Now we see that \(\mathrm{diam}(C)\subset \mathrm{diam}(U)\circ {\varepsilon }^{k+1}\subset \delta \circ {\varepsilon }^{n+1}\). \(\square \)
Propositions 2.2 and 2.3 imply:
Corollary 2.4
A coarse space \((X,\mathcal{E })\) has asymptotic dimension \(\mathrm{asdim}(X)\le n\) for some \(n\in {\omega }\) if and only if for any \(\varepsilon \in \mathcal{E }\) there are \(\delta \in \mathcal{E }\) and a coloring \(\chi :X\rightarrow n+1\) such that any \(\chi \)monochrome \(\varepsilon \)chain \(C\subset X\) has \(\mathrm{diam}(C)\subset \delta \).
This corollary can be generalized as follows (cf. [6]).
Proposition 2.5
A coarse space \((X,\mathcal{E })\) has \(\mathrm{asdim}(X)\le n\) for some \(n\in {\omega }\) if and only if for any entourage \(\varepsilon \in \mathcal{E }\) there is an entourage \(\delta \in \mathcal{E }\) such that for any finite set \(F\subset X\) there is a coloring \(\chi :F\rightarrow n+1\) such that each \(\chi \)monochrome \(\varepsilon \)chain \(C\subset F\) has \(\mathrm{diam}(C)\subset \delta \).
Proof
This proposition will follow from Corollary 2.4 as soon as for any \({\varepsilon }\in \mathcal{E }\) we find \(\delta \in \mathcal{E }\) and a coloring \(\chi :X\rightarrow n+1\) such that each \(\chi \)monochrome \({\varepsilon }\)chain in \(X\) has diameter less that \(\delta \).
By our assumption, there is an entourage \(\delta \in \mathcal{E }\) such that for every finite subset \(F\subset X\) there is a coloring \(\chi _F:F\rightarrow n+1\) such that each \(\chi _F\)monochrome \({\varepsilon }\)chain in \(F\) has diameter less that \(\delta \). Extend \(\chi _F\) to a coloring \(\tilde{\chi }_F:X\rightarrow n+1\).
Let \(\mathcal{F }\) denote the family of all finite subsets of \(X\), partially ordered by the inclusion relation \(\subset \). The colorings \(\tilde{\chi }_F\), \(F\in \mathcal{F }\), can be considered as elements of the compact Hausdorff space \(K=\{0,\dots ,n\}^X\) endowed with the Tychonoff product topology. The compactness of \(K\) implies that the net \(\{\tilde{\chi }_F\}_{F\in \mathcal{F }}\) has a cluster point \(\chi \in K\), see [8, 3.1.23]. The latter means that for each finite set \(F_0\in \mathcal{F }\) and a neighborhood \(O(\chi )\subset K\) there is a finite set \(F\in \mathcal{F }\) such that \(F\supset F_0\) and \(\tilde{\chi }_F\in O(\chi )\).
We claim that the coloring \(\chi :X\!\rightarrow \! n+1\) has the required property: each \(\chi \)monochrome \({\varepsilon }\)chain \(C\subset X\) has \(\mathrm{diam}(X)\subset \delta \). Observe that the finite set \(C\) determines a neighborhood \(O_C(\chi )\!=\!\{f\in K:fC\!=\!\chi C\}\), which contains a coloring \(\tilde{\chi }_F\) for some finite set \(F\supset C\). The choice of the coloring \(\chi _F\!=\!\tilde{\chi }_FF\) guarantees that the set \(C\subset F\) has \(\mathrm{diam}(C)\subset \delta \). \(\square \)
Proposition 2.5 admits the following selfgeneralization.
Theorem 2.6
A coarse space \((X,\mathcal{E })\) has \(\mathrm{asdim}(X)\le n\) for some \(n\in {\omega }\) if and only if for any entourage \(\varepsilon \in \mathcal{E }\) there is an entourage \(\delta \in \mathcal{E }\) such that for any finite \({\varepsilon }\)connected subset \(F\subset X\) there is a coloring \(\chi :F\rightarrow n+1\) such that each \(\chi \)monochrome \(\varepsilon \)chain \(C\subset F\) has \(\mathrm{diam}(C)\subset \delta \).
Finally, let us prove Addition Theorem for the asymptotic dimension. For metrizable spaces this theorem is well known; see [14, 9.13] or [5, 9.7.1].
Theorem 2.7
Proof
Only the case of finite \(n=\max \{\mathrm{asdim}(A),\mathrm{asdim}(B)\}\) requires the proof. Without loss of generality the sets \(A\) and \(B\) are disjoint. To show that \(\mathrm{asdim}(A\cup B)\le n\) we shall apply Corollary 2.4. Fix any entourage \({\varepsilon }\in \mathcal{E }\). Since \(\mathrm{asdim}(A)\le n\) there are an entourage \(\delta _A\in \mathcal{E }\) and a coloring \(\chi _A:A\rightarrow n+1\) such that each \(\chi \)monochrome \({\varepsilon }\)chain in \(A\) has diameter less that \(\delta _A\). Since \(\mathrm{asdim}(B)\le n\), for the entourage \({\varepsilon }_B={\varepsilon }\circ \delta _A\circ {\varepsilon }\) there are an entourage \(\delta _B\in \mathcal{E }\) and a coloring \(\chi _B:A\rightarrow \{0,\dots ,n\}\) such that each \(\chi \)monochrome \({\varepsilon }_B\)chain in \(B\) has diameter less that \(\delta _B\).
The union of the colorings \(\chi _A\) and \(\chi _B\) yields the coloring \(\chi :A\cup B\rightarrow \{0,\dots ,n\}\) such that \(\chi A=\chi _A\) and \(\chi B=\chi _B\). We claim that each \(\chi \)monochrome \({\varepsilon }\)chain \(C=\{x_0,\dots ,x_m\}\subset A\cup B\) has \(\mathrm{diam}(C)\subset \delta \) where \(\delta =\delta _A\circ {\varepsilon }\circ \delta _B\circ {\varepsilon }\circ \delta _A\). Without loss of generality, the points \(x_0,\dots ,x_m\) of the chain \(C\) are pairwise distinct.
If \(C\subset A\), then \(C\), being a \(\chi _A\)monochrome \({\varepsilon }\)chain in \(A\) has \(\mathrm{diam}(C)\subset \delta _A\subset \delta \) and we are done. So, we assume that \(C\not \subset A\). In this case \(b=C\cap B\ge 1\) and we can choose a strictly increasing sequence \(0\le k_1<k_2<\cdots <k_b\le m\) such that \(\{x_{k_1},\dots ,x_{k_b}\}=C\cap B\). Then \(\{x_0,\dots ,x_{k_11}\}\), being a \(\chi _A\)monochrome \({\varepsilon }\)chain in \(A\), has diameter less that \(\delta _A\). Consequently, the \({\varepsilon }\)chain \(\{x_0,\dots ,x_{k_1}\}\) has diameter less that \(\delta _A\circ {\varepsilon }\subset {\varepsilon }_B\). By the same reason the \({\varepsilon }\)chain \(\{x_{k_b},\dots ,x_m\}\) has diameter less that \({\varepsilon }\circ \delta _A\subset {\varepsilon }_B\) and for every \(1\le i<b\) the \({\varepsilon }\)chain \(\{x_{k_i},\dots ,x_{k_{i+1}}\}\subset \{x_{k_i}\}\cup A\cup \{x_{k_{i+1}}\}\) has diameter less than \({\varepsilon }\circ \delta _A\circ {\varepsilon }={\varepsilon }_B\). Then \(\{x_{k_1},\dots ,x_{k_b}\}\), being a \(\chi _B\)monochrome \({\varepsilon }_B\)chain in \(B\), has diameter less that \(\delta _B\). Now we see that the \({\varepsilon }\)chain \(C=\{x_0,\dots ,x_m\}\) has \(\mathrm{diam}(C)\subset \delta _A\circ {\varepsilon }\circ \delta _B\circ {\varepsilon }\circ \delta _A=\delta \). \(\square \)
The characterization Theorem 2.6 will be applied to prove the following theorem which was known [7, 2.1] in the context of countable groups.
Theorem 2.8
Proof
Let \(n=\sup \,\{\mathrm{asdim}(H):H {\text{ is } \text{ a } \text{ compactly } \text{ generated } \text{ subgroup } \text{ of }} G\}.\) It is clear that \(n\le \mathrm{asdim}(G)\). The reverse inequality \(\mathrm{asdim}(G)\le n\) is trivial if \(n=\infty \). So, we assume that \(n<\infty \). To prove that \(\mathrm{asdim}(G)\le n\), we shall apply Theorem 2.6. Let \(\mathcal{E }\) be the left coarse structure of the topological group \(G\). Given any entourage \({\varepsilon }\in \mathcal{E }\), we should find an entourage \(\delta \in \mathcal{E }\) such that for each finite \({\varepsilon }\)connected subset \(F\subset G\) there is a coloring \(\chi :F\rightarrow n+1\) such that each \(\chi \)monochrome \({\varepsilon }\)chain \(C\subset F\) has \(\mathrm{diam}(C)\subset \delta \).
By the definition of the coarse structure \(\mathcal{E }\), for the entourage \({\varepsilon }\in \mathcal{E }\) there is a compact subset \(K_{\varepsilon }=K_{\varepsilon }^{1}\subset G\) such that \({\varepsilon }\subset \{(x,y)\in G^2:x\in yK_{\varepsilon }\}\). Let \(H\) be the subgroup of \(G\) generated by the compact set \(K_{\varepsilon }\), \(\mathcal{E }_H\) be the left coarse structure of \(H\), and \({\varepsilon }_H=\{(x,y)\in H^2:x\in yK_{\varepsilon }\}\in \mathcal{E }_H\). Since \(\mathrm{asdim}_{col}(H)=\mathrm{asdim}(H)\le n\), by Proposition 2.2, there is a coloring \(\chi _H:H\rightarrow n+1\) and an entourage \(\delta _H\in \mathcal{E }_H\) such that each \(\chi _H\)monochrome \({\varepsilon }\)chain \(C\subset H\) has diameter \(\mathrm{diam}(C)\subset \delta _H\). By the definition of the coarse structure \(\mathcal{E }_H\), there is a compact subset \(K_\delta =K_\delta ^{1}\ni 1_H\) of \(H\) such that \(\{(x,y)\in H\times H:x\in yK_\delta \}\).
We claim that the entourage \(\delta =\{(x,y)\in G\times G:x\in yK_\delta \}\) satisfies our requirements. Let \(F\) be a finite \({\varepsilon }\)connected subset of \(G\). Then for each point \(x_0\in F\) we get \(F\in x_0 H\) and hence \(x_0^{1}F\subset H\). So, we can define a coloring \(\chi :F\rightarrow n+1\) letting \(\chi (x)=\chi _H(x_0^{1}x)\) for \(x\in F\). If \(C\subset F\) is a \(\chi \)monochrome \({\varepsilon }\)chain, then \(x_0^{1}C\) is a \(\chi _H\)monochrome \({\varepsilon }_H\)chain in \(H\) and hence \(\mathrm{diam}(x_0^{1}C)\subset \delta _H\). The latter means that for any points \(c,c^{\prime }\in C\) we get \((x_0^{1}c,x_0^{1}c^{\prime })\in \delta _H\subset \{(x,y)\in H\times H:x\in yK_\delta \}\) and hence \(x_0^{1}c\in x_0^{1}c^{\prime }K_\delta \) and \(c\in c^{\prime }K_\delta \), which means that \((c,c^{\prime })\in \delta \) and hence \(\mathrm{diam}(C)\subset \delta \). \(\square \)
3 Proof of Proposition 1.4
 1.
First we prove that \(\mathrm{asdim}(X)=\mathrm{asdim}(Y)\). Actually, this fact is known [14, p.129] and we present a proof for the convenience of the reader. By the symmetry, it suffices to show that \(\mathrm{asdim}(X)\le \mathrm{asdim}(Y)\). This inequality is trivial if \(n=\mathrm{asdim}(Y)\) is infinite. So, assume that \(n<\infty \). By Propositions 2.2 and 2.3, the inequality \(\mathrm{asdim}(X)\le n\) will be proved as soon as for each \({\varepsilon }_X\in \mathcal{E }_X\) we find \(\delta _X\in \mathcal{E }_X\) and a coloring \(\chi _X:X\rightarrow n+1\) such that each \(\chi _X\)monochrome \({\varepsilon }_X\)chain \(C\subset X\) has diameter \(\mathrm{diam}(C)\subset \delta _X\).
 2.
The second statement of Proposition 1.4, follows Claims 3.1 and 3.4 proved below.
Claim 3.1
A subset \(A\subset X\) and its image \(f(A)\subset Y\) have the same asymptotic dimension \(\mathrm{asdim}(A)=\mathrm{asdim}(f(A))\).
Proof
This claim follows from Proposition 1.4(1) proved above, since \(A\) and \(f(A)\) are coarsely equivalent. \(\square \)
Claim 3.2
A subset \(A\subset X\) is large in \(X\) if and only if its image \(f(A)\) is large in \(Y\).
Proof
If \(A\) is large in \(X\), then \(B(A,{\varepsilon }_X)=X\) for some \({\varepsilon }_X\in \mathcal{E }_X\). Since \(f\) is coarse, there exists \({\varepsilon }_Y\in \mathcal{E }_Y\) such that for each \((x_0,x_1)\in {\varepsilon }_X\) we get \((f(x_0),f(x_1))\in {\varepsilon }_Y\). It follows that \(B(f(A),{\varepsilon }_Y)\supset f(Y)\) and \(B(f(A),{\varepsilon }_Y\circ \eta _Y)=B(B(f(A),{\varepsilon }_Y),\eta _Y)\supset B(f(X),\eta _Y)=Y\), which means that \(f(A)\) is large.
Now assume conversely that the set \(f(A)\) is large in \(Y\). Then \(g\circ f(A)\) is large in \(X\). Since \(g\circ f(A)\subset B(A,\eta _X)\), we conclude that \(A\) in large in \(X\). So, \(A\) is large in \(X\) if and only if \(f(A)\) is large in \(Y\). \(\square \)
Claim 3.3
A subset \(A\subset X\) is small if and only if for each entourage \({\varepsilon }_X\in \mathcal{E }_X\) the set \(B(A,{\varepsilon }_x)\) is small.
Proof
The “if” part is trivial. To prove the “only if” part, assume that the set \(A\) is small in \(X\). To show that \(B(A,{\varepsilon }_X)\) is small in \(X\), it is necessary to check that for each large subset \(L\subset X\) the complement \(L\setminus B(A,{\varepsilon }_X)\) is large in \(X\). Consider the set \(L^{\prime }=(L\setminus B(A,{\varepsilon }_X))\cup A\) and observe that \(L\subset B(L^{\prime },{\varepsilon }_X)\) and hence \(L^{\prime }\) is large in \(X\). Since \(A\) is small, the set \(L^{\prime }\setminus A=L\setminus B(A,{\varepsilon }_X)\) is large in \(X\). \(\square \)
Claim 3.4
A subset \(A\subset X\) is small in \(X\) if and only if its image \(f(A)\) is small in \(Y\).
Proof
Assume that \(A\) is small in \(X\). To prove that \(f(A)\) is small in \(Y\), we need to check that for any large subset \(L\subset Y\) the complement \(L\setminus f(A)\) is large in \(Y\). Claim 3.2 implies that the set \(g(L)\) is large in \(X\). By Claim 3.3, the set \(B(A,\eta _X)\) is small in \(X\) and hence the complement \(g(L)\setminus B(A,\eta _X)\) remains large in \(X\). By Claim 3.2 \(f(g(L)\setminus B(A,\eta _X))\) is large in \(Y\). We claim that \(f(g(L)\setminus B(A,\eta _X))\subset B(L\setminus f(A),\eta _Y)\). Indeed, given point \(y\in f(g(L)\setminus B(A,\eta _X))\), find a point \(x\in g(L)\setminus B(A,\eta _X)\) such that \(y=f(x)\) and a point \(z\in L\) such that \(x=g(z)\). We claim that \(z\notin f(A)\). Assuming conversely that \(z\in f(A)\), we get \(x=g(z)\in g\circ f(A)\subset B(A,\eta _X)\), which contradicts the choice of \(x\). So, \(z\in L\setminus f(A)\) and \(y=f\circ g(z)\in B(z,\eta _Y)\subset B(L\setminus f(A),\eta _Y)\).
Taking into account that the set \(f(g(L)\setminus B(A,\eta _X))\subset B(L\setminus f(A),\eta _Y)\) is large in \(Y\), we conclude that the set \(L\setminus f(A)\) is large in \(Y\) and hence \(f(A)\) is small in \(Y\).
Now assume that the set \(f(X)\) is small in \(Y\). Then the set \(g\circ f(A)\) is small in \(X\) and so are the sets \(B(g\circ f(A),\eta _X)\supset A\). \(\square \)
4 Proof of Theorem 1.6
Let \(G\) be a topological group and \(\mathcal{E }\) be its left coarse structure. The inclusion \(\mathcal{D }_{<}(G)\subset \mathcal{S }(G)\) will follow as soon as we prove that each nonsmall subset \(A\subset G\) has asymptotic dimension \(\mathrm{asdim}(A)=\mathrm{asdim}(G)\). We divide the proof of this fact into 3 steps.
Claim 4.1
There is an entourage \({\varepsilon }_A\in \mathcal{E }\) such that the set \(G{\setminus } B(A,{\varepsilon }_A)\) is not large in \(G\).
Proof
Since \(A\) is not small, there is a large set \(L\subset X\) such that the complement \(L{\setminus } A\) is not large. Since \(L\) is large in \(X\), there is an entourage \({\varepsilon }_A\in \mathcal{E }\) such that \(B(L,{\varepsilon }_A)=G\). We claim that the set \(G{\setminus } B(A,{\varepsilon }_A)\) is not large. Assuming the opposite, we can find an entourage \(\delta \in \mathcal{E }\) such that \(B(G{\setminus } B(A,{\varepsilon }_A),\delta )=G\). Then for each \(x\in G\) the ball \(B(x,\delta )\) meets \(G\setminus B(A,{\varepsilon }_A)\) at some point \(y\). By the choice of \({\varepsilon }_A\), the ball \(B(y,{\varepsilon }_A)\) meets the large set \(L\) at some point \(z\). It follows from \(y\notin B(A,{\varepsilon }_A)\) that \(z\in L\cap B(y,{\varepsilon })\subset L\cap (X{\setminus } A)=L{\setminus } A\) and hence \(x\in B(L{\setminus } A,{\varepsilon }_A\circ \delta )\), which means that \(L\setminus A\) is large in \(X\). This is a required contradiction. \(\square \)
Claim 4.2
\(\mathrm{asdim}(B(A,{\varepsilon }_A))=\mathrm{asdim}(A)\).
Proof
Observe that the identity embedding \(i:A\rightarrow B(A,{\varepsilon }_A)\) is a coarse equivalence. The coarse inverse \(j:B(A,{\varepsilon }_A)\rightarrow A\) to \(i\) can be defined by choosing a point \(j(x)\in B(x,{\varepsilon }_A)\cap A\) for each \(x\in B(A,{\varepsilon }_A)\). Now we equality \(\mathrm{asdim}(B(A,{\varepsilon }_A))=\mathrm{asdim}(A)\) follows from the invariance of the asymptotic dimension under coarse equivalences, see Proposition 1.4. \(\square \)
Claim 4.3
\(\mathrm{asdim}(G)=\mathrm{asdim}(A)\).
Proof
The inequality \(\mathrm{asdim}(A)\le \mathrm{asdim}(G)\) is trivial. So, it suffices to check that \(\mathrm{asdim}(G)\le n\) where \(n=\mathrm{asdim}(A)=\mathrm{asdim}(B(A,{\varepsilon }_A))\). If \(n\) is infinite, then there is nothing to prove. So, we assume that \(n\in {\omega }\).
For the proof of the inequality \(\mathrm{asdim}(G)\le n\), we shall apply Theorem 2.6. Given any \({\varepsilon }\in \mathcal{E }\) we should find \(\delta \in \mathcal{E }\) such that for each finite \({\varepsilon }\)connected subset \(F\subset G\) there is a coloring \(\chi :F\rightarrow n+1\) such that each \(\chi \)monochrome \({\varepsilon }\)chain \(C\subset F\) has \(\mathrm{diam}(C)\subset \delta \). By the definition of the left coarse structure \(\mathcal{E }\) we lose no generality assuming that \({\varepsilon }=\{(x,y)\in G\times G:x\in yK_{\varepsilon }\}\) for some compact subset \(K_{\varepsilon }=K_{\varepsilon }^{1}\subset G\) containing the neutral element \(1_G\) of \(G\). In this case the entourage \({\varepsilon }\) is left invariant in the sense that for each pair \((x,y)\in {\varepsilon }\) and each \(z\in G\) the pair \((zx,zy)\) belongs to \({\varepsilon }\).
Since \(\mathrm{asdim}_{col}(B(A,{\varepsilon }_A))=\mathrm{asdim}(B(A,{\varepsilon }_A))\le n\), for the entourage \({\varepsilon }\in \mathcal{E }\), there are an entourage \(\delta \in \mathcal{E }\) and a coloring \(\chi _A:B(A,{\varepsilon }_A)\rightarrow n+1\) such that each \(\chi \)monochrome \({\varepsilon }\)chain \(C\subset B(A,{\varepsilon }_A)\) has \(\mathrm{diam}(C)\subset \delta \), see Proposition 2.2. By the definition of the left coarse structure \(\mathcal{E }\), we lose no generality assuming that \(\delta =\{(x,y)\in G\times G:x\in yK_\delta \}\) for some compact set \(K_\delta =K_\delta ^{1}\ni 1_G\) of \(G\), which implies that the entourage \(\delta \) is left invariant.
Now take any finite \({\varepsilon }\)connected subset \(F\subset G\). Replacing \(F\) by \(F\cup F^{1}\cup \{1_G\}\) we can assume that \(F=F^{1}\ni 1_G\). Since the set \(G\setminus B(A,{\varepsilon }_A)\) is not large, there is a point \(z\notin (G\setminus B(A,{\varepsilon }_A))F\). Then \(zF^{1}\) is disjoint with \(G\setminus B(A,{\varepsilon }_A)\) and hence \(zF=zF^{1}\subset B(A,{\varepsilon }_A)\). So, it is legal to define a coloring \(\chi :F\rightarrow n+1\) by the formula \(\chi (x)=\chi _A(zx)\) for \(x\in F\). Taking into account the left invariance of the entourages \({\varepsilon }\) and \(\delta \), it is easy to see that each \(\chi \)monochrome \({\varepsilon }\)chain \(C\subset F\) has diameter \(\mathrm{diam}(C)\subset \delta \). By Propositions 2.2 and 2.3, \(\mathrm{asdim}(G)=\mathrm{asdim}_{col}(G)\le n=\mathrm{asdim}(A)\). \(\square \)
5 Proof of Theorem 1.2
We need to prove that a subset \(A\subset \mathbb{R }^n\) is small if and only if it has asymptotic dimension \(\mathrm{asdim}(A)<\mathrm{asdim}(\mathbb{R }^n)=n\). The “if” part of this characterization follows from the inclusion \(\mathcal{D }_{<}(\mathbb{R }^n)\subset \mathcal{S }(\mathbb{R }^n)\) proved in Theorem 1.6. To prove the “only if” part, we need to recall some (standard) notions of Combinatorial Topology [10, 12].
On the Euclidean space \(\mathbb{R }^n\) we shall consider the metric generated by the supnorm \(\Vert x\Vert =\max _{i\in n}x(i)\).
Claim 5.1
\(\bigcap _{i=0}^n B(St^{\prime }_\Delta (v_i),{\varepsilon })\subset B(b_\Delta ,n{\varepsilon })\) for each positive real number \({\varepsilon }\).
Proof
In fact, \(n\)dimensional simplexes can be alternatively defined as images of the standard \(n\)dimensional simplex \(\Delta \) under injective affine maps \(f:\Delta \rightarrow \mathbb{R }^n\).
A map \(f:\Delta \rightarrow \mathbb{R }^n\) is called affine if \(f(tx+(1t)y)=tf(x)+(1t)f(y)\) for any points \(x,y\in \Delta \) and a real number \(t\in [0,1]\). It is wellknown that each affine function \(f:\Delta \rightarrow \mathbb{R }^n\) is uniquely defined by its restriction \(f\Delta ^{(0)}\) to the set \(\Delta ^{(0)}=\{v_i\}_{i\le n}\) of vertices of \(\Delta \).
A map \(f:\Delta \rightarrow \mathbb{R }^n\) will be called \(b_\Delta \) affine if for every \(i\le n\) the restriction \(f\mathrm{conv}(\{b_\Delta \}{\cup }\Delta ^{(0)}{\setminus }\{v_i\})\) is affine. A \(b_\Delta \)affine function \(f:\Delta \rightarrow \mathbb{R }^n\) is uniquely determined by its restriction \(f\Delta ^{(0)}\cup \{b_\Delta \}\).
Claim 5.2
For any \(n\)dimensional simplex \(\sigma \) in \(\mathbb{R }^n\) there is a real constant \(L\) such that each \(b_\Delta \)affine function \(f:\Delta \rightarrow \sigma \) with \(f(\Delta ^{(0)})=\sigma ^{(0)}\) and \(f(b)\in \frac{1}{2}b_\sigma +\frac{1}{2}\sigma \) is bijective, biLipschitz and has \(\mathrm{Lip}(f)\cdot \mathrm{Lip}(f^{1})\le L\).
This claim can be easily derived from the fact that each \(b_\Delta \)affine function \(f:\Delta \rightarrow \sigma \) with \(f(\Delta ^{(0)})=\sigma ^{(0)}\) is Lipschitz and its Lipschitz constant \(\mathrm{Lip}(f)\) depends continuously on \(f(b_\Delta )\).
Claim 5.3
For any \(n\)dimensional simplex \(\sigma \) in \(\mathbb{R }^n\) there is a real constant \(L\) such that for each point \(b^{\prime }\in \frac{1}{2}b_\sigma +\frac{1}{2}\sigma \) and each \({\varepsilon }>0\) we get \(\sigma \cap \bigcap _{v\in \sigma ^{(0)}}B(\mathcal{S }t^{\prime }_{\sigma ,b^{\prime }}(v),{\varepsilon })\subset B(b^{\prime },L{\varepsilon })\).
Proof
Now consider the binary unit cube \(K=\{0,1\}^n\subset \mathbb{R }^n\) endowed with the partial ordering \(\le \) defined by \(x\le y\) iff \(x(i)\le y(i)\) for all \(i<n\). Given two vectors \(x,y\in \{0,1\}^n\), we write \(x<y\) if \(x\le y\) and \(x\ne y\).
Now we are able to prove the “only if” part of Theorem 1.2. Assume that a subset \(A\subset \mathbb{R }^n\) is small. Then there is a function \(\varphi :(0,\infty )\rightarrow (0,\infty )\) such that for each \(\delta \in (0,\infty )\) and a point \(x\in \mathbb{R }^n\) there is a point \(y\in \mathbb{R }^n\) with \(B(y,\delta )\subset B(x,\varphi (\delta ))\setminus A\). The inequality \(\mathrm{asdim}(A)<n\) will follow as soon as given any \(\delta <\infty \) we construct a cover \(\mathcal{U }\) of \(A\) with finite \(\mathrm{mesh}(\mathcal{U })=\sup _{U\in \mathcal{U }}\mathrm{diam}(U)\) such that each \(\delta \)ball \(B(a,\delta )\), \(a\in A\), meets at most \(n\) elements of the cover \(\mathcal{U }\).
By Claim 5.3, there is a constant \(L\) such that for each simplex \(\sigma \in \mathcal{T }\), each point \(b^{\prime }\in \frac{1}{2} b_\sigma +\frac{1}{2}\sigma \) and each \({\varepsilon }>0\) we get \(\sigma \cap \bigcap _{v\in \sigma ^{(0)}}B(\mathcal{S }t^{\prime }_{\sigma ,b^{\prime }}(v),{\varepsilon })\subset B(b^{\prime },L{\varepsilon })\).
 (1)
\(B\big (b_\sigma ,{\varepsilon }\varphi (L\delta )\big )\subset \frac{1}{2}b_\sigma +\frac{1}{2}\sigma \);
 (2)
for any \(b^{\prime }\in \frac{1}{2}b_\sigma +\frac{1}{2}\sigma \) and any vertex \(v\in \sigma ^{(0)}\) the \(2{\varepsilon }\delta \)neighborhood \(B(\mathcal{S }t^{\prime }_{\sigma ,b^{\prime }}(v),2{\varepsilon }\delta )\) lies in the \(\mathcal{T }\)star \(\mathcal{S }t_\mathcal{T }(v)\) of \(v\).

\((1_{\varepsilon })\) \(B\big (b_\sigma ,\varphi (L\delta )\big )\subset \frac{1}{2}b_\sigma +\frac{1}{2}\sigma \);

\((2_{\varepsilon })\) for any \(b^{\prime }\in \frac{1}{2}b_\sigma +\frac{1}{2}\sigma \) and any vertex \(v\in \sigma ^{(0)}\) the \(2\delta \)neighborhood \(B(\mathcal{S }t^{\prime }_{\sigma ,b^{\prime }}(v),2\delta )\) lies in the \(\widetilde{\mathcal{T }}\)star \(\mathcal{S }t_{\widetilde{\mathcal{T }}}(v)\) of \(v\).
Assume conversely that there are a point \(a\in A\) and a set \(V\subset {\varepsilon }^{1}\mathbb{Z }^n\) of cardinality \(V=n+1\) such that \(B(a,\delta )\cap \mathcal{S }t^{\prime }(v)\ne \emptyset \) for each \(v\in V\). Then \(a\in \bigcap _{v\in V}B(\mathcal{S }t^{\prime }(v),\delta )\). It follows from \(a\in \bigcap _{v\in V}B(\mathcal{S }t^{\prime }(v),\delta )\subset \bigcap _{v\in V}\mathcal{S }t_{\widetilde{\mathcal{T }}}(v)\) that \(V\) coincides with the set \(\sigma ^{(0)}\) of vertices of some simplex \(\sigma \in \widetilde{\mathcal{T }}\) and \(a\) lies in the interior of the simplex \(\sigma \).
Next, we show that \(a\in B(\mathcal{S }t^{\prime }_{\sigma ,b_\sigma ^{\prime }}(v),\delta )\) for each \(v\in V\). In the opposite case, \(a\in B(\mathcal{S }t^{\prime }_{\tau ,b_\tau ^{\prime }}(v),\delta )\subset B(\tau ,\delta )\) for some simplex \(\tau \in \widetilde{\mathcal{T }}\setminus \{\sigma \}\) such that \(v\in \tau ^{(0)}\setminus \sigma ^{(0)}\). Choose a vertex \(u\in \sigma ^{(0)}\setminus \tau ^{(0)}\) and observe that the condition \((2_{\varepsilon })\) implies that \(a\in B(\mathcal{S }t^{\prime }(u),\delta )\cap B(\tau ,\delta )=\emptyset \), which is a contradiction.
6 Proof of Theorem 1.7
 (1)
\(\mathcal{S }(G)=\mathcal{D }_<(G)\);
 (2)
\(G\) is compactly generated;
 (3)
\(G\) is coarsely equivalent to a Euclidean space \(\mathbb{R }^n\) for some \(n\in {\omega }\).
To prove that \((2){\Rightarrow }(3)\), assume that the group \(G\) is compactly generated. By Theorem 24 [11, p.85], \(G\) is topologically isomorphic to the direct sum \(\mathbb{R }^n\times \mathbb{Z }^m\times K\) for some \(n,m\in {\omega }\) and a compact subgroup \(K\subset G\). Since the projection \(\mathbb{R }^n\times \mathbb{Z }^m\times K\rightarrow \mathbb{R }^n\times \mathbb{Z }^m\) and the embedding \(\mathbb{Z }^n\times \mathbb{Z }^m\rightarrow \mathbb{R }^n\times \mathbb{Z }^m\) are coarse equivalences, we conclude that \(G\) is coarsely equivalent to \(\mathbb{Z }^{n+m}\) and to \(\mathbb{R }^{n+m}\).
To prove that \((1){\Rightarrow }(2)\), assume that \(\mathcal{S }(G)=\mathcal{D }_{<}(G)\). First we prove that \(G\) has finite asymptotic dimension. By the Principal Structure Theorem 25 [11, p.26], \(G\) contains an open subgroup \(G_0\) that is topologically isomorphic to \(\mathbb{R }^n\times K\) for some \(n\in {\omega }\) and some compact subgroup \(K\) of \(G_0\). The subgroup \(G_0\) has asymptotic dimension \(\mathrm{asdim}(G_0)=\mathrm{asdim}(\mathbb{R }^n)=n<\infty \). If \(\mathrm{asdim}(G)=\infty \), then the quotient group \(G/G_0\) has infinite asymptotic dimension and hence has infinite free rank. Then the group \(G/G_0\) contains a subgroup isomorphic to the free abelian group \(\oplus ^{\omega }\mathbb{Z }\) with countably many generators. It follows that \(G\) also contains a discrete subgroup \(H\) isomorphic to \(\oplus ^{\omega }\mathbb{Z }\). Replacing \(H\) by a smaller subgroup, if necessary, we can assume that \(H\) has infinite index in \(G\) and hence is small in \(G\). Since \(\mathrm{asdim}(H)=\infty =\mathrm{asdim}(G)\), we conclude that \(\mathcal{S }(G)\ne \mathcal{D }_<(G)\), which is a desired contradiction showing that \(\mathrm{asdim}(G)<\infty \).
By Theorem 2.8, there is a compactly generated subgroup \(H\subset G\) with \(\mathrm{asdim}(H)=\mathrm{asdim}(G)\). Since \(H\notin \mathcal{D }_{<}(G)=\mathcal{S }(G)\), the subset \(H\) is not small in \(G\). Repeating the proof of Claim 4.1, we can show that the set \(G\setminus B(H,{\varepsilon })\) is not large for some entourage \({\varepsilon }\in \mathcal{E }\). By the definition of the left coarse structure \(\mathcal{E }\), there is a compact subset \(K\subset G\) such that \(B(H,{\varepsilon })\subset HK\). We claim that \(K^{1}HK=G\). Assuming the opposite, we can find a point \(x\in G\setminus K^{1}HK\) and consider the finite set \(F=\{x,x^{1},xx^{1}\}=F^{1}\). Since the set \(G\setminus HK\) is not large, there is a point \(z\in (G\setminus HK)F\). For this point \(z\) we get \(zF\cap (G\setminus HK)=\emptyset \) and hence \(z\in zF\subset HK\). Then \(x\in z^{1}zF\subset z^{1}HK\subset K^{1}HHK=K^{1}HK\), which is a contradiction. Now the compact generacy of the subgroup \(H\) implies the compact generacy of the group \(G=K^{1}HK\).
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