Asymptotic dimension and small subsets in locally compact topological groups

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Introduction
In this paper we study the interplay between the ideal S(X ) of small subsets of a coarse space X and the ideal D < (X ) of subsets of asymptotic dimension less than asdim(X ) in X . We show that these two ideals coincide in spaces that are coarsely equivalent to R n , in particular, they coincide in each compactly generated locally compact abelian group.
Let us recall that a coarse space is a pair (X, E) consisting of a set X and a coarse structure E on X , which is a family of subsets of X × X (called entourages) satisfying the following axioms: (A) each ε ∈ E contains the diagonal X = {(x, y) ∈ X 2 : x = y} and is symmetric in the sense that ε = ε −1 where ε −1 = {(y, x) : (x, y) ∈ ε}; (B) for any entourages ε, δ ∈ E there is an entourage η ∈ E that contains the composition δ • ε = {(x, z) ∈ X 2 : ∃y ∈ X with (x, y) ∈ ε and (y, z) ∈ δ}; (C) a subset δ ⊂ X 2 belongs to E if X ⊂ δ = δ −1 ⊂ ε for some ε ∈ E.
A family B of subsets of X 2 is a base of a (unique) coarse structure if and only if it satisfies the axioms (A), (B).
Each subset A of a coarse space (X, E) carries the induced coarse structure E A = {ε ∩ A 2 : ε ∈ E}. Endowed with this structure, the space (A, E A ) is called a subspace of (X, E).
For an entourage ε ⊂ X 2 , a point x ∈ X , and a subset A ⊂ X let B(x, ε) = {y ∈ X : (x, y) ∈ ε} be the ε-ball centered at x, B(A, ε) = a∈A B(a, ε) be the ε-neighborhood of A in X , and diam(A) = A × A be the diameter of A. For a family U of subsets of X we put mesh(U) = U ∈U 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) ∈ X 2 : d(x, y) < ε} where 0 ≤ ε < ∞. 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) ∈ G 2 : x ∈ y K } 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 left-invariant continuous metric d on G which is proper in the sense that each closed ball B(e, R) = {x ∈ G : d(x, e) ≤ R} is compact. In particular, the coarse structure on R n , generated by the Euclidean metric coincides with the left coarse structure of the Abelian topological group R n . Now we recall the definitions of large and small sets in coarse spaces. Such sets were introduced in [4] and studied in [13, §11] and [2]. A subset A of a coarse space (X, E) is called • large if B(A, ε) = X for some ε ∈ E; • small if for each large set L ⊂ X the set L \ A remains large in X .
It follows that the family S(X ) of small subsets of a coarse space (X, E) is an ideal. A subfamily I ⊂ P(X ) of the power-set of a set X is called an ideal if I is additive (in the sense that A ∪ B ∈ I for all A, B ∈ I) and downwards closed (which means that A ∩ B ∈ I for all A ∈ I and B ⊂ X ).
Small sets can be considered as coarse counterparts of nowhere dense subsets in topological spaces, see [2]. It is well-known [8, 7.4.18] that the ideal of nowhere dense subsets in a Euclidean space R n coincides with the ideal generated by closed subsets of topological dimension < n in 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 asdim(X ) of a coarse space X .

Definition 1.1
The asymptotic dimension asdim(X ) of a coarse space (X, E) is the smallest number n ∈ ω such that for each entourage ε ∈ E there is a cover U of X such that mesh(U) ⊂ δ for some δ ∈ E and each ε-ball B(x, ε), x ∈ X , meets at most n + 1 sets U ∈ U. If such a number n ∈ ω does not exist, then we put asdim(X ) = ∞.
In Theorem 2.7 we shall prove that asdim(A ∪ B) ≤ max{asdim(A), asdim(B)} for any subspaces A, B of a coarse space X . This implies that for every number n ∈ ω ∪ {∞} the family {A ⊂ X : asdim(A) < n} is an ideal in P(X ). In particular, the family is an ideal in P(X ). According to [5, 9.8.4], asdim(R n ) = n for every n ∈ ω.
The main result of this paper is: 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: It should be mentioned that the class of coarse spaces X with S( Two coarse spaces X, Y are called coarsely equivalent if there is a coarse equivalence f : X → Y .

Proposition 1.4
Assume that coarse spaces X, Y are coarsely equivalent. Then This proposition will be proved in Sect. 3. Combined with Theorem 1.2 it implies: 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 D < (X ) ⊂ S(X ).
We recall that a topological group G is compactly generated if G is algebraically generated by some compact subset K ⊂ G. Theorem 1.7 For an Abelian locally compact topological group X the following conditions are equivalent: (2) X is compactly generated; (3) X is coarsely equivalent to the Euclidean space R n for some n ∈ ω.
This theorem will be proved in Sect. 6. Remark 1. 8 The equivalence (1)⇔(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 ⊂ F 2 has infinite index in F 2 and hence is small, yet asdim(Z ) = asdim(F 2 ) = 1.
A less trivial example is the wreath product A Z of a non-trivial finite abelian group A and Z. The group A Z has asymptotic dimension 1 (see [9]) and the subgroup Z is small in A Z and has asdim(Z) = 1 = asdim(A Z). Let us recall that the group A Z consists of ordered pairs ((a i ) i∈Z , n) ∈ ⊕ Z A) × Z and the group operation on A Z is defined by The group A Z is finitely-generated and meta-abelian but is not finitely presented, see [3]. Groups which are coarsely equivalent to abelian groups were studied in [1]. Problem 1.9 Is S(X ) = D < (X ) for each connected Lie group X ? For each discrete polycyclic group X ?

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, E) be a coarse space, ε ∈ E and A ⊂ X . We shall say that A has diameter less than In this case the finite set C = {x 0 , . . . , x m } also will be called an ε-chain. A set C ⊂ X is called ε-connected if any two points x, y ∈ C can be linked by an ε-chain x = x 0 , . . . , x m = y. The maximal ε-connected subset C(x, ε) ⊂ X containing a given point x ∈ X is called the ε-connected component of x. It consists of all points y ∈ X that can be linked with x by an ε-chain We shall study the interplay between the asymptotic dimension introduced in Definition 1.1 and the following modification: The colored asymptotic dimension asdim col (X ) of a coarse space (X, E) is the smallest number n ∈ ω such that for every entourage ε ∈ E there is a cover U of X such that mesh(U) ⊂ δ for some δ ∈ E and U can be written as the union U = i∈n+1 U i of n + 1 many ε-disjoint subfamilies U i . If such a number n ∈ ω does not exist, then we put asdim col (X ) = ∞.
Without lost of generality we can assume that the cover U = i∈n+1 U i in the above definition consists of pairwise disjoint sets. In this case we can consider the coloring χ : X → n + 1 = {0, . . . , n} such that χ −1 (i) = U i for every i ∈ n + 1. For this coloring every χ-monochrome ε-connected subset C ⊂ X lies in some U ∈ U and hence has diameter is a singleton. Thus we arrive to the following useful characterization of the colored asymptotic dimension. E) has asdim col (X ) ≤ n for some number n ∈ ω if and only if for any ε ∈ E there are a coloring χ : X → n + 1 and an entourage δ ∈ E such that each χ-monochrome ε-chain C ⊂ X has diam(C) ⊂ δ.
Proof The "only if" part follows from the above discussion. To prove the "if" part, for every ε ∈ E we need to construct a cover U = i∈n+1 U i such that mesh(U) ∈ E and each family U i is ε-disjoint. By our assumption, there is a coloring χ : X → n + 1 and an entourage δ ∈ E such that each χ-monochrome ε-chain C ⊂ X has diam(C) ⊂ δ.
For each x ∈ X let C χ (x, ε) be the set of all points y ∈ X that can be linked with x by a χ-monochrome ε-chain 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].
Proof To prove that asdim(X ) ≤ asdim col (X ), put n = asdim col (X ) and take any entourage x ∈ X , meets at most one set of each family U i . Assuming that B(x, ε) meets two distinct sets U, V ∈ U i , we can find points u ∈ U and v ∈ V with (x, u), (x, v) ∈ ε and conclude that (u, v) ∈ ε • ε, which is not possible as U i is ε • ε-disjoint. Now we see that the ball B(x, ε) meets at most n + 1 element of the cover U and hence asdim(X ) ≤ n.
The proof of the inequality asdim col (X ) ≤ asdim(X ) is a bit longer. If the dimension n = asdim(X ) is infinite, then there is nothing to prove. So, we assume that n ∈ ω. To prove that asdim col (X ) ≤ n, fix an entourage ε ∈ E. Let ε 0 = X and ε k+1 = ε k • ε for k ∈ ω. Since asdim(X ) ≤ n, for the entourage ε n+1 ∈ E we can find a cover U of X such that δ = mesh(U) ∈ E and each ε n+1 -ball B(x, ε n+1 ) meets at most n + 1 many sets U ∈ U. For every i ≤ n +1 and x ∈ X consider the subfamily In such a way we have defined a coloring χ : To finish the proof it suffices to show that any χ-monochrome ε-chain Propositions 2.2 and 2.3 imply: E) has asymptotic dimension asdim(X ) ≤ n for some n ∈ ω if and only if for any ε ∈ E there are δ ∈ E and a coloring χ : X → n + 1 such that any χ-monochrome ε-chain C ⊂ X has diam(C) ⊂ δ.
This corollary can be generalized as follows (cf. [6]). E) has asdim(X ) ≤ n for some n ∈ ω if and only if for any entourage ε ∈ E there is an entourage δ ∈ E such that for any finite set F ⊂ X there is a coloring χ : Proof This proposition will follow from Corollary 2.4 as soon as for any ε ∈ E we find δ ∈ E and a coloring χ : X → n + 1 such that each χ-monochrome ε-chain in X has diameter less that δ.
By our assumption, there is an entourage δ ∈ E such that for every finite subset F ⊂ X there is a coloring χ F : F → n +1 such that each χ F -monochrome ε-chain in F has diameter less that δ. Extend χ F to a coloringχ F : X → n + 1.
Let F denote the family of all finite subsets of X , partially ordered by the inclusion relation ⊂. The coloringsχ F , F ∈ F , can be considered as elements of the compact Hausdorff space K = {0, . . . , n} X endowed with the Tychonoff product topology. The compactness of K implies that the net {χ F } F∈F has a cluster point χ ∈ K , see [8, 3.1.23]. The latter means that for each finite set We claim that the coloring χ : X → n + 1 has the required property: The choice of the coloring χ F =χ F |F guarantees that the set C ⊂ F has diam(C) ⊂ δ.
Proposition 2.5 admits the following self-generalization. Theorem 2.6 A coarse space (X, E) has asdim(X ) ≤ n for some n ∈ ω if and only if for any entourage ε ∈ E there is an entourage δ ∈ E such that for any finite ε-connected subset F ⊂ X there is a coloring χ : 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 For any subspaces A, B of a coarse space
Proof Only the case of finite n = max{asdim(A), asdim(B)} requires the proof. Without loss of generality the sets A and B are disjoint. To show that asdim(A ∪ B) ≤ n we shall apply Corollary 2.4. Fix any entourage ε ∈ E. Since asdim(A) ≤ n there are an entourage δ A ∈ E and a coloring χ A : A → n + 1 such that each χ-monochrome ε-chain in A has diameter less that δ A . Since asdim(B) ≤ n, for the entourage ε B = ε • δ A • ε there are an entourage δ B ∈ E and a coloring χ B : A → {0, . . . , n} such that each χ-monochrome ε B -chain in B has diameter less that δ B .
The union of the colorings χ A and χ B yields the coloring χ : A∪ B → {0, . . . , n} such that χ|A = χ A and χ|B = χ B . We claim that each χ-monochrome ε-chain C = {x 0 , . . . , Without loss of generality, the points x 0 , . . . , x m of the chain C are pairwise distinct.
If C ⊂ A, then C, being a χ A -monochrome ε-chain in A has diam(C) ⊂ δ A ⊂ δ and we are done. So, we assume that C ⊂ A. In this case b = |C ∩ B| ≥ 1 and we can choose a strictly increasing sequence 0 ≤ k 1 Then {x 0 , . . . , x k 1 −1 }, being a χ A -monochrome ε-chain in A, has diameter less that δ A . Consequently, the ε-chain {x 0 , . . . , x k 1 } has diameter less that δ A • ε ⊂ ε B . By the same reason the ε-chain {x k b , . . . , x m } has diameter less that ε • δ A ⊂ ε B and for every 1 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. Proof Let n = sup {asdim(H ) : H is a compactly generated subgroup ofG}. It is clear that n ≤ asdim(G). The reverse inequality asdim(G) ≤ n is trivial if n = ∞. So, we assume that n < ∞. To prove that asdim(G) ≤ n, we shall apply Theorem 2.6. Let E be the left coarse structure of the topological group G. Given any entourage ε ∈ E, we should find an entourage δ ∈ E such that for each finite ε-connected subset F ⊂ G there is a coloring χ : By the definition of the coarse structure E, for the entourage ε ∈ E there is a compact Let H be the subgroup of G generated by the compact set K ε , E H be the left coarse structure of H , and ε H = {(x, y) ∈ H 2 : x ∈ y K ε } ∈ E H . Since asdim col (H ) = asdim(H ) ≤ n, by Proposition 2.2, there is a coloring χ H : H → n + 1 and an entourage δ H ∈ E H such that each χ H -monochrome ε-chain C ⊂ H has diameter diam(C) ⊂ δ H . By the definition of the coarse structure E H , there is a compact subset K δ = K −1 We claim that the entourage δ = {(x, y) ∈ G × G : x ∈ y K δ } satisfies our requirements. Let F be a finite ε-connected subset of G. Then for each point x 0 ∈ F we get F ∈ x 0 H and hence x −1 0 F ⊂ H . So, we can define a coloring χ : F → n + 1 letting The latter means that for any points c, c ∈ C we get and c ∈ c K δ , which means that (c, c ) ∈ δ and hence diam(C) ⊂ δ.

Proof of Proposition 1.4
Let f : X → Y be a coarse equivalence between two coarse spaces (X, E X ) and (Y, E Y ). Then there is a coarse map g : 1. First we prove that asdim(X ) = 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 asdim(X ) ≤ asdim(Y ). This inequality is trivial if n = asdim(Y ) is infinite. So, assume that n < ∞. By Propositions 2.2 and 2.3, the inequality asdim(X ) ≤ n will be proved as soon as for each ε X ∈ E X we find δ X ∈ E X and a coloring χ X : X → n + 1 such that each χ X -monochrome ε X -chain C ⊂ X has diameter diam(C) ⊂ δ X .
Since the map f : X → Y is coarse, for the entourage ε X there is an entourage Since the function g : Y → X is coarse, for the entourage δ Y there is an entourage δ X such that {(g(y), g(y )) : (y, y ) ∈ δ Y } ⊂ δ Y . Put δ X = η X • δ X • η Y and consider the coloring Then the set C X = g(C Y ) has diameter diam(C X ) ⊂ δ X . Now take any two points c, c ∈ C X and observe that the pairs (c, g • f (c)) and (c , g • f (c )) belong to the entourage η X . Consequently, which means that the ε X -chain C X has diameter diam(C X ) ⊂ δ X . So, asdim(X ) ≤ n.

Claim 3.1 A subset A ⊂ X and its image f (A) ⊂ Y have the same asymptotic dimension asdim(A) = asdim( f (A)).
Proof This claim follows from Proposition 1.4(1) proved above, since A and f (A) are coarsely equivalent.

Claim 3.2 A subset A ⊂ X is large in X if and only if its image f (A) is large in Y .
Proof

Claim 3.3 A subset A ⊂ X is small if and only if for each entourage
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, ε X ) is small in X , it is necessary to check that for each large subset L ⊂ X the complement L \ B(A, ε X ) is large in X . Consider the set L = (L \ B(A, ε X ))∪ A and observe that L ⊂ B(L , ε X ) and hence L is large in X . Since A is small, the set L \ A = L \ B(A, ε X ) is large in X .

Claim 3.4 A subset A ⊂ 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 ⊂ Y the complement L \ 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, η X ) is small in X and hence the complement g(L) \ B(A, η X ) remains large in X . By Claim 3. B(A, η X )), find a point x ∈ g(L) \ B(A, η X ) such that y = f (x) and a point z ∈ L such that x = g(z). We claim that z / ∈ f (A). Assuming conversely that z ∈ f (A), we Taking into account that the set Now assume that the set f (X ) is small in Y . Then the set g • f (A) is small in X and so are the sets B (g • f (A), η X ) ⊃ A.

Proof of Theorem 1.6
Let G be a topological group and E be its left coarse structure. The inclusion D < (G) ⊂ S(G) will follow as soon as we prove that each non-small subset A ⊂ G has asymptotic dimension asdim(A) = asdim(G). We divide the proof of this fact into 3 steps.

Claim 4.1 There is an entourage ε A ∈ E such that the set G\B(A, ε A ) is not large in G.
Proof Since A is not small, there is a large set L ⊂ X such that the complement L\A is not large. Since L is large in X , there is an entourage ε A ∈ E such that B(L , ε A ) = G. We claim that the set G\B (A, ε A ) is not large. Assuming the opposite, we can find an entourage δ ∈ E such that B (G\B(A, ε A ) , which means that L \ A is large in X . This is a required contradiction.
For the proof of the inequality asdim(G) ≤ n, we shall apply Theorem 2.6. Given any ε ∈ E we should find δ ∈ E such that for each finite ε-connected subset F ⊂ G there is a coloring χ : F → n +1 such that each χ-monochrome ε-chain C ⊂ F has diam(C) ⊂ δ. By the definition of the left coarse structure E we lose no generality assuming that ε = {(x, y) ∈ G × G : x ∈ y K ε } for some compact subset K ε = K −1 ε ⊂ G containing the neutral element 1 G of G. In this case the entourage ε is left invariant in the sense that for each pair (x, y) ∈ ε and each z ∈ G the pair (zx, zy) belongs to ε.
Since asdim col (B(A, ε A )) = asdim(B(A, ε A )) ≤ n, for the entourage ε ∈ E, there are an entourage δ ∈ E and a coloring χ A : B(A, ε A ) → n + 1 such that each χ-monochrome ε-chain C ⊂ B(A, ε A ) has diam(C) ⊂ δ, see Proposition 2.2. By the definition of the left coarse structure E, we lose no generality assuming that δ = {(x, y) ∈ G × G : x ∈ y K δ } for some compact set K δ = K −1 δ 1 G of G, which implies that the entourage δ is left invariant. Now take any finite ε-connected subset F ⊂ G. B(A, ε A ). So, it is legal to define a coloring χ : F → n + 1 by the formula χ(x) = χ A (zx) for x ∈ F. Taking into account the left invariance of the entourages ε and δ, it is easy to see that each χ-monochrome ε-chain C ⊂ F has diameter diam(C) ⊂ δ. By Propositions 2.2 and 2.3, asdim(G) = asdim col (G) ≤ n = asdim(A).

Proof of Theorem 1.2
We need to prove that a subset A ⊂ R n is small if and only if it has asymptotic dimension asdim(A) < asdim(R n ) = n. The "if" part of this characterization follows from the inclusion D < (R n ) ⊂ S(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 R n we shall consider the metric generated by the sup-norm x = max i∈n |x(i)|.
By the standard n-dimensional simplex we understand the compact convex subset  nε) for each positive real number ε. Proof Given any vector x ∈ n i=0 B (St (v i ), ε), for every i ≤ n we can find a vector

It is clear that
On the other hand, Now we are going to generalize Claim 5.1 to arbitrary simplexes. By an n-dimensional simplex in R n we understand the convex hull σ = conv(σ (0) ) of an affinely independent subset σ (0) ⊂ R n of cardinality |σ (0) | = n + 1. Each point v ∈ σ (0) is called a vertex of the simplex σ . The arithmetic mean of the vertices is called the barycenter of the simplex σ . By ∂σ we denote the boundary of the simplex σ in R n . Observe that the homothetic copy 1 2 In fact, n-dimensional simplexes can be alternatively defined as images of the standard n-dimensional simplex under injective affine maps f : → R n .
A map f : for any points x, y ∈ and a real number t ∈ [0, 1]. It is well-known that each affine function f : → R n is uniquely defined by its restriction f | (0) to the set (0) = {v i } i≤n of vertices of .
A map f : Given an n-dimensional simplex σ ⊂ R n and a point b ∈ σ \ ∂σ in its interior, fix a b -affine function f : → σ such that f ( (0) ) = σ (0) and It is easy to see that the set St σ,b (v) does not depend on the choice of the b -affine function f .
Now consider the binary unit cube K = {0, 1} n ⊂ R n endowed with the partial ordering ≤ defined by x ≤ y iff x(i) ≤ y(i) for all i < n. Given two vectors x, y ∈ {0, 1} n , we write x < y if x ≤ y and x = y.
For every increasing chain v 0 < v 1 < . . . < v n of points of the binary cube K = {0, 1} n , consider the simplex conv{v 0 , . . . , v n } and let T K be the (finite) set of these simplexes. Next, consider the family T = {σ + z : σ ∈ T K , z ∈ Z n } of translations of the simplexes from the family T K , and observe that T = R n . For each point v ∈ Z n let be the T -star of v in the triangulation T of the space R n . Now we are able to prove the "only if" part of Theorem 1.2. Assume that a subset A ⊂ R n is small. Then there is a function ϕ : (0, ∞) → (0, ∞) such that for each δ ∈ (0, ∞) and a point x ∈ R n there is a point y ∈ R n with B(y, δ) ⊂ B(x, ϕ(δ)) \ A. The inequality asdim(A) < n will follow as soon as given any δ < ∞ we construct a cover U of A with finite mesh(U) = sup U ∈U diam(U ) such that each δ-ball B(a, δ), a ∈ A, meets at most n elements of the cover U.
By Claim 5.3, there is a constant L such that for each simplex σ ∈ T , each point b ∈ Lε). Given any δ < ∞, choose ε > 0 so small that for any simplex σ ∈ T the following conditions hold:

Now consider the closed cover
of the space R n and observe that for each simplex σ ∈ T we get By the choice of the function ϕ, for each simplex σ ∈ T , there is a point b σ ∈ R n such that For every point v ∈ 1 ε Z n consider the set and observe that U = {St (v) : v ∈ 1 ε Z n } is a cover of the Euclidean space R n . It follows that It remains to check that each ball B(a, δ), a ∈ A, meets at most n sets U ∈ U.
Assume conversely that there are a point a ∈ A and a set V ⊂ ε −1 Z n of cardinality Given an Abelian locally compact topological group G endowed with its left coarse structure, we need to prove the equivalence of the following statements: (1) S(G) = D < (G); (2) G is compactly generated; (3) G is coarsely equivalent to a Euclidean space R n for some n ∈ ω.
To prove that (2)⇒(3), assume that the group G is compactly generated. By Theorem 24 [11, p.85], G is topologically isomorphic to the direct sum R n × Z m × K for some n, m ∈ ω and a compact subgroup K ⊂ G. Since the projection R n × Z m × K → R n × Z m and the embedding Z n × Z m → R n × Z m are coarse equivalences, we conclude that G is coarsely equivalent to Z n+m and to R n+m .
To prove that (1)⇒ (2), assume that S(G) = 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 R n × K for some n ∈ ω and some compact subgroup K of G 0 . The subgroup G 0 has asymptotic dimension asdim(G 0 ) = asdim(R n ) = n < ∞. If asdim(G) = ∞, 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 ⊕ ω Z with countably many generators. It follows that G also contains a discrete subgroup H isomorphic to ⊕ ω 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 asdim(H ) = ∞ = asdim(G), we conclude that S(G) = D < (G), which is a desired contradiction showing that asdim(G) < ∞.
By Theorem 2.8, there is a compactly generated subgroup H ⊂ G with asdim(H ) = asdim(G). Since H / ∈ D < (G) = S(G), the subset H is not small in G. Repeating the proof of Claim 4.1, we can show that the set G \ B (H, ε) is not large for some entourage ε ∈ E. By the definition of the left coarse structure E, there is a compact subset K ⊂ G such that B(H, ε) ⊂ H K . We claim that K −1 H K = G. Assuming the opposite, we can find a point x ∈ G\K −1 H K and consider the finite set F = {x, x −1 , x x −1 } = F −1 . Since the set G\H K is not large, there is a point z ∈ (G \ H K )F. For this point z we get z F ∩ (G \ H K ) = ∅ and hence z ∈ z F ⊂ H K . Then x ∈ z −1 z F ⊂ z −1 H K ⊂ K −1 H H K = K −1 H K , which is a contradiction. Now the compact generacy of the subgroup H implies the compact generacy of the group G = K −1 H K .
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