New York Journal of Mathematics

New York J. Math. 12 (2006) 249–256.

Asymptotic dimension of coarse spaces

Bernd Grave

Abstract. We consider asymptotic dimension in the general setting of coarse spaces and prove some basic properties such as monotonicity, a formula for the asymptotic dimension of finite unions and estimates for the asymptotic dimension of the product of two coarse spaces.

Contents

1. Coarse geometry 249 2. Asymptotic dimension 250 References 256

1. Coarse geometry

Originally (cf. [Gro93]) coarse geometry was studied in the concrete setting of metric spaces. But it turned out that, similar to infinitesimal scale geometry (i.e., topology), there is an axiomatic description of large scale geometry.

Coarse concepts have applications, e.g., in geometric group theory. As a striking example, the strong Novikov conjecture for many groups can be proved using methods from coarse geometry. This started out using the large scale geometry of metric spaces, but one now observes that using general abstract coarse structures offers greater flexibility which is frequently very useful. Compare [Yu98], [Wri04] and [Mit01].

To make this paper self-contained I include a short discussion of the basic concepts of coarse geometry. More information can be found in Roe’s textbook [Roe03] and in [Gra06].

Definition 1 (Coarse structure). Let X be a set. A collection E of subsets of X×X is called a coarse structure, and the elements of E will be called entourages, if the following axioms are fulfilled:

(a) A subset of an entourage is an entourage.

(b) A finite union of entourages is an entourage.

(c) The diagonal Δ_{X} := {(x, x) | x ∈ X} is an entourage.

249^{ISSN 1076-9803/06}

Received February 2, 2006.

Mathematics Subject Classification. 51F99, 20F69.

Key words and phrases. asymptotic dimension, coarse geometry, coarse structure.

(d) The inverse E^{−1} of an entourage E is an entourage:

E^{−1} := {(y, x) ∈ X × X | (x, y) ∈ E}.

(e) The composition E_{1}E_{2} of entourages E_{1} and E_{2} is an entourage:

E_{1}E_{2} := {(x, z) ∈ X × X | ∃_{y∈X}(x, y) ∈ E_{1} and (y, z) ∈ E_{2}}.

The pair (X, E) is called a coarse space.

A coarse space is called connected if every point of X × X is contained in an entourage.

Definition 2 (Bounded sets). Let (X, E) be a coarse space, A ⊆ X and E ∈ E. We define E[A] := {x ∈ X | (x, a) ∈ E for some a ∈ A}. For a point x ∈ X we will write E(x) instead of E[{x}]. Sets of the form E(x) with x ∈ X and E ∈ E are called bounded.

Definition 3. Let X be a set and M a collection of subsets of X × X. Since any intersection of coarse structures on X is itself a coarse structure, we can make the following definitions. We call the smallest coarse structure containing M, i.e., the intersection of all coarse structures containing M, the coarse structure generated by M. In the same way, define the connected coarse structure generated by M.

Definition 4. Let (X, E_{X}), (Y, E_{Y}) be coarse spaces and f: X → Y a map. • We call f coarsely proper if the inverse image of each bounded set is bounded.

• We call f coarsely uniform if the image of each entourage under the map f × f: X × X → Y × Y is an entourage.

• We call f a coarse map if it is coarsely proper and coarsely uniform.

• We call f a coarse embedding if f is coarsely uniform and the inverse image of an entourage under f × f is an entourage.

Definition 5 (Coarse equivalence). We call a map f: X → Y a coarse equivalence if f is coarsely uniform and there exists a coarsely uniform map g: Y → X such that g ∘ f is close^{1} to id_{X} and f ∘ g is close to id_{Y}.

2. Asymptotic dimension

In [Roe03], John Roe defined asymptotic dimension for coarse spaces in general. We are now going to give another definition of asymptotic dimension for coarse spaces which generalizes a different characterisation of asymptotic dimension for metric spaces.

Definition 6 (Asymptotic dimension of coarse spaces). Let (X, E) be a coarse space.

• Let L ∈ E be an entourage and U a cover of X. We say that U has appetite L if for all x ∈ X there exists a set U ∈ U such that L(x) ⊆ U.

• We call a cover U uniformly bounded if Δ_{U} := ⋃_{U∈U} U × U is an entourage.

• Let n ∈ ℕ. We say asdim(X, E) ≤ n if for every^{2} entourage L ∈ E there exists a cover U of X such that:

1Let (X, E) be a coarse space and S a set. The maps f: S → X and g: S → X are called close if {(f(s), g(s)) | s ∈ S)} is an entourage.

2An entourage L ∈ E is called symmetric if L = L^{−1}. We need to consider only symmetric entourages which contain the diagonal, because for any entourage L ∈ E we have L ⊆ L ∪ L^{−1} ∪ Δ_{X} ∈ E.

(1) The multiplicity μ(U) is at most n + 1.

(2) U has appetite L.

(3) U is uniformly bounded.

The following is a redraft of Roe’s definition of asymptotic dimension.

Definition 7. Let (X, E) be a coarse space. We say asdim_{Roe}(X, E) ≤ n if for every entourage L ∈ E there is a cover U of X such that:

(1) U = U_{1} ∪ ⋯ ∪ U_{n+1}.

(2) Each of the families U_{1}, …, U_{n+1} is L-disjoint (i.e., whenever A, B ∈ U_{i} and A ≠ B, then A × B ∩ L = ∅).

(3) U is uniformly bounded.

There is a small difference between Definition 7 and the definition given in [Roe03]. In Roe’s original definition the cover U is supposed to be countable. We will not make any assumptions on the cardinality of U.

A third version of asymptotic dimension will appear in Theorem 9.

Definition 8. asdim_{fam}(X, E) ≤ n if for every entourage L ∈ E there is a cover U of X such that:

(1) U = U_{1} ∪ ⋯ ∪ U_{n+1}, where each of the families U_{i} consists of disjoint sets.

(2) U has appetite L.

(3) U is uniformly bounded.

Theorem 9. Let (X, E) be a coarse space. Then

asdim(X, E) = asdim_{Roe}(X, E) = asdim_{fam}(X, E).

Proof. We first prove asdim_{Roe} ≥ asdim_{fam}. Assume asdim_{Roe}(X, E) = n ∈ ℕ. Let L be a symmetric entourage which contains the diagonal. For L^{2} := LL ∈ E, there exists a cover U as in Definition 7. Since A × B ∩ L^{2} = ∅ is equivalent to L[A] ∩ L[B] = ∅, the cover U_{L} := {L[U] | U ∈ U} meets all conditions required in Definition 8. Note that ⋃_{U∈U} L[U] × L[U] ⊆ L (⋃_{U∈U} U × U) L^{−1} ∈ E.

In a second step we have to prove asdim_{fam} ≥ asdim, but this is obvious, since condition (1) of Definition 8 implies condition (1) of Definition 6.

It remains to prove asdim ≥ asdim_{Roe}. For this purpose we need to construct a uniformly bounded cover V consisting of L-disjoint families from a uniformly bounded cover U with appetite L^{n+1}. The idea is to take all intersections of n + 1 sets from U as one family, the intersections of exactly n sets from U as a second family, etc. However, we have to ensure these families to be L-disjoint.

Assume that asdim(X, E) = n ∈ ℕ. Let L ∈ E be a symmetric entourage that contains the diagonal. Let U be a uniformly bounded cover of X with appetite L^{n+1} and multiplicity at most n + 1. For an entourage E and U ⊆ X we define Int_{E}(U) := {x ∈ X | E(x) ⊆ U}. Observe that E_{1} ⊆ E_{2} implies Int_{E_{2}}(U) ⊆ Int_{E_{1}}(U). Some more definitions are needed to get V:

U_{i} := {U_{1} ∩ ⋯ ∩ U_{i} | U_{1}, …, U_{i} ∈ U pairwise distinct}
S_{i} := ⋃_{U∈U_{i}} Int_{L^{n+2−i}}(U) and S_{n+2} = ∅
V_{i} := {Int_{L^{n+2−i}}(U)\S_{i+1} | U ∈ U_{i}}
V := V_{1} ∪ ⋯ ∪ V_{n+1}.

Since {Int_{L^{n+1}}(U) | U ∈ U} is a cover of X, so is V. Actually, V is a refinement of the cover U. Therefore V is uniformly bounded.

It remains to prove that each of the families V_{1}, …, V_{n+1} is L-disjoint. Let A, B ∈ V_{i} such that A ≠ B. There are A_{1}, …, A_{i}, B_{1}, …, B_{i} ∈ U such that A = Int_{L^{n+2−i}}(A_{1} ∩ ⋯ ∩ A_{i})\S_{i+1} and B = Int_{L^{n+2−i}}(B_{1} ∩ ⋯ ∩ B_{i})\S_{i+1}. The sets A_{1}, …, A_{i} are supposed to be pairwise distinct as are the sets B_{1}, …, B_{i}.

Let (a, b) ∈ A × B ∩ L and observe the following facts:

(1)  a, b ∉ S_{i+1}
(2)  a ∈ A ⊆Int_{L^{n+1−i}}(A_{1} ∩ ⋯ ∩ A_{i})
(3)  b ∈ B ⊆Int_{L^{n+1−i}}(B_{1} ∩ ⋯ ∩ B_{i})
(4)  a ∈ L[B] ⊆ L[Int_{L^{n+2−i}}(B_{1} ∩ ⋯ ∩ B_{i})]
(5)  b ∈ L[A] ⊆ L[Int_{L^{n+2−i}}(A_{1} ∩ ⋯ ∩ A_{i})].

Since

L[Int_{L^{j}}(U)] = {x | ∃_{y∈X}L^{j}(y) ⊆ U, x ∈ L(y)} ⊆ {x | L^{j−1}(x) ⊆ U}
= Int_{L^{j−1}}(U),

we get the following conclusions from (4) and (5):

a ∈ Int_{L^{n+1−i}}(B_{1} ∩ ⋯ ∩ B_{i})
b ∈ Int_{L^{n+1−i}}(A_{1} ∩ ⋯ ∩ A_{i}).

Finally a, b ∈ Int_{L^{n+2−(i+1)}}(A_{1} ∩⋯∩A_{i} ∩B_{1} ∩⋯∩B_{i}). Since A ≠ B, we know that the set {A_{1}, …, A_{i}, B_{1}, …, B_{i}} contains at least i + 1 elements. Thus a, b ∈ S_{i+1}, but this is a contradiction to (1). □

Theorem 10. If f: (X, E) → (Y, F) is a coarse embedding, then

asdim(X, E) ≤ asdim(Y, F).

Proof. Suppose that n := asdim(Y, F) < ∞. Let E ∈ E be an entourage and set F := f × f(E). Note that E ⊆ (f × f)^{−1}(F). There is a uniformly bounded cover U of Y with appetite F and multiplicity at most n + 1. The inverse image of U is a uniformly bounded cover of X with appetite E and the same multiplicity as U. □

Corollary 11 (Monotonicity of asymptotic dimension). Let (X, E) be a coarse space and A ⊆ X. Note that the inclusion map is a coarse embedding. Hence

asdim(A, E|_{A}) ≤ asdim(X, E).

Corollary 12 (Coarse invariance of asymptotic dimension). Given a coarse equivalence f: (X, E_{X}) → (Y, E_{Y}), we have

asdim(X, E_{X}) = asdim(Y, E_{Y}).

Definition 13. Let (X, E) be a coarse space and A ⊆ X. We call A a substantial part of X if asdim(A, E|_{A}) = asdim(X, E).

For a coarsely uniform map (X, E) → (Y, F) which is also injective, there is no relation between the asymptotic dimensions of (X, E) and (Y, F).

Example 14. Let (X, E) be a coarse space. Observe that the power set P(X×X) is a coarse structure on X. The map id: (X, E) → (X, P(X×X)) is coarsely uniform, but asdim(X, E) ≥ 0 = asdim(X, P(X ×X)).

Example 15. Let n ∈ {2, 3, … }. By E_{·} we denote the coarse structure coming from the one-point compactification^{3} of ℝ^{n} and by E_{eucl} the bounded coarse structure^{4} corresponding to the euclidean metric of ℝ^{n}. It follows that the map id: (ℝ^{n}, E_{eucl}) → (ℝ^{n}, E_{·}) is coarse, but asdim(ℝ^{n}, E_{eucl}) = n > 1 = asdim(ℝ^{n}, E_{·}).

Proof. Since E_{eucl} ⊆ E_{·}, the map id is coarsely uniform. A set B is bounded with respect to E_{eucl} if and only if B is precompact. The same is true for E_{·}. Thus id is coarsely proper.

It remains to prove asdim(ℝ^{n}, E_{·}) = 1. For this we refer to Example 9.7 of [Roe03]. □

Proposition 16. Let (X, E) be a coarse space and E_{cn} the connected coarse structure generated by E. Then asdim(X, E) = asdim(X, E_{cn}).

Proof. Suppose n := asdim(X, E_{cn}) < ∞. Let E ∈ E ⊆ E_{cn}. There is a cover U of X with appetite E and multiplicity at most n + 1 which is uniformly bounded with respect to E_{cn}. Each U ∈ U can be written as the disjoint union of finitely many sets U_{1}, …, U_{k} which are bounded with respect to E and such that the union of any two of the sets U_{1}, …, U_{k} is not bounded with respect to E. Define comp(U) := {U_{1}, …, U_{k}} and observe that U′ := ⋃_{U∈U} comp(U) is a cover of X with multiplicity at most n+1. Furthermore, U′ has appetite E and is uniformly bounded with respect to E. Hence asdim(X, E) ≤ n.

Set n := asdim(X, E). Let E ∈ E_{cn} be a symmetric entourage. This implies that E = E′ ∪ (A_{1} × A_{σ(1)}) ∪ ⋯ ∪ (A_{k} × A_{σ(k)}) with E′ ∈ E, k ∈ ℕ, A_{1}, …, A_{k} bounded subsets of X (not necessarily pairwise distinct) and σ a permutation of the set {1, …, k} with the additional property σ ∘ σ = id. Set M := Δ_{X} ∪ A_{1}^{2} ∪ ⋯ ∪ A_{k}^{2} and observe that E′′ := (E′ ∪ Δ_{X})M ∈ E. Let U′′ be a cover of X which is uniformly bounded with respect to E and which has appetite E′′ and multiplicity at most n + 1. Note that there are sets U_{1}, …, U_{k} ∈ U′′ such that E′[A_{i}] ∪ A_{i} ⊆ U_{i}. We define the cover U := U′′ ∪ {U_{1} ∪ ⋯ ∪ U_{k}}\{U_{1}, …, U_{k}} of X. Observe that U is uniformly bounded with respect to E_{cn} and has multiplicity at most n + 1.

Moreover, U has appetite E. To see this, let x ∈ X. If x ∈ A_{i}, then

E(x) = E′(x) ∪ A_{σ(i)} ⊆ U_{i} ∪ U_{σ(i)} ⊆ U_{1} ∪ ⋯ ∪ U_{k}.

If x ∉ {A_{1} ∪ ⋯ ∪ A_{k}}, then E(x) = E′(x) ⊆ E′′(x). □

Proposition 17 (Asymptotic dimension of finite unions). Let (X, E) be a coarse space and A, B ⊆ X with A ∪ B = X. Then

asdim(X, E) = max{asdim(A, E|_{A}), asdim(B, E|_{B})}.

E_{X} := {E ⊆ X × X | E ⊆ X × X ∪ Δ_{X}} E_{d} := {E ⊆ X × X | E ⊆ Δ_{r} for some r > 0}.

3Let X be a Hausdorff space and X a compactification of X, i.e., X is a dense and open subset of the compact set X. The collection of all subsets E ⊆ X×X, whose closure meets the boundary (X×X)\(X×X) only in the diagonal, is a connected coarse structure on X.

4Let (X, d) be a metric space. Set Δ_{r} := {(x, y) ∈ X × X | d(x, y) < r} and define It is easy to verify that E_{d} is the (connected) coarse structure generated by {Δ_{r} | r > 0}. It is called the bounded coarse structure corresponding to the metric space (X, d).

Proof. The proof of ≥ follows from monotonicity. To see ≤ , we generalize an argument of Bell and Dranishnikov (see [BD01]).

Let n be the maximum of asdim(A, E|_{A}) and asdim(B, E|_{B}) and take a symmetric entourage L ∈ E which contains Δ_{X}. For U ⊆ P(X) and V ⊆ X we define

N_{L}(V, U) := V ∪ ⋃_{\begin{matrix}U∈U \\ L∩U×V ≠∅ \end{matrix}} U.

There is a uniformly bounded cover U = U_{1} ∪⋯∪U_{n+1} of A consisting of L-disjoint families U_{i}. Moreover, there is a uniformly bounded cover V = V_{1} ∪ ⋯ ∪ V_{n+1} of B consisting of (LΔ_{U}LΔ_{U}L)-disjoint families V_{i}. For i ∈ {1, …, n + 1} set

W_{i} := {N_{L}(V, U_{i}) | V ∈ V_{i}} ∪ {U ∈ U_{i} | L ∩ U × V = ∅ for all V ∈ V_{i}}.

Observe that N_{L}(V, U_{i}) ⊆ Δ_{U}L[V]. Hence, we get a uniformly bounded cover W = W_{1} ∪ ⋯ ∪ W_{n+1} of X where W_{i} is L-disjoint for 1 ≤ i ≤ n + 1. This proves asdim(X, E) ≤ n. □

Proposition 18 (Asymptotic dimension of coproducts). Let Λ be any set and let (X_{λ}, E_{λ}) be a coarse space for every λ ∈ Λ. Define X := ∐_{λ∈Λ}X_{λ}. If (X, E) is

the coproduct in the category of coarse spaces and uniformly bounded maps, then asdim (X, E) = sup_{λ∈Λ} asdim(X_{λ}, E_{λ}).

Proof. Set n := sup_{λ∈Λ} asdim(X_{λ}, E_{λ}). Monotonicity of asymptotic dimension implies

asdim(X, E) ≥ n.

We will now prove asdim(X, E) ≤ n. Take an entourage L ∈ E which contains Δ_{X}. Then there are λ_{1}, …, λ_{k} ∈ Λ and L_{λ_{i}} ∈ E_{λ_{i}} such that L = L_{λ_{1}}∪⋯∪L_{λ_{k}}∪Δ_{X}. For i ∈ {1, …, k} choose a uniformly bounded cover U_{λ_{i}} of X_{λ_{i}} with appetite L_{λ_{i}} and multiplicity at most asdim(X_{λ_{i}}, E_{λ_{i}}) + 1. For λ ∈ Λ\{λ_{1}, …, λ_{k}} set U_{λ} := {{x} | x ∈ X_{λ}}. The union U := ⋃_{λ∈Λ} U_{λ} is a uniformly bounded cover of X with appetite L whose multiplicity does not exceed n + 1. □

Definition 19. For i ∈ {1, …, k} let (X_{i}, E_{i}) be a coarse space. By p_{i}: X_{1} × ⋯ × X_{k} → X_{i} we denote the projection to the i-th factor. The product coarse structure is defined as follows:

E_{1} ∗ ⋯ ∗ E_{k} := {E ⊆ (X_{1} × ⋯ × X_{k})^{2} | (p_{i} × p_{i})(E) ∈ E_{i} for i ∈ {1, …, k}} .

If (X, E) is a coarse space, we will sometimes write E^{∗k} for the product coarse structure on X^{k}.

It is easy to prove that E_{1} ∗ ⋯ ∗ E_{k} actually is a coarse structure. The product coarse structure E_{1} ∗ ⋯ ∗ E_{k} is connected if and only if the coarse structures E_{i} are connected. Moreover, we have the following formulas:

Δ_{X_{1}×···×X_{k}} = Δ_{X_{1}} × ⋯ × Δ_{X_{k}}
(E_{1} × ⋯ × E_{k})(E′_{1} × ⋯ × E′_{k}) = E_{1}E′_{1} × ⋯ × E_{k}E′_{k}.

One remark on our notation: If E ⊆ X × X, we should not confuse the composition E^{k} = E ⋯ E and the product E^{×k} = E × ⋯ × E.

Proposition 20 (Asymptotic dimension of products). Let (X, E_{X}) and (Y, E_{Y}) be coarse spaces. Then

asdim(X × Y, E_{X} ∗ E_{Y}) ≤ asdim(X, E_{X}) + asdim(Y, E_{Y}),
asdim(X, E_{X}) ≤ asdim(X × Y, E_{X} ∗ E_{Y}) if Y ≠ ∅.

Proof. Compare [Roe03] for the special case of bounded coarse structures.

Set n := asdim(X) and m := asdim(Y). Let E ∈ E_{X} ∗ E_{Y}. There are symmetric entourages E_{X} ∈ E_{X} and E_{Y} ∈ E_{Y} containing the diagonals Δ_{X} and Δ_{Y} respectively such that E ⊆ E_{X} × E_{Y}.

There is a uniformly bounded cover U of X with appetite E_{X}^{n+m+1} and multiplicity μ(U) ≤ n + 1. There is also a uniformly bounded cover V of Y with appetite E_{Y}^{n+m+1} and multiplicity μ(V) ≤ m + 1. We get a uniformly bounded cover U × V := {U × V | U ∈ U, V ∈ V} of X × Y with appetite E^{n+m+1} and multiplicity ≤ (n + 1) · (m + 1) = n · m + n + m + 1. Thus, we need to improve the multiplicity.

We proceed similar as in the proof of Theorem 9 and begin with some definitions. Let k ∈ {2, …, n + m + 2}. Set

A_{k} := {U_{1} ∩ ⋯ ∩ U_{p} × V_{1} ∩ ⋯ ∩ V_{q} |
p + q = k, U_{i} ∈ U, V_{i} ∈ V pairwise distinct }
B_{k} := ⋃_{A∈A_{k}} Int_{E^{n+m+3−k}}(A) and B_{n+m+3} := ∅
W_{k} := {Int_{E^{n+m+3−k}}(U) \ B_{k+1} | U ∈ A_{k}}
W := W_{2} ∪ ⋯ ∪ W_{n+m+2}.

Notice that W is a uniformly bounded cover of X × Y consisting of the n + m + 1 disjoint families W_{2}, …, W_{n+m+2}. It remains to prove that W_{k} is E-disjoint for k ∈ {2, …, n + m + 2}.

For this purpose let M, N ∈ W_{k} with M ≠ N and suppose M ×N ∩ E ≠ ∅. Choose ((x_{M}, y_{M}), (x_{N}, y_{N})) ∈ M×N ∩ E. There are p_{M}, q_{M} ∈ ℕ with p_{M} +q_{M} = k and M_{1}, …, M_{p_{M}} ∈ U, M′_{1}, …, M′_{q_{M}} ∈ V such that

M = Int_{E^{n+m+3−k}}(\underbrace{M_{1} ∩ ⋯ ∩ M_{p_{M}} × M′_{1} ∩ ⋯ ∩ M′_{q_{M}}}_{=:MM′}) \ B_{k+1}.

Similarly, there are p_{N}, q_{N} ∈ ℕ with p_{N} + q_{N} = k and sets N_{1}, …, N_{p_{N}} ∈ U and N′_{1}, …, N′_{q_{N}} ∈ V such that

N = Int_{E^{n+m+3−k}}(\underbrace{N_{1} ∩ ⋯ ∩ N_{p_{N}} × N′_{1} ∩ ⋯ ∩ N′_{q_{N}}}_{=:NN′}) \ B_{k+1}.

Observe that the following relations hold:

(x_{M}, y_{M}) ∉ B_{k+1}
(x_{M}, y_{M}) ∈ M ⊆ Int_{E^{n+m+2−k}}(MM′)
(x_{M}, y_{M}) ∈ E[N] ⊆ E[Int_{E^{n+m+3−k}}(NN′)] ⊆ Int_{E^{n+m+2−k}}(NN′).

It follows that (x_{M}, y_{M}) ∈ Int_{E^{n+m+3−(k+1)}}(MM′ ∩ NN′). Since M ≠ N, the set {M_{1}, …, M_{p_{M}} , M′_{1}, …, M′_{q_{M}} , N_{1}, …, N_{p_{N}} , N′_{1}, …, N′_{q_{N}} } contains at least k + 1 256 Bernd Grave

different elements. Hence (x_{M}, y_{M}) ∈ B_{k+1}. But this is a contradiction to what we found before. □

The equality asdim(X × Y) = asdim(X) + asdim(Y) is not true in general. Compare [BL] and Corollary 5.9 of [Gra06].

References

[BD01] Bell, G.; Dranishnikov, A. On asymptotic dimension of groups. Algebr. Geom. Topol. 1 (2001), 57–71 (electronic). MR1808331 (2001m:20062), Zbl 1008.20039.

[BL] Buyalo, Sergei; Lebedeva, Nina. Dimensions of locally and asymptotically self-similar spaces. math.GT/0509433.

[Gra06] Grave, Bernd. Coarse geometry and asymptotic dimension, Mathematica Gottingensis (2006).

[Gro93] Gromov, Mikhael. Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993. MR1253544 (95m:20041), Zbl 0841.20039.

[Mit01] Mitchener, Paul D. Coarse homology theories, Algebr. Geom. Topol. 1 (2001), 271– 297 (electronic). MR1834777 (2002k:55012), Zbl 0978.58011. Addendum. Algebr. Geom. Topol. 3 (2003), 1089–1101. MR2012966 (2004i:55007).

[Roe03] Roe, John. Lectures on coarse geometry, University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003. MR2007488 (2004g:53050), Zbl 1042.53027.

[Wri04] Wright, Nick. The coarse Baum-Connes conjecture via C_{0} coarse geometry. J. Funct. Anal. 220 (2005), no. 2, 265–303. MR2119281 (2006d:58023), Zbl 1073.46046.

[Yu98] Yu, Guoliang. The Novikov conjecture for groups with finite asymptotic dimension. Ann. of Math. (2) 147 (1998), 325–355. MR1626745 (99k:57072), Zbl 0911.19001.

Mathematisches Institut, Universität Göttingen, Bunsenstr. 3–5, 37073 Göttingen, GER-MANY berndgrave@gmail.com

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