New York Journal of Mathematics New York J. Math.

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 strik- ing 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 con- cepts 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 collectionEof subsets ofX×X is called acoarse structure, and the elements ofE will be calledentourages, 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.

Received February 2, 2006.

Mathematics Subject Classification. 51F99, 20F69.

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

ISSN 1076-9803/06

249

(d) The inverseE⁻¹ of an entourageE is an entourage:

E⁻¹:={(y, x)∈X×X |(x, y)∈E}.

(e) The compositionE₁E₂of entourages E₁andE₂ is an entourage:

E₁E₂:={(x, z)∈X×X| ∃y∈X(x, y)∈E₁and (y, z)∈E₂}. The pair (X,E) is called acoarse 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 defineE[A] :={x∈X |(x, a)∈E for somea∈A}. For a pointx∈X we will write E(x) instead of E[{x}]. Sets of the form E(x) with x∈X and E ∈ E are calledbounded.

Definition 3. LetX be a set and Ma collection of subsets ofX×X. Since any intersection of coarse structures onX is itself a coarse structure, we can make the following definitions. We call the smallest coarse structure containingM, i.e., the intersection of all coarse structures containing M, the coarse structure generated byM. In the same way, define theconnected coarse structure generated by M. Definition 4. Let (X,EX), (Y,EY) be coarse spaces andf:X →Y a map.

• We callfcoarsely properif 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 callf acoarse map if it is coarsely proper and coarsely uniform.

• We callf acoarse embedding iff is coarsely uniform and the inverse image of an entourage underf×f is an entourage.

Definition 5 (Coarse equivalence). We call a mapf:X →Y a coarse equivalence if f is coarsely uniform and there exists a coarsely uniform map g: Y → X such thatg◦f is close¹ to id_X andf◦gis 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.

• LetL∈ E be an entourage andU a cover ofX. We say thatU hasappetite Lif for allx∈X there exists a set U ∈ U such thatL(x)⊆U.

• We call a coverU uniformly bounded if Δ_U:=

U∈UU×U is an entourage.

• Let n ∈ N. We say asdim(X,E) ≤n if for every² entourage L ∈ E there exists a coverU ofX such that:

1Let (X,E) be a coarse space andSa set. The mapsf:S→Xandg:S→Xare calledclose if{(f(s), g(s))|s∈S)}is an entourage.

2An entourageL∈ E is called symmetric ifL=L⁻¹. We need to consider only symmetric entourages which contain the diagonal, because for any entourageL∈ Ewe haveL⊆L∪L⁻¹∪ Δ_X∈ E.

(1) The multiplicity μ(U) is at mostn+ 1.

(2) U has appetiteL.

(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 entourageL∈ E there is a coverU ofX such that:

(1) U =U1 ∪ · · · ∪ Un+1.

(2) Each of the familiesU1, . . . ,Un+1 isL-disjoint (i.e., wheneverA, B∈ Uiand A=B, thenA×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 ofU.

A third version of asymptotic dimension will appear in Theorem9.

Definition 8. asdim_fam(X,E)≤nif for every entourageL∈ E there is a coverU ofX such that:

(1) U=U₁ ∪ · · · ∪ U_n+1, where each of the familiesU_iconsists 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. LetL be a symmetric entourage which contains the diagonal. For L² :=LL ∈ E, there exists a cover U as in Definition 7. Since A×B ∩ L² =∅ is equivalent to L[A]∩L[B] =∅, the cover UL :={L[U]|U ∈ U} meets all conditions required in Definition8. Note that

U∈UL[U]×L[U]⊆L

U∈UU×U

L⁻¹∈ E.

In a second step we have to prove asdim_fam ≥asdim, but this is obvious, since condition (1) of Definition8 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 coverU with appetiteLⁿ⁺¹. The idea is to take all intersections ofn+ 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 beL-disjoint.

Assume that asdim(X,E) =n∈N. LetL ∈ E be a symmetric entourage that contains the diagonal. Let U be a uniformly bounded cover of X with appetite Lⁿ⁺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 E₁ ⊆ E₂ implies Int_E2(U) ⊆ Int_E1(U). Some more definitions are needed to getV:

U_i:={U₁∩ · · · ∩U_i|U₁, . . . , U_i∈ U pairwise distinct} S_i:=

U∈Ui

Int_Ln+2−i(U) and S_n+2=∅ V_i:={Int_Ln+2−i(U)\S_i+1|U ∈ U_i}

V :=V1∪ · · · ∪ Vn+1.

Since{Int_Ln+1(U)|U ∈ U}is a cover ofX, so isV. Actually,V is a refinement of the coverU. ThereforeV is uniformly bounded.

It remains to prove that each of the families V1, . . . ,Vn+1 is L-disjoint. Let A, B ∈ Vi such that A = B. There are A₁, . . . , A_i, B₁, . . . , B_i ∈ U such that A= Int_Ln+2−i(A₁∩ · · · ∩A_i)\S_i+1andB = Int_Ln+2−i(B₁∩ · · · ∩B_i)\S_i+1. The sets A₁, . . . , A_i are supposed to be pairwise distinct as are the setsB₁, . . . , B_i.

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

a, b ∈S_i+1 (1)

a ∈ A⊆Int_Ln+1−i(A₁∩ · · · ∩A_i) (2)

b ∈ B⊆Int_Ln+1−i(B₁∩ · · · ∩B_i) (3)

a ∈ L[B]⊆L[Int_Ln+2−i(B₁∩ · · · ∩B_i)]

(4)

b ∈ L[A]⊆L[Int_Ln+2−i(A₁∩ · · · ∩A_i)].

(5) Since

L[Int_Lj(U)] =

x| ∃_y∈XL^j(y)⊆U, x∈L(y)

⊆ {x|L^j−1(x)⊆U}

= Int_Lj−1(U),

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

a∈Int_Ln+1−i(B₁∩ · · · ∩B_i) b∈Int_Ln+1−i(A₁∩ · · · ∩A_i).

Finallya, b∈Int_Ln+2−(i+1)(A₁∩· · ·∩A_i∩B₁∩· · ·∩B_i). SinceA=B, we know that the set{A₁, . . . , A_i, B₁, . . . , B_i} contains at leasti+ 1 elements. Thusa, 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)<∞. LetE ∈ E be an entourage and set F :=f×f(E). Note thatE⊆(f ×f)⁻¹(F). There is a uniformly bounded cover U ofY with appetiteF and multiplicity at mostn+ 1. The inverse image ofU is a uniformly bounded cover ofX with appetiteEand the same multiplicity asU. Corollary 11 (Monotonicity of asymptotic dimension). Let (X,E) be a coarse space andA⊆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 equiv- alencef: (X,EX)→(Y,EY), we have

asdim(X,E_X) = asdim(Y,E_Y).

Definition 13. Let (X,E) be a coarse space and A⊆X. We callAa substantial part ofX 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 setP(X×X) is a coarse structure onX. 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 com- ing from the one-point compactification³ of Rⁿ and by Eeucl the bounded coarse structure⁴ corresponding to the euclidean metric of Rⁿ. It follows that the map id : (Rⁿ,Eeucl)→(Rⁿ,E·) is coarse, but asdim(Rⁿ,Eeucl) =n >1 = asdim(Rⁿ,E·).

Proof. SinceEeucl⊆ E·, the map id is coarsely uniform. A setB is bounded with respect to E_eucl if and only ifB is precompact. The same is true for E_·. Thus id is coarsely proper.

It remains to prove asdim(Rⁿ,E_·) = 1. For this we refer to Example 9.7 of

[Roe03].

Proposition 16. Let (X,E)be a coarse space andEcn the connected coarse struc- ture generated byE. Thenasdim(X,E) = asdim(X,Ecn).

Proof. Supposen := asdim(X,E_cn)<∞. Let E ∈ E ⊆ E_cn. There is a coverU of X with appetiteE and multiplicity at mostn+ 1 which is uniformly bounded with respect to E_cn. Each U ∈ U can be written as the disjoint union of finitely many sets U₁, . . . , U_k which are bounded with respect to E and such that the union of any two of the sets U₁, . . . , U_k is not bounded with respect to E. De- fine comp(U) :={U₁, . . . , U_k}and observe thatU :=

U∈Ucomp(U) is a cover of X with multiplicity at mostn+ 1. Furthermore,Uhas appetiteEand is uniformly bounded with respect toE. Hence asdim(X,E)≤n.

Set n := asdim(X,E). Let E ∈ E_cn be a symmetric entourage. This implies that E =E∪

A₁×A_σ(1)

∪ · · · ∪

A_k×A_σ(k)

withE ∈ E, k∈ N, A₁, . . . , A_k bounded subsets ofX (not necessarily pairwise distinct) andσa permutation of the set{1, . . . , k}with the additional propertyσ◦σ= id. SetM := Δ_X∪A²₁∪ · · · ∪A²_k and observe that E := (E ∪Δ_X)M ∈ E. Let U be a cover of X which is uniformly bounded with respect toE and which has appetiteE and multiplicity at mostn+ 1. Note that there are setsU₁, . . . , U_k∈ Usuch thatE[A_i]∪A_i⊆U_i. We define the coverU :=U∪ {U₁∪ · · · ∪U_k}\{U₁, . . . , U_k}ofX. Observe thatU is uniformly bounded with respect toEcnand has multiplicity at mostn+ 1.

Moreover,U has appetiteE. To see this, letx∈X. Ifx∈A_i, then E(x) =E(x)∪A_σ(i)⊆U_i∪U_σ(i)⊆U₁∪ · · · ∪U_k.

Ifx ∈ {A₁∪ · · · ∪A_k}, thenE(x) =E(x)⊆E(x).

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

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

3LetXbe a Hausdorff space andXa compactification ofX, i.e.,Xis a dense and open subset of the compact setX. The collection

E_X:={E⊆X×X|E⊆X×X∪Δ_X}

of all subsetsE⊆X×X, whose closure meets the boundary (X×X)\(X×X) only in the diagonal, is a connected coarse structure onX.

4Let (X, d) be a metric space. Set Δ_r:={(x, y)∈X×X|d(x, y)< r}and define Ed:={E⊆X×X|E⊆Δ_r for somer >0}.

It is easy to verify thatEdis the (connected) coarse structure generated by {Δr|r >0}. It is called thebounded coarse structurecorresponding 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]).

Letnbe the maximum of asdim(A,E|A) and asdim(B,E|B) and take a symmetric entourageL∈ E which contains Δ_X. ForU ⊆ P(X) andV ⊆X we define

N_L(V,U) := V ∪

U∈U L∩U×V=∅

U.

There is a uniformly bounded coverU =U₁∪· · ·∪U_n+1ofAconsisting ofL-disjoint families U_i. Moreover, there is a uniformly bounded coverV =V₁∪ · · · ∪ V_n+1 of B consisting of (LΔ_ULΔ_UL)-disjoint familiesV_i. Fori∈ {1, . . . , n+ 1}set

W_i:=

N_L(V,U_i)|V ∈ V_i

∪

U ∈ U_i|L∩U ×V =∅ for allV ∈ V_i . Observe that N_L(V,U_i) ⊆ Δ_UL[V]. Hence, we get a uniformly bounded cover W=W₁∪ · · · ∪ W_n+1 ofX whereW_i isL-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∈Λasdim(X_λ,Eλ).

Proof. Setn:= sup

λ∈Λasdim(X_λ,Eλ). 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 areλ₁, . . . , λ_k∈Λ andL_λi ∈ E_λisuch thatL=L_λ1∪· · ·∪L_λk∪Δ_X. For i ∈ {1, . . . , k} choose a uniformly bounded cover U_λi of X_λi with appetite L_λi and multiplicity at most asdim(X_λi,E_λi) + 1. For λ ∈ Λ\{λ₁, . . . , λ_k} set Uλ :={{x} |x∈X_λ}. The union U :=

λ∈ΛUλ is a uniformly bounded cover of X with appetiteLwhose multiplicity does not exceedn+ 1.

Definition 19. Fori∈ {1, . . . , k}let (X_i,E_i) be a coarse space. Byp_i:X₁× · · · × X_k →X_i we denote the projection to thei-th factor. Theproduct coarse structure is defined as follows:

E₁∗ · · · ∗ E_k :=

E⊆(X₁× · · · ×X_k)²|(p_i×p_i)(E)∈ E_i fori∈ {1, . . . , k} . If (X,E) is a coarse space, we will sometimes write E^∗k for the product coarse structure onX^k.

It is easy to prove thatE1∗ · · · ∗ Ek actually is a coarse structure. The product coarse structureE1∗ · · · ∗ Ek is connected if and only if the coarse structuresEi are connected. Moreover, we have the following formulas:

Δ_{X1×···×Xk} = Δ_X1× · · · ×Δ_Xk (E₁× · · · ×E_k)(E₁ × · · · ×E_k) =E₁E₁ × · · · ×E_kE_k.

One remark on our notation: IfE⊆X×X, we should not confuse the compo- sitionE^k=E· · ·E and the productE^×k =E× · · · ×E.

Proposition 20 (Asymptotic dimension of products). Let(X,EX)and(Y,EY)be coarse spaces. Then

asdim(X×Y,EX∗ EY)≤asdim(X,EX) + asdim(Y,EY), asdim(X,E_X)≤asdim(X×Y,E_X∗ E_Y) ifY =∅. 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 sym- metric entourages E_X ∈ E_X and E_Y ∈ E_Y containing the diagonals Δ_X and Δ_Y respectively such thatE⊆E_X×E_Y.

There is a uniformly bounded cover U of X with appetite E^n+m+1_X and mul- tiplicity 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 Theorem9and begin with some definitions.

Letk∈ {2, . . . , n+m+ 2}. Set

A_k:={U₁∩ · · · ∩U_p×V₁∩ · · · ∩V_q|

p+q=k, U_i∈ U, V_i ∈ V pairwise distinct} B_k:=

A∈Ak

Int_En+m+3−k(A) and B_n+m+3:=∅ W_k:={Int_En+m+3−k(U)\B_k+1|U ∈ A_k}

W:=W2∪ · · · ∪ Wn+m+2.

Notice thatW is a uniformly bounded cover ofX×Y consisting of then+m+ 1 disjoint families W2, . . . ,Wn+m+2. It remains to prove that Wk is E-disjoint for k∈ {2, . . . , n+m+ 2}.

For this purpose let M, N ∈ Wk with M =N and suppose M×N ∩ E =∅. Choose ((x_M, y_M),(x_N, y_N))∈M×N∩E. There arep_M, q_M ∈Nwithp_M+q_M =k andM₁, . . . , M_pM ∈ U,M₁, . . . , M_q

M ∈ V such that

M = Int_En+m+3−k(M₁∩ · · · ∩M_pM×M₁ ∩ · · · ∩M_qM

=:MM

)\B_k+1.

Similarly, there arep_N, q_N ∈Nwith p_N +q_N =k and setsN₁, . . . , N_pN ∈ U and N₁, . . . , N_qN ∈ V such that

N= Int_En+m+3−k(N₁∩ · · · ∩N_pN ×N₁∩ · · · ∩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_En+m+2−k(MM)

(x_M, y_M)∈E[N] ⊆ E[Int_En+m+3−k(NN)] ⊆ Int_En+m+2−k(NN).

It follows that (x_M, y_M) ∈ Int_En+m+3−(k+1)(MM ∩NN). Since M = N, the set{M₁, . . . , M_pM, M₁, . . . , M_qM, N₁, . . . , N_pN, N₁, . . . , N_qN}contains at leastk+ 1

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].

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[email protected]

This paper is available via http://nyjm.albany.edu/j/2006/12-15.html.