ON THE DYNAMIC ASYMPTOTIC DIMENSION OF ´ETALE GROUPOIDS

. We investigate the dynamic asymptotic dimension for ´etale groupoids introduced by Guentner, Willett and Yu. In particular, we establish several permanence properties, including estimates for products and unions of groupoids. We also establish invariance of the dynamic asymptotic dimension under Morita equivalence. In the second part of the article, we consider a canonical coarse structure on an ´etale groupoid, and compare the asymptotic dimension of the resulting coarse space with the dynamic asymptotic dimension of the underlying groupoid.


Introduction
Dynamic asymptotic dimension is a dimension theory for dynamical systems introduced by Guentner, Willett and Yu in [11] in the general framework of étale groupoids.Since its inception, the concept has found numerous applications: it provides upper bounds on the nuclear dimension of the resulting groupoid C * -algebras [11], even in the presence of a twist [6].It is also closely related to the diagonal dimension of sub-C * -algebras recently introduced in [13].In [3] it was shown that the groupoid homology groups of a totally disconnected étale groupoid vanish in all degrees exceeding the dynamic asymptotic dimension of the groupoid.Estimates on the dynamic asymptotic dimension (denoted henceforth by dad(•)) are known for many concrete classes of examples (see e.g.[1,5,7,11]).However, the basic theory is still not very well developed in the literature.The aim of this article is to remedy this situation and provide a set of general results to compute the dynamic asymptotic dimension of étale groupoids.
To this end we prove several permanence properties of dynamic asymptotic dimension, which are summarised in the theorem below.Let us remark that it is straightforward to obtain multiplicative estimates for the dynamic asymptotic dimension of a product of groupoids.The main difficulty in proving (3) lies in reducing this crude estimate and obtain an additive estimate.
Results of this kind have a long history: let us mention the classical result [8] in topological dimension theory, the results of Dranishnikov-Bell for Gromov's asymptotic dimension [2] and the results in [4] for Assuad-Nagata dimension.Our proof is more closely modelled on the beautiful approach presented in [4], who in turn attribute some of the underlying ideas to Kolmogorov and Ostrand.We should mention that Pilgrim in [15,16] independently proved similar results for the transformation group case.However, the most natural context for dynamic asymptotic dimension is the world of groupoids.
In the second part of this article, we investigate the relation between the dynamic asymptotic dimension of a groupoid G and its asymptotic dimension, when equipped with a canonical coarse structure denoted by E G .It was recently shown in [5] that free actions Γ X of discrete groups on zero-dimensional second countable compact Hausdorff spaces satisfy dad(Γ ⋉ X) ∈ {asdim(Γ), ∞}.In fact it is believed, that such a result should be true beyond the zero-dimensional setting.Evidence towards this can be found in the results in [1,16].The most natural context for the notion of dynamic asymptotic dimension however is the world of étale groupoids, and in this generality nothing seems to be known so far.Hence the second part of this article is concerned with initiating the study of this question more generally.Our main contribution is the following result.
Theorem B. Let G be a σ-compact principal étale groupoid with compact and totally disconnected unit space, and let E G be the canonical coarse structure on G. Then asdim(G, E G ) ≤ dad(G).
Acknowledgement: I am grateful to Anna Duwenig and Rufus Willett for helpful comments on an earlier version of this paper.

Permanence properties of dynamic asymptotic dimension
Given a groupoid G, we will denote its set of units by G 0 and let r, s : G → G 0 denote the range and source map of G, respectively.Throughout this text we will only deal with locally compact Hausdorff groupoids that are étale, meaning that the range and source maps are local homeomorphisms.
Note, that open subgroupoids of étale groupoids are automatically étale again.Given a groupoid G and a subset A ⊆ G 0 , the restriction of G to A is the subgroupoid We will first list some elementary results.To make sense of the statement of the following lemma, recall that given a groupoid G with a non-compact unit space, we can form the Alexandrov groupoid by considering G + := G ∪ (G 0 ) + , that is we take the Alexandrov or one-point compactification of the unit space adding no further arrows (confer [12,Definition 7.7] for details).The following Lemma below can be proven in an elementary way just from the definition above (see [6] for details).
Lemma 2.2.Let G be an étale Hausdorff groupoid.Then the following hold: (1) dad(H) ≤ dad(G) for all closed étale subgroupoids H ⊆ G, (2) dad(G| U ) ≤ dad(G) for all open subsets U ⊆ G 0 , and Here is another useful permanence property that can be proven directly.
Proof.We may assume that d := lim inf dad(G n ) < ∞ since otherwise there is nothing to show.Let K ⊆ G be an open, relatively compact subset.We may assume without loss of generality that s(K) ∪ r(K) ⊆ K. Since all the By definition of the lim inf there exists N ≥ N ′ such that dad(G N +1 ) ≤ d.
Hence there exist open subsets U 0 , . . ., U d ⊆ G 0 N +1 covering s(K)∪r(K) such that the closure of the subgroupoid 2.1.Morita invariance.Besides the obvious notion of isomorphism for étale groupoids, there are various notions of (Morita) equivalence in the literature and most of them are equivalent to each other.Let us recall the formulation that will be most useful for our purposes.Given an étale groupoid G and a surjective local homeomorphism ψ : X → G 0 from another locally compact Hausdorff space X onto G 0 , we can define the ampliation, or blow-up of G with respect to ψ as It is routine to check that G ψ with the multiplication (x, g, y)(y, h, z) = (x, gh, z) and inverse map (x, g, y) −1 = (y, g, x) is an étale groupoid in its own right when equipped with the relative topology from X × G × X.
We will say that two étale groupoids G and H are equivalent if they admit isomorphic blow-ups, i.e. if there exist locally compact Hausdorff spaces X and Y together with surjective local homeomorphisms ψ : X → G 0 and φ : Y → H 0 such that G ψ ∼ = H ϕ .
We refer the reader to [9, Section 3] for a detailed overview of other notions of (Morita) equivalence and in particular [9, Proposition 3.10], where it is proved that many of the most common notions coincide.
To prove that the dynamic asymptotic dimension is invariant under Morita equivalence, we first need another permanence property that is interesting in its own right.Recall, that a continuous homomorphism π : G → H between two étale groupoids is called locally proper, if its restriction to G| C is proper for every compact subset C ⊆ G 0 .Examples include the inclusion map of a closed étale subgroupoid H ֒→ G, the inclusion map G| U ֒→ G for an open subset U ⊆ G 0 , or the projection map G ⋉ X → G, where G ⋉ X is the transformation groupoid associated with a continuous action of an étale groupoid G on a locally compact Hausdorff space X.Proposition 2.4.Let G and H be étale groupoids and π : G → H a continuous and locally proper homomorphism.Then dad(G) ≤ dad(H).
Proof.We may assume, that dad(H) = d < ∞, since otherwise there is nothing to prove.Let K ⊆ G be an open relatively compact subset.Then π(K) is a relatively compact subset of H. Let C be an open relatively compact subset of H with π(K) ⊆ C. By assumption we may find open subsets U 0 , . . ., U d of H 0 which cover s(C) ∪ r(C), such that for each 0 ≤ i ≤ d the the subgroupoid . Since π is locally proper, the latter is a compact subgroupoid of G.This completes the proof.
Note that an application of this result to the examples mentioned above gives an alternative way to prove items (1) and (2) in Lemma 2.2 above.We are now ready to proceed with the proof of the main result of this section: Proposition 2.5.The dynamic asymptotic dimension is invariant under (Morita) equivalence.
Proof.Let ψ : X → G 0 be a surjective local homeomorphism and let be the ampliation of G with respect to ψ.We are going to show dad(G ψ ) = dad(G).
It is routine to check that the canonical projection π : G ψ → G is locally proper.Hence Proposition 2.4 immediately yields the inequality dad(G ψ ) ≤ dad(G).
For the reverse inequality suppose d := dad(G ψ ) < ∞ and let K ⊆ G be an open relatively compact subset.Using the assumption that ψ is a local homeomorphism, find C ⊆ X open and relatively compact such that s(K) ∪ r(K) ⊆ ψ(C).Then the set This verifies the inequality dad(G) ≤ dad(G ψ ) and completes the proof.

2.2.
A union theorem.This section is dedicated to the following result: Theorem 2.6.Let G be an étale groupoid and U a finite open cover.Then The main technical observation needed in the proof is isolated in the following Lemma: Lemma 2.7.Let G be an étale groupoid and Proof.
Armed with this observation we have the following cases: (3) In all other cases, there must exist indices s, t such that g s is the first element such that r(g s ) ∈ V 1 and g t is the last element such that s(g t ) ∈ V 1 .In this case Proof of Theorem 2.6.It follows from Lemma 2.2 that dad(G| U ) ≤ dad(G) for all U ∈ U .Hence we only need to verify the inequality dad(G) ≤ max U ∈U dad(G| U ). Inductively, it will be enough to show this for a cover by two open sets U 0 and U 1 .Moreover, using Morita invariance, we may pass to the groupoid G[U ] to assume that the U 0 and U 1 are disjoint, i.e. a clopen cover.Given an open, relatively compact subset K ⊆ G the set K ∩ G| U 0 is open and relatively compact in G| U 0 .So by assumption, we may find an open cover V 0,0 , . . ., V 0,d of (s an open cover of s(K) ∪ r(K) and Lemma 2.7 implies that for each 0 ≤ i ≤ d the groupoid K ∩ G| V i is relatively compact.
Note, that we can also combine Theorem 2.6 with Proposition 2.3 to obtain an estimate for infinite covers.We will leave the details to the reader.

2.3.
A product theorem.The goal of this subsection is to establish an estimate for the dynamic asymptotic dimension of the product of two étale groupoids.
Theorem 2.8.Let G and H be étale Hausdorff groupoids.Then Note, that one can find a multiplicative estimate for dad(G × H) in a straightforward fashion from the definitions.The proof of the additive formula in Theorem 2.8 however is more involved. Let Note that a d-dimensional control function for G exists if and only if dad(G) ≤ d.The following definition and results are inspired by [4].In order to state them let us introduce the following terminology: A collection We note that a (d, d)-dimensional control function is just a d-dimensional control function in the sense of the previous definition.There is also a version of the dimension control function for continuous homomorphisms: To prove the main technical proposition in this subsection we need the following facts about n-fold covers: Lemma 2.11.Let X be a compact Hausdorff space.
(1) A collection {U 0 , . . ., U d } of open subsets of X is an n-fold cover of X if and only if Proof.
Conversely, assume for contradiction that there exists an x ∈ X contained in at most n − 1 members of the cover.Then By assumption x must then be contained in U i for some i ∈ F c which contradicts our choice of F .
(2) Using the first part of this Lemma, i which already form a cover of X.Another application of item (1) concludes the proof.
The following proposition shows how to obtain (d, k)-dimensional control functions for every k ≥ d starting from a (d, d)-dimensional control function.This is the main technical ingredient needed to prove the main results.
This means that g = g n • • • g 1 for g 1 , . . ., g n ∈ K such that s(g j ), r(g j ) ∈ U ′ i = KU i .It follows that for each k, there exists a h j ∈ K such that s(h j ) = s(g j ) and r(h j ) ∈ U i .If we set g ′ j := h j+1 g j h −1 j ∈ K 3 .Note that s(g ′ j ), r(g ′ j ) ∈ U i and hence it follows that For a 1-cover this can be found using [11,Lemma 7.4].In the general case we can also follow the proof of this result, but use Lemma 2.11 above, when shrinking covers.
The additional open set needed at stage k + 1 will be the set where S runs through the subsets of {0, . . ., k} of cardinality k+1 for all 1 ≤ l ≤ n.Then there exist subsets S 1 , . . ., S n+1 ⊆ {0, . . ., k} of cardinality k + 1 − d such that s(g l ) ∈ Observe, that S 1 = . . .= S n+1 .Indeed, suppose for contradiction that there is some index 1 ≤ l ≤ n such that S l+1 = S l .Then we may assume without loss of generality that there exists an i ∈ S l \ S l+1 .Since i ∈ S l we have Since S l+1 = S l for all 1 ≤ l ≤ n, s(g l ), r(g l ) ∈ V j ⊆ U j for all j ∈ S l+1 = S l and hence To complete the proof we will show that x ∈ U ′ k+1 .To see this note that the assumption together with the fact that By our choices of the cover (U i ) i above, the set {i | x ∈ U i } has cardinality at least dad +1 (H) and similarly, the set {i | y ∈ V i } has cardinality at least dad +1 (G).Since both of them are subsets of {0, . . ., k}, their intersection cannot be empty, which proves our claim.To complete the proof note that Finally, consider the case that G 0 and H 0 are merely locally compact.Note that G 0 and H 0 are open in their respective one-point compactifications and (G In [18,Proposition 2.3] the authors prove a multiplicative formula for the nuclear dimension of tensor products of C * -algebras.Combining Theorem 2.8 with the main results in [11,6] yields improved estimates for the nuclear dimension of tensor products of groupoid C * -algebras: Corollary 2.13.Let (G 1 , Σ 1 ) and (G 2 , Σ 2 ) be two twisted étale groupoids.Then Proof.We only need to note that ) and apply the results mentioned above.
Remark 2.14.It seems reasonable to expect that there is a general Hurewicz type result for the dynamic asymptotic dimension.To be a little more precise: If π : G → H is a continuous homomorphism between two étale groupoids, we can define the dynamic asymptotic dimension of π as We have the following examples: With these two examples in mind we conjecture that for any continuous groupoid homomorphism π : G → H we have dad(G) ≤ dad(π) + dad(H).
Proof.The assumptions imply that the partial action admits a Hausdorff globalisation.In other words, there exists a free global action of Γ on some locally compact Hausdorff space Y containing X as an open subset and such that Γ ⋉ X is Morita equivalent to Γ ⋉ Y .Note that dim(Y ) = 0 as well by [10,Proposition 3.2].Hence Proposition 2.5 implies that dad(Γ ⋉ X) = dad(Γ ⋉ Y ) and by [5,Theorem 10.7], the latter is bounded above by asdim(Γ).
Of course the collection D γ need not always be closed.Nevertheless, we should expect the same upper bound on the dynamic asymptotic dimension as the following examples shows: Example 2.16.Let X be a zero-dimensional metrisable space and θ : U → V a homeomorphism between two open sets such that θ generates a free partial action of Z.Let be the associated transformation groupoid.Using [10,Corollary 3.6] there exist partial actions θ (k) for k ∈ N such that each θ (k) admits a Hausdorff globalisation, and such that Z ⋉ θ X can be written as an increasing union Thus, for each k ∈ N there exists a zero-dimensional Hausdorff space Y k , and a (global for all k ∈ N and hence dad(Z ⋉ θ X) ≤ 1 as well by Proposition 2.3.

Asymptotic dimension
In this second part of the article we will compare the dynamic asymptotic dimension of an étale groupoid G with the classical asymptotic dimension of G with respect to a canonical coarse structure on G. Coarse structures on étale groupoids have been studied before by other authors, see for example [14,Remark 4.15].
Let us first specify which coarse structure we want to consider on a σcompact étale groupoid G: Let E G be the collection of subsets of Then E is a coarse structure on G.The elements of E G are called controlled sets.Note, that the coarse structure on G also induces a coarse structure E G x on each of the range fibres G x by intersecting each controlled set Example 3.1.Let Γ ⋉ X be the transformation groupoid for an action of a countable discrete group Γ on a locally compact space X.Restricting the canonical coarse structure considered above to any range fibre (Γ ⋉ X) x and identifying it with Γ in the canonical way, gives rise to the coarse structure on Γ described by Roe in [17,Example 2.13].
Let us now recall the definition of asymptotic dimension: If E is a controlled set for a coarse space (X, E), then a family Moreover, X is said to have asymptotic dimension at most d if d is the smallest number with the following property: For any controlled set E there exists a controlled set F and a cover U of X which is F -bounded and admits a decomposition Since the asymptotic dimension of a subspace is at most the asymptotic dimension of the ambient space, we have the obvious estimate (1) sup In the case of a transformation groupoid Γ ⋉ X considered in Example 3.1 all the range fibres canonically identify with the group Γ itself and hence the asymptotic dimension of (Γ ⋉ X, E Γ⋉X ) coincides with the asymptotic dimension of Γ with respect to the canonical coarse structure.However, the reverse of inequality (1) may fail, because even if asdim(G x ) < ∞ for all x ∈ G 0 the sets F x obtained from the definition of asymptotic dimension that are controlling the size of the members of the cover, may grow in an uncontrollable manner as x varies across G 0 .Another example where we do get an equality is the following: where we adopt the convention that Q 0 = G 0 .We say that G is treeable G admits a graphing such that every g ∈ G \ G 0 has a unique (reduced Treeable groupoids are in some sense the analogues of free groups in the world of groupoids.Indeed, if S denotes a free generating set for F n and we are given an action F n X on a compact space, then F n ⋉ X is a treeable groupoid with Q = S × X. We are now going to show that asdim(G, E G ) = 1 for any treeable groupoid G. Let Q be a graphing as in the definition above.This graphing induces a length function ℓ : G → [0, ∞) given as the length of the unique reduced factorisation of g as a product of elements in Q.The function ℓ is continuous since Q is open, and controlled and proper since Q is relatively compact.We denote by be open and relatively compact.Then there exists an N ∈ N such that K ⊆ B N .We consider the "annuli" If we only consider those annuli indexed by even (resp.odd) numbers A 2k (resp.A 2k+1 ) then these families are pairwise K-disjoint.However, they are not yet uniformly bounded.Hence we need to further subdivide each annulus.For k ≥ 2 we define an equivalence relation on A k by setting g ∼ k h if g and h have the same past up to distance N (k − 1) from the origin, i.e. if g = g 1 • • • g m and h = h 1 • • • h l are the unique reduced factorisations of G with elements in Q, then g i = h i for all i ≤ N (k − 1).This is clearly an equivalence relation on A k and we denote the equivalence class of Hence the diameter of each equivalence class is uniformly bounded by 4N .Since A 0 and A 1 are also bounded we can set As we have seen, the even and odd annuli are already N -disjoint, so we only have to check for N -disjointness within each A k for k ≥ 2. So fix k ≥ 2 and let g, h ∈ A k be such that they do not agree on the first N (k − 1) elements in the unique factorisation.Let j ∈ {1, . . ., N (k − 1)} be the minimal number such that g j = h j in the unique factorisations of g and h.Then we compute This finishes the proof.
The following Proposition gives the first half of Theorem B. Proof.We may assume that d := dad(G) < ∞ since otherwise there is nothing to prove.Let E be a controlled set for G. Since G 0 is compact, there exists a relatively compact open subset Using the assumption, find open subsets U 0 , . . ., U d covering G 0 such that the subgroupoid H i := K ∩ G| U i of G is relatively compact (and open) for every i = 0, . . ., d.
Then F is by its very definition a controlled set for G. Let ∼ i denote the equivalence relation on G U i given by g ∼ i h :⇔ r(g) = r(h) and g −1 h ∈ H i .
Then each Then there exist g 0 , h 0 ∈ G U i such that g 0 ∼ i g and h 0 ∼ i h such that (g 0 , h 0 ) ∈ E. In particular, we have g −1 0 h 0 ∈ K, which implies g 0 ∼ i h 0 and hence Moreover, the collection Remark 3.4.We remark that our assumption that G 0 is compact in the previous proposition cannot be relaxed.Consider for example the action of the integers Z on R by translation.This action is free and proper.Using this it is not hard to show that dad(Z ⋉ R) = 0. On the other hand, asdim(Z) = In what follows we want to prove the converse in the case that G 0 is zero-dimensional.The proof is inspired by the recent article [5].
We need a variant of dynamic asymptotic dimension that keeps track of the size of the subgroupoids obtained in the definition.Given open relatively compact subsets K, L ⊆ G, we say that an open subgroupoid H ⊆ G has (K, L)-dad at most d if there exists a cover of H 0 ∩ (s(K) ∪ r(K)) by open sets U 0 , . . ., U d such that K ∩ G| U i ⊆ L for all 0 ≤ i ≤ d.
Similarly, a coarse space (X, E) has (E, F ) − asdim at most d if there exists a cover U of X which is F -bounded and admits a decomposition Proof.This follows inductively from Lemma 2.7.
As a simple application of this Lemma, we obtain: Lemma 3.6.Let G be an étale groupoid.Suppose Assume that there exists an increasing sequence The next Lemma is well-known: Lemma 3.7.Let G be a compact principal ample groupoid.Then there exists a clopen fundamental domain Y * ⊆ H 0 , i.e.Y * meets each H-orbit in H 0 exactly once.
The key technicalities for the main result of this section are contained in the following Lemma.The main obstacle in generalising the proof presented in [5] for group actions to the case of general groupoids is that the decompositions of the fibres G x obtained from the finite asymptotic dimension assumption are not uniform as x varies.In contrast, if we take a decomposition of Γ = T 0 ⊔ . . .⊔ T d as in the definition of asymptotic dimension at most d, then T i × X gives a decomposition of the transformation groupoid, which works uniformly over X. Lemma 3.8.Let G be a pricipal, ample groupoid and let Y ⊆ G 0 be a clopen subset.Assume further, that for given Proof.Fix x ∈ Y for the moment.We can use the assumption to obtain a partition G x = T x 0 ⊔ . . .⊔ T x d where each T x i further decomposes as such that each D x i,j is L-bounded and D x i,j 1 and D x i,j 2 are K-disjoint for all j 1 = j 2 .Intersecting each component of this partition with H K (Y ) gives a partition of H K (Y ) x with the same properties.Let D x i,j := D x i,j ∩ H K (Y ) and T x i := T x i,j ∩ H K (Y ) denote these intersections.Our first goal is to show that we can make this decomposition work uniformly over a neighbourhood V x of x.Note that since H K (Y ) is compact, only finitely many of the sets D x i,j are non-empty and each of them has to be finite.For fixed i, j write D x i,j = {g 1 , . . ., g n }.Using continuity of the product map we can find a compact open bisection U k around each g k such that U −1 k U l ⊆ L. We may also assume that the U k are pairwise disjoint and all have the same range (if that's not the case, replace U k by U k ∩r −1 ( l r(U l ))).Let D x,i,j := n l=1 U l (we put the x in the subscript to indicate that this set still depends on x, but need not be a subset of G x ).By construction D x,i,j is compact open and L-bounded.By enumerating the set {j | D x i,j = ∅} we can successively shrink the sets D x,i,j to make sure that they remain K-disjoint.Let V x := i,j r(D x,i,j ).Then V x is a clopen neighbourhood of x.Replace each D x,i,j by D x,i,j ∩ r −1 (V x ) and let T x,i := i D x,i,j .Then for each y ∈ V x H K (Y ) y = T y x,0 ⊔ . . .⊔ T y x,d and each T y x,i further decomposes as For y = x we recover the original decomposition found in the beginning, i.e.D x x,i,j = D x i,j .Using that Y is compact we can thus partition Y by finitely many sets V 1 , . . .V p such that for each k ∈ {1, . . ., p} there exists a partition Then U 0 , . . ., U d are clearly clopen subsets of Y .We have to show that K ∩ G| U i is contained in L. To this end write an arbitrary g ∈ K ∩ G| U i as g = g 1 g 2 • • • g n with g k ∈ K and s(g k ), r(g k ) ∈ U i .It follows that for each 1 ≤ k ≤ n there exist h k , h ′ k ∈ T i with r(h k ), r(h ′ k ) ∈ Y * and s(g k ) = s(h k ) and r(g k ) = s(h ′ k ).Note that the set {r(h k ), r(h ′ k ) | 1 ≤ k ≤ n} is contained in a single H K (Y )-orbit.Since Y * meets each H K (Y )-orbit exactly once we must have r(h 1 ) = r(h ′ 1 ) = . . .= r(h k ) = r(h ′ k ).Hence h −1 k h ′ k g k ∈ Iso(G) = G 0 , so using principality of G we get g k = (h ′ k ) −1 h k .Moreover, since g k ∈ K, the elements h k and h ′ k are in the same D i,j .So we can write Note further, that since s(h k (h ′ k+1 ) −1 ) = r(h ′ k+1 ) = r(h k ) = r(h k (h ′ k+1 ) −1 ) we can use principality again to conclude that h k = h ′ k+1 .In particular, there exists a unique j such that h k , h ′ k ∈ D i,j for all 1 ≤ k ≤ n.Putting these two facts together we obtain g = (h ′ 1 ) −1 h n ∈ D −1 i,j D i,j ⊆ L as desired.
We can now proceed with the proof of the second half of Theorem B.
If G 0 is compact, equality holds.
Proof.We first prove the result under the additional assumption that G 0 is compact.Let dad(G) ≤ D and d = asdim(G).Let K be a compact open subset of G.We want to show dad(G) ≤ d.Inductively find an increasing sequence (K i ) i of compact opens such that K ∪ G 0 ⊆ K 0 and such that G has (K 15 i , K i+1 )-asdim at most d for every i.Now use the assumption dad(G) ≤ D to find clopen subsets X 0 , . . .X D such that is a compact open subgroupoid.Apply Lemma 3.8 to each X i to see that (K 15  i , K i+1 ) − dad(G| X i ) ≤ d.Hence Lemma 2.7 implies that (K 0 , K 5 D ) − dad(G) ≤ d.As we started with an arbitrary K, this implies that dad(G) ≤ d as desired.
If G 0 is just locally compact and σ-compact, then there exists a nested sequence of compact open sets W n ⊆ G 0 covering G 0 .Let G n := G| Wn be the restriction of G to W n .Then (G n ) n is a nested sequence of compact open subgroupoids of G and hence dad(G n ) ≤ dad(G) < ∞.By the first part of this proof, dad(G n ) ≤ asdim(G n , E Gn ) ≤ asdim(G, E G ) and hence the result follows from Proposition 2.3.Remark 3.10.Let us remark that the assumption dad(G) < ∞ in the previous theorem cannot be dropped: Consider for example the free group F 2 .Since F 2 is residually finite, it admits a decreasing sequence N 1 ⊇ N 2 ⊇ . . . of finite index normal subgroups such that k∈N N k = {e}.For each k ∈ N there is a canonical surjective group homomorphism F 2 /N k+1 → F 2 /N k .Let X be the inverse limit of the sequence F 2 /N 1 ← F 2 /N 2 ← • • • , which as a topological space is a Cantor set.The group F 2 acts from the left on each quotient F 2 /N k and this induces a free and minimal action on X.The space X also admits a unique Borel probability measure µ, induced by the uniform probability measures on the (finite) quotients F 2 /N k .It follows that the transformation groupoid F 2 ⋉X is non-amenable, and hence in particular it cannot have finite dynamic asymptotic dimension by [11,Corollary 8.25].On the other hand we have asdim(F 2 ⋉ X) = asdim(F 2 ) = 1.
Given an étale groupoid G and a subset K ⊆ G, we will write K for the subgroupoid of G generated by K.If K is open, then this subgroupoid is automatically open.With this in mind we can recall the definition of dynamic asymptotic dimension given in [11, Definition 5.1].Definition 2.1.Let G be an étale Hausdorff groupoid and d ∈ N. We say that G has dynamic asymptotic dimension at most d, if for every open relatively compact subset K ⊆ G, there exists a cover of s(K) ∪ r(K) by d + 1 open sets U 0 , . . ., U d such that for each 0 ≤ i ≤ d, the groupoid

Proposition 2 . 12 .
Let G be an étale groupoid with compact unit space.If G admits a d-dimensional control function D G , set D (d) G := D G and inductively define functions D (k)

Proof.
We proceed by induction on k ≥ d.Since G admits a d-dimensional control function by assumption, the base case k = d is obvious.Suppose now that D (k) G is a (d, k)-dimensional dimension function and let K ∈ O c (G) be an open relatively compact subset.Then K 3 ∈ O c (G) as well.Hence the induction hypothesis provides a fold by the induction hypothesis.Fix x ∈ G 0 .If it belongs to k + 2 − d among the sets U ′ 0 , . . ., U ′ k we are done.So let us assume that it belongs exactly to k + 1 − d of the sets U ′ 0 , . . ., U ′ k .
Moreover, x ∈ i∈S V i and our hypothesis implies x / ∈ i∈{0,...,k}\S KV i , which together exactly means that x ∈ U ′ k+1 .Proof of Theorem 2.8.We will first prove the result in the case that G 0 and H 0 are compact.We may assume that dad(G) and dad(H) are both finite.Set k := dad(G) + dad(H).By Proposition 2.12 we may find a (dad(G), k)−dimensional dimension function D G for G and a (dad(H), k)dimensional dimension function D H for H. Now let C ⊆ G × H be open and relatively compact.Since increasing C only makes the problem harder, we may assume that C = K × L for open, relatively compact subsets