Almost elementariness and fiberwise amenability for étale groupoids
aa r X i v : . [ m a t h . OA ] D ec ALMOST ELEMENTARINESS AND FIBERWISE AMENABILITYFOR ´ETALE GROUPOIDS
XIN MA AND JIANCHAO WU
Abstract.
In this paper, we introduce two new types of approximation proper-ties for ´etale groupoids, almost elementariness and (ubiquitous) fiberwise amenabil-ity , inspired by Matui’s and Kerr’s notions of almost finiteness. In fact, we showthat, in their respective scopes of applicability, both notions of almost finitenessare equivalent to the conjunction of our two properties. Our new properties stemfrom viewing ´etale groupoids as coarse geometric objects in the spirit of geomet-ric group theory. Fiberwise amenability is a coarse geometric property of ´etalegroupoids that is closely related to the existence of invariant measures on unitspaces and corresponds to the amenability of the acting group in a transformationgroupoid. Almost elementariness may be viewed as a better dynamical analogueof the regularity properties of C ∗ -algebras than almost finiteness, since, unlikethe latter, the former may also be applied to the purely infinite case. To supportthis analogy, we show almost elementary minimal groupoids give rise to tracially Z -stable reduced groupoid C ∗ -algebras. In particular, the C ∗ -algebras of minimalsecond countable amenable almost finite groupoids in Matui’s sense are Z -stable. Contents
1. Introduction 12. Preliminaries 83. Amenability of extended coarse spaces 114. Coarse geometry of ´etale groupoids 155. Fiberwise amenability 216. Almost elementary ´etale groupoids and groupoid strict comparison 307. Almost elementariness and almost finiteness 398. Small boundary property and a nesting form of almost elementariness 469. Tracial Z -stability 56References 671. Introduction
Groupoids are a generalization of groups where the multiplication operation isallowed to be only partially defined. The study of topological groupoids lies at thecrossroads of group theory, dynamics, geometry, topology, mathematical physics,and operator algebras, largely thanks to their Swiss-knife-like ability to handle manymathematical objects, such as groups, group actions, equivalence relations on topo-logical spaces, nonperiodic tilings, etc., through a unifying framework.
Mathematics Subject Classification.
A recurring theme in many of these mathematical topics is the various waysin which infinite structures may be approximated by finite structures. Analysisoften enters the picture this way. An influential poster child of this theme is thenotion of amenability in group theory, together with its many reincarnations in otherfields, such as injectivity in von Neumann algebra theory, nuclearity in C ∗ -algebraictheory, topological amenability in topological dynamics, and metric amenabilityin coarse geometry, etc. For topological groupoids usually assumed to belocally compact, σ -compact and Hausdorff, and sometimes also ´etale , which is agroupoid generalization of having discretized time, as opposed to continuous time,in a classical topological dynamical system the richness of their structures isreflected in a number of different yet intricately related approximation propertiesthat realize this theme.Among the strongest approximation properties for topological groupoids is thenotion of AF groupoids ([Ren80]). The terminology AF was borrowed from op-erator algebras and was originally short for approximately finite . This propertyapplies to ample (i.e., totally disconnected) ´etale groupoids and demands that ev-ery compact subset of a topological groupoid is contained in a subgroupoid that is elementary namely, it is isomorphic to a principal (i.e., not containing non-trivial subgroups) finite groupoid, typically “fattened up” with topological spaces.The way these elementary groupoids embed reminds one of Kakutani-Rokhlin tow-ers, a fundamental tool in measure-theoretic and topological dynamics. Thanks tothe transparent and rigid structure of elementary groupoids, AF groupoids are wellstudied and classified ([Kri80, GPS04, GMPS08]), though the notion is relativelyrestrictive.Inspired by the Følner set approach to (group-theoretic) amenability, Matui[Mat12] introduced a more general notion called almost finite groupoids , which,like AF groupoids, also applies to ample ´etale groupoids, but it only demandsthat every compact subset of a topological groupoid is almost contained in an el-ementary subgroupoid in a Følner-like sense. Almost finiteness strikes a remark-able balance between applicability and utility: it was shown to be enjoyed by transformation groupoids arising from free actions on the Cantor set by Z n (latergeneralized in [KS20]; see below), as well as those arising from aperiodic tilings([IWZ19]); on the other hand, this notion has found applications in the homologytheory of ample groupoids, topological full groups, and the structure theory and K -theory of reduced groupoid C ∗ -algebras, and deep connections to an increas-ing number of other important properties have been established (see, for example,[Mat15, Mat17, Nek19, Ker20, Suz20]).Focusing on the case of transformation groupoids (arising from topological dynam-ical systems , that is, groups acting on topological spaces), Kerr [Ker20] presented aninsightful perspective that links almost finiteness with regularity properties in theclassification and structure theory of C ∗ -algebras, extending the link between AFgroupoids and AF C ∗ -algebras, as well as paralleling the link between hyperfiniteness for ergodic probability-measure-preserving equivalence relations and hyperfinitenessfor II factors. Here, “regularity properties” refers to a handful of natural and in-trinsic properties of C ∗ -algebras pioneered by Winter that arose from Elliott’s classi-fication program of simple separable nuclear C ∗ -algebras and played pivotal roles in Almost finiteness should not be abbreviated as AF or confused with almost AF groupoids ([Phi05])!
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 3 its eventual success: in short, they were the missing piece needed to characterize the C ∗ -algebras classifiable via the Elliott invariant [GLN15, EGLN15, TWW17, Phi00].Moreover, as predicted by Toms and Winter, this handful of regularity proper-ties turn out to be (mostly) equivalent for simple separable nuclear C ∗ -algebras([Rør04, MS12, Win12, SWW15, TWW17, CET + tracial Z -stability ([HO13]) that Kerr found to be the closest in spiritto Matui’s almost finiteness.Furthermore, Kerr did not merely translate Matui’s almost finiteness into thelanguage of group actions this would just mean that we have a group actionon a totally disconnected space that admits partitions into open (in fact, clopen)Rokhlin towers modelled on Følner sets. He installed an upgrade to its applicability:while the elementary subgroupoids in Matui’s almost finiteness are required to coverthe entire unit space, Kerr’s almost finiteness allows a “small” remainder to be leftuncovered. This slight relaxation has immense consequences: while for actions ona totally disconnected (i.e., zero-dimensional) space, this “small” remainder canalways be absorbed into the Rokhlin towers and we thus recover Matui’s definition,yet Kerr’s version now also applies to actions on higher-dimensional spaces (thoughin this case we also need to explicitly require that the levels of the Rokhlin towers allhave tiny diameters). This idea of approximation modulo a small remainder is wellaligned with the understanding of tracial Z -stability as an analogue of the McDuffproperty that allows for a tracially small error in a way similar to Lin’s earlierdefinition of tracially AF C ∗ -algebras ([Lin01b, Lin01a]).This pivotal upgrade, however, depends on the precise meaning of “small” sets.There are two natural ways to describe them: The first is measure-theoretic: namely,a set is “small” if, with regard to every invariant measure, its measure is smallerthan a predetermined positive number ε . The other is topological: roughly speaking,a set is “small” if it is dynamically subequivalent to a (predetermined) nonemptyopen set , i.e., roughly speaking, the “small” set is able to be disassembled and thentranslated, piece by piece via the group action, into “non-touching” positions insidethe nonempty open set, where “non-touching” means the closures of the translatedpieces do not intersect. It is clear that the second method yields a stricter senseof smallness, and it turns out to be a desirable property of a topological dynamicalsystem for the two methods to agree. This is the essence of what Kerr dubbed dynamical (strict) comparison , after the analogous property of strict comparison ofpositive elements in a C ∗ -algebra, which is also among the aforementioned handfulof C ∗ -algebraic regularity properties.Thus Kerr’s almost finiteness comes in two flavors for higher-dimensional spaces:the ordinary one uses dynamical subequivalence to express smallness of the remain-der and an auxiliary notion called almost finiteness in measure , which uses invari-ant measures instead. Kerr and Szab´o [KS20] showed that the former condition isequivalent to the conjunction of the latter and dynamical strict comparison, whilethe latter condition is equivalent to the small boundary property (which, in turn,is closely related to mean dimension zero as well as C ∗ -regularity properties; see Kerr’s definition actually requires the remainder to be dynamically subequivalent to a smallportion of the unit space of the open elementary subgroupoid, instead of a predeterminednonempty open set. This makes it work better in the non-minimal setting.
XIN MA AND JIANCHAO WU [GK10, EN17, Niu19a, Niu19b, Niu20]). To cement the link to C ∗ -algebraic regu-larity properties, Kerr [Ker20] proved that a free minimal almost finite action ona compact metrizable space by a countable discrete amenable group gives rise to atracially Z -stable crossed product C ∗ -algebra. This was applied in [KS20] to showthat any free minimal action on a finite-dimensional metric space by a group withsubexponential growth produces a classifiable crossed product C ∗ -algebra. Based onthese evidences, Kerr suggested, at least in the case of actions by amenable groups,almost finiteness may be understood as a dynamical regularity property.This great confluence of ideas from topological dynamicals, topological groupoidtheory, and operator algebras opened the gate to a plethora of new connectionsand applications. At the same time, it also left us with a number of unansweredquestions and problems. Q1.
Do almost finite groupoids in Matui’s sense always give rise to groupoid C ∗ -algebras satisfying regularity properties such as tracial Z -stability?Kerr’s result above answered this in the affirmative for transformation groupoids,but it remained largely unclear beyond that case. Kerr’s method appears difficult togeneralize, for it makes use of the fact that the levels in the Rokhlin towers witnessingalmost finiteness are labeled by group elements, which allows one to apply Ornsteinand Weiss’ theory of quasi-tilings to carefully manipulate the towers. Ito, Whittakerand Zacharias [IWZ19] managed to extend Kerr’s method and result to the caseof ´etale groupoids from aperiodic tilings, exploiting the fact that ´etale groupoidsfrom aperiodic tilings are reductions of transformation groupoids associated to R n -actions. Nevertheless, this method appears unsuitable for groupoids without obviousunderlying group structures. Q2.
Is there a groupoid regularity property that works both for groupoids withinvariant measures on their unit spaces and for those without?Both Matui’s and Kerr’s notions of almost finiteness imply the existence of in-variant measures on the unit space of a groupoid and thus the existence of traces intheir groupoid C ∗ -algebras. While this is often a useful feature, it does not line upwith the fact that C ∗ -algebraic regularity properties also applies to purely infinitealgebras , which do not have any traces. Thus one would hope a more relaxed no-tion could include groupoids without invariant measures on their unit spaces. Wepoint out that the correspondence between dynamical strict comparison and pureinfiniteness was established by the first author [Ma20]. Q3.
Related to the previous item: Can we isolate from Matui’s definiton of almostfiniteness an “amenability property” that is responsible for the existence of invariantmeasures?Both Kerr’s and Matui’s definitions made explicit use of Følner sets or a Følner-type condition this is the immediate reason for the existence of invariantmeasures. While Kerr’s notion forces the acting group to be amenable, it suggestedan analogous kind of amenability property for ´etale groupoids might hide behindMatui’s notion. A possible candidate was topological amenability, but this would bethe wrong target, as it does not imply the existence of invariant measures in general,and a number of examples have shown almost finiteness does not imply topologicalamenability ([ABBL20, Ele18]).
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 5
The present paper is intended to address these questions. We summarize ourresults, with the following standing assumption.
Assumption:
The groupoids below are σ -compact, locally compact, Hausdorff,´etale topological groupoids.In addition, many of our results focus on minimal groupoids, as it appears to us atthe moment there lacks a clear vision of the landscape of regularity properties fornon-simple C ∗ -algebra and non-minimal groupoids.Motivated by Q3, we introduce a notion termed fiberwise amenability for ´etalegroupoids (Definition 5.4; see also Proposition 5.5). It simply demands, for any ε and any compact subset K in our ´etale groupoid G , there is a nonempty finite subset F in G such that | KF || F | ≤ ε. Essentially, F is what one may call a Følner set . This notion satisfies the followingbasic properties: S1 (Remark 5.6) . In the case of transformation groupoids, fiberwise amenabilityis equivalent to the amenability of the acting groups (rather than the topologicalamenability of the actions). S2 (Proposition 5.18) . In the case of coarse groupoids of metric spaces, fiberwiseamenability is equivalent to the metric amenability of the underlying metric spaces. S3 (Proposition 5.9) . When the unit space is compact, it implies the existence ofinvariant probability measures on it.While this notion is often easy to verify in concrete examples, yet given thetypically non-homogeneous structure of groupoids, it appears too weak by itselffor many purposes for example, it is not hard to see that ´etale groupoidswith noncompact unit spaces are always fiberwise amenable. This is what led us tointroduce a stronger variant termed ubiquitous fiberwise amenability (Definition 5.4),which requires, in addition to the existence of a single Følner set for each choice of(
K, ε ) as above, that such Følner sets can be found in every source fiber and nearevery element of G in a uniform sense. Although one may quickly see that this isstrictly stronger than fiberwise amenability in general, we show: S4 (Theorem 5.13) . For minimal groupoids, ubiquitous fiberwise amenability isequivalent to fiberwise amenability.To prove the above statement and apply these notions, we develop a way to view´etale groupoids from the lens of coarse geometry, akin to how countable groups aretreated as metric spaces in geometric group theory. S5 (Theorem 4.10 and Definition 4.12) . Up to coarse equivalence, there is a canon-ical right-invariant extended metric on an ´etale groupoid G , which is induced froma proper continuous length function on G .This extended metric may be characterized by the description that a prototypicalbounded neighborhood of an arbitrary set E in G looks like (a subset of) the productset KE for some compact subset K in G . Here “extended” means points may haveinfinite distances. In this case, two elements of G have a finite distance if and only ifthey are on the same source fiber. This entails that our canonical extended metric XIN MA AND JIANCHAO WU rarely induces the topology of G in the usual sense; rather, they are related in thatthe metric changes continuously from one source fiber to another. In fact, we provea local slice lemma (Lemma 5.10) that details how the canonical extended metriccan be locally trivialized. We then observe: S6 (Proposition 5.5) . Both fiberwise amenability and ubiquitous fiberwise amenabil-ity are coarse geometric properties of ´etale groupoids.More precisely, these notions only depend on the coarse geometry of the canonicalright-invariant extended metric. This enables us to use metric techniques to showthat ubiquitous fiberwise amenability has an a priori stronger reformulation: foreach choice of (
K, ε ), we can convert any finite subset of G into a ( K, ε )-Følner setby enlarging it within a uniformly bounded distance (Theorem 5.15). This strongerform of ubiquitous fiberwise amenability plays an important role in our discussion of groupoid strict comparison (Definition 6.2), the natural generalization of dynamicalstrict comparison to the groupoids setting.Motivated by Q2, we introduce, for (minimal) ´etale groupoids with compact unitspaces, a new regularity property and approximation property called almost elemen-tariness , which generalizes both Matui’s and Kerr’s almost finiteness (c.f., Section 7).In fact, we show: S7 (Theorem 7.8) . For transformation groupoids, Kerr’s almost finiteness is equiva-lent to the conjunction of almost elementariness and ubiquitous fiberwise amenabil-ity (i.e., the amenability of the acting group). S8 (Theorem 7.4 and Proposition 7.6) . For groupoids with totally disconnectedunit spaces, Matui’s almost finiteness is also equivalent to the conjunction of almostelementariness and ubiquitous fiberwise amenability.Our definition replaces the Følner-type conditions in Matui’s and Kerr’s almostfiniteness by a new condition that requires the elementary subgroupoids in the ap-proximation to be extendable to a larger elementary subgroupoid. This conditionagain draws inspiration from coarse geometry, e.g., from how, in the definition ofthe asymptotic dimension of a metric space, we ask for a cover that is able to growor shrink by a predetermined large distance without losing its desired structure, inthis case, the chromatic number as well as the property of being a cover. Similarly,in our definition of almost elementary groupoids, we would like our elementary sub-groupoid to be able to grow by a predetermined distance (as measure by compactsubsets in the groupoid) without losing its elementariness (alternatively, it shouldbe able to shrink without jeopardizing the smallness of the remainder). Replacingthe Følner-type conditions by this new condition allows our notion to break freefrom fiberwise amenability and apply to groupoids both with and without invariantprobability measures on their unit spaces. This inclusiveness with regard to invari-ant measures underlies the fact that the groupoid C ∗ -algebras of almost elementarygroupoids include both stably finite algebras and purely infinite ones , a fact thatmakes almost elementariness a candidate for a closer analogue of C ∗ -algebraic reg-ularity properties.To support this analogy with C ∗ -algebraic regularity properties, we establishseveral connections. First of all, we have This is one reason why we decided not to use the word “finite” in the name of our new notion,after a helpful suggestion of George Elliott to the second author.
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 7 S9 (Remark 6.18) . For minimal almost elementary groupoids, almost elementarinessimplies groupoid strict comparison.As indicated above, ubiquitous fiberwise amenability and the coarse geometryof ´etale groupoids play important roles in this result. As a consequence of thisinteraction between these properties, we have the following link between the measurestructure and the geometric structure of a minimal almost elementary groupoid:
S10 (Remark 6.18) . For minimal almost elementary groupoids, fiberwise amenabil-ity is equivalent to the existence of invariant probability measures on the unit spaces.We also establish a direct link to tracial Z -stablility, providing an affirmativeanswer to Q1. Recall that to show a C ∗ -algebra A is tracially Z -stable, we need toproduce order zero maps from arbitrarily large matrix algebras into A with approx-imately central images and with only “tracially small” defects from being unital.We also remind the reader of our standing assumption above. S11 (Theorem 9.7) . Let G be a second countable minimal groupoid with a compactunit space. Suppose G is almost elementary (e.g., almost finite). Then the reducedgroupoid C ∗ -algebra C ∗ r ( G ) is tracially Z -stable.We remark that if G is also ample, one can drop the assumption of second count-ability in the above (see Corollary 9.9). This answers Q1 above. In addition, our the-orem generalizes Kerr’s result on almost finite actions of amenable groups ([Ker20])in several ways: our theorem can be applied to groupoids without obvious underly-ing group structures and without topological amenability; even when restricted totransformation groupoids, our result can now deal with possibly non-free actions bypossibly nonamenable groups. For example, using a result of [OS20], S11 can beimmediately applied to all minimal actions on the Cantor set by the infinite dihedralgroup Z ⋊ Z , regardless of freeness (c.f., Example 9.13).Moreover, as indicated in Q1, our proof necessarily takes an approach differ-ent from Kerr’s. In place of the Ornstein-Weiss tiling theory of amenable groups,we develop a “nesting” form of almost elementariness, which is an approximation(modulo a small remainder) of the groupoid G by not one, but two open elemen-tary subgroupoids in a nested position, a notion reminiscent of how multi-matrixalgebras embed into each other. Thus passing from approximation by one elemen-tary subgroupoid to approximation by a nesting of two (or perhaps more) is akin tohow, from the local definition of an AF algebra, one can produce a tower of nestedmulti-matrix algebras organized by a Bratteli diagram. Indeed, an open elementarysubgroupoid of G will induce an order zero map from a multi-matrix algebra into C ∗ r ( G ), and if another open elementary subgroupoid is nested in the first one, thenwe have an embedding between two multi-matrix algebras. By arranging the nest-ing to have a large multiplicity , we can ensure this embedding has a large relativecommutant. This will essentially be the source of the desired large matrix algebratogether with an order zero map into C ∗ r ( G ) with an approximately central image.The existence of the remainders in these approximations unfortunately makes theproof appear technically complicated, but their smallness will eventually guaranteethat the resulting order zero map only has a “tracially small” defect from beingunital.The following is a direct consequence of S11 by combining results in [CET + XIN MA AND JIANCHAO WU
S12 (Corollary 9.10) . Let G be a second countable amenable minimal ´etale groupoidwith a compact unit space. Suppose G is almost elementary. Then C ∗ r ( G ) is unitalsimple separable nuclear and Z -stable and thus has nuclear dimension one. Inaddition, in this case C ∗ r ( G ) is classified by its Elliott invariant. Finally, if M ( G ) = ∅ ,then C ∗ r ( G ) is quasidiagonal; if M ( G ) = ∅ , then C ∗ r ( G ) is a unital Kirchberg algebra.We will also provide several explicit examples in the last section. Acknowledgements . The authors would like to thank David Kerr for his helpful com-ments during a number of discussions, as well as Guoliang Yu and Zhizhang Xie fortheir hospitality during the authors’ visits to Texas A&M University, where a largepart of the work was completed.2.
Preliminaries
In this section we recall some basic backgrounds on coarse geometry, ´etale groupoidsand C ∗ -algebras.In this paper, there are two types of metric spaces under consideration. One con-cerns usual topological metrizable spaces focus on local behavior while another arecoarse metric spaces from large scale geometric point of view. However, even thesetwo types have different nature, as metric spaces, they share some same notations.Let ( X, d ) be a metric space equipped with the metric d . We denote by B d ( x, R )the open ball B d ( x, R ) = { y ∈ X : d ( x, y ) < R } and by ¯ B d ( x, R ) the closed ball¯ B d ( x, R ) = { y ∈ X : d ( x, y ) ≤ R } . Let A be a subset of X . We write B d ( A, R )and ¯ B d ( A, R ) for analogous meaning. If the metric is clear, we write B ( A, R ) and¯ B ( A, R ) instead for simplification. We refer to [NY12] as a standard reference fortopics of large scale geometry.We refer to [Ren80] and [Sim17] as references for groupoids and we record severalfundamental definitions and results for locally compact Hausdorff ´etale groupoidshere.
Definition 2.1. A groupoid G is a set equipped with a distinguished subset G (2) ⊂G × G , called the set of composable pairs , a product map G (2) → G , denoted by( γ, η ) γη and an inverse map G → G , denoted by γ γ − such that the followinghold(i) If ( α, β ) ∈ G (2) and ( β, γ ) ∈ G (2) then so are ( αβ, γ ) and ( α, βγ ). In addition,( αβ ) γ = α ( βγ ) holds in G .(ii) For all α ∈ G one has ( γ, γ − ) ∈ G (2) and ( γ − ) − = γ .(iii) For any ( α, β ) ∈ G (2) one has α − ( αβ ) = β and ( αβ ) β − = α .Every groupoid is equipped with a subset G (0) = { γγ − : γ ∈ G} of G . We refer toelements of G (0) as units and to G (0) itself as the unit space . We define two maps s, r : G → G (0) by s ( γ ) = γ − γ and r ( γ ) = γγ − , respectively, in which s is calledthe source map and r is called the range map.When a groupoid G is endowed with a locally compact Hausdorff topology underwhich the product and inverse maps are continuous, the groupoid G is called alocally compact Hausdorff groupoid. A locally compact Hausdorff groupoid G iscalled ´etale if the range map r is a local homeomorphism from G to itself, whichmeans for any γ ∈ G there is an open neighborhood U of γ such that r ( U ) is openand r | U is a homeomorphism. In this case, since the map of taking inverses is an LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 9 involutive homeomorphism on G that intertwines r and s , thus the source map s isalso a local homeomorphism. A set B is called a bisection if there is an open set U in G such that B ⊂ U and the restriction of the source map s | U : U → s ( U ) and therange map r | U : U → r ( U ) on U are both homeomorphisms onto open subsets of G (0) . It is not hard to see a locally compact Hausdorff groupoid is ´etale if and onlyif its topology has a basis consisting of open bisections. We say a locally compactHausdorff ´etale groupoid G is ample if its topology has a basis consisting of compactopen bisections. Example 2.2.
Let X be a locally compact Hausdorff space and Γ be a discretegroup. Then any action Γ y X by homeomorphisms induces a locally compactHausdorff ´etale groupoid X ⋊ Γ := { ( γx, γ, x ) : γ ∈ Γ , x ∈ X } equipped with the realtive topology as a subset of X × Γ × X . In addition, ( γx, γ, x )and ( βy, β, y ) are composable only if βy = x and( γx, γ, x )( βy, β, y ) = ( γβy, γβy, y ) . One also defines ( γx, γ, x ) − = ( x, γ − , γx ) and announces that G (0) := { ( x, e Γ , x ) : x ∈ X } . It is not hard to verify that s ( γx, γ, x ) = x and r ( γx, γ, x ) = γx . Thegroupoid X ⋊ Γ is called a transformation groupoid .The following are several basic properties of locally compact Hausdorff ´etalegroupoids whose proofs could be found in [Sim17].
Proposition 2.3.
Let G be a locally compact Hausdorff ´etale groupoid. Then G (0) is a clopen set in G . Proposition 2.4.
Let G be a locally compact Hausdorff ´etale groupoid. Suppose U and V are open bisections in G . Then U V = { αβ ∈ G : ( α, β ) ∈ G (2) ∩ U × V } isalso an open bisection. It is also convenient to define, for n = 1 , , . . . , the set of composable n -tuples G ( n ) = { ( x , . . . , x n ) ∈ G n : s ( x i ) = r ( x i +1 ) for i = 1 , , . . . , n − } and the n -ary multiplication map δ ( n ) : G ( n ) → G , ( x , . . . , x n ) x · · · x n . Corollary 2.5.
Let G be a locally compact Hausdorff ´etale groupoid. Then for any n ∈ { , , . . . } , the n -ary multiplication map is a local homeomorphism. We also record the following useful fact about local homeomorphisms.
Lemma 2.6.
Let f : X → Y be a local homeomorphism between topological spaceswith Y being Hausdorff. Then for any y ∈ Y and any compact subset K ⊆ X , thereare an open neighborhood U of y in Y and a finite family of open subsets V , . . . , V n in X such that(1) the map f restricts to a homeomorphism between V i and U , for any i ∈{ , . . . , n } , and(2) we have f − ( U ) ∩ K ⊆ V ∪ . . . ∪ V n . Proof.
Since f is a local homeomorphism, we know for any x ∈ f − ( y ), there areopen neighborhoods V x of x and U x of y such that f restricts to a homeomor-phism between V x and U x . Since the collection (cid:8) f − ( Y \ { y } ) , V x : x ∈ f − ( y ) (cid:9) form an open cover of K , by compactness, there are x , . . . , x n ∈ f − ( y ) suchthat (cid:8) f − ( Y \ { y } ) , V x i : i = 1 , . . . , n (cid:9) form a finite open cover of K . Let L = K \ S ni =1 V x i , which is a closed subset of K and thus also compact; so is the im-age f ( L ). Observe that L ⊆ f − ( Y \ { y } ), i.e., y f ( L ). Since a Hausdorffspace has separation between a point and a compact set, there is an open neigh-borhood W of y in Y such that W ∩ f ( L ) = ∅ . Let U = W ∩ ( T ni =1 U x i ) andlet V i = (cid:16) f | V xi (cid:17) − ( U ), for i = 1 , . . . , n . They clearly satisfy the first condi-tion. As for the second condition, we observe that f − ( U ) ∩ L = ∅ and thus f − ( U ) ∩ K = f − ( U ) ∩ ( L ∪ V ∪ . . . ∪ V n ) ⊆ V ∪ . . . ∪ V n . (cid:3) For any set D ⊂ G (0) , Denote by G D := { γ ∈ G : s ( γ ) ∈ D } , G D := { γ ∈ G : r ( γ ) ∈ D } , and G DD := G D ∩ G D . For the singleton case D = { u } , we write G u , G u and G uu instead for simplicity. Inthis situation, we call G u a source fiber and G u a range fiber . In addition, each G uu is a group, which is called the isotropy at u . We also denote byIso( G ) = [ u ∈G (0) G uu = { x ∈ G : s ( x ) = r ( x ) } the isotropy of the groupoid G . We say a groupoid G is principal if Iso( G ) = G (0) .A groupoid G is called topologically principal if the set { u ∈ G (0) : G uu = { u }} isdense in G (0) . The groupoid G is also said to be effective if Iso( G ) o = G (0) . Recallthat effectiveness is equivalent to topological principalness if G is second countable(See [Sim17, Lemma 4.2.3]). Therefore, effectiveness is equivalent to the topologicalfreeness of an action of a countable discrete group acting on a compact metrizablespace by looking at the corresponding transformation groupoid.A subset D in G (0) is called G - invariant if r ( G D ) = D , which is equivalent to thecondition G D = G D . Note that G| D := G DD is a subgroupoid of G with the unit space D if D is a G -invariant set in G (0) . A groupoid G is called minimal if there are noproper non-trivial closed G -invariant subsets in G (0) .Let G be a locally compact Hausdorff ´etale groupoid. We define a convolutionproduct on C c ( G ) by ( f ∗ g )( γ ) = X αβ = γ f ( α ) g ( β )and an involution by f ∗ ( γ ) = f ( γ − ) . These two operations make C c ( G ) a ∗ -algebra. Then the reduced groupoid C ∗ -algebra C ∗ r ( G ) is defined to be the completion of C c ( G ) with respect to the norm k·k r induced by all regular representation π u for u ∈ G (0) , where π u : C c ( G ) → B ( ℓ ( G u ))is defined by π u ( f ) η = f ∗ η and k f k r = sup u ∈G (0) k π u ( f ) k . It is well known thatthere is a C ∗ -algebraic embedding ι : C ( G (0) ) → C ∗ r ( G ). On the other hand, E : C c ( G ) → C c ( G (0) ) defined by E ( a ) = a | G (0) extends to a faithful canonicalconditional expectation E : C ∗ r ( G ) → C ( G (0) ) satisfying E ( ι ( f )) = f for any f ∈ C ( G (0) ) and E ( ι ( f ) aι ( g )) = f E ( a ) g for any a ∈ C ∗ r ( G ) and f, g ∈ C ( G (0) ). LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 11
As a typical example, it can be verified that for the transformation groupoid inExample 2.2, the reduced groupoid C ∗ -algebra is isomorphic to the reduced crossedproduct C ∗ -algebra of the dynamical system. The following are some standard factson reduced groupoid C ∗ -algebras that could be found in [Sim17]. Throughout thepaper, the notation supp o ( f ) for a function f on a topological space X denotes theopen support { x ∈ X : f ( x ) = 0 } of f . In addition, we write supp( f ) the usualsupport supp o ( f ) of f . We say an open set O in a topological space X is precompact if O is compact. Proposition 2.7.
Let G be a locally compact Hausdorff ´etale groupoid. Any f ∈ C c ( G ) can be written as a sum f = P ni =0 f i such that there are precompact openbisections V , . . . , V n such that V ⊂ G (0) and V i ∩ G (0) = ∅ for all < i ≤ n as wellas supp( f i ) ⊂ V i for any ≤ i ≤ n . Proposition 2.8.
Let G be a locally compact Hausdorff ´etale groupoid. Suppose U, V are open bisections and f, g ∈ C c ( G ) such that supp( f ) ⊂ U and supp( g ) ⊂ V .Then supp( f ∗ g ) ⊂ U · V and for any γ = αβ ∈ U · V one has ( f ∗ g )( γ ) = f ( α ) g ( β ) . Proposition 2.9.
Let G be a locally compact Hausdorff ´etale groupoid. For f ∈ C c ( G ) , one has k f k ∞ ≤ k f k r . If f is supported on a bisection, then one has k f k ∞ = k f k r . Let G be a locally compact Hausdorff ´etale groupoid. Suppose U is an openbisection and f ∈ C c ( G ) + such that supp( f ) ⊂ U . Define functions s ( f ) , r ( f ) ∈ C ( G (0) ) by s ( f )( s ( γ )) = f ( γ ) and r ( f )( r ( γ )) = f ( γ ) for γ ∈ supp( f ). Since U is abisection, so is supp( f ). Then the functions s ( f ) and r ( f ) are well-defined functionssupported on s (supp( f )) and r (supp( f )), respectively. Note that s ( f ) = ( f ∗ ∗ f ) / and r ( f ) = ( f ∗ f ∗ ) / .The Jiang-Su algebra Z , introduced in [JS99] by Jiang and Su, is a infinite di-mensional unital nuclear simple separable C ∗ -algebra, but KK-equivalent to C inthe sense of Kasparov. We say a C ∗ -algebra A is Z - stable if A ⊗ Z ≃ A .Finally, throughout the paper, we write B ⊔ C to indicate that the union of sets B and C is a disjoint union. In addition, we denote by F i ∈ I B i for the disjoint unionof the family { B i : i ∈ I } . We also denote by ⌈·⌉ the ceiling function and by ⌊·⌋ thefloor function from [0 , ∞ ) to N .3. Amenability of extended coarse spaces
In this section, we recall and study the amenability of (uniformly locally finite) ex-tended metric spaces from a coarse geometric point of view. In particular, we intro-duce a strengthening of the notion of metric amenability called ubiquitous (metric)amenability , which will be a central tool in our investigation of coarse structuresof groupoids. In particular, we prove a pair of lemmas at the end of the sectionthat display how ubiquitous amenability and non-amenability lead to constrastingbehaviors on bounded enlargements of arbitrary finite subsets in metric spaces.
Definition 3.1.
Recall an extended metric space is a metric space in which themetric is allowed to take the value ∞ . An extended metric space admits a uniquepartition into ordinary metric spaces, called its coarse connected components , suchthat two points have finite distance if and only if they are in the same coarse con-nected component. An extended metric space is called locally finite if any boundedset has finite cardinality. Let (
X, d ) be a locally finite extended metric space and A be a subset of X . Forany R > A :(i) outer R -boundary : ∂ + R A = { x ∈ X \ A : d ( x, A ) ≤ R } ;(ii) inner R -boundary : ∂ − R A = { x ∈ A : d ( x, X \ A ) ≤ R } ;(iii) R -boundary : ∂ R A = { x ∈ X : d ( x, A ) ≤ R and d ( x, X \ A ) ≤ R } . Remark 3.2.
Let (
X, d ) be an extended metric space. Suppose A ⊂ X and R > ∂ + R A ⊂ ¯ B ( ∂ − R A, R ) and ∂ − R A ⊂ ¯ B ( ∂ + R A, R ).The following concept of amenability of extended metric spaces was introducedin [BW92] by Block and Weinberger and further studied in [ALLW18] by Ara, Li,Lled´o, and the second author.
Definition 3.3.
Let (
X, d ) be a extended locally finite metric space.(i) For
R > ǫ >
0, a finite non-empty set F ⊂ X is called ( R, ǫ )-Følner ifit satisfies | ∂ R F || F | ≤ ǫ. We denote by Føl(
R, ǫ ) the collection of all (
R, ǫ )-Følner sets.(ii) The space (
X, d ) is called amenable if, for every
R > ǫ >
0, thereexists an (
R, ǫ )-Følner set.The following elementary lemma shows that Følner sets can always be “localized”to a single coarse connected component.
Lemma 3.4.
Let ( X, d ) be a extended locally finite metric space and let X i ,i ∈ I ,be its coarse connected components. Fix R, ε > and let F be an ( R, ǫ ) -Følner setof X . Write F i = F ∩ X i for each i ∈ I . Then there is i ∈ I such that F i is alsoan ( R, ǫ ) -Følner set.Proof. Suppose for any i ∈ I , F i were not an ( R, ǫ )-Følner set, i.e., either F i = ∅ or | ∂F i | > ǫ | F i | . Observe that ` ∂F = F i ∈ I ∂F i and only finitely many among the F i ’sare non-empty. Thus we would have | ∂F | = X i ∈ I | ∂F i | > X i ∈ I ǫ | F i | = ǫ | F | , a contradiction to the assumption that F is an ( R, ǫ )-Følner set. (cid:3)
In this paper, we also need the following stronger version of this amenability.
Definition 3.5.
An extended metric space (
X, d ) is called ubiquitously amenable (or ubiquitously metrically amenable ) if, for every
R > ǫ >
0, there exists an
S > x ∈ X , there is an ( R, ǫ )-Følner set F in the ball ¯ B ( x, S ).An extended metric space ( X, d ) is called uniformly locally finite if for any R > R . i.e., sup x ∈ X | ¯ B ( x, R ) | < ∞ . To simplify the notation, for uniformly locally finite space (
X, d ), we define anfunction N : R + → N by(3.1) N ( r ) = sup x ∈ X | ¯ B ( x, r ) | . This notion also appeared as bounded geometry in the literature.
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 13
The following criteria for establishing amenability for uniformly locally finite ex-tended metric spaces is straightforward but useful.
Proposition 3.6.
Let ( X, d ) be a uniformly locally finite extended metric space.The following are equivalent.(i) For any R, ǫ > there is a finite set F ⊂ X such that | ∂ R F | ≤ ǫ | F | .(ii) For any R, ǫ > there is a finite set F ⊂ X such that | ∂ + R F | ≤ ǫ | F | .(iii) For any R, ǫ > there is a finite set F ⊂ X such that | ∂ − R F | ≤ ǫ | F | .(iv) For any R, ǫ > there is a finite set F ⊂ X such that | ¯ B ( F, R ) | ≤ (1 + ǫ ) | F | .Proof. The proof is straightforward by observing that for any
R > F ⊂ X one has ∂ R F = ∂ + R F ∪ ∂ − R F and ∂ + R F = ¯ B ( F, R ) \ F as well as the facts | ∂ + R F | ≤ N ( R ) | ∂ − R F | and | ∂ − R F | ≤ N ( R ) | ∂ + R F | by Remark 3.2. (cid:3) The following lemma is useful in establishing Proposition 3.8 and 3.9.
Lemma 3.7.
Suppose that ( X, d ) is a uniformly locally finite extended metric space.Let s > and n ∈ N . Then for any finite set F ⊂ X satisfying | F | ≥ n · N ( s ) , thereexist distinct points x , . . . , x n ∈ F such that, for all i, j ∈ { , . . . , n } with i = j , onehas d ( x i , x j ) > s . In particular there exists at least n many disjoint balls ¯ B ( x i , s/ for i = 1 , . . . , n in ¯ B ( F, s/ .Proof. We choose points x , . . . , x n by induction. First pick x ∈ F . Suppose that x , . . . , x k has been defined such that d ( x i , x j ) > s for all 1 ≤ i = j ≤ k . Thenobserve that | k [ i =1 ¯ B ( x i , s ) | ≤ k X i =1 | ¯ B ( x i , s ) | ≤ k · N ( s ) . Then choose x k +1 ∈ F \ S ki =1 ¯ B ( x i , s ) whose cardinality satisfies | F \ k [ i =1 ¯ B ( x i , s ) | ≥ ( n − k ) · N ( s ) . This finishes the proof. (cid:3)
The following proposition shows that Følner sets appear quite repetitively inubiquitously amenable and uniformly locally finite spaces.
Proposition 3.8.
Let ( X, d ) be a ubiquitously amenable and uniformly locally finiteextended metric space. Then for all r, ǫ > there exists an S > such that for allfinite sets M ⊂ X there exists a finite set F with M ⊂ F ⊂ ¯ B ( M, S ) and | ∂ + r F || F | ≤ ǫ. Proof.
Suppose the contrary, i.e., there exist r, s >
X, d ) is ubiquitously amenable, for the r, ǫ > s ′ > x ∈ X there is an finite set F x ⊂ ¯ B ( x, s ′ ) such that | ∂ + r F x || F x | ≤ ǫ/ . Now define n = ⌈ N ( r ) ǫ ⌉ · N (2 s ′ ) , ˆ S = ⌈ log (1+ ǫ ) n ⌉ · r and S = ˆ S + s ′ + 1 . Then for the S there is a finite set M in X such that for every finite F with M ⊂ F ⊂ ¯ B ( M, S ) always satisfies | ∂ + r F || F | > ǫ. Thus in particular one has | ¯ B ( F, r ) || F | > ǫ. Then by induction we define F = M and F k +1 = ¯ B ( F k , r ) for k ∈ N . Thisimplies that | F k +1 | = | ¯ B ( F k , r ) | > (1 + ǫ ) | F k | whenever F k ⊂ ¯ B ( M, S ). Thus if F k ⊂ ¯ B ( M, S ) one has | ¯ B ( M, kr ) || M | ≥ | F k || F | > (1 + ǫ ) k . In particular, by the definition of ˆ S one has | ¯ B ( M, ˆ S ) || M | ≥ (1 + ǫ ) ⌈ log (1+ ǫ ) n ⌉ > n = ⌈ N ( r ) ǫ ⌉ · N (2 s ′ ) . Now write m = ⌈ N ( r ) /ǫ ⌉ · | M | . Then Lemma 3.7 implies that there are distinctpoints x , . . . , x m ∈ ¯ B ( M, ˆ S ) such that d ( x i , x j ) > s ′ . Then we write F = M ∪ ( F mi =1 F x i ) ⊂ ¯ B ( M, S ) because all F x i ⊂ ¯ B ( x i , s ′ ). Then we have | ∂ + r ( F ) || F | ≤ | ∂ + r ( M ) | + P mi =1 | ∂ + r ( F x i ) | P mi =1 | F x i | = | ∂ + r ( M ) || M | · | M | P mi =1 | F x i | + m X i =1 | ∂ + r ( F x i ) || F x i | · | F x i | P mi =1 | F x i | . Because | ∂ + r ( M ) | ≤ | ¯ B ( M, r ) | ≤ N ( r ) | M | and | M | P mi =1 | F x i | ≤ | M | m ≤ ǫ N ( r ) , one has | ∂ + r ( M ) || M | · | M | P mi =1 | F x i | ≤ ǫ . On the other hand, by the definition of all F x i one has m X i =1 | ∂ + r ( F x i ) || F x i | · | F x i | P mi =1 | F x i | ≤ ǫ · m X i =1 | F x i | P mi =1 | F x i | = ǫ . This implies that | ∂ + r ( F ) || F | ≤ ǫ, which is a contradiction. This finishes the proof. (cid:3) In contrast, we show below a paradoxical phenomenon in non-amenable extendedmetric spaces that can be considered as the polar opposite of the above.
Proposition 3.9.
Let ( X, d ) be a uniformly locally finite extended metric space,which is not amenable. For all n ∈ N and R > there exists an S > such that forall finite set M in X there are at least n | M | many disjoint R -balls, ¯ B ( x i , R ) : i =1 , . . . , n | M | , contained in ¯ B ( M, S ) . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 15
Proof.
Let n ∈ N and R >
0. Since (
X, d ) is not amenable, there is an ǫ > r > F in X one has | ¯ B ( F, r ) || F | > ǫ. Choose a k ∈ N such that (1 + ǫ ) k ≥ n · N (2 R ). Then define F = M and F k +1 =¯ B ( F k , r ) for k ∈ N . for all finite set M ⊂ X one has | ¯ B ( M, kr ) || M | ≥ (1 + ǫ ) k ≥ n · N (2 R ) . We write S = kr + R . Then Lemma 3.7 implies that there exist distinct points x , . . . , x n | M | ∈ ¯ B ( M, kr ) such that d ( x i , x j ) > R for all i = j ∈ { , . . . , n | M |} .In particular there exists at least n | M | disjoint balls ¯ B ( x i , R ) for i = 1 , . . . , n in¯ B ( M, S ). (cid:3) Coarse geometry of ´etale groupoids
A fundamental and motivating fact in coarse geometry is that one can always as-sign a length function to a countable discrete group, which induces a (right-)invariantproper metric on the group, in a way unique up to coarse equivalence for afinitely generate group, this amounts to taking the graph metric of the Cayley graph(after fixing a set of generators). In this procedure, the amenability of the group it-self is equivalent to the metric amenability mentioned in last section of the resultingmetric space.Motivated by this, one may establish a similar framework for locally compact σ -compact Hausdorff ´etale groupoids, realizing them as extended metric spaces byequipping metrics to all the source (or range) fibers in a uniform and invariantmanner. We remark that fiberwise defined Caylay graph has also been considered in[Nek19] for ample groupoids to study topological full groups. We start our discussionwithout the topological structure. Definition 4.1.
An extended metric on a groupoid G is • invariant (or, more precisely, right-invariant ) if, for any x, y, z ∈ G with s ( x ) = s ( y ) = r ( z ), we have ρ ( x, y ) = ρ ( xz, yz ); • fiberwise (or, more precisely, source-fiberwise ) if, for any x, y ∈ G , we have ρ ( x, y ) = ∞ if and only if s ( x ) = s ( y ).Just as in the case of groups, it is more efficient to encode invariant metrics bylength functions. To the best knowledge of the authors, the discussion of lengthfunctions on ´etale groupoids first appeared in [OY19, Definition 2.21], with ideasfrom J.-L. Tu. Our terminology differs slightly. Definition 4.2.
Recall a length function on a groupoid G is a function ℓ : G → [0 , ∞ )satisfying, for any x, y ∈ G ,(i) ℓ ( x ) = 0 if and only if x ∈ G (0) ,(ii) (symmetricity) ℓ ( x ) = ℓ ( x − ), and(iii) (subadditivity) ℓ ( xy ) ≤ ℓ ( x ) + ℓ ( y ) if x and y are composable in G .On a groupoid G , there is a canonical one-to-one correspondence between lengthfunctions and invariant fiberwise extended metrics. On the one hand, given any length function ℓ on G , we associate an extended metric ρ ℓ by declaring, for x, y ∈ G , ρ ℓ ( x, y ) = ( ℓ ( xy − ) , s ( x ) = s ( y ) ∞ , s ( x ) = s ( y ) . On the other hand, given any invariant fiberwise extended metric ρ on G , we asso-ciate a function ℓ ρ : G → [0 , ∞ ) , g ρ ( g, s ( g )) , which does not take the value ∞ since ρ is fiberwise. Lemma 4.3.
On a groupoid G , the above assignments give rise to a pair of bijectionsbetween length functions and invariant fiberwise extended metrics.Proof. It is routine to verify that ρ ℓ as defined above is indeed an extended metric,where positive definiteness and symmetricity of ℓ lead to those of ρ ℓ and subad-ditivity leads to the triangle inequality. It is also clear that ρ ℓ is invariant andfiberwise. On the other hand, to verify that ℓ ρ is a length function, we see, with thehelp of invariance, the same correspondence between the conditions in the oppositedirection. (cid:3) Now we focus on ´etale groupoids. We show, in analogy with the case of groups,that a locally compact σ -compact Hausdorff ´etale groupoid determines, up to coarseequivalence, a canonical invariant fiberwise extended metric that enjoys the followingproperties. Definition 4.4.
Let ℓ : G → [0 , ∞ ) be a length function on an ´etale groupoid G .For any subset K ⊆ G , we write ℓ ( K ) = sup x ∈ K ℓ ( x ) . We say ℓ is • proper if, for any K ⊂ G \ G (0) , ℓ ( K ) < ∞ implies that K is precompact, • controlled if, for any K ⊂ G , ℓ ( K ) < ∞ is implied by that K is precompact,and • coarse if it is both proper and controlled. • continuous if it is a continuous function with regard to the topology of G .Two length functions ℓ , ℓ are said to be coarsely equivalent if for any r >
0, wehave sup { ℓ ( x ) : ℓ ( x ) ≤ r } < ∞ and sup { ℓ ( x ) : ℓ ( x ) ≤ r } < ∞ . It is straightforward to see that coarse equivalence of length functions is indeedan equivalence relation, and a continuous length function is controlled. We may alsoexpress coarse equivalence using control functions , as is common in coarse geometry.
Lemma 4.5.
Two length functions ℓ , ℓ are coarsely equivalent if and only if thereare non-decreasing unbounded functions f + , f − : [0 , ∞ ) → [0 , ∞ ) (sometimes re-ferred to as control functions ) such that (4.1) f − ( ℓ ( x )) ≤ ℓ ( x ) ≤ f + ( ℓ ( x )) for any x ∈ G . Moreover, we may also assume f + (0) = f − (0) = 0 in the above. LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 17
Proof.
Assuming there are non-decreasing unbounded functions f + , f − : [0 , ∞ ) → [0 , ∞ ) satisfying (4.1), then for any r >
0, we havesup { ℓ ( x ) : ℓ ( x ) ≤ r } ≤ sup { f + ( ℓ ( x )) : ℓ ( x ) ≤ r } ≤ f + ( r ) < ∞ , andsup { ℓ ( x ) : ℓ ( x ) ≤ r } ≤ sup { ℓ ( x ) : f − ( ℓ ( x )) ≤ r } ≤ sup { s : f − ( s ) ≤ r } < ∞ , thanks to the unboundedness of f − .On the other hand, assuming ℓ , ℓ are coarsely equivalent as above, we maydefine f + ( r ) = sup { r, ℓ ( x ) : ℓ ( x ) ≤ r } and f − ( r ) = inf { r, ℓ ( x ) : ℓ ( x ) ≥ r } for r ∈ [0 , ∞ ). It is immediate that both functions are nondecreasing, f + is un-bounded, f + (0) = f − (0) = 0, and (4.1) is satisfied. To see f − is unbounded, weobserve that for any r, s ≥
0, we have f − ( r ) < s if and only if either r < s orthere is x ∈ G such that ℓ ( x ) < s but ℓ ( x ) ≥ r , the latter possibility implying r ≤ sup { ℓ ( x ) : ℓ ( x ) < s } . This shows that f − − ([0 , s ]) is bounded for any s ≥ f − is unbounded. (cid:3) Remark 4.6.
Under the correspondence of Lemma 4.3, coarse equivalence of twolength functions ℓ and ℓ translates to coarse equivalence of their induced extendedmetrics ρ ℓ and ρ ℓ , that is, we havesup { ρ ℓ ( x, y ) : ρ ℓ ( x, y ) ≤ r } < ∞ and sup { ρ ℓ ( x, y ) : ρ ℓ ( x, y ) ≤ r } < ∞ , or equivalently, there are nondecreasing unbounded functions f + , f − : [0 , ∞ ) → [0 , ∞ ) (sometimes referred to as control functions ) such that f − ( ρ ℓ ( x, y )) ≤ ρ ℓ ( x, y ) ≤ f + ( ρ ℓ ( x, y )) for any x, y ∈ G , where we adopt the convention that f + ( ∞ ) = f − ( ∞ ) = ∞ . Lemma 4.7.
Any two coarse length functions on an ´etale groupoid G are coarselyequivalent to each other.Proof. Given two coarse length functions ℓ , ℓ on G , we see that for any r >
0, by theproperness of ℓ , the set (cid:8) g ∈ G \ G (0) : ℓ ( g ) ≤ r (cid:9) is precompact, and thus by thefacts that ℓ is controlled and ℓ ( G (0) ) = { } , we have sup { ℓ ( g ) : ℓ ( g ) ≤ r } < ∞ .Similarly, we have sup { ℓ ( g ) : ℓ ( g ) ≤ r } < ∞ , as desired. (cid:3) Remark 4.8.
A coarse length function on an ´etale groupoid G is bounded if andonly if G \ G (0) is compact. This follows directly from Definition 4.4 and the factthat G (0) is open in G .To prove the existence of coarse continuous length functions, we make use of thefollowing simple topological fact. We include the proof for completeness. Recall acontinuous function g between two topological spaces is called proper if g − ( K ) iscompact for any compact set K . Lemma 4.9.
Let X be a σ -compact, locally compact and Hausdorff space. Thenthere exists a continuous proper map g : X → [0 , ∞ ) .Proof. Choose a sequence of compact subsets K ⊂ K ⊂ . . . ⊂ X with K i ⊆ K o i +1 for each i and X = S ∞ i =0 K i . It follows that the closed sets K , ∂K , ∂K , . . . are disjoint, allowing us to define g ( K ) = { } and g ( ∂K i ) = { i } for i = 1 , , . . . . Applying the Tietze extension theorem to the compact Hausdorffspaces K i +1 \ K o i , for i = 0 , , , . . . , we obtain a continuous function g : X → [0 , ∞ )mapping K i +1 \ K o i into [ i, i + 1], for i = 0 , , , . . . , which implies it is proper. (cid:3) We remark that such a continuous proper function g is bounded if and only if X is bounded. Theorem 4.10.
Up to coarse equivalence, any σ -compact locally compact Hausdorff´etale groupoid has a unique coarse continuous length function.Proof. Uniqueness up to coarse equivalence follows from Lemma 4.7. It remains toshow existence. To get started, we choose a continuous function f : G → { }∪ [1 , ∞ )such that f − (0) = G (0) , f ( x ) = f (cid:0) x − (cid:1) for any x ∈ G , and f | G\G (0) is proper, i.e.,for any r ≥
1, the inverse image f − ([1 , r ]) is compact. Indeed, to construct f , wefirst observe that since G (0) is a clopen subset of G , the complement G \ G (0) is also σ -compact, locally compact and Hausdorff, which enables us to apply Lemma 4.9to obtain a proper continuous function g : G \ G (0) → [0 , ∞ ) and then define f ( x ) = ( , x ∈ G (0) g ( x )+ g ( x − ) , x ∈ G \ G (0) , which clearly satisfies all the requirements.We define a function ℓ : G → [0 , ∞ ) by ℓ ( x ) = inf k X j =1 f ( y j ) : k ∈ N and y , . . . , y k ∈ G \ G (0) such that x = y . . . y k s ( x ) for x ∈ G , where the degenerate case of k = 0 corresponds to x = s ( x ) and ℓ ( x ) = 0.It is immediate that the function ℓ defined above is a length function on G .To study ℓ , we describe an equivalent definition of it in terms of the sets G ( n ) of composable n -tuples and the n -ary multiplication maps δ ( n ) . For n = 1 , , . . . ,define˚ G ( n ) = G ( n ) ∩ (cid:16) G \ G (0) (cid:17) n = n ( x , . . . , x n ) ∈ G ( n ) : x , . . . , x n ∈ G \ G (0) o , ˚ δ ( n ) = δ ( n ) | ˚ G ( n ) : ˚ G ( n ) → G , ( x , . . . , x n ) x · · · x n ,f ( n ) : G ( n ) → [0 , ∞ ) , ( x , . . . , x n ) k X j =1 f ( x j ) , ˚ f ( n ) = f ( n ) | ˚ G ( n ) : ˚ G ( n ) → [0 , ∞ ) . Combining these definitions, we have ℓ ( x ) = inf { f ( x ) } ∪ ∞ [ j =2 ˚ f ( j ) (cid:18)(cid:16) ˚ δ ( j ) (cid:17) − ( { x } ) (cid:19) for any x ∈ G . Moreover, observe that for n = 1 , , . . . , the range of ˚ f ( n ) is contained in [ n, ∞ ) and,for any r ≥
1, we have (cid:16) ˚ f ( n ) (cid:17) − ([1 , r ]) ⊆ (cid:0) f − ([1 , r ]) (cid:1) n . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 19
Hence for any x ∈ G and N ∈ N satisfying ℓ ( x ) ≤ N , we may remove large valuesthat do not affect the infimum and obtain(4.2) ℓ ( x ) = inf { f ( x ) } ∪ N [ j =2 ˚ f ( j ) (cid:18)(cid:0) f − ([1 , N ]) (cid:1) j ∩ (cid:16) ˚ δ ( j ) (cid:17) − ( { x } ) (cid:19) . To show that ℓ is proper, let K ⊂ G \ G (0) and suppose ℓ ( K ) < N for some N ∈ N . Then for any x ∈ K , since f ( x ) ≥
1, it follows from (4.2) that there is n ∈ { , . . . , N } such that the set (cid:0) f − ([1 , N ]) (cid:1) n ∩ (cid:16) ˚ δ ( n ) (cid:17) − ( { x } )is non-empty. Therefore we have K ⊆ N [ j =1 f − ([1 , N ]) · · · f − ([1 , N ]) | {z } j , the latter set being a finite union of products of compact sets, and thus compact.This shows that ℓ defined above is proper.To show that ℓ is continuous, it suffices to prove that for any x ∈ G , there is anopen neighborhood U of x such that ℓ is continuous when restricted to U . To thisend, we let N = ⌈ ℓ ( x ) + 1 ⌉ . Thus for any y in the open neighborhood f − ([0 , N ))of x , the formula (4.2) applies with y in place of x . Observe that for j = 1 , , . . . N ,by Corollary 2.5 and the fact that ˚ G ( j ) is a clopen subset of G ( j ) , we know ˚ δ ( j ) isa local homeomorphism. Applying Lemma 2.6 with ˚ δ ( j ) , x , and (cid:0) f − ([1 , N ]) (cid:1) j inplace of f , y , and K , we may find an open neighborhood U ( j ) of x inside f − ([0 , N ))and a finite family of open subsets V ( j )1 , . . . , V ( j ) m j in ˚ G ( j ) such that ˚ δ ( j ) restricts to ahomeomorphism between V ( j ) i and U ( j ) , for any i ∈ { , . . . , m j } , and we have (cid:0) f − ([1 , N ]) (cid:1) j ∩ (cid:16) ˚ δ ( j ) (cid:17) − (cid:16) U ( j ) (cid:17) ⊆ V ( j )1 ∪ . . . ∪ V ( j ) m j . Writing η ( j ) i : U ( j ) → V ( j ) i for the inverse of ˚ δ ( j ) | V ( j ) i , for i = 1 , . . . , m j . Then forany y ∈ U ( j ) , we have (cid:0) f − ([1 , N ]) (cid:1) j ∩ (cid:16) ˚ δ ( j ) (cid:17) − ( { y } ) = (cid:0) f − ([1 , N ]) (cid:1) j ∩ n η ( j ) i ( y ) : i = 1 , . . . , m j o . Let U = U ∩ . . . ∩ U N . Then, for any y ∈ U , we may rewrite (4.2) as(4.3) ℓ ( y ) = min n f ( y ) , (cid:16) ˚ f ( j ) ◦ η ( j ) i (cid:17) ( y ) : j = 1 , . . . , N, i = 1 , . . . , m j o . Hence on the open neighborhood U of x , ℓ is equal to the minimum of finite numberof continuous functions, and is thus itself continuous, as desired.The controlledness of ℓ follows directly from its continuity. Therefore ℓ is a coarsecontinuous length function. (cid:3) Remark 4.11.
If a σ -compact locally compact Hausdorff ´etale groupoid G is alsoample, then we can choose a coarse length function ℓ that is locally constant. Indeed,when carrying out the proof of Theorem 4.10, we observe that we can choose thefunction g and thus also the function f to be locally constant, by choosing anincreasing sequence of open compact subsets K ⊂ K ⊂ . . . ⊂ G\G (0) with G\G (0) = S ∞ i =0 K i and then defining g ( x ) = min { i : x ∈ K i } for all x ∈ G \G (0) . It then follows from (4.3) that ℓ is locally equal to the minimum of a finite collection of locally finitefunctions, and thus is itself locally constant. Definition 4.12.
Let G be a σ -compact locally compact Hausdorff ´etale groupoid.The unique-up-to-coarse-equivalence invariant fiberwise extended metric induced bya coarse continuous length function on G will be called a canonical extended metric.We will abuse notation and denote any such metric by ρ or ρ G . Example 4.13.
Let X be a σ -compact locally compact Hausdorff space and Γ bea countable group that acts on X by homeomorphisms. Then the transformationgroupoid X ⋊ Γ (c.f., Example 2.2) is also σ -compact. To construct a coarse con-tinuous length function on X ⋊ Γ, we may fix a proper length function ℓ Γ on Γ anda continuous proper function g : X → [0 , ∞ ), and then define ℓ X ⋊ Γ : X ⋊ Γ → [0 , ∞ ) , ( γx, γ, x ) ℓ Γ ( γ ) (1 + max { g ( γx ) , g ( x ) } ) . Note that when X is compact, then we may simply choose g = 0 and thus ℓ X ⋊ Γ ( γx, γ, x ) = ℓ Γ ( γ )for any ( γx, γ, x ) ∈ X ⋊ Γ. Lemma 4.14.
Any canonical extended metric ρ on a σ -compact locally compactHausdorff ´etale groupoid G is uniformly locally finite.Proof. Let ℓ be the coarse length function that induces ρ by Lemma 4.3. Fix R > ℓ is proper, the set L R = (cid:8) z ∈ G \ G (0) : ℓ ( z ) ≤ R (cid:9) is precompact. Then thereis a finite family { V , . . . , V m } of precompact open bisections such that L R ⊂ m [ i =1 V i . Then, for every x ∈ G , one has¯ B ( x, R ) = { y ∈ G : ρ ( y, x ) ≤ R } = { y ∈ G : ℓ ( yx − ) ≤ R } ⊂ { x }∪ L R x ⊂ { x }∪ m [ i =1 V i x. This implies that | ¯ B ( x, R ) | ≤ m + 1. Since R was arbitrarily chosen and m onlydepends on R , thus ρ is uniformly locally finite. (cid:3) Remark 4.15.
For readers familiar with abstract coarse spaces in terms of en-tourages (c.f., [Roe03, Chapter 2]), we point out that the coarse structure on G (as a set) determined by any canonical extended metric can be directly defined asfollows: a subset E of G × G is an entourage if and only if there is a precompactsubset K of G such that for any ( x, y ) ∈ E , we have either x = y or x ∈ Ky . Thisconstruction is different from, but related to, the notion of coarse structures on agroupoid (c.f., [HPR97, TWY18]), in that our entourages are subsets of G × G with G embedded as the diagonal, instead of subsets of G with G (0) playing the role ofa diagonal, but on the other hand, our coarse structure can be viewed as inducedfrom the smallest coarse structure on the groupoid G (generated by the relativelycompact subsets and G (0) ) via the canonical translation action of G on itself.A lot of the contents in this paper may be handled with this abstract coarse struc-ture, which would have the advantage of circumventing the somewhat inconvenientfact that the canonical extended metric is only unique up to coarse equivalence.The definition and some basic properties even extend beyond the case of σ -compact LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 21
Hausdorff ´etale groupoids. However, we opt to stick to the language of metricssince it is more intuitive while σ -compact Hausdorff groupoids are prevalent in themain applications we have in mind (indeed, it is necessary to ensure that C ∗ r ( G ) isseparable). 5. Fiberwise amenability
In this section, we introduce the notions of fiberwise amenability and ubiquitousfiberwise amenability for ´etale groupoids, with inspirations from (uniform) metricamenability (c.f., Definition 3.3 and 3.5). Fiberwise amenability is closely relatedto the existence of invariant measures on the unit space of an ´etale groupoid. As amotivating example, a transformation groupoid is (uniformly) fiberwise amenable ifand only if the acting group is amenable. Ubiquitous fiberwise amenability will playan important auxiliary role when we discuss groupoid strict comparison and almostelementariness for minimal groupoids in the later sections. For this purpose, we showin the later half of this section that for minimal groupoids, fiberwise amenability isalso equivalent to the a priori stronger notion of ubiquitous fiberwise amenability.We first define boundary sets in groupoids in analogy with Definition 3.1.
Definition 5.1.
Let G be a groupoid. For any subsets A, K ⊆ G , we define thefollowing boundary sets:(i) left outer K -boundary : ∂ + K A = ( KA ) \ A = { yx ∈ G \ A : y ∈ K, x ∈ A } ;(ii) left inner K -boundary : ∂ − K A = A ∩ ( K − ( G \ A )) = { x ∈ A : yx ∈ G \ A for some y ∈ K } ;(iii) left K -boundary : ∂ K A = ∂ + K A ∪ ∂ − K A .Observe that if A as above is contained in a single source fiber, then KA and allthese boundary sets are also contained in this source fiber. This is the reason forthe terminology fiberwise amenability . Remark 5.2.
For any subsets
A, K ⊆ G , it is straightforward to see ∂ + K A ⊂ K∂ − K A and ∂ − K A ⊂ K − ∂ + K A .The following concept is analogous to the metric case, too. Definition 5.3.
Let G be a locally compact ´etale groupoid. For any subset K ⊆ G and ǫ >
0, a finite non-empty set F ⊂ X is called ( K, ǫ )-Følner if it satisfies | ∂ K F || F | ≤ ǫ. We denote by Føl(
K, ǫ ) the collection of all (
K, ǫ )-Følner sets.This leads to a natural definition of fiberwise amenability.
Definition 5.4.
Let G be a locally compact ´etale groupoid.(1) We say G is fiberwise amenable if for any compact subset K of G and any ǫ >
0, there exists a (
K, ǫ )-Følner set.(2) We say G is ubiquitously fiberwise amenable if and only if for any compactsubset K of G and any ǫ >
0, there exists a compact subset L of G such thatfor any unit u ∈ G (0) , there is a ( K, ǫ )-Følner set in Lu ∪ { u } .Since the groupoids we focus on are σ -compact and come equipped with extendedmetric structure in a somewhat canonical way (c.f., Definition 4.12), we may also reformulate Definition 5.4 using the canonical extended metric. This will establisha connection with Section 3 and enable us to apply the results there. Proposition 5.5.
Let G be a σ -compact locally compact Hausdorff ´etale groupoidand let ( G , ρ ) be the extended metric space induced by a coarse length function ℓ .(1) The groupoid G is fiberwise amenable if and only if for any compact subset K of G and any ǫ > , there exists a nonempty finite subset F in G satisfying | KF || F | ≤ ǫ, if and only if ( G , ρ ) is amenable in the sense of Definition 3.3..(2) The groupoid G is ubiquitously fiberwise amenable if and only if for anycompact subset K of G and any ǫ > , there exists a compact subset L of G such that for any unit u ∈ G (0) , there is a nonempty finite subset F in Lu ∪ { u } satisfying | KF || F | ≤ ǫ, if and only if ( G , ρ ) is ubiquitously amenable in the sense of Definition 3.5.Proof. We prove the second statement, the first being similar. To prove the threeconditions are equivalent, we first observe that the equivalence of the first two followsfrom Remark 5.2.To prove the last condition is equivalent to the rest, we first observe that byProposition 3.6, the ubiquitous fiberwise amenability of G is equivalent to that forany R > ǫ >
0, there is
S > x ∈ G , there is a nonemptyfinite subset F in B ρ ( x, S ) such that | ¯ B ρ ( F, R ) | ≤ (1 + ǫ ) | F | . Making use of theright-invariance of ρ and replacing x by r ( x ) and F by F x − if necessary, we seethat, without loss of generality, we may replace, in the above, the quantifier x ∈ G by x ∈ G (0) .Now, to prove the ubiquitous fiberwise amenability of G implying ubiquitousamenability of ( G , ρ ), we fix arbitrary R > ǫ >
0, from which we define K tobe the closure of (cid:8) x ∈ G \ G (0) : ℓ ( x ) ≤ R (cid:9) , which is precompact by the propernessof ℓ , and then our assumption provides us a compact subset L of G such thatfor any unit u ∈ G (0) , there is a nonempty finite subset F in Lu ∪ { u } satisfying | KF | ≤ (1 + ǫ ) | F | ; thus setting S = sup x ∈ L ℓ ( x ), which is finite as ℓ is controlled,we see that for any x ∈ G (0) , there is a nonempty finite subset F in B ρ ( x, S ) suchthat | ¯ B ρ ( F, R ) | ≤ (1 + ǫ ) | F | , as desired. The reverse direction follows the samearguments. (cid:3) Remark 5.6.
Let α : Γ y X be an action of a countable discrete group Γ on acompact space X . We denote by X ⋊ α Γ the transformation groupoid of this action α . When we equip Γ with a proper length function ℓ Γ and X ⋊ α Γ with the inducedlength function ℓ X ⋊ α Γ : ( γx, γ, x ) ℓ Γ ( γ ) (c.f., Example 4.13), each source fiber( X ⋊ α Γ) x = { ( γx, γ, x ) : γ ∈ Γ } , for x ∈ X , becomes isometric to Γ. Therefore X ⋊ α Γ is fiberwise amenable if and only if Γ is amenable.In particular, the group Γ as a groupoid is fiberwise amenable if and only if Γ isamenable. In addition, it follows from the homogeneity of a group to see that Γ isamenable if and only if it is ubiquitously fiberwise amenable .
Remark 5.7.
Fiberwise amenability is not an interesting property for groupoids G with noncompact unit spaces, for it is automatically satisfied in this case. Indeed, LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 23 for any compact subset K of G , if we choose an arbitrary point u in G (0) \ s ( K ),then Ku = ∅ and thus { u } becomes a ( K, Definition 5.8.
A measure on the unit space of a locally compact Hausdorff ´etalegroupoid G is invariant if µ ( r ( U )) = µ ( s ( U )) for any measuable bisection U . Wewrite M ( G ) for the collection of all invariant regular Borel probability measures on G (0) . Proposition 5.9.
Let G be a fiberwise amenable, σ -compact, locally compact Haus-dorff ´etale groupoid with a compact unit space. Then M ( G ) = ∅ .Proof. It suffices to show that there is a Borel probability measure µ on G (0) such that µ ( r ( f )) = µ ( s ( f )) for all function f ∈ C c ( G ) + whose support supp( f ) is a compactbisection. We may also assume k f k ≤
1. Write K = supp( f ) for simplicity. Notethat r ( f ) , s ( f ) are functions supported on r ( K ) and s ( K ), respectively. Now, wework in the metric space ( G , ρ ) defined above. First, define R = sup y ∈ K ∪ K − ℓ ( y ) < ∞ . Then one has ( K ∪ K − ) x ⊂ ¯ B ρ ( x, R ) for all x ∈ G .Since G is fiberwise amenable, for a decreasing sequence { ǫ n : n ∈ N } convergingto 0, we can choose a sequence of finite sets { F n ⊂ G : n ∈ N } such that for all n ∈ N one has | ¯ B ρ ( F n , n ) | < (1 + ǫ n / N ( R )) | F n | . Now for each n ∈ N we define µ n = 1 | F n | X x ∈ F n δ r ( x ) which are probability measures on G (0) . Suppose that µ is a w ∗ -cluster point of { µ n : n ∈ N } and in fact we may assume µ n → µ in the w ∗ -topology by passing tosubsequences. We show that µ ∈ M ( G ) by estimating the following | µ ( r ( f )) − µ ( s ( f )) | ≤ | µ ( r ( f )) − µ n ( r ( f )) | + | µ n ( r ( f )) − µ n ( s ( f )) | + | µ ( s ( f )) − µ n ( s ( f )) | . Since K is a bisection, one has | µ n ( r ( f )) − µ n ( s ( f )) | = | | F n | ( X x ∈ F n r ( f )( r ( x )) − X x ∈ F n s ( f )( r ( x ))) | = | | F n | ( X x ∈ F n r ( f )( r ( x )) − X x ∈ F n r ( f )( r ( Kx ))) | = | | F n | ( X x ∈ F n r ( f )( r ( x )) − X x ∈ KF n r ( f )( r ( x ))) | . ≤ | F n | | X x ∈ KF n ∆ F n r ( f )( r ( x )) | ≤ k r ( f ) k | KF n ∆ F n || F n | . Now for n > R , since K is a bisection, observe that F n ∩ KF n = { γ ∈ F n : K − γ ∈ F n } = F n \ ∂ − K − F n . Then because ∂ − K − F n ⊂ K · ∂ + K − F n , one has | F n \ KF n | = | ∂ − K − F n | ≤ N ( R ) | ∂ + K − F n | ≤ ǫ n | F n | . This shows that | KF n ∆ F n | = | KF n \ F n | + | F n \ KF n | ≤ ǫ n | F n | . Now for every ǫ > n > R big enough such that | µ ( r ( f )) − µ n ( r ( f )) | <ǫ/ | µ ( s ( f )) − µ n ( s ( f )) | < ǫ/ ǫ n < ǫ/
6. This implies that | µ ( r ( f )) − µ ( s ( f )) | < ǫ. This establishes µ ( s ( f )) = µ ( r ( f )) as desired. (cid:3) In the rest of the section, we show that for minimal groupoids, fiberwise amenabil-ity is equivalent to the a priori stronger notion of ubiquitous fiberwise amenability.The strategy to show the former implies the latter, roughly speaking, is: on the onehand, a Følner set on a single source fiber, is always able to “permeate” horizon-tally to nearby fibers; on the other hand, the recurrence behavior guaranteed byminimality allows every source fiber to “pick up” a Følner set from this permeationevery so often, thus resulting in ubiquitous fiberwise amenability.To explain how this “permeation” arises, it is convenient to use the followingresult about the existence of local trivializations that almost preserve the metric.
Lemma 5.10 ( Local Slice Lemma ) . Let G be a σ -compact locally compact Hausdorff´etale groupoid and let ρ be a canonical extended metric on G induced by a coarsecontinuous length function ℓ as in Definition 4.12. Let u ∈ G (0) . Then for any R, ε > , there are a number S ∈ [ R, R + ε ) , an open neighborhood V of u in G (0) ,an open set W in G , and a homeomorphism f : ¯ B ρ ( u, S ) × V → W such that(1) f ( u, v ) = v for any v ∈ V ,(2) f ( x, u ) = x for any x ∈ ¯ B ρ ( u, S ) ,(3) f (cid:0) ¯ B ρ ( u, S ) × { v } (cid:1) = ¯ B ρ ( v, S ) for any v ∈ V , and(4) | ρ ( x, y ) − ρ ( f ( x, v ) , f ( y, v )) | < ε for any x, y ∈ ¯ B ρ ( u, S ) and v ∈ V .Proof. By Lemma 4.14, the “open” ball B ρ ( u, R + ε ), i.e., the set { x ∈ G u : ℓ ( x ) < R + ε } ,is finite, and thus ℓ ( B ρ ( u, R + ε )) = max { ℓ ( x ) : x ∈ B ρ ( u, R + ε ) } < R + ε. Hence we may choose S ∈ [ R, R + ε ) ∩ (cid:0) ℓ ( B ρ ( u, R + ε )) , R + ε (cid:1) , e.g., S = max (cid:26) R, ℓ ( B ρ ( u, R + ε )) + R + ε (cid:27) , which guarantees ¯ B ρ ( u, S ) = B ρ ( u, R + ε ) and thus S > ℓ (cid:0) ¯ B ρ ( u, S ) (cid:1) .For each x ∈ ¯ B ρ ( u, S ), choose an open bisection U x containing x and let f x : s ( U x ) → U x be the inverse of the homeomorphism s | U x . With out loss of generality,we may assume U u = G (0) and f u is the identity map. Define L = ℓ − ([0 , S ]) \ [ x ∈ ¯ B ρ ( u,S ) U x and U = G (0) \ s ( L ) . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 25
Unpacking the definition and using the fact ¯ B ρ ( v, S ) = ℓ − ([0 , S ]) ∩ s − ( v ) for any v ∈ G (0) , we have(5.1) U = v ∈ G (0) : ¯ B ρ ( v, S ) ⊆ [ x ∈ ¯ B ρ ( u,S ) U x and, in particular, u ∈ U . Since ℓ is proper and continuous, we see that L is compactand hence U is an open neighborhood of u in G (0) . Define a continuous map f : ¯ B ρ ( u, S ) × U → G , ( x, v ) f x ( v ) . It follows from the construction of the f x ’s that(1) f ( u, v ) = v for any v ∈ U , and(2) f ( x, u ) = x for any x ∈ ¯ B ρ ( u, S ).We also have ( s ◦ f ) ( x, v ) = v for any ( x, v ) ∈ ¯ B ρ ( u, S ) × U . It then follows from(5.1) that(5.2) f (cid:0) ¯ B ρ ( u, S ) × { v } (cid:1) ⊇ ¯ B ρ ( v, S ) for any v ∈ U. Now we define a finite collection of continuous maps g xy : U → [0 , ∞ ) , v ρ ( f ( x, v ) , f ( y, v )) , for x, y ∈ ¯ B ρ ( u, S ) and define η = min (cid:26) ε, S − ℓ (cid:0) ¯ B ρ ( u, S ) (cid:1) , ρ ( x, y )2 : x, y ∈ ¯ B ρ ( u, S ) with x = y (cid:27) . Note that η > S . By continuity, there exists an open neighborhood V of u inside U such that | g xy ( u ) − g xy ( v ) | < η for any v ∈ V and x, y ∈ ¯ B ρ ( u, S ).This choice implies the following:(3) For any x, y ∈ ¯ B ρ ( u, S ) and v ∈ V , since g xy ( u ) = ρ ( x, y ), we have | ρ ( x, y ) − ρ ( f ( x, v ) , f ( y, v )) | < ε. (4) For any x ∈ ¯ B ρ ( u, S ) and v ∈ V , we have ℓ ( f ( x, v )) = ρ ( f ( x, v ) , f ( u, v )) = g xu ( v ) < g xu ( u ) + η = ℓ ( x ) + η ≤ ℓ (cid:0) ¯ B ρ ( u, S ) (cid:1) + η ≤ S , and thus combinedwith (5.2), we have f (cid:0) ¯ B ρ ( u, S ) × { v } (cid:1) = ¯ B ρ ( v, S ) . • For any v ∈ V and any x, y ∈ ¯ B ρ ( u, S ) with x = y , we have ρ ( f ( x, v ) , f ( y, v )) = g xy ( v ) > g xy ( u ) − η = ρ ( x, y ) − η > f ( x, v ) = f ( y, v ). This impliesthat the collection (cid:8) f x ( V ) : x ∈ ¯ B ρ ( u, S ) (cid:9) of open sets is disjoint and f is ahomeomorphism onto its image when restricted to ¯ B ρ ( u, S ) × V .Defining W = f (cid:0) ¯ B ρ ( u, S ) × V (cid:1) and restricting f to ¯ B ρ ( u, S ) × V thus completesthe construction. (cid:3) The existence of local slices as in Lemma 5.10 allows us to “clone” a Følner setin every nearby source fiber.
Lemma 5.11.
Let G be a σ -compact locally compact Hausdorff ´etale groupoid andlet ρ be a canonical extended metric on G induced by a coarse continuous lengthfunction ℓ as in Definition 4.12. Let R, S, ε > and u ∈ G (0) . Then there is anopen neighborhood V of u in G (0) such that whenever there exist v ∈ V and an ( R, ε ) -Følner set in ¯ B ρ ( v , S ) , then for any v ∈ V , there is an ( R, ε ) -Følner set in ¯ B ρ ( v, S + ε ) . Proof.
Let R ′ = S + R + ε and η = 12 min (cid:8) ε, ρ ( x, y ) − R : x, y ∈ ¯ B ρ ( u, R ′ ) with ρ ( x, y ) > R (cid:9) . Applying Lemma 5.10 with u , R ′ , and η in place of u , R , ε , we obtain a number S ′ ∈ [ R ′ , R ′ + η ), an open neighborhood V of u in G (0) , an open set W in G , and ahomeomorphism f : ¯ B ρ ( u, S ′ ) × V → W such that(1) f ( u, v ) = v for any v ∈ V ,(2) f ( x, u ) = x for any x ∈ ¯ B ρ ( u, S ′ ),(3) f (cid:0) ¯ B ρ ( u, S ′ ) × { v } (cid:1) = ¯ B ρ ( v, S ′ ) for any v ∈ V , and(4) | ρ ( x, y ) − ρ ( f ( x, v ) , f ( y, v )) | < η for any x, y ∈ ¯ B ρ ( u, S ′ ) and v ∈ V .Now assuming that there is an ( R, ε )-Følner set F in ¯ B ρ ( v , S ), we then define, forany v ∈ V , the bijections τ v : ¯ B ρ ( u, S ) → ¯ B ρ ( v, S ) , x f ( x, v )and the set F v = τ v ◦ τ − v ( F ) . For any v ∈ V , we claim that F v is the desired ( R, ε )-Følner set in ¯ B ρ ( v, S + ε ).Indeed, it follows from condition (4) that ℓ ( F v ) < ℓ (cid:0) τ − v ( F ) (cid:1) + η ≤ ℓ ( F ) + 2 η ≤ S + ε and thus F v ⊆ ¯ B ρ ( v, S + ε ). On the other hand, to see F v is an ( R, ε )-Følner setjust like F , it suffices to show that ∂ + R F v ⊆ τ v ◦ τ − v (cid:0) ∂ + R F (cid:1) and ∂ − R F v ⊆ τ v ◦ τ − v (cid:0) ∂ − R F (cid:1) . To prove the former containment, we observe that for any y ∈ ∂ + R F v , since ℓ ( y ) ≤ ℓ ( F v ) + R ≤ S + R + ε ≤ S ′ , it is in the range of τ v . Let x = τ v ◦ τ − v ( y ). Since y F v , we have x F . It remains to show that ρ ( x, F ) ≤ R . Suppose this werenot the case, i.e., for any z ∈ F , we have ρ ( x, z ) > R . Then by our choice of η , wewould have ρ ( y, F v ) = min (cid:8) ρ (cid:0) τ v ◦ τ − v ( x ) , τ v ◦ τ − v ( z ) (cid:1) : z ∈ F (cid:9) > min { ρ ( x, z ) − η : z ∈ F } ≥ R, contradictory to the fact that y ∈ ∂ + R F v . This shows ∂ + R F v ⊆ τ v (cid:0) ∂ + R F (cid:1) .To prove the latter containment, we observe that any y ∈ ∂ − R F v is in F v andthus we may define x = τ v ◦ τ − v ( y ) in F . It remains to show that ρ ( x, G \ F ) ≤ R .Suppose this were not the case. Then by the decomposition G\ F v = (cid:0) G \ ¯ B ρ ( v, S ′ ) (cid:1) ∪ (cid:0) ¯ B ρ ( v, S ′ ) \ F v (cid:1) and our choice of S and η , we would have ρ ( y, G \ F v ) = inf { ρ ( y, w ) : w ∈ G \ F v } = inf (cid:8) ρ ( y, w ) , ρ (cid:0) τ v ◦ τ − v ( x ) , τ v ◦ τ − v ( z ) (cid:1) : w ∈ G \ ¯ B ρ ( v, S ′ ) , z ∈ ¯ B ρ ( u, S ′ ) \ F (cid:9) > min (cid:8) S ′ − ℓ ( y ) , ρ ( x, z ) − η : z ∈ ¯ B ρ ( u, S ) \ F (cid:9) ≥ R, contradictory to the fact that y ∈ ∂ − R F v . This shows ∂ − R F v ⊆ τ v (cid:0) ∂ − R F (cid:1) and com-pletes the proof. (cid:3) The following lemma underlies the recurrence behavior of minimal groupoids withcompact unit spaces.
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 27
Lemma 5.12.
Let G be a minimal locally compact Hausdorff ´etale groupoid. Let K and V be subsets of G (0) such that K is compact and V is non-empty and open.Then there are precompact open bisections V , . . . , V n such that S ni =1 r ( V i ) ⊆ V and K ⊆ S ni =1 s ( V i ) .Proof. Since G is minimal, for any u ∈ K , there is a v ∈ V and x ∈ G such that r ( x ) = v and s ( x ) = u . Then since G is locally compact ´etale, there is a precompactopen bisection V x such that x ∈ V x ⊂ r − ( V ). This implies that v = r ( x ) ∈ r ( V x ) ⊂ V and u = s ( x ) ∈ s ( V x ). In addition, all such s ( V x )’s form an open cover of K . Bycompactness, there are finitely many precompact open bisection V , . . . , V n such that K ⊆ S ni =1 s ( V i ). In addition, our construction also implies S ni =1 r ( V i ) ⊆ V . (cid:3) Now we are ready to establish the equivalence of fiberwise amenability and ubiq-uitously fiberwise amenability for minimal groupoids.
Theorem 5.13.
Let G be a σ -compact locally compact Hausdorff ´etale groupoid.Suppose G is minimal. Then G is fiberwise amenable if and only if it is ubiquitouslyfiberwise amenable.Proof. The “if” direction follows directly from the definitions. To show the “onlyif” direction, we let ρ be a canonical extended metric on G induced by a coarsecontinuous length function ℓ as in Definition 4.12. By Proposition 5.5, it sufficesto show, assuming the extended metric space ( G , ρ ) is amenable, that it is alsoubiquitously amenable, i.e., for every R > ε >
0, there exists an
S > x ∈ G , there is an ( R, ε )-Følner set F in the ball ¯ B ρ ( x, S ). To thisend, given R, ε >
0, since we assume ( G , ρ ) is amenable, we know there exists an( R, ε )-Følner set F in G . By Lemma 3.4, we may assume without loss of generalitythat F is contained in a single source fiber G u for some u ∈ G (0) . By Lemma 5.11,there is an open neighborhood V of u in G (0) such that for any v ∈ V , there is an( R, ε )-Følner set F v in ¯ B ρ (cid:0) v, ℓ ( F ) + ε (cid:1) . Let K = s (cid:16) ℓ − ([0 , R ]) \ G (0) (cid:17) , which is a compact subset of G (0) , as ℓ is a continuous proper length function. ByLemma 5.12, there are precompact open bisections V , . . . , V n such that S ni =1 r ( V i ) ⊆ V and K ⊆ S ni =1 s ( V i ). Let S = ℓ ( F ) + ε + max (cid:8) ℓ ( V i ) : i = 1 , . . . , n (cid:9) , which is finite since all the sets involved are precompact. Now, for any x ∈ G , weneed to construct an ( R, ε )-Følner set F in ¯ B ρ ( x, S ). There are two cases: • If r ( x ) K , then ¯ B ρ ( x, R ) = ¯ B ρ ( r ( x ) , R ) x = (cid:0) G r ( x ) ∩ ℓ − ([0 , R ]) (cid:1) x = { r ( x ) x } = { x } by our choice of K , and thus we may set F = { x } , which isan ( R, • If r ( x ) ∈ K , then we may choose i x ∈ { , . . . , n } such that r ( x ) ∈ s ( V i x ).Let z ∈ V i x be such that r ( x ) = s ( z ). Note that r ( z ) ∈ V and thus we havean ( R, ε )-Følner set F r ( z ) in ¯ B ρ (cid:0) r ( z ) , ℓ ( F ) + ε (cid:1) . Let F = F r ( z ) zx , whichis also an ( R, ε )-Følner set by the right-invariance of ρ (see Lemma 4.3).Finally, since for any y ∈ F r ( z ) , we have ρ ( yzx, x ) = ρ ( yz, r ( x )) = ℓ ( yz ) ≤ ℓ (cid:0) F r ( z ) (cid:1) + ℓ ( V i x ) ≤ ℓ ( F ) + ε + ℓ ( V i x ) ≤ S , we conclude that F is in theball ¯ B ρ ( x, S ). (cid:3) Corollary 5.14.
Let G be a σ -compact locally compact Hausdorff ´etale groupoid.Suppose G is minimal and G (0) is noncompact. Then G is ubiquitously fiberwiseamenable.Proof. This follows from Theorem 5.13 and Remark 5.7. (cid:3)
The following theorem, as an application of Theorem 5.13, shows a dichotomy onamenability against paradoxicality for locally compact σ -compact Hausdorff ´etaleminimal groupoids on compact spaces, which are main objects for our study in thefollowing sections. Theorem 5.15.
Let G be a locally compact Hausdorff σ -compact minimal ´etalegroupoid. Equip G with the metric ρ as in Definition 4.12. Then we have thefollowing dichotomy.(1) If G is fiberwise amenable then for all R, ǫ > there is a compact set K with G (0) ⊂ K ⊂ G such that for all compact set L ⊂ G and all unit u ∈ G (0) there is a finite set F u satisfying Lu ⊂ F u ⊂ KLu and ¯ B ρ ( F u , R ) ≤ (1 + ǫ ) | F u | . (2) If G is not fiberwise amenable then for all compact set L ⊂ G and n ∈ N thereis a compact set K ⊂ G such that for all compact set M ⊂ G and all u ∈ G (0) ,the set KM u contains at least n | M u | many disjoint sets of the form Lγu ,i.e., there exists a disjoint family { Lγ i u ⊂ KM u : i = 1 , . . . , n | M u |} .Proof. As usual, let ℓ be a coarse continuous length function and ρ the inducedcanonical metric. Suppose that G is fiberwise amenable. Theorem 5.13 shows that G is in fact ubiquitously fiberwise amenable. Let R, ǫ >
0. Proposition 3.8 showsthat there is an
S > L ⊂ G and u ∈ G (0) there is afinite set F u satisfying Lu ⊂ F u ⊂ ¯ B ρ ( Lu, S ) and | ¯ B ρ ( F u , R ) | ≤ (1 + ǫ ) | F u | . On the other hand, For this
S >
0, define a compact set K = { z ∈ G : ℓ ( z ) ≤ S } .It is straightforward to see for all x ∈ G one has ¯ B ρ ( x, S ) ⊂ Kx . This implies that¯ B ρ ( Lu, S ) ⊂ KLu for all compact set L ⊂ G and u ∈ G (0) . This establishes (1).Now suppose that G is not fiberwise amenable. Let L be a compact subset of G and n ∈ N . Define R = sup y ∈ L ℓ ( y ) < ∞ and thus Lx ⊂ ¯ B ρ ( x, R ) for all x ∈ G .Then Proposition 3.9 shows that there is an S > F in G there are at least n | F | many R -balls contained in ¯ B ρ ( F, S ). In particular,this holds for the finite set F = M u whenever M ⊂ G is compact and u ∈ G (0) .On the other hand, for K = { z ∈ G : ℓ ( z ) ≤ S } , one has ¯ B ρ ( x, S ) ⊂ Kx for all x ∈ G . Therefore one has KM u contains at least n | M u | many disjoint R -balls,say, { ¯ B ρ ( γ i , R ) : i = 1 , . . . , n | M u |} . Now since Lγ i u = Lγ i ⊂ ¯ B ρ ( γ i , R ) for each i ≤ n | M u | , the family { Lγ i u : i ≤ n | M u |} is disjoint and KM u contains Lγ i u forall i = 1 , . . . , n | M u | . This establishes (2). (cid:3) In the rest of the section, we indicate a further connection between metric amenabil-ity discussed in 3 and fiberwise amenability via the construction of coarse groupoids.We shall not need this in the following sections.
Definition 5.16 ([STY02, 3.2]) . Let (
Y, d ) be a uniformly locally finite extendedmetric space. The coarse groupoid G ( Y,d ) associated to ( Y, d ) is defined as follows:
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 29 • for any r ≥
0, we define E r = { ( y, z ) ∈ Y × Y : d ( y, z ) ≤ r } ; • as a topological space, we have G ( Y,d ) = S r ≥ E r inside β ( Y × Y ), Stone- ˇCechcompactification of Y × Y ; • we have G (0)( Y,d ) = E ∼ = βY ; • the range and source maps are, respectively, the unique extensions of thefirst and second factor maps Y × Y → Y ; • the multiplication is the unique extension of the composition map G (2)( Y,d ) ∩ (( Y × Y ) × ( Y × Y )) → ( Y × Y ), (( y, z ) , ( z, w )) ( y, z ) ◦ ( z, w ) = ( y, w ), aswe notice that E r ◦ E s ⊆ E r + s for any r, s ≥ Y, d ) implies that this groupoid is locally compact,Hausdorff, principal and ´etale (c.f., [STY02, Proposition 3.2]). It is also σ -compactby definition. Remark 5.17.
There is a canonical length function ℓ on G ( Y,d ) defined by extendingthe metric d : Y × Y → [0 , ∞ ] to β ( Y × Y ) and observing that it takes finite valueson G ( Y,d ) . This length function is continuous by definition. It is also proper sinceby the density of Y × Y inside β ( Y × Y ) , we have, for any r >
0, the open set ℓ − ([0 , r )) is contained in ℓ − ([0 , r )) ∩ ( Y × Y ), which in turn is contained in thecompact set E r . Moreover, observe that for any y ∈ Y viewed as a unit of G ( Y,d ) ,the source fiber (cid:0) G ( Y,d ) (cid:1) y , under the invariant fiberwise extended metric induced by ℓ , is isometric to ( Y, d ). Proposition 5.18.
Let ( Y, d ) be a uniformly locally finite extended metric space.Then it is amenable (respectively, ubiquitously amenable) if and only if G ( Y,d ) isfiberwise amenable (respectively, ubiquitously fiberwise amenable).Proof. Let ℓ be the canonical length function on G ( Y,d ) given in Remark 5.17 andlet ρ be the induced invariant fiberwise extended metric. By Proposition 5.5, itsuffices to show ( Y, d ) is amenable (respectively, ubiquitously amenable) if and onlyif (cid:0) G ( Y,d ) , ρ (cid:1) is. We observed that ( Y, d ) is isometric to (cid:16)(cid:0) G ( Y,d ) (cid:1) y , ρ (cid:17) for any y ∈ Y ;thus ( Y, d ) embeds isometrically into (cid:0) G ( Y,d ) , ρ (cid:1) as some of the coarse connectedcomponents. It follows that the amenability of ( Y, d ) implies that of (cid:0) G ( Y,d ) , ρ (cid:1) ,while the ubiquitous amenability of the latter implies that of the former.Now we assume (cid:0) G ( Y,d ) , ρ (cid:1) is amenable and show so is ( Y, d ). Given
R, ε >
0, weapply Lemma 3.4 to obtain a unit u ∈ βY and an ( R, ε )-Følner set F in (cid:0) G ( Y,d ) (cid:1) u ,and then apply Lemma 5.11 together with the density of Y in βY to obtain y ∈ Y and an ( R, ε )-Følner set F ′ in (cid:0) G ( Y,d ) (cid:1) y . Since each (cid:0) G ( Y,d ) (cid:1) y is isometric to ( Y, d )and R and ε were chosen arbitrarily, this shows ( Y, d ) is amenable.Finally we assume (
Y, d ) is ubiquitously amenable and show so is (cid:0) G ( Y,d ) , ρ (cid:1) . Thusgiven R, ε >
0, there exists
S > y ∈ Y , there exists an ( R, ε )-Følner set in ¯ B d ( y, S ). Now given x ∈ G ( Y,d ) , we claim there exists an ( R, ε )-Følnerset F in ¯ B ρ ( x, S + ε ). Indeed, if s ( x ) ∈ Y , then this follows from the fact that (cid:0) G ( Y,d ) (cid:1) s ( x ) is isometric to ( Y, d ). If s ( x ) ∈ βY \ Y instead, then by Lemma 5.11,there is an open neighborhood V of r ( x ) in G (0)( Y,d ) such that the existence of an( R, ε )-Følner set F in ¯ B ρ ( r ( x ) , S + ε ) is implied by the existence of an ( R, ε )-Følnerset in ¯ B ρ ( v , S ) for some v ∈ V , but the latter condition holds as soon as we pick v ∈ Y ∩ V by the density of Y in βY . It follows that F x is an (
R, ε )-Følner set in¯ B ρ ( x, S + ε ), as desired. (cid:3) We find it intriguing that while Lemma 5.11 (which depends on the local slicelemma) is used in the above proof for both amenability and ubiquitous amenability,the two instances occur in opposite directions.6.
Almost elementary ´etale groupoids and groupoid strictcomparison
In this section, we introduce two regularity properties of ´etale groupoids, groupoidstrict comparison and almost elementariness . Both are central to our analysis.
Definition 6.1.
Let G be a locally compact Hausdorff ´etale groupoid.(i) Let K be a compact subset of G (0) and V an open subset of G (0) . We write K ≺ G V if there are open bisections A , . . . , A n such that K ⊂ S ni =1 s ( A i ), F ni =1 r ( A i ) ⊂ V .(ii) Let U, V be open subsets of G (0) . We write U - G V if K ≺ G V for everycompact subset K ⊂ U .(iii) If X ⋊ α Γ is a transformation groupoid for an action α : Γ y X of countablediscrete group Γ on a compact metrizable space X , we write ≺ α instead of ≺ X ⋊ α Γ , and - α instead of - X ⋊ α Γ , for the sake of simplicity.We remark that if U is compact and open and V is open, then U ≺ G V if andonly if U - G V . We also point out that for any open sets U, V in G (0) , it isnot hard to verify that if U - G V then µ ( U ) ≤ µ ( V ) holds for all µ ∈ M ( G ) (c.f.Definition 5.8). The notion of groupoid strict comparison below is a partial converseof this condition. Definition 6.2.
Let G be a locally compact Hausdorff ´etale groupoid. We say G has groupoid strict comparison (or simply groupoid comparison or comparison ) if,for any open sets U, V ⊂ G (0) , we have U - G V whenever µ ( U ) < µ ( V ) for all µ ∈ M ( G ). If a transformation groupoid X ⋊ α Γ of an action α : Γ y X describedin Definition 6.1(iii) has groupoid strict comparison, we say α has dynamical strictcomparison .From now on, we mainly focus on groupoids with compact metrizable unit spaces.If the groupoid under consideration is also σ -compact, then it is necessarily secondcountable. Remark 6.3.
It is not hard to see our dynamical strict comparison defined abovefor transformation groupoid X ⋊ α Γ of an action an action α : Γ y X describedin Definition 6.1(iii) is equivalent to the dynamical strict comparison (c.f., [Ker20,Definition 3.2]) defined directly for the action α : Γ y X .On the other hand, for ample groupoids, it is helpful to work with a simplifiedversion of groupoid strict comparison. Definition 6.4.
Let G be a locally compact Hausdorff ´etale ample groupoid. Wesay G has groupoid strict comparison for compact open sets for any compact opensets U, V in G (0) one has U - G V whenever µ ( U ) < µ ( V ) for any µ ∈ M ( G ). LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 31
Remark 6.5.
We remark that in an ample groupoid G , if U, V are compact open setsin G (0) and satisfying U - G V , using a similar argument in [Ker20, Definition 3.5],one actually has that there is a collection { A , . . . , A n } of compact open bisectionssuch that U = F ni =1 s ( A i ) and F ni =1 r ( A i ) ⊂ V The following lemma proved by Kerr in [Ker20, Lemma 3.3] is very useful. Werecord this here for completeness.
Lemma 6.6 (Kerr) . Let X be a compact metrizable space with a compatible metric d and let Ω be a weak* closed subset of M ( X ) , which is the set consisting of allBorel regular probability measures on X . Let A be a closed set and O be an openset in X such that µ ( A ) < µ ( O ) for all µ ∈ Ω . Then there exists an η > such thatthe sets O − η = { x ∈ X : d ( x, X \ O ) > η } , and A + η = ¯ B ( A, η ) = { x ∈ X : d ( x, A ) ≤ η } satisfy µ ( A + η ) + η ≤ µ ( O − η ) for all µ ∈ Ω . When the groupoid G is ample and G (0) is compact metrizable, the two comparisonproperties introduced in Definitions 6.2 and 6.4 coincides. Proposition 6.7.
Let G be a locally compact Hausdorff ´etale ample groupoid witha compact metrizable unit space. Then G has groupoid strict comparison if and onlyif G has groupoid strict comparison for compact open sets.Proof. It suffices to show the “if” part. Suppose G has groupoid strict comparisonfor compact open sets. Now, let O, W be open sets in G (0) such that µ ( O ) < µ ( W )for any µ ∈ M ( G ). Now since G is ample, for any compact set K ⊂ O there is acompact open set N such that K ⊂ N ⊂ O and also satisfies that µ ( N ) < µ ( W )for all µ ∈ M ( G ). Then Lemma 6.6 allows us to find an η > W − η such that W − η ⊂ W − η ⊂ W − ( η/ ⊂ W and µ ( N ) < µ ( W − η ) for all µ ∈ M ( G ).Now, choose another compact open set P such that W − η ⊂ P ⊂ W . Note thatone has µ ( N ) < µ ( P ) for any µ ∈ M ( G ), which implies N - G P since we haveassumed that G has groupoid strict comparison for compact open sets. In addition,this establishes O - G W since K ⊂ N ⊂ O and P ⊂ W . (cid:3) The following definition of mutisections was introduced by Nekrashevych in [Nek19,Definition 3.1]
Definition 6.8.
A finite set of bisections T = { C i,j : i, j ∈ F } with a finite indexset F is called a multisection if it satisfies(1) C i,j C j,k = C i,k for i, j, k ∈ F ;(2) { C i,i : i ∈ F } is a disjoint family of subsets of G (0) .We call all C i,i the levels of the multisection T . All C i,j ( i = j ) are called laddersof the multisection T .We say a multisection T = { C i,j : i, j ∈ F } open ( closed ) if all bisections C i,j are open (closed). In addition, we call a finite disjoint family of multisections C = {T l : l ∈ I } a castle , where I is a finite index set. If all multisections in C are open (closed) then we say the castle C is open (closed) . We also explicitly write C = { C li,j : i, j ∈ F l , l ∈ I } , which satisfies the following(i) { C li,j : i, j ∈ F l } is a multisection;(ii) C li,j C l ′ i ′ ,j ′ = ∅ if l = l ′ .Let C = { C li,j : i, j ∈ F l , l ∈ I } be a castle. Any certain level in a multisection in C is usually referred to as a C -level. Analogously, any ladder in in a multisection in C is usually referred as a C -ladder. We remark that the disjoint union H C = [ C = G l ∈ I G i,j ∈ F l C li,j of bisections in C is an elementary groupoid. From this point of view, we denote by C (0) = { C li,i : i ∈ F l , l ∈ I } . Sometimes we will talk about multisections inside C .We denoted by C l = { C li,j : i, j ∈ F l } for each index l ∈ I . Similarly, each C -ladder C li,j in C l for i = j is also called a C l -ladder and any C -level C li,i is also referred as a C l -level. Finally, we write ( C l ) (0) = { C li,i : i ∈ F l } . Let C and D be two castles, wesay C is sub-castle of D if C ⊂ D .Suppose G (0) is a compact. Let C = { C li,j : i, j ∈ F l , l ∈ I } be a castle and K be acompact set in G with G (0) ⊂ K . We say that C is K - extendable if there is anothercastle D = { D li,j : i, j ∈ E l , l ∈ I } with C ⊂ D such that K · G i,j ∈ F l C li,j ⊂ G i,j ∈ E l D li,j where E l ⊂ F l and C li,j = D li.j if i, j ∈ E l for all l = 1 , . . . m . In this case, we alsosay that C is K -extendable to D . Definition 6.9.
Let G be a locally compact Hausdorff ´etale groupoid with a compactunit space. We say that G is almost elementary if for any compact set K satisfying G (0) ⊂ K ⊂ G , any non-empty open set O in G (0) and any open cover V there areopen castles C = { C li,j : i, j ∈ F l , l ∈ I } and D = { D li,j : i, j ∈ E l , l ∈ I } satisfying(i) C is K -extendable to D ;(ii) every D -level is contained in an open set V ∈ V ;(iii) G (0) \ F l ∈ I F i ∈ F l C li,i ≺ G O .Now we show the first property of almost elementary groupoids when it is minimal. Proposition 6.10.
Let G be a minimal locally compact Hausdorff ´etale groupoid ona compact space. Suppose that G is almost elementary. Then G is effective.Proof. Suppose the contrary that Iso( G ) o \ G (0) = ∅ . Then it is an open set and thusthere is a precompact bisection V such that V ⊂ V ⊂ Iso( G ) o \ G (0) because G islocally compact Hausdorff ´etale. Then define an open set O = s ( V ) = r ( V ) ⊂ G (0) .Since G is additionally assumed to be minimal, there are precompact open bisections V , . . . , V n such that(i) all V k are also bisections;(ii) S nk =1 r ( V k ) ⊂ O (iii) V = { s ( V ) , . . . , s ( V n ) } is an open cover of G (0) . We point out that in [Nek19, Definition 3.1], the author worked with ample groupoids andassumed multisections are clopen; we do not make such an assumption.
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 33
Note that (ii) above implies that V · V k = ∅ for all k ≤ n . In addition, for all k ≤ m and λ ∈ V k and γ ∈ V with r ( λ ) = s ( γ ), one has γλ = λ . Otherwise, γλ = λ implies that γ = γr ( λ ) = γλλ − = λλ − = r ( λ ) ∈ G (0) . But this is a contradiction for our original assumption on γ ∈ V ⊂ Iso( G ) o \ G (0) .Now we define a compact set K = ( S nk =1 V · V k ) ∪ ( S nk =1 V k ) ∪ G (0) . Since G is almostelementary, for K and V , there are open castles C = { C li,j : i, j ∈ F l , l ∈ I } and D = { D li,j : i, j ∈ E l , l ∈ I } satisfying(i) C is K -extendable to D and(ii) every D -level is contained in an open set s ( V k ) ∈ V .Now consider a C -level C li,i and a unit u ∈ C li,i for an l ∈ I and an i ∈ F l . First wehave D li,i = C li,i ⊂ s ( V k ) for some k ≤ n . Let λ ∈ V k and γ ∈ V such that s ( λ ) = u and r ( λ ) = s ( γ ). Then observe that { γλu, λu } ⊂ ( V · V k ∪ V k ) u ⊂ Ku ⊂ [ j ∈ E l D lj,i u. Since γλ = λ and the subgroupoid H D = S D is principal, there are different j = j ∈ E l such that { λ } = { λu } = D lj ,i u and { γλ } = { γλu } = D lj ,i u. On the other hand, observe that r ( γλ ) = r ( λ ). But this implies that D j ,j ∩ D j ,j = ∅ , which is a contradiction since they are different D -levels. (cid:3) Remark 6.11.
We remark that in Proposition 6.10, we cannot expect the strongerproperty of principality instead of effectiveness. Recall that for transformationgroupoids, principality corresponds to the freeness of the action while effectivenesscorresponds to topological freeness. It was shown in [OS20] that all minimal actionson the Cantor set by the infinite dihedral group
Z ⋊ Z give rise to transformationgroupoids that are almost finite in Matui’s sense (and thus almost elementary; c.f.,Theorem 7.4), but some of these actions in particular, some odometer actionsare not free.If the unit space G (0) is also metrizable, we then remark that there is an equivalentdefinition of almost elementariness by considering closed bisections. Proposition 6.12.
Let G be a locally compact Hausdorff ´etale groupoid on a compactmetrizable unit space. G is almost elementary if and only for any compact set K satisfying G (0) ⊂ K ⊂ G , any non-empty open set O and any open cover V of G (0) there is an open castle C = { C li,j : i, j ∈ F l , l ∈ I } and a closed castle A = { A li,j : i, j ∈ F l , l ∈ I } satisfying(i) A li,j and C li,j are open and A li,j ⊂ C li,j for all i, j ∈ F l and all l ∈ I ;(ii) C is K -extendable to an open castle D = { D li,j : i, j ∈ E l , l ∈ I } ;(iii) every D -level is contained in an open set V ∈ V ;(iv) G (0) \ F ml =1 F i ∈ F l A li,i ≺ G O . Proof.
It is not hard to see that if a groupoid G satisfies the conditions (i)-(iv) abovethen G is almost elementary. Thus it suffices to show the converse.Now suppose that G is almost elementary. Write T = G (0) \ F l ∈ I F i ∈ F l C li,i forsimplicity. Since one has T ≺ G O , there are bisections { U , . . . , U n } such that T ⊂ S nk =1 s ( U k ) and F nk =1 r ( U k ) ⊂ O . Now fix a compatible metric d on G (0) . Thenthere is a δ > B d ( T, δ ) ⊂ S nk =1 s ( U k ). For any η > P we denote by P − η the open set { u ∈ G (0) : d ( u, G (0) \ P ) > η } as in Lemma 6.6.Now for each l and j ∈ F l choose an η > C lj,j ) − η ⊂ C lj,j and for all l and j ∈ F l one has C lj,j \ ( C lj,j ) − η ⊂ B d ( T, δ ) . Fix an l ∈ I and an i l ∈ F l . Denote by B lj,j = ( C lj,j ) − η for simplicity. Define A li l ,i l = [ j ∈ F l r ( C li l ,j B lj,j ) , which is a subset of C li l ,i l . In addition for i, j ∈ F l we define bisections A lj,i l = C lj,i l A li l ,i l , and A lj,i = A lj,i l · ( A li,i l ) − Observe that A li l ,i l = [ j ∈ F l r ( C li l ,j B lj,j ) ⊂ C li ,i and A lj,i l = ( s | C lj,il ) − ( A li l ,i l ) . In addition, one has A lj,i = A lj,i · ( A li,i ) − Now we claim the castle A = { A li,j : i, j ∈ F l , l ∈ I } satisfies the condition (i)-(iv)above. In fact, by our construction, it suffices to verify (iv) for A . By our definition,one has B li,i ⊂ A li,i and thus C li,i \ A li,i ⊂ C li,i \ ( C li,i ) − η ⊂ B d ( T, δ ) . Now write T ′ = G (0) \ F l ∈ I F i ∈ F l A li,i and thus T ′ ⊂ T ∪ G l ∈ I G i ∈ F l ( C li,i \ ( C li,i ) − η ) ⊂ B d ( T, δ ) ⊂ n [ k =1 s ( U k )while one has n G k =1 r ( U k ) ⊂ O. This establishes (iv) for A . (cid:3) The following is a preliminary general result having the similar flavor to Lemma6.6 established by the first author in [Ma19, Lemma 3.2]. See also [Ker20, Lemma9.1].
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 35
Lemma 6.13.
Let X be a compact metrizable space with a compatible metric d and Ω a weak*-closed subset of M ( X ) . Suppose λ > and A is a closed subset of X such that µ ( A ) < λ for all µ ∈ Ω . Then there is a δ > such that µ ( A + δ ) < λ forall µ ∈ Ω where A + δ = ¯ B d ( A, δ ) = { x ∈ X : d ( x, A ) ≤ δ } . Remark 6.14.
In Definition 6.9, when G (0) is metrizable, we remark that forany ǫ > C = { C li,j : i, j ∈ F l , l ∈ I } satisfying µ ( F l ∈ I F i,i ∈ F l C li,i ) > − ǫ as well. This is because one can choose a non-empty openset O with µ ( O ) < ǫ for any µ ∈ M ( G ) by Lemma 6.13 and make G (0) \ S C (0) ≺ G O a priori. The same argument shows that one can also ask the castle { A li,j : i, j ∈ F l , l ∈ I } in Proposition 6.12 above satisfying that µ ( F l ∈ I F i,i ∈ F l A li,i ) > − ǫ . Remark 6.15.
Furthermore, if G is ample and G (0) is metrizable, in the definitionof almost elementariness, one can choose C and D to be compact open castles.This can be done by refining Proposition 6.12 by finding a compact open castle B = { B li,j : i, j ∈ F l , l ∈ I } such that A li,j ⊂ B li,j ⊂ C li,j . Then define a castle B ′ = { D li,j B lj,j : i, j ∈ E l , l ∈ I } . Then B and B ′ are what we want. In addition,similarly, for any given ǫ , one can make µ ( S l ∈ I S i ∈ F l B li,i ) > − ǫ for any µ ∈ M ( G ).In the remaining part of this section, when G (0) is metrizable, we will show that if G is almost elementary and minimal then it has groupoid strict comparison. Thereare two metrics involved in the proof of the following propositions. When the unitspace G (0) is metrizable, we usually fix an metric d on it. On the other hand, wemay view G as a coarse metric space with a canonical extended metric ρ by a coarselength function ℓ as in Definition 4.12. We begin with the fiberwise amenable case. Proposition 6.16.
Let G be a fiberwise amenable minimal locally compact Hausdorff´etale groupoid with a compact metrizable unit space. If G is almost elementary then G has groupoid strict comparison.Proof. It suffices to show M ≺ G N for any compact set M and open set N in G (0) satisfying µ ( M ) < µ ( N ). Let d be a compatible metric on G (0) . First one has M ( G ) = ∅ by Proposition 5.9. Then the function from M ( G ) to [0 , ∞ ) defined by µ µ ( N ) − µ ( M ) is lower semi-continuous. Then the compactness of M ( G ) showsthat there is an η > µ ( N ) − µ ( M ) ≥ η for all µ ∈ M ( G ). This implies N \ M is open and non-empty. Let y ∈ N \ M and choose a δ > C = ¯ B d ( y, δ ) ⊂ N \ M and µ ( C ) < η/ µ ∈ M ( G ) by Lemma 6.13. Define O = N \ C and thus one has µ ( O ) = µ ( N ) − µ ( C ) > µ ( M ) + η/ µ ∈ M ( G ). Then choose η ′ ≤ η and define O − η ′ = { u ∈ G (0) : d ( u, G (0) \ O ) > η ′ } and M + η ′ = ¯ B d ( M, η ′ )such that M + η ′ ∩ C = O − η ′ ∩ C = ∅ and µ ( M + η ′ ) + η ′ ≤ µ ( O − η ′ ) by Lemma 6.6.Now we claim that there are R , ǫ > u ∈ G (0) and all finite F ⊂ G u one has that if | ¯ B ρ ( F, R ) | ≤ (1 + ǫ ) | F | then one has( ⋆ ) 1 | F | X x ∈ F M + η ′ ( r ( x )) + η ′ / ≤ | F | X x ∈ F O − η ′ ( r ( x )) . Suppose the contrary, for all n ∈ N and ǫ n > { ǫ n } decreasing to 0, there are u n ∈ G (0) and finite set F n ⊂ G u n such that | ¯ B ρ ( F n , n ) | ≤ (1 + ǫ n ) | F n | and µ n ( M + η ′ ) + η ′ / > µ n ( O − η ′ ) , where µ n = | F n | P x ∈ F n δ r ( x ) . Then the argument in Proposition 5.9 shows that anycluster point µ of { µ n } is an invariant probability measure. Then the PortmanteauTheorem shows that µ ( O − η ′ ) + η ′ / ≤ lim inf n →∞ µ n ( O − η ′ ) + η ′ / ≤ lim sup n →∞ µ n ( M + η ′ ) + η ′ ≤ µ ( M + η ′ ) + η ′ . This is a contradiction. Therefore, the claim above holds.Now since G is fiberwise amenable, for R , ǫ > G (0) ⊂ K ⊂ G satisfying the first part of Theorem 5.15. Then since G (0) is compact, we fix an open cover V of G (0) containing open sets with diameter lessthan min { η ′ , δ/ } . Because G is also almost elementary, for the compact set K , theopen cover V and open ball B d ( y, δ ) ⊂ C , Proposition 6.12 implies that there is anopen castle C = { C li,j : i, j ∈ F l , l ∈ I } and a closed castle A = { A li,j : i, j ∈ F l , l ∈ I } satisfying(1) A li,j and C li,j are open and A li,j ⊂ C li,j for all i, j ∈ F l and all l = 1 , . . . , m ;(2) C is K -extendable to an open castle D = { D li,j : i, j ∈ E l , l ∈ I } ;(3) every D -level is contained in an open set V ∈ V ; and(4) G (0) \ F l ∈ I F i ∈ F l A li,i ≺ G B d ( y, δ ).For each l ∈ I we define a compact set L l = S i,j ∈ F l A li,j in G . Then Theorem 5.15(1)shows that for all u ∈ G (0) there is an finite set T u such that L l u ⊂ T u ⊂ KL l u and | ¯ B ρ ( T u , R ) | < (1 + ǫ ) | T u | . Now for each l ∈ I we fix an i l ∈ F l and choose a u l ∈ A li l ,i l . In addition, we write T l = T u l for simplicity. Then note that L l u l ⊂ T l ⊂ KL l u l ⊂ S i ∈ E l D li,i l u l . Thenfor each l ∈ I we define S l = { D li,i ∈ D (0) : i ∈ E l and T l ∩ D li,i l = ∅} and also define(i) P l, = { D li,i ∈ S l : D li,i ∩ M = ∅} and(ii) P l, = { D li,i ∈ S l : D li,i ∩ O − η ′ = ∅} .Since every member V ∈ V has diameter less than min { η ′ , δ/ } , this shows thatdiam d ( D li,i ) ≤ min { η ′ , δ/ } for all D -levels D li,i . Then one has(i) P l, ⊂ Q l, := { D li,i ∈ S l : D li,i ⊂ M + η ′ } ;(ii) P l, ⊂ Q l, := { D li,i ∈ S l : D li,i ⊂ O } ;Then, for all l ∈ I we claim | P l, | < | P l, | . Suppose the contrary. Then there is an l such that | P l, | ≥ | P l, | and thus | Q l, | ≥ | P l, | . Then we have |{ r ( x ) : x ∈ T l and r ( x ) ∈ M + η ′ }| ≥ | Q l, | ≥ | P l, | ≥ |{ r ( x ) : x ∈ T l and r ( x ) ∈ O − η ′ }| , and thus one has X x ∈ T l M + η ′ ( r ( x )) ≥ X x ∈ T l O − η ′ ( r ( x ))which is a contradiction to the inequality ( ⋆ ) above because T l satisfies | ¯ B ρ ( T l , R ) | < (1 + ǫ ) | T l | . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 37
This establishes our claim | P l, | < | P l, | . Now, since | P l, | < | P l, | ≤ | Q l, | holds for each l ∈ I , we choose an injection φ l : P l, → Q l, . Observe that all sets in P l, and Q l, are D -levels in the samemultisection D l . This implies that for any U ∈ P l, there is a bisection ˆ U such that s ( ˆ U ) = U and r ( ˆ U ) = φ l ( U ). This implies that G l ∈ I G P l, - G G l ∈ I G Q l, ⊂ O On the other hand, recall that G i ∈ F l C i,i l · u l = G i ∈ F l A i,i l · u l = L l · u l ⊂ T l · u l . Then one has { C li,i : i ∈ F l } ⊂ S l . Thus G (0) \ G l ∈ I G S l ⊂ G (0) \ G l ∈ I G i ∈ F l C li,i ⊂ G (0) \ G l ∈ I G i ∈ F l A li,i ≺ G B d ( y, δ ) ⊂ C. We write R = G (0) \ F ml =1 F S l for simplicity. So one has R ∩ M ≺ G B d ( y, δ ) ⊂ C. Recall that C ∩ O = ∅ . Finally since M ∩ ( G (0) \ R ) = M ∩ ( G l ∈ I G S l ) ⊂ G l ∈ I G P l, holds and G l ∈ I G P l, - G G l ∈ I G Q l, ⊂ O obtained above, we have verified that M ≺ G N . This shows that G has groupoidstrict comparison. (cid:3) We say a topological space X is perfect if X has no isolated points. Let G bea locally compact Hausdorff ´etale minimal groupoid with an infinite compact unitspace. Then G (0) is necessarily perfect. Suppose the contrary, let { u } be an openset in G (0) . Since G (0) is compact and G is minimal, one can find finitely many openbisections O , . . . , O n such that G (0) = S ni =1 s ( O i ) and S ni =1 r ( O i ) ⊂ { u } . But thisimplies G (0) is finite, which is a contradiction to our assumption. The followingproposition shows that if G stated above is almost elementary but not fiberwiseamenable then it has paradoxical flavor by using Theorem 5.15(2) in the sense thatevery two non-empty open sets in the unit space can be compared in the sense of6.1. In particular it still has groupoid strict comparison. Proposition 6.17.
Let G be a non-fiberwise amenable minimal locally compact ´etalegroupoid with a compact metrizable unit space. Suppose G is almost elementary then G has groupoid strict comparison and M ( G ) = ∅ .Proof. We first show G (0) ≺ G O for any non-empty open set O ⊂ G (0) . Since G (0) isperfect, one can choose disjoint non-empty open subsets U, V of O Let η > V − η = { u ∈ G (0) : d ( u, G (0) \ V ) > η } is non-empty. Since G is minimal thereare precompact open bisections { W , . . . , W n } such that(i) W k is also a bisection for each k = 1 , . . . , n ;(ii) S nk =1 s ( W k ) = G (0) ; (iii) S nk =1 r ( W k ) ⊂ V − η .We write W for the cover { s ( W ) , . . . , s ( W n ) } of G (0) . Define L = S nk =1 W k ∪ G (0) .Then Theorem 5.15 shows that there is a compact set G (0) ⊂ K ⊂ G such that forall compact set M ⊂ G and all u ∈ s ( M ) one has KM u contains at least | M u | many disjoint non-empty sets of form Lγu . We choose another open cover V of G (0) finer than W and contains open sets with diameter less than η . Because G is almostelementary, Proposition 6.12 implies that there is an open castle C = { C li,j : i, j ∈ F l , l ∈ I } and a closed castle A = { A li,j : i, j ∈ F l , l ∈ I } satisfying(i) A li,j and C li,j are open and A li,j ⊂ C li,j for all i, j ∈ F l and all l ∈ I ;(ii) C is K -extendable to an open castle D = { D li,j : i, j ∈ E l , l ∈ I } ;(iii) every D -level is contained in a member of V ;(iv) G (0) \ F l ∈ I F i ∈ F l A li,i ≺ G U .Define M = F ml =1 F i,j ∈ F l A li,j . For each l we choose i l ∈ F l and u l ∈ A i l ,i l . Notethat | M u l | = F l for each l ∈ I . Thus, by our choice of K , for each i ∈ I , there is afamily { γ li ∈ G : i ∈ F l , r ( γ li ) = u l } with the following properties(i) { Lγ li u l : i ∈ F l } is a disjoint family.(ii) F i ∈ F l Lγ li u l ⊂ KM u l ⊂ F i,j ∈ E l D i,j u l = F i ∈ E l D i,i l u l Then for each l ∈ I we choose a bijection as follows. ϕ l : { C li,i l : i ∈ F l } → { γ li ∈ G : i ∈ F l } . Therefore, in particular, one has G i ∈ F l Lϕ l ( C li,i l ) u l ⊂ G j ∈ E l D j,i l u l . Now since G (0) ⊂ L , for each i ∈ F l there is a j i ∈ E l such that ϕ l ( C li,i l ) u l = D lj i ,i l u l .Observe that r ( D lj i ,i l ) = D lj i ,j i ⊂ s ( W k ji ) for some k j i ≤ n because the cover V isfiner than W . Thus, one has r ( W k ji D lj i ,i l ) ⊂ V − η , which implies that r ( W k ji ϕ l ( C li,i l ) u l ) = r ( W k ji D lj i ,i l u l ) ∈ V − η . On the other hand, since W k ji ϕ l ( C li,i l ) u l ∈ Lϕ l ( C li,i l ) u l ⊂ F t ∈ E l D lt,i l u l , there is a t k ji ∈ E l such that W k ji ϕ l ( C li,i l ) u l = D lt kji ,i l u l and thus one has D lt kji ,t kji ∩ V − η = ∅ . For all i ∈ F l we write f l ( i ) for t k ji ∈ E l obtained above for simplicity. Then we have D lf l ( i ) ,f l ( i ) ⊂ V since the diameter of all D -levels are less than η . In addition, sincethe family { Lϕ l ( C i ) u l : i ∈ F l } is disjoint for each l , the member in { f l ( i ) : i ∈ F l } are distinct. This shows that( ∗ ) m G l =1 G i ∈ F l C li,i - G m G l =1 G i ∈ F l D lf l ( i ) ,f l ( i ) ⊂ V. On the other hand, one has( ∗∗ ) G (0) \ G l ∈ I G i ∈ F l C li,i ⊂ G (0) \ G l ∈ I G i ∈ F l A li,i ≺ G U LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 39
Recall that U and V are disjoint subset of O . Then combining ( ∗ ) and ( ∗∗ ), onehas G (0) ≺ G O as desired.Then it is straightforward to see M ( G ) = ∅ . Otherwise, suppose µ ∈ M ( G ) andlet O and O be two disjoint non-empty sets in G (0) . Then for each i = 1 ,
2, since G (0) ≺ G O i , one has 1 = µ ( G (0) ) ≤ µ ( O i ) ≤
1. But this is a contradiction because2 = µ ( O ⊔ O ) ≤
1. Then since one also has U ⊂ G (0) ≺ G O for any non-emptyopen sets U, O , the groupoid G has groupoid strict comparison. (cid:3) Remark 6.18.
Let G be a minimal locally compact σ -compact ´etale groupoid ona compact metrizable space. We remark that Proposition 6.17 above shows that if G is almost elementary and non-fiberwise amenable then G is purely infinite in thesense of [Ma20, Definition 3.5]. See also Lemma 3.10 and the discussion before it in[Ma20]. Therefore, if G is assumed topologically principal, we know in priori thatthe reduced groupoid C ∗ -algebra C ∗ r ( G ) in this case is strongly purely infinite byCorollary 1.1 in [Ma20]. In particular it is Z -stable. On the other hand, combiningProposition 5.9 and 6.17, we obtain that if G is almost elementary, we have M ( G ) = ∅ if and only if G is fiberwise amenable.We end this section by listing the following theorem as a direct corollary of Propo-sition 6.16 and 6.17. Theorem 6.19.
Let G be a minimal locally compact ´etale groupoid on a compactmetrizable unit space. Suppose that G is almost elementary. Then G has groupoidstrict comparison. Almost elementariness and almost finiteness
In this section we show the relation among almost finiteness, pure infinitenessdefined in [Ma20] and our almost elementariness. Recall that the original notion ofalmost finiteness was introduced by Matui in [Mat12] for ample groupoid. In [Ker20],Kerr generalized this notion in the case of transformation groupoid generating byactions of amenable groups on compact metrizable spaces. We begin by recallingMatui’s notion as follows (see [Mat12, Definition 6.2]).
Definition 7.1 (Matui) . Let G be a locally compact Hausdorff ´etale ample groupoidon a compact unit space. G is called almost finite if for any compact set K in G and ǫ > H of G with H (0) = G (0) such that for any u ∈ G (0) one has | K H u \ H u | < ǫ |H u | . It was proved by Matui in [Mat12, Lemma 6.7] that almost finite second countablelocally compact Hausdorff ´etale ample groupoid has groupoid strict comparison forcompact open sets. In fact, his proof still works in the setting that the groupoid isonly σ -compact. Thus we have the following result. Proposition 7.2 (Matui) . Let G be a locally compact σ -compact Hausdorff ´etaleample groupoid with a compact unit space. If G is almost finite then G has groupoidstrict comparison for compact open sets. We first show that almost finiteness implies fiberwise amenability.
Proposition 7.3.
Let G be a locally compact σ -compact Hausdorff ´etale amplegroupoid on a compact space. If G is almost finite then G is fiberwise amenablewith respect to the metric ρ . Proof.
Let ℓ be a coarse continuous length function for G and ρ be the canonicalmetric induced by ℓ (c.f., Definition 4.12). Let u ∈ G (0) and R, ǫ >
0. Define K = { z ∈ G : ℓ ( z ) ≤ R } , which is a compact set in G . In general, for any y ∈ G onehas ¯ B ρ ( y, R ) = { x ∈ G : ρ ( x, y ) ≤ R } = { x ∈ G : ℓ ( xy − ) ≤ R } ⊂ Ky and therefore for any finite set F ⊂ G one has¯ B ρ ( F, R ) = [ y ∈ F ¯ B ρ ( y, R ) ⊂ [ y ∈ F Ky = KF.
Now, since G is almost finite, for the K and the ǫ above, there is a compact openelementary subgroupoid H of G with H (0) = G (0) such that for any v ∈ G (0) one has | K H v \ H v | < ǫ |H v | . Now define F = H u , which is a finite set in G . Then one has | ¯ B ρ ( F, R ) | ≤ | KF | < (1 + ǫ ) | F | . This shows that ( G , ρ ) is fiberwise amenable by Proposition 3.6. (cid:3) Let K be a compact set in G and F ⊂ G u for some u ∈ G (0) . Recall ∂ + K F = KF \ F and ∂ − K F = { x ∈ F : Kx ∩ ( G \ F ) = ∅} , which satisfy ∂ + K F ⊂ K · ∂ − R F and ∂ − K F ⊂ K − · ∂ + K F . Now we show that almost finiteness implies almost elementariness. Alsorecall the function N : [0 , ∞ ) → N given by N ( r ) = sup x ∈G | ¯ B ρ ( x, r ) | . Theorem 7.4.
Let G be a locally compact σ -compact Hausdorff almost finite mini-mal ample groupoid on a compact space. Then G is almost elementary.Proof. Let K be a compact set in G with G (0) ⊂ K , O a non-empty open set in G (0) and V an open cover of G (0) . First, choose a non-empty clopen set U ⊂ O .In addition since U is non-empty and G is minimal, there is an ǫ > µ ( U ) > ǫ for any µ ∈ M ( G ). Furthermore, by choosing a finer cover, one can alsoassume all members in V are clopen.Let ρ be the canonical metric induced by a coarse continuous length function ℓ as usual. Denote by R = sup { ℓ ( y ) : y ∈ K ∪ K − } < ∞ . Note that for any x ∈ G one has ( K ∪ K − ) x ⊂ ¯ B ρ ( x, R ) because ρ ( yx, x ) = ℓ ( y ) forany y ∈ K ∪ K − . To establish the almost elementariness, without loss of generality,we may assume K = F nk =0 O k where each of O k is a compact open bisection and O = G (0) . Let H be the compact open principal elementary subgroupoid satisfyingalmost finiteness for ( K, ǫ/ N ( R )) with the fundamental domain decomposition H = F l ∈ I F i,j ∈ F l C li,j such that for any u ∈ G (0) one has | K H u \ H u | < ( ǫ/ N ( R )) |H u | .In addition, by a standard chopping process for clopen sets, one may assume thatthe partition { C li,i : i ∈ F l , l ∈ I } , as an open cover of G (0) , is finer than V .Fix a u ∈ G (0) and define F = H u , which satisfies | ∂ + K F | < ( ǫ/ N ( R )) | F | . Define M = { γ ∈ F : K · γ ⊂ F } . Note that M = F \ ∂ − K F . Then because ∂ − K F ⊂ K − · ∂ + K F ⊂ ¯ B ρ ( ∂ + K F, R ), one has | ∂ − K F | ≤ N ( R ) | ∂ + K F | and thus | M | = | F | − | ∂ − K F | ≥ | F | − N ( R ) | ∂ + K F | ≥ (1 − ǫ ) | F | . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 41
Now we claim M = T nk =0 ( O − k · F ). Indeed, for any η ∈ T nk =0 ( O − k · F ) ⊂ F onehas for any 0 ≤ k ≤ n there is an γ k ∈ O k and α k ∈ F such that η = γ − k α k . Thenbecause all O k are bisections, one has Kη = n G k =0 O k γ − k α k = { α , . . . , α n } ⊂ F. This shows T nk =0 ( O − k · F ) ⊂ M . For the reverse direction, for any γ ∈ M , thedefinition of M implies Kγ = F nk =0 O k γ ⊂ F . Then for each 0 ≤ k ≤ n , since each O k is a bisection, there is a unique η k ∈ O k such that η k γ ∈ F . This implies that γ ∈ O − k · F for all 0 ≤ k ≤ n and thus M ⊂ T nk =0 ( O − k · F ). This establishes theclaim. Now define T = T nk =0 ( O − k · H ) ⊂ H . Since G is ample and all O − k · C li,j arestill compact open bisections, one can choose finitely many compact open bisections N , . . . , N m such that(i) T = F mp =1 N p ;(ii) for any 1 ≤ p , p ≤ m either s ( N p ) = s ( N p ) or s ( N p ) ∩ s ( N p ) = ∅ and(iii) for any 1 ≤ p ≤ m one has N p ⊂ C li,j for some l ∈ I and i, j ∈ F l .On the other hand, by the analysis above, for any u ∈ G (0) one has KT u ⊂ H u and | T u | = | n \ i =0 ( O − i · H u ) | ≥ (1 − ǫ ) |H u | . In particular, one has s ( T ) = G (0) . Now, for any u ∈ G (0) , denoted by C l u i u ,i u theunique level in H such that u ∈ C l u i u ,i u . Define a function f : G (0) → F l ∈ I P ( F l ) by f ( u ) = { i ∈ F l u : T u ∩ C l u i,i u = ∅} , where P ( F l ) denotes the power set of F l . This is equivalent to say T u = F i ∈ f ( u ) C l u i,i u u. We claim that f is locally constant. Indeed, let u ∈ C l u i u ,i u and denote by J u = { p ≤ m : u ∈ s ( N p ) } . For each p ∈ J u , by (iii) above for N p , one can choose aunique i p ∈ F l u such that N p ⊂ C l u i p ,i u . Note that f ( u ) = { i p : p ∈ J u } . Define W = s ( N p ) for some p ∈ J u , which is a compact open neighborhood of u . For any w ∈ W ⊂ C l u i u ,i u one has T w = m G p =1 N p w = G p ∈ J u N p w = G p ∈ J u C l u i p ,i u w = G i ∈ f ( u ) C l u i,i u w = G i ∈ f ( u ) C l w i,i w w. This implies that f ( w ) = f ( u ) holds for any w ∈ W and thus f is locally constantand thus continuous.Now for each l ∈ I fix an i l ∈ F l . For any S ⊂ F l define a compact open set W Si l ,i l = f − ( { S } ) ∩ C li l ,i l (could be empty). Then the collection { W Si l ,i l : S ⊂ F l } forms a compact open partition of C li l ,i l . Then for any S ⊂ F l and i ∈ F l define W Si,i l = C li,i l W Si l ,i l . In addition, for any i, j ∈ S define W Si,j = W Si,i l ( W Sj,i l ) − . It isobvious to see that the collection H S = { W Si,j : i, j ∈ F l } is a multisection. In fact,observe that {H S : S ⊂ F l , l ∈ I } is a castle and form a decomposition of H inthe sense that F S ⊂ F l W Si,j = C li,j for any l ∈ I and i, j ∈ F l . Therefore, we abusethe notation by writing H = {H S : S ⊂ F l , l ∈ I } . Define a subcastle W ⊂ H by W = { W Si,j : i, j ∈ S, S ⊂ F l , l ∈ I } By our construction, For each S ⊂ F l the level W Si,i ⊂ C li,i and thus W Si,i is asubset of a member of V . Now, we claim W is K -extendable to H . It suffices toshow K W u ⊂ H for any u ∈ W (0) . First assume u ∈ W Si l ,i l for some l ∈ I and S ⊂ F l . Then one has W u = G i ∈ S W Si,i l u = G i ∈ S C li,i l u. Since u ∈ W Si l ,i l = f − ( { S } ) ∩ C li l ,i l , one has f ( u ) = S and thus W u = T u . Therefore, K W u = KT u ⊂ H u ⊂ H . Now if v ∈ W Si,i ⊂ C li,i for some i ∈ S , S ⊂ F l and l ∈ I .Then there is a γ ∈ W Si l ,i such that s ( γ ) = v and r ( γ ) ∈ W Si l ,i l . Then K W v = K ( G j ∈ S W Sj,i v ) = K ( G j ∈ S W Sj,i l W Si l ,i v ) = K ( G j ∈ S W Sj,i l r ( γ ) γ ) = KT · r ( γ ) · γ ⊂ H since KT · r ( γ ) ⊂ H by the argument above and the fact γ ∈ W Si l ,i ⊂ C li l ,i ⊂ H .This establishes that W is K -extendable to H .In addition, since { W Si l ,i l : S ⊂ F l } forms a compact open partition of C li l ,i l , forany µ ∈ M ( G ) one has µ ( C li l ,i l ) = X S ⊂ F l µ ( W Si l ,i l ) . Note that for any l ∈ I and S ⊂ F l , if f − ( { S } ) = ∅ then there is a u such that | S | = | f ( u ) | = | T u | > (1 − ǫ ) |H u | = (1 − ǫ ) | F l | . This implies that for any µ ∈ M ( G ) one has µ ( G S ⊂ F l G i ∈ S W Si,i ) = X S ⊂ F l X i ∈ S µ ( W Si,i ) = X S ⊂ F l | S | µ ( W Si l ,i l ) ≥ (1 − ǫ ) | F l | µ ( C li l ,i l ) . Thus for any µ ∈ M ( G ) one has µ ( G l ∈ I G S ⊂ F l G i ∈ S W Si,i ) = X l ∈ I µ ( G S ⊂ F l G i ∈ S W Si,i ) ≥ (1 − ǫ ) X l ∈ I | F l | µ ( C li l ,i l )= (1 − ǫ ) µ ( H (0) ) = 1 − ǫ. This shows that µ ( G (0) \W (0) ) ≤ ǫ < µ ( U ) for any µ ∈ M ( G ). Since U and G (0) \W (0) are compact open, Proposition 7.2 implies that G (0) \ W (0) ≺ G U ⊂ O . Thus G isalmost elementary. (cid:3) Then we show the reverse direction. We begin with the following lemma.
Lemma 7.5.
Let G be a locally compact σ -compact Hausdorff minimal groupoidequipped with the canonical metric ρ such that G (0) is infinite and compact. Thenfor any N > there is a R > such that for any x ∈ G one has | ¯ B ρ ( x, R ) | > N .Proof. First, for any u ∈ G (0) , there is a R u > | ¯ B ρ ( u, R u ) | > N . Thenthe Local Slice Lemma (Lemma 5.10) implies that there is an open neighborhood V u of u in G (0) , a number S u ∈ [ R u , R u + 1) and an open set W in G as well as ahomeomorphism f : ¯ B ρ ( u, S ) × V u → W such that f ( ¯ B ρ ( u, S u ) × { v } ) = ¯ B ρ ( v, S u )for any v ∈ V u . This implies that | ¯ B ρ ( v, S u ) | > N for any v ∈ V u . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 43
Now since G (0) is compact, choose a finite subcover { V , . . . , V n } of { V u : u ∈ G (0) } together with the corresponding S , . . . , S n . Define R = max { S , . . . , S n } . For any v ∈ G (0) , there is a V i such that v ∈ V i and thus one has | ¯ B ρ ( v, R ) | ≥ | ¯ B ρ ( v, S i ) | > N. Now for any x ∈ G , one has | ¯ B ρ ( x, R ) | = | ¯ B ρ ( r ( x ) , R ) | > N because the right-invariance of ρ . (cid:3) Proposition 7.6.
Let G be a locally compact σ -compact Hausdorff almost elemen-tary fiberwise amenable minimal ample groupoid on a compact metrizable space.Then G is almost finite.Proof. As usual, denote by ℓ be a coarse continuous length function on G and by ρ its induced canonical metric. Since G is ample, we may assume ℓ is locally constantby Remark 4.11. Let K be a comapct set in G and ǫ >
0. We will show below thatthere is a compact open elementary groupoid H satifying the condition in almotfiniteness with respect to K and ǫ . Without loss of any generality, one may assume G (0) ⊂ K .Now choose R > S = ℓ − ([0 , R ]) such that KA ⊂ ¯ B ρ ( A, R ) ⊂ SA.
Recall N ( R ) = sup x ∈G | ¯ B ( x, R ) | < ∞ . Define a positive number N ∈ N with N ≥ N ( R ) /ǫ and a δ > δ < min { ǫ/ , / (2 N − } . Then Lemma 7.5implies that there is a R > | ¯ B ρ ( A, R ) | > N (1+ δ ) and SA ⊂ ¯ B ρ ( A, R )for any finite A . Then for R and δ , Theorem 5.15 (1) yields a compact set G (0) ⊂ M such that for any compact set L ⊂ G and any unit u ∈ G (0) there is a finite set A u satisfying Lu ⊂ A u ⊂ M Lu and ¯ B ρ ( A u , R ) ≤ (1 + δ ) | A u | . Note that | A u | > N by the choice of R above.Since G is almost elementary, Remark 6.15 implies that there are compact opencastles C = { C li,j : i, j ∈ F l , l ∈ I } and D = { D li,j : i, j ∈ E l , l ∈ I } such that C is SM -extendable to D and µ ( G (0) \ S C (0) ) < δ for any µ ∈ M ( G ). Then since ℓ is locally constant, for any D li,j ∈ D and any x ∈ D li,j , there is a compact openbisection W x such that x ∈ W x ⊂ D li,j and ℓ is constant on W x . Therefore, by astandard chopping technique, without loss of generality, one may assume that ℓ isconstant on all D li,j ∈ D .Now, for each l ∈ I we fixe an i l ∈ F l and a unit u ∈ C li l ,i l . Then the choice of M implies that there is a finite set A u such that [ j ∈ F l C lj,i l u ⊂ A u ⊂ M ( [ j ∈ F l C lj,i l u )and | SA u | ≤ | ¯ B ρ ( A u , R ) | ≤ (1 + δ ) | A u | . Since C is SM -extendable to D , one has A u ⊂ KA u ⊂ ¯ B ρ ( A, R ) ⊂ SA u ⊂ SM ( [ j ∈ F l C lj,i l u ) ⊂ [ j ∈ E l D lj,i l u. Define T u = { j ∈ E l : A u ∩ D lj,i l = ∅} . Note that | T u | = | A u | . For any v ∈ C li l ,i l ,define A v = S i ∈ T u D lj,i l v . In addition, we define f : S i ∈ E l D lj,i l v → S i ∈ E l D lj,i l u byclaiming that if D lj,i l v = { x } then { f ( x ) } = D lj,i l u . Since all D lj,i l are bisections, f is bijective. Furthermore, one has f ( A v ) = A u . Now if x ∈ ¯ B ρ ( F v , R ) where { x } = D lj ,i l v for some j ∈ E l then there is a y ∈ A v with { y } = D lj ,i l v such that ρ ( x, y ) = r ≤ R . Note that yx − ∈ D lj ,j and ℓ ( yx − ) = r ≤ R . Now since f ( y ) f ( x ) − ∈ D lj ,j as well and ℓ is constant on D lj ,j , one has f ( x ) ∈ ¯ B ρ ( A u , R )because f ( y ) ∈ A u . Therefore we have verified that f ( ¯ B ρ ( A v , R )) ⊂ ¯ B ρ ( A u , R ).Then one has | ¯ B ρ ( A v , R ) | ≤ | ¯ B ρ ( A u , R ) | ≤ (1 + δ ) | A u | = (1 + δ ) | A v | since f is bijective and f ( A v ) = A u . Then define T l = T u ⊂ E l .Define C ′ = { D li,j : i, j ∈ T l , i ∈ I } , which is still a compact open castle satisfying S C (0) ⊂ S C ′ (0) . Denote by P = G (0) \ S C ′ (0) and for each l ∈ I choose T ′ l ⊂ T l with (1 / N ) | T l | ≤ | T ′ l | < (1 /N ) | T l | . This is possible since | T l | > N . Denote by W = F l ∈ I F i ∈ T ′ l D li,i and we have µ ( P ) < δ < (1 / N )(1 − δ ) ≤ µ ( W )for any µ ∈ M ( G ) by our choice of δ . Then since G has groupoid strict comparisonfor compact open sets by Proposition 6.7, one actually has P - G W . Then Remark6.5 implies that there is a collection { O , . . . , O m } of compact open bisections suchthat P = F mk =1 s ( O k ) and F mk =1 r ( O k ) ⊂ W . Then using the standard choppingtechnique, after decomposition for all D li,j ∈ C ′ and O k , one may assume that forany k ≤ m there is a unique l k ∈ I and a unique j k ∈ T ′ l k such that r ( O k ) = D l k j k ,j k .Now, for each l ∈ I define Q l = { k ≤ m : r ( O k ) = D lj k ,j k } and H l = T ′ l ⊔ Q l .Observe that | Q l | ≤ | T ′ l | ≤ (1 /N ) | T l | and H l = T ′ l if l is not equal to any l k . Nowfor any k ∈ H l \ T ′ l , we define D lj,k = D lj,j k O k for any j ∈ T ′ l and D lk,j = ( D lj,k ) − .In addition, for k , k ∈ H l \ T ′ l , we define D lk ,k = D lk ,j D lj,k for a j ∈ T ′ l . Thendefine a castle A = { D li,j : i, j ∈ H l , l ∈ I } such that H = S A is an elementarygroupoid and H (0) = G (0) .Finally, for any u ∈ H (0) , then there is an l ∈ I and an j u ∈ H l such that H u = S i ∈ H l D li,j u u . Then choose v ∈ C li l ,i l such that { v } = r ( D li l ,j u u ) and therefore |H u | = |H v | since H u = ( H v ) · z , where { z } = D li l ,j u u . In addition, one actually has | ¯ B ρ ( H u, R ) | = | ¯ B ρ ( H v, R ) | by considering the bijective map x xz from ¯ B ρ ( H v, R ) to ¯ B ρ ( H u, R ).Denote by B v = S j ∈ Q l D li,i l v and thus | B v | = | Q l | ≤ (1 /N ) | T l | = (1 /N ) | A v | . Using H v = A v ⊔ B v , one has | K H u | ≤ | ¯ B ρ ( H u, R ) | = | ¯ B ρ ( H v, R ) | ≤ | ¯ B ρ ( A v , R ) | + | ¯ B ρ ( B v , R ) |≤ (1 + δ ) | A v | + X x ∈ B v | ¯ B ρ ( x, R ) |≤ (1 + δ ) | A v | + (1 /N ) N ( R ) | A v | < (1 + ǫ ) |H u | LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 45
The final inequality is because N ≥ N ( R ) /ǫ and δ < ǫ/
2. Thus G is almostfinite. (cid:3) Then we will show almost finiteness introduced by Kerr in [Ker20] for a free action α : Γ y X of a countable discrete amenable group Γ on a compact metrizable space X is equivalent to that the transformation groupoid X ⋊ α Γ of the action α isfiberwise amenable and almost elementary. We first recall the definition of thealmost finiteness in the sense of Kerr (see [Ker20, Definition 8.2]). Let S ⊂ Γ be afinite subset and V be a set in X . We say ( S, V ) is a tower if { sV : s ∈ S } is adisjoint family. A tower is called open if V is open. Similar to the groupoid case,a finite family { ( S i , V i ) : i ∈ I } of towers is called a castle if sV i ∩ tV j = ∅ for any s ∈ S i and t ∈ S j and different i, j ∈ I . A castle is called open if all towers insideare open. Definition 7.7 (Kerr) . We say a free action α : Γ y X of a countable discreteamenable group Γ on a compact metrizable space X is almost finite if for every n ∈ N , finite set K ⊂ Γ and δ > { ( S i , V i ) : i ∈ I } in which all shapes S i are ( K, δ )-invariantin the sense that | T t ∈ K t − S i | ≥ (1 − ǫ ) | S i | and all levels sV i for s ∈ S i havediameter less than δ ,(ii) sets S ′ i ⊂ S i such that | S ′ i | < | S i | /n and X \ G i ∈ I S i V i ≺ α G i ∈ I S ′ i V i . Theorem 7.8.
Let X ⋊ α Γ be the transformation groupoid of a minimal free action α : Γ y X of a countable discrete amenable group Γ on a compact metrizable space X . Then X ⋊ α Γ is fiberwise amenable and almost elementary if and only if α isalmost finite.Proof. Suppose X ⋊ α Γ describe above is fiberwise amenable and almost elementary.Then Remark 5.6 shows that Γ is amenable, which is necessary for α to be almostfinite. In addition, since X ⋊ α Γ is minimal and almost elementary, Theorem 6.19shows that X ⋊ α Γ has dynamical strict comparison. Then Remark 6.3 shows thatthe action α : Γ y X has dynamical strict comparison in the sense of [Ker20,Definition 3.2]. On the other hand, Proposition 3.8 in [Ma19] shows that α : Γ y X has the small boundary property. Therefore, Theorem A in [KS20] shows that α isalmost finite.For the inverse direction, suppose α is almost finite. Let O be an non-empty openset in X , K a compact set in X ⋊ α Γ and U an open cover of X . Denote by l U theLebesgue number for U and choose finitely many group element γ , . . . , γ n ∈ Γ suchthat K ⊂ K ′ := S ni =1 { ( γ i x, γ i , x ) : x ∈ X } . Write F = { γ , . . . , γ n } for simplicity.Now choose δ > µ ( O ) > δ for all µ ∈ M ( G ). Choose 0 < ǫ < l U and n ∈ N such that (1 − ǫ )(1 − /n ) > − δ . Then almost finiteness of α implies thatthere are(i) an open castle S = { ( S i , V i ) : i ∈ I } whose shapes S i are ( F, ǫ )-invariant andall levels sV i for s ∈ S i have diameter less than ǫ , and(ii) sets S ′ i ⊂ S i such that | S ′ i | < | S i | /n and X \ G i ∈ I S i V i ≺ α G i ∈ I S ′ i V i . Now, for each i ∈ I , define T i = T t ∈ F t − S i , which satisfies F T i ⊂ S i . Since S i is( F, ǫ )-invariant, one has | T i | ≥ (1 − ǫ ) | S i | . In addition, since each | S ′ i | ≤ | S i | /n , (ii)above implies that µ ( X \ G i ∈ I S i V i ) ≤ µ ( G i ∈ I S ′ i V i ) ≤ (1 /n ) µ ( G i ∈ I S i V i ) ≤ /n for all µ ∈ M ( X ⋊ α Γ). This implies that µ ( F i ∈ I S i V i ) ≥ − /n for any µ ∈ M ( X ⋊ α Γ) and thus one has µ ( G i ∈ I T i V i ) ≥ (1 − ǫ )(1 − /n ) > − δ for any µ ∈ M ( X ⋊ α Γ). Therefore, one has µ ( X \ F i ∈ I T i V i ) < µ ( O ) for any µ ∈ M ( X ⋊ α Γ) and this implies X \ F i ∈ I T i V i ≺ α O since α has dynamical strictcomparison by Theorem 9.2 in [Ker20].In addition, by our definition, T = { ( T i , V i ) : i ∈ I } is K ′ -extandable to S = { ( S i , V i ) : i ∈ I } and thus K -extendable to S . Finally, since each level sV i in S hasdiameter ǫ < l U and thus sV i is contained in some member of U . Thus, we haveestablished that X ⋊ α Γ is almost elementary. Finally, since Γ is amenable, Remark6.3 shows that X ⋊ α Γ is fiberwise amenable. (cid:3)
To end this section, we record an example due to Gabor Elek. This example in-dicates that our ubiquitous fiberwise amenability in general not necessarily implies(topologically) amenability of groupoids. However, in the transformation groupoidcases, this is a well-known fact that any action of an amenable group is (topologi-cally) amenable.
Example 7.9.
In [Ele18, Theorem 6], Elek constructed a class of groupoids, called geometric groupoid by using so-called stable actions. They are locally compact Haus-dorff (second countable) ´etale minimal principal almost finite ample groupoids butnot (topologically) amenable. However, Proposition 7.3 and Theorem 5.13 impliesthat Elek’s geometric groupoids are ubiquitous fiberwise amenable.8.
Small boundary property and a nesting form of almostelementariness
In this section, we establish a nesting version of the almost elementariness, whichis the main tool in investigating the structure of reduced groupoid C ∗ -algebras. Westart with the following lemmas. Lemma 8.1.
Let G be a locally compact Hausdorff ´etale groupoid on a compactspace. Let K be a compact set in G and M be a precompact open bisection such that M ⊂ K . Let C = { C i,j : i, j ∈ F } be an open multisection that is K -extendableto an open multisection D = { D i,j : i, j ∈ E } and C k,k be a C -level such that C k,k ⊂ s ( M ) . Then there are open castles A = { A li,j : i, j ∈ F l , l ∈ E } and B = { B li,j , i, j ∈ E l , l ∈ E } satisfying(1) index sets F l = F and E l = E for every l ∈ E ;(2) A li,j ⊂ C i,j and B li,j ⊂ D i,j for all i, j ∈ E l and l ∈ E .(3) A is K -extendable to B .(4) C k,k = F l ∈ E A lk,k .(5) For any A lk,k one has M A lk,k = B ll,k ∈ B . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 47 (6) S A (0) = S C (0) and S B (0) = S D (0) .Proof. Since C is K -extendable to D , because M ⊂ K , one has M · G i,j ∈ F C i,j ⊂ G i,j ∈ E D i,j . In particular, we have
M C k,k ⊂ G j ∈ E D j,k . Now for any u ∈ C k,k , since each D l,k is a bisection and C k,k ⊂ s ( M ), there is aunique j u ∈ E such that M u ∈ D j u ,k . Then for each l ∈ E , define O l = { u ∈ C k,k : M u ∈ D l,k } (may be the empty set).We claim that O l is open for any l ∈ E . Indeed for a non-empty O l , let u ∈ O l and let γ ∈ D l,k be such that { γ } = M u . Then one can choose an open bisection N ⊂ M ∩ D l,k such that γ ∈ N . Note that s ( N ) ∩ C k,k is an open neighborhood of u .For any v ∈ s ( N ) ∩ C k,k , because N and M are bisections, one has M v = N v = D l,k v and thus u ∈ s ( N ) ∩ C k,k ⊂ O l . This shows that O l is open.On the other hand, since M and all D l,k are bisections, one has if l = l ∈ E then O l ∩ O l = ∅ . This implies that C k,k = F l ∈ E O l . Define A lk,k = O l . Then forany i, j ∈ F define A li,k = C i,k · A lk,k and A li,j = A li,k · ( A lj,k ) − . By our definition, it isnot hard to see A = { A li,j : i, j ∈ F, l ∈ E } is a castle. Similarly, for any i ∈ E define B li,k = D i,k · A lk,k and B li,j = B li,k · ( B lj,k ) − . Observe that B = { B li,j : i, j ∈ E, l ∈ E } is also a castle. By our construction, it is not hard to see (1), (2), (3), (4) and (6)above hold. Now for any A lk,k ⊂ C k,k one has M A lk,k = D l,k A lk,k = B ll,k . This establishes (5). (cid:3)
Then we take a a groupoid version of the small boundary property into our pictureto select levels of a castle. The small boundary property for dynamical systems wasfirst introduced by Lindenstrauss and Weiss in [LW00]. The following is a directgroupoid analogue.
Definition 8.2.
Let G be a locally compact Hausdorff ´etale groupoid on a compactspace. G is said to have groupoid small boundary property (GSBP for short)if forany u ∈ G (0) and any open neighborhood U of u with u ∈ U ⊂ G (0) there is anotheropen neighborhood V such that u ∈ V ⊂ V ⊂ U and µ ( ∂V ) = 0 for any µ ∈ M ( G ).We then shows that the almost elementariness implies the groupoid small bound-ary property. First if M ( G ) = ∅ then G satisfies the groupoid small boundaryproperty trivially. Therefore, it suffices to show the case that M ( G ) = ∅ . We beginwith the following concept. Definition 8.3.
Let X be a compact Hausdorff space. Let Ω be a weak*-closedsubset of M ( X ). We say X has Ω -small boundary property if for any x ∈ X andopen neighborhood U of x there is another open neighborhood V of x such that x ∈ V ⊂ V ⊂ U and µ ( ∂V ) = 0 for any µ ∈ Ω.Note that if α : Γ y X be an action of a countable discrete group Γ on X Thenthe small boundary property in the sense of Lindenstrauss and Weiss is nothing but M Γ ( X )-small boundary property. The following result is an equivalent approx-imation form of Ω-small boundary property. This result was first established inthe case of the original small boundary property for Ω = M Γ ( X ) by G´arbo Szab´o(c.f., [Ma19, Proposition 3.8]). However, the same proof would establish the casefor general weak*-closed set Ω in M ( X ) and thus we omit the proof. Proposition 8.4.
Let X be a compact metrizable space with a compatible metric d .Let Ω be a weak*-closed subset of M ( X ) . Suppose for any ǫ, δ > there is a disjointcollection U of open sets such that max U ∈U diam d ( U ) < δ and µ ( X \ S U ) < ǫ forany µ ∈ Ω . Then X has Ω -small boundary property. In our groupoid case, note that M ( G ) is a weak*-closed set in M ( G (0) ). Thenusing Proposition 8.4, we have the following result. Theorem 8.5.
Let G be a locally compact Hausdorff ´etale minimal groupoid on acompact metrizable space. Then if G is almost elementary then G has groupoid smallboundary property.Proof. As we mentioned above, it suffices to show the case that M ( G ) = ∅ . Let ǫ, δ >
0. Choose an open cover V of G (0) such that each member U ∈ U has thediameter less than δ . In addition, as usual, choose an open set O such that µ ( O ) < ǫ for any µ ∈ M ( G ) by Lemma 6.13. Since G is almost elementary, for the V and O one has a castle C = { C li,j : i, j ∈ F l .l ∈ I } such that(i) each C -level C li,i is contained in some V ∈ V and(ii) G (0) \ S C (0) ≺ G O .For the disjoint collection C (0) of C -levels, one has max i ∈ F l ,l ∈ I diam( C li,i ) < δ and µ ( G (0) \ C ) < ǫ for any µ ∈ M ( G ). Then Proposition 8.4 shows that G has M ( G )-small boundary property, which is exactly the groupoid small boundary property. (cid:3) Then based on Lemmas above, we have the following characterization of almostelementariness.
Theorem 8.6.
Let G be a locally compact Hausdorff minimal ´etale groupoid with acompact metrizable unit space. Then G is almost elementary if and only if for anyprecompact open bisections U , . . . , U n satisfying U = G (0) and µ ( ∂s ( U i )) = 0 forany i = 0 , . . . , n and any µ ∈ M ( G ) , non-empty open set O in G (0) and open cover U there are open castles C and D satisfying(1) C is K -extendable to D , where K = S nk =0 U k ;(2) every D -level is contained in an open set U ∈ U ;(3) for any C -level C and ≤ p ≤ n either C ⊂ s ( U p ) or C ∩ s ( U p ) = ∅ and if C ⊂ s ( U p ) then there exists a D ∈ D such that U p · C = D ;(4) G (0) \ S C (0) ≺ G O .Proof. Fix a compatible metric d on G (0) . For “if” part, suppose G satisfies assump-tions above. First note that the same proof of Theorem 8.5 implies that G has theGSBP. Now for any compact set K ′ with G (0) ⊂ K ′ ⊂ G , there are precompact openbisections G (0) = V , . . . , V n such that K ′ ⊂ S ni =0 V i . Then the GSBP implies thatthere are precompact open bisections G (0) = U , . . . , U n such that U i ⊂ U i ⊂ V i and µ ( ∂s ( U i )) = 0 for any i = 0 , . . . , n and µ ∈ M ( G ) as well as K ′ ⊂ S ni =0 U i . Now, LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 49 take castles C and D satisfying (1)-(4) above. Then C is also K ′ -extendable to D .This shows that G is almost elementary.For “only if” part, first suppose K = S nk =0 U k where all U i are precompact openbisections satisfying U = G (0) and µ ( ∂s ( U i )) = 0 for any i = 0 , . . . , n and any µ ∈ M ( G ). Let O be an open set in G (0) and U an open cover G (0) . Minimality of G implies that there is an ǫ > µ ( O ) > ǫ for all µ ∈ M ( G ).Define S = S ni =0 ∂s ( U i ) Now Lemma 6.13 implies that there is a δ > µ ( ¯ B d ( S, δ )) < ǫ for any µ ∈ M ( G ). Now choose an open cover V of G (0) , whichis finer than U and each member in V is of diameter less than δ . Now for each0 ≤ i ≤ n define T i = { u ∈ G (0) : d ( u, G (0) \ s ( U i )) ≥ δ } ∪ { u ∈ G (0) : d ( u, s ( U i )) ≥ δ } In addition, define R = T ni =0 T i . Note that G (0) \ R ⊂ ¯ B ( S, δ ) and thus µ ( R ) ≥ − ǫ for any µ ∈ M ( G ). Now since G is almost elementary, there are open castles A and B such that(i) A is K -extendable to B ;(ii) every B -level is contained in an open set V ∈ V ;(iii) G (0) \ S A (0) ≺ G O .Then for any A -level A ∈ A (0) , since diam d ( A ) < δ , if A ∩ R = ∅ then A ∩ S = ∅ and thus either A ⊂ s ( U i ) or A ∩ s ( U i ) = ∅ for any 0 ≤ i ≤ n .Now define A (0)0 = { A ∈ A (0) : A ∩ S = ∅} . Observe that µ ( S A (0)0 ) ≥ − ǫ for any µ ∈ M ( G ). Now we define A = { A ∈ A : s ( A ) , r ( A ) ∈ A (0)0 } . Thensicne A ⊂ A one has A is K -extendable to B as well. In addition, observe that µ ( G (0) \ S A (0)0 ) < µ ( O ) for any µ ∈ M ( G ). This implies that G (0) \ S A (0)0 ≺ G O by Theorem 6.19. We proceed by induction on bisections U , . . . , U n to establishdesired castles. First, for U = G (0) , observe that (1)-(4) holds trivially for A and B and multisection U . Define A = A and B = B . Now suppose we have definedcastles A k and B k for 0 ≤ k < n such that(i) A k is K -extendable to B k , where K = S np =0 U p ;(ii) every B k -level is contained in an open set V ∈ V ;(iii) for any A k -level A and p ≤ n either A ⊂ s ( U p ) or A ∩ s ( U p ) = ∅ .(iv) if A ⊂ s ( U p ) for some p ≤ k then there exists a B k -ladder B such tat U p · A = B with s ( B ) = A ;(v) S A (0) k = S A (0)0 Now for U k +1 , we write A k = { A mi,j : i, j ∈ S m , m ∈ J } and B k = { B mi,j : i, j ∈ T m , m ∈ J } explicitly. First (iii) above says that for any A mi,i ⊂ s ( U p ) for some p ≤ k there is a B mj,i ∈ B k such that( ⋆⋆ ) U p A mi,i = B mj,i . Note that by our assumption one has either A mi,i ⊂ s ( U k +1 ) or A mi,i ∩ s ( U k +1 ) = ∅ .For each m ∈ J define F m = { i ∈ S m : A mi,i ⊂ s ( U k +1 ) } . Denoted by N m = | F m | andfix an enumeration F m = { i , . . . , i N m } .For any m ∈ J , fix multisections A mk = { A mi,j : i, j ∈ S m } and B mk = { B mi,j : i, j ∈ T m } inside A k and B k , respectively. Then for any i ∈ F m , apply Lemma 8.1to multisections A mk , B mk and A mi ,i ⊂ U k +1 to decompose A mk and B mk to castles A mk,i = { A m,l i,j : i, j ∈ S m , l ∈ T m } and B mk,i = { B m,l i,j : i, j ∈ T m , l ∈ T m } satisfy-ing the corresponding properties (1)-(6) in Lemma 8.1 and in particular note that U k +1 A m,l i ,i = B m,l l ,i and S ( A mk ) (0) = S ( A mk,i ) (0) as well as S ( B mk ) (0) = S ( B mk,i ) (0) .In addition, for each l ∈ T m and i ∈ S m one still has that either A m,l i,i ⊂ s ( U k +1 )or A m,l i,i ∩ s ( U k +1 ) = ∅ . Now apply Lemma 8.1 to any multisection A m,l k,i = { A m,l i,j : i, j ∈ S m } and B m,l k,i = { B m,l i,j : i, j ∈ T m } as well as A m,l i ,i ⊂ s ( U k +1 )with j ∈ F m and i = i to decompose multisections A m,l k,i and B m,l k,i to castles A m,l k, { i ,i } = { A m,l ,l i,j : i, j ∈ S m , l ∈ T m } and B m,l k, { i ,i } = { B m,l ,l i,j : i, j ∈ T m , l ∈ T m } satisfying the corresponding properties (1)-(6) in Lemma 8.1. In particular,one has that U k +1 A m,l ,l i ,i = B m,l ,l l ,i and S ( A m,l k,i ) (0) = S ( A m,l ,l k, { i ,i } ) (0) as well as S ( B m,l k,i ) (0) = S ( B m,l ,l k, { i ,i } ) (0) . In addition, for each l ∈ T m one has U k +1 A m,l ,l i ,i = U k +1 A m,l i ,i A m,l ,l i ,i = B m,l l ,i A m,l ,l i ,i = B m,l ,l l ,i . Then we can do the same decomposition process for all multisections in A m,l k, { i ,i } and B m,l k, { i ,i } for another index i ∈ F m such that i = i , i . In addition, we repeatthis process by induction, for all l , . . . , l N m ∈ T m , to obtain disjoint multisections A m,l ,...,l N k,F m = A m,l ,...,l Nm k, { i ,...,i Nm } = { A m,l ,...,l Nm i,j : i, j ∈ S m } and B m,l ,...,l N k,F m = B m,l ,...,l Nm k, { i ,...,i Nm } = { B m,l ,...,l Nm i,j : i, j ∈ T m } such that for any m ∈ J and l , . . . , l N m ∈ T m one has(i) A m,l ,...,l Nm i,j ⊂ A mi,j for any i, j ∈ S m and B m,l ,...,l Nm i,j ⊂ B mi,j for any i, j ∈ T m ;(ii) A m,l ,...,l N k,F m is K -extendable to B m,l ,...,l N k,F m ;(iii) every B m,l ,...,l N k,F m -level is contained in an open set V ∈ V ;(iv) U k +1 A m,l ,...,l N i p ,i p = B m,l ,...,l N l p ,i p for any i p ∈ F m ;(v) S l ,...,l Nm ∈ T m S ( A m,l ,...,l Nm k,F m ) (0) = S ( A mk ) (0) and(vi) S l ,...,l Nm ∈ T m S ( B m,l ,...,l Nm k,F m ) (0) = S ( B mk ) (0) Now define A k +1 = G m ∈ J G l ,...,l Nm ∈ T m A m,l ,...,l N k,F m and B k +1 = G m ∈ J G l ,...,l Nm ∈ T m B m,l ,...,l N k,F m . By our definition of A k +1 and B k +1 , it is straightforward to see that A k +1 is K -extendable to B k +1 . In addition, each B k +1 -level is contained in a member of thecover V and thus a member in U . Furthermore, by our construction, since each A k +1 -level A is contained in a A k -level, then for any p ≤ n either A ⊂ s ( U p ) or A ∩ s ( U p ) = ∅ . Finally one still has S A (0) k +1 = S A (0)0 .Now suppose a A k +1 -level A m,l ,...,l Nm i,j satisfies A m,l ,...,l Nm i,i ⊂ s ( U p ) for some p ≤ k .Then using ( ⋆⋆ ) and (i) above there is a j ∈ T m such that U p A m,l ,...,l Nm i,i = U p A mi,i A m,l ,...,l Nm i,i = B mj,i A m,l ,...,l Nm i,i = B m,l ,...,l Nm j,i . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 51
Therefore, combing (iv) above one actually has that if a A k +1 -level A ⊂ s ( U p ) forsome p ≤ k + 1 then there is a B k +1 -ladder B such that U p A = B . This finishes theinductive step for U k +1 .Now define C = A n and D = B n . The argument above has established conditions(1)-(3). Finally since S C (0) = S A (0)0 and G (0) \ A (0)0 ≺ G O , one has G (0) \ C (0)0 ≺ G O as well. (cid:3) Another equivalent condition established in the following theorem for almost el-ementariness is called the nesting form of the almost elementariness. We need thefollowing definition.
Definition 8.7.
Let C = { C i,j : i, j ∈ F } and D = { D m,n : , m, n ∈ E } be twomultisections. Let N ∈ N . We say C is nesting in D with multiplicity at least N ifthe following holds.(i) For any C -level C i,i ∈ C (0) there is a D -level D n,n ∈ D (0) with C i,i ⊂ D n,n .(ii) { r ( D m,n C ) : C ⊂ D n,n , C ∈ C (0) } = { C ∈ C (0) : C ⊂ D m,m } holds for any m, n ∈ E .(iii) |{ C ∈ C (0) : C ⊂ D m,m }| > N holds for one m ∈ E .We remark that (ii) and (iii) in fact imply |{ C ∈ C (0) : C ⊂ D m,m }| > N holds forany m ∈ E . Definition 8.8.
Let C = { C l : l ∈ I } and D = { D k : k ∈ J } be two castles in which C l and D k are multisections. Let N ∈ N . We say C is nesting in D with multiplicityat least N if the following holds.(i) for any multisection C l in C there is a unique multisection D k in D such that C l is nesting in D with multiplicity at least N .(ii) for any multisection D k in D there is at least one multisection C l in C suchthat C l is nesting in D with multiplicity at least N . Theorem 8.9.
Let G be a locally compact σ -compact minimal ´etale groupoid on acompact metrizable unit space. Then G is almost elementary if and only if for anycompact set K satisfying G (0) ⊂ K ⊂ G , any non-empty open set O in G (0) , anyopen cover V and any integer N ∈ N there are open castles A , B , C and D such that(1) Both ¯ C = { C : C ∈ C} and ¯ D = { D : D ∈ D} are compact castles;(2) A is K -extendable to B and ¯ C is K -extendable to ¯ D ;(3) B is nested in D with multiplicity at least N ;(4) A is nested in C with multiplicity at least N ;(5) any D -level is contained in a member of V ;(6) G (0) \ S A (0) ≺ G O .Proof. It is straightforward to see that if G satisfies the conditions above then G isalmost elementary. Now we establish the converse. Fix a compatible metric d on G (0) . Since G is minimal and G (0) is compact, one can choose an ǫ > µ ( O ) > ǫ for all µ ∈ M ( G ). Now because G is almost elementary, Proposition6.12 implies that there are open castles C + = { C km,n : m, n ∈ T k , k ∈ J } and D + = { D km,n : m, n ∈ S k , k ∈ J } such that(i) ¯ C + = { C km,n : m, n ∈ T k , k ∈ J } and ¯ D + = { D km,n : m, n ∈ S k , k ∈ J } arecompact castles;(ii) ¯ C + is K -extendable to ¯ D + ; (iii) every ¯ D + -level is contained in an open set V ∈ V .(iv) µ ( F k ∈ J F m ∈ T k C km,m ) ≥ − ǫ for any µ ∈ M ( G ) by remark 6.14.In addition, since G is almost elementary, applying the GSBP in the proof of Propo-sition 6.12, one can make C + and D + additionally satisfy(v) µ ( ∂D km,m ) = 0 for any µ ∈ M ( G ), m ∈ S k and k ∈ J .Write F = G (0) \ S C (0)+ for simplicity. By Lemma 6.13 one can choose a δ > M = B d ( F, δ ) satisfying µ ( M ) < ǫ for any µ ∈ M ( G ). Then thecollection U = { C km,m : m ∈ T k , k ∈ J } ∪ { M } forms an open cover of G (0) . Let N ∈ N .Let C km,m be a C + -level and u ∈ G (0) . Define H k,u,m = { γ ∈ G : s ( γ ) = u, r ( γ ) ∈ C km,m } . Because G is minimal, the unit space G (0) is perfect in this case by thediscussion after Proposition 6.16. This implies that each H k,u,m is infinite. Thus, onecan choose a P k,u,m ⊂ H k,u,m such that | P k,u,m | > N and r ( P k,u,m ) consists distinctunits in C km,m . Since G is Hausdorff, there are open bisections { U k,u,m,γ : γ ∈ P k,u,m } such that γ ∈ U k,u,m,γ for each γ ∈ P k,u,m and the collection { r ( U k,u,m,γ ) : γ ∈ P k,u,m } is a disjoint family of compact subsets of C km,m . In addition, since G isalmost elementary and thus has the GSBP, by shrinking each U k,u,m,γ , one mayassume µ ( s ( U k,u,m,γ )) = 0 for any µ ∈ M ( G ).Then since all C + -levels are disjoint, note that actually { r ( U k,u,m,γ ) : γ ∈ P k,u,m , m ∈ F k , k ∈ J } is a disjoint family. Define O u = \ k ∈ J \ m ∈ T k \ γ ∈ P k,u,m s ( U k,u,m,γ ) , which is an open neighborhood of u . Then { O u : u ∈ G (0) } forms a cover of G (0) .Then the compactness of G (0) implies there is a finite subcover O = { O u , . . . , O u p } .Denoted by W = O ∨ U . Define a compact set H =( G k ∈ J G m,n ∈ S k D kn,m ) · ( [ k ∈ J [ m ∈ T k p [ q =1 [ γ ∈ P k,uq,m U k,u q ,m,γ ∪ G (0) ) ∪ G (0) =( G k ∈ J G n ∈ S k m ∈ T k p [ q =1 [ γ ∈ P k,uq,m D kn,m · U k,u q ,m,γ ) ∪ ( G k ∈ J G m,n ∈ S k D kn,m ) ∪ G (0) . Note that for any bisection D kn,m U k,u q ,m,γ ⊂ H and µ ∈ M ( G ) one also has µ ( s ( D kn,m U k,u q ,m,γ )) = µ ( s ( U k,u q ,m,γ )) = 0 . Now, for H and the cover W , Theorem 8.6 implies that there are open castles A ′ = { A ′ li,j : i, j ∈ F ′ l , l ∈ I ′ } and B ′ = { B ′ li,j : i, j ∈ E ′ l , l ∈ I ′ } such that(i) A ′ is H -extendable to B ′ ;(ii) each B ′ -level is contained in a member of W ;(iii) if a A ′ -level A ⊂ s ( D kn,m U k,u q ,m,γ ) then there is a B ∈ B ′ such that D kn,m U k,u q ,m,γ A = B .(iv) if a A ′ -level A ⊂ s ( D kn,m ) then there is a B ∈ B ′ such that D kn,m A = B .(v) µ ( F l ∈ I ′ F i ∈ F ′ l A ′ li,i ) > − ǫ for any µ ∈ M ( G ).Now we define required castles A and B as sub-castles of B ′ . First define R l = { i ∈ F ′ l : A ′ li,i is contained in a C + -level } and a castle A ′′ = { A ′ li,j : i, j ∈ R l , l ∈ I ′ } . LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 53
Because each B ′ -level is contained in a member of W , which is a refinement of U ,for any µ ∈ M ( G ), one has µ ( G l ∈ I G i ∈ F ′ l \ R ′ l A ′ li,i ) ≤ µ ( M ) < ǫ. Therefore, for any µ ∈ M ( G ), one has µ ( [ ( A ′′ ) (0) ) = µ ( G l ∈ I G i ∈ R l A ′ li,i ) > − ǫ. On the other hand, Since any B ′ -level is contained in a member of W and thus alsoin one member in O . Then for any l ∈ I and A ′′ -leverl A ′ li,i there is an O u q suchthat A ′ li,i ⊂ O u q . This shows that A ′ li,i ⊂ s ( U k,u q ,m,γ ) = s ( D kn,m · U k,u q ,m,γ ) for any k ∈ J , m ∈ T k , n ∈ S k and γ ∈ P k,u q ,m such that r ( U k,u q ,m,γ ) ⊂ C km,m . Therefore,in these case, by (iii) for A ′ and B ′ above one may assume there is a j ∈ E ′ l suchthat( ⋆ ) D kn,m · U k,u q ,m,γ A ′ li,i = B ′ lj ,i . On the other hand, any A ′′ -level A ′ li,i is contained in some C + -level C km,m . Then (iv)for A ′ and B ′ above shows that for any n ∈ S k there is a j ∈ E ′ l such that( ⋆⋆ ) D kn,m A ′ li,i = B ′ lj ,i . Now, for any multisection D k + = { D km,n : m, n ∈ S k } together with its sub-multisection C k + = { C km,n : m, n ∈ T k } and any l ∈ I ′ , by ( ⋆ ) and ( ⋆⋆ ), one candefine index sets Q Al,k and Q Bl,k as subsets of E ′ l by claiming for any j ∈ E ′ l , j ∈ Q Al,k if there exists i ∈ R l , m, n ∈ T k , q ≤ p, γ ∈ P k,u q ,m with A ′ li,i ⊂ C km,m ∩ O u q such that B ′ lj,i = C kn,m A ′ li,i or B ′ lj,i = C kn,m U k,u q ,m,γ A ′ li,i and j ∈ Q Bl,k if there exists i ∈ R l , m ∈ T k , n ∈ S k , q ≤ p, γ ∈ P k,u q ,m with A ′ li,i ⊂ C km,m ∩ O u q such that B ′ lj,i = D kn,m A ′ li,i or B ′ lj,i = D kn,m U k,u q ,m,γ A ′ li,i . Now for j , j ∈ Q Al,k ⊂ E ′ l define A l,kj ,j = B ′ lj ,j . Similarly, j , j ∈ Q Bl,k ⊂ E ′ l define B l,kj ,j = B ′ lj ,j . Then we define following multisections for k ∈ J and l ∈ I ′ by A l,k = { A l,kj ,j : j , j ∈ Q Al,k } and B l,k = { B l,kj ,j : j , j ∈ Q Bl,k } . Note that some A l,k and B l,k may be empty. Thus, we refine our castle D + by firstdefining the index set I = { k ∈ J : there exist an m ∈ T k and a A ′′ -level A ′ li,i such that A ′ li,i ⊂ C km,m . } and define C = { C kn,m ∈ C + : m, n ∈ T k , k ∈ I } and D = { D kn,m ∈ D + : m, n ∈ S k , k ∈ I } . Then we define A = { A l,kj ,j : j , j ∈ Q Al,k , k ∈ I, l ∈ I ′ } and B = { B l,kj ,j : j , j ∈ Q Bl,k , k ∈ I, l ∈ I ′ } . Now, we prove the castle A , B , C and D satisfying our requirements. First (1) and(5) are clear by the definition of these castles. For (2), note that C and D consists ofmultisections in C + and D + with the same index set I ⊂ J , respectively. Then since¯ C + is K -extendable to ¯ D + , the castle ¯ C is K -extendable to ¯ D as well. In order to showthat A is K -extendable to B , first fix a θ ∈ A l,kj ,j ∈ A . By definition, A l,kj ,j = B ′ lj ,j where j , j ∈ Q Al,k ⊂ E ′ l . Note that first A l,kj ,j = B ′ lj ,i A ′ li ,i ( B ′ lj ,i ) − for some i , i ∈ R l such that B ′ lj ,i = C kn ,m U k,u q ,m ,γ A ′ li ,i or B ′ lj ,i = C kn ,m A ′ li ,i for some n , m ∈ T k , q ≤ p, γ ∈ P k,u q ,m with A ′ li ,i ⊂ C km ,m ∩ O u q by thedefinition of Q Al,k . Then r ( θ ) ∈ C kn ,n and since C is K -extendable to D , suppose B ′ lj ,i = C kn ,m U k,u q ,m ,γ A ′ li ,i then there is an n ∈ S k such that Kθ = Kr ( θ ) θ ∈ D kn,n C kn ,m U k,u q ,m ,γ A ′ li ,i A ′ li ,i ( B ′ lj ,i ) − = D kn,m U k,u q ,m ,γ A ′ li ,i A ′ li ,i ( B ′ lj ,i ) − = B l,ij ,i A ′ li ,i ( B ′ lj ,i ) − = B l,kj ,j for some j ∈ Q Bl,k . On the other hand, if B ′ lj ,i = C kn ,m A ′ li ,i then same argumentabove shows that there is a j ∈ Q Bl,k such that Kθ ∈ B l,kj ,j . This shows that A is K -extendable to B and establishes (2).We now establish (3) and (4). We begin with showing A is nested in C withmultiplicity at least N . First, fix a k ∈ I, l ∈ I ′ in A . Look at a multisection A l,k = { A l,kj ,j : j , j ∈ Q Al,k } . By the definition of A , it is straightforward that any A l,k -level is contained in a C k -level, where the multisection C k = { C km,n : m, n ∈ T k } .Then for a C k -level C kn,n such that there is an A ′ -level A ′ li ,i ⊂ C kn,n by definition of I . Then one has { A ∈ ( A l,k ) (0) : A ⊂ C kn,n } = { r ( B ′ lj,i ) : B ′ lj,i = C kn,m U k,u q ,m,γ A ′ li,i or B ′ lj,i = C kn,m A ′ li,i , A ′ li,i ⊂ C km,m ∩ O u q ,γ ∈ P k,u q ,m , q ≤ p, and m ∈ T k , i ∈ R l } , ( † )and in particular contains the set { r ( B ′ lj,i ) : B ′ lj,i = A ′ li ,i or B ′ lj,i = U k,u q ,n,γ A ′ li ,i , A ′ li ,i ⊂ O u q , q ≤ p, γ ∈ P k,u q ,m } . This thus implies( †† ) |{ A ∈ ( A l,k ) (0) : A ⊂ C kn,n }| > N because | P k,u q ,n | ≥ N and { r ( U k,u q ,m,γ ) : γ ∈ P k,u q ,n } is a disjoint family of subsetsof C kn,n for any q ≤ p such that A ′ li ,i ⊂ O u q . Now for another m ∈ T k , by ourdefinition of A and ( † ), one has { A ∈ ( A l,k ) (0) : A ⊂ C km,m } = { r ( C km,n A ) : A ∈ ( A l,k ) (0) , A ⊂ C kn,n } . This shows that multisection A k,l is nested in C k with multiplicity at least N for k ∈ I and l ∈ I ′ . Therefore, one has that the castle A is nested in the castle C bythe definition of the index set I .Similarly, to show B l,k = { B l,kj ,j : j , j ∈ Q Bl,k } is nested in D k = { D km,n : m, n ∈ S k } for the k ∈ I and l ∈ I ′ , first note that any B l,k -level is contained in a D k -level LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 55 by definition. Then since the level D kn,n = C kn,n for any n ∈ T k above and any A -levelis also a B -level by K -extendability of A to B and C to D , respectively, one has |{ B ∈ B (0) l,k : B ⊂ D kn,n }| > N for the n ∈ T k established in ( †† ) above. Finally, it suffices to observe by definitionof B and ( † ) again that { B ∈ B (0) l,k : B ⊂ D km,m } = { r ( D km,n B ) : B ∈ A (0) l,k , B ⊂ D kn,n } for any m ∈ S k . This shows that multisection B k,l is nested in D k with multiplicityat least N for the k ∈ I and l ∈ I ′ . Thus the castle B is nested in the castle D . Thisestablishes (3) and (4) as desired.Finally, we establish (6). Note that A ′′ ⊂ A because R l ⊂ S k ∈ I Q Al,k . Thus forany µ ∈ M ( G ) one has µ ( [ A (0) ) ≥ µ ( [ ( A ′′ ) (0) ) > − ǫ. This implies that µ ( O ) > ǫ ≥ µ ( G (0) \ S A (0) ) for any µ ∈ M ( G ). Then Theorem6.19 shows that G (0) \ S A (0) ≺ G O . This establishes (6). (cid:3) Remark 8.10.
We remark that if the compact set K in Theorem 8.9 is a union ofcompact bisections, say, K = S ni =0 O i where each O i is a precompact open bisectionwith O = G (0) and µ ( s ( O i )) = 0 for any 0 ≤ i ≤ n and µ ∈ M ( G ), Theorem 8.6implies that the castles C and D can be chosen furthermore satisfying that(1) for any 0 ≤ i ≤ n and C ∈ C (0) either C ⊂ s ( O i ) or C ∩ s ( O i ) = ∅ and(2) whenever a C -level C ⊂ s ( O i ) for some i ≤ n then there is a D ∈ D suchthat s ( D ) = C and O i C = D .Indeed, we do this by adjusting the beginning of the proof of Theorem 8.9, First,for ǫ > V there, using Theorem 8.6 one obtains open castles C ′ and D ′ satisfying(i) C ′ is K -extendable to D ′ , where K = S ni =0 O i ;(ii) every D ′ -level is contained in an open set V ∈ V ;(iii) for any C ′ -level C ′ and 0 ≤ i ≤ n either C ′ ⊂ s ( O i ) or C ′ ∩ s ( O i ) = ∅ and if C ′ ⊂ s ( O i ) then there exists a D ′ ∈ D ′ such that O i · C = D ;(iv) µ ( S C ′ (0) ) > − ǫ for any µ ∈ M ( G ).Then write C ′ = { C ′ km,n : m, n ∈ T k , k ∈ J } and D ′ = { D ′ km,n : m, n ∈ S k , k ∈ J } explicitly and use the shrinking technique in Proposition 6.12 and the GSBP for C ′ and D ′ , one obtains castles C + = { C km,n : m, n ∈ T k , k ∈ J } and D + = { D km,n : m, n ∈ S k , k ∈ J } such that(i) ¯ C + = { C km,n : m, n ∈ T k , k ∈ J } and ¯ D + = { D km,n : m, n ∈ S k , k ∈ J } arecompact castles;(ii) D km,n ⊂ D ′ km,n for any k ∈ J and m, n ∈ S k ;(iii) ¯ C + is K -extendable to ¯ D + ;(iv) every ¯ D + -level is contained in an open set V ∈ V ;(v) µ ( F k ∈ J F m ∈ T k C km,m ) ≥ − ǫ for any µ ∈ M ( G ) by remark 6.14.(vi) µ ( ∂D km,m ) = 0 for any µ ∈ M ( G ), m ∈ S k and k ∈ J .Therefore, by (ii) for ¯ D + and thus ¯ C + , for any C + -level C and any 0 ≤ i ≤ n either C ⊂ s ( O i ) or C ∩ s ( O i ) = ∅ . Write C = C km,m explicitly and suppose C km,m ⊂ s ( O i ) for some i ≤ n . Then necessarily C ′ km,m ⊂ s ( O i ) by (ii) for C + and (iii) for C ′ .Therefore there is a D ′ kn,m ∈ D ′ such that O i C ′ km,m = D ′ kn,m . This implies that O i C km,m = O i C ′ km,m C km,m = D ′ kn,m C km,m = D kn,m C km,m = D kn,m . This thus establishes original conditions for C + and D + in Theorem 8.9 and addi-tional conditions (1) and (2) above. Therefore, we may arrange the castles C and D satisfying (1) and (2) because they are proper subcastles of C + and D + , respectively. Remark 8.11.
We also remark that in the case that the groupoid G is ample, thenone can drop the assumption of metrizability of the unit space in Proposition 8.6and Theorem 8.9. The metric on the unit space G (0) is used in two ways. First, oneapplies Proposition 8.5 to obtain the GSBP to eliminate all levels intersecting theboundary of the sources or the ranges of the given precompact bisections. Second,one need to apply groupoid strict comparison to prove the results, which only wasestablished in general in the case that the unit space is metrizable. However, ifthe groupoid is ample, one can circumvent GSBP by using compact open castlesby Remark 6.15. Then a standard chopping technique ensures that there is nolevels intersecting the source or the range of a given compact open bisection andtheir complement. In addition, comparison for compact open sets established inProposition 7.2 does not require the metrizability of the unit space. Therefore, oneactually has the following result by adapting the proof of Theorem 8.9 and Remark8.10. Theorem 8.12.
Let G be a locally compact σ -compact minimal ample ´etale groupoidon a compact space. Suppose G is almost elementary. Then for any compact set K = S ni =0 M i in which all M i are compact open bisections and M = G (0) , anynon-empty compact open set O in G (0) , any open cover V and any integer N ∈ N there are compact open castles A , B , C and D such that(1) for any ≤ i ≤ n and C ∈ C (0) either C ⊂ s ( M i ) or C ∩ s ( M i ) = ∅ and(2) whenever a C -level C ⊂ s ( M i ) for some i ≤ n then there is a D ∈ D suchthat s ( D ) = C and M i C = D .(3) A is K -extendable to B and C is K -extendable to D ;(4) B is nested in D with multiplicity at least N ;(5) A is nested in C with multiplicity at least N ;(6) any D -level is contained in a member of V ;(7) G (0) \ S A (0) ≺ G O . (cid:3) Tracial Z -stability In this section, we investigate structure properties of the reduced C ∗ -algebra C ∗ r ( G ) of an almost elementary groupoid G . In particular, we will show C ∗ r ( G ) istracially Z -stable in the sense of [HO13, Definition 2.1].Let A, B be C ∗ -algebras. Denote by A + the set of all positive elements in A .Recall a c.p.c. map ψ : A → B is said to be order zero if for any a, b ∈ A + with ab =0, we have ψ ( a ) ψ ( b ) = 0 as well. For a, b ∈ A + , a is said to be Cuntz-subequivalent to b , denoted by a - b , if there is a sequence { x n ∈ A : n ∈ N } such that lim n →∞ k a − x n bx ∗ n k = 0. We write a ∼ b if a - b and b - a . The following concept of tracial Z -stability was introduced by Hirshberg-Orovitz in [HO13, Definition 2.1]. LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 57
Definition 9.1. [Hirshberg-Orovitz] A unital C ∗ -algebra A is said tracially Z -stable if A = C and for any finite set F ⊂ A , ǫ >
0, any non-zero positive element a ∈ A + and n ∈ N there is an order zero c.p.c. map ψ : M n ( C ) → A such that the followinghold:(1) 1 A − ψ (1 n ) - a ;(2) for any x ∈ M n ( C ) with k x k = 1 and any y ∈ F one has k [ ψ ( x ) , y ] k < ǫ. In addition, it was proved in [HO13] that the tracial Z -stability is equivalent to Z -stability in the case that the C ∗ -algebra A under consideration is unital simpleseparable nuclear. For our case, we will use the nesting form of almost elementarinessestablished in Theorem 8.9. We begin with the following result, which is a groupoidversion of [Phi14, Lemma 7.9]. Lemma 9.2.
Let G be a locally compact Hausdorff ´etale effective groupoid on acompact sapce. For any non-zero element a ∈ C ∗ r ( G ) + there is a non-zero function g ∈ C ( G (0) ) + such that g - a . If G is assumed ample, then g can be chosen to besupported on a compact open set in G (0) .Proof. Let a ∈ C ∗ r ( G ) + \ { } . Without loss of generality, one may assume k a k =1. Let ǫ ≤ (1 / k E ( a ) k where E : C ∗ r ( G ) → C ( G (0) ) is the canonical faithfulexpectation. Then there is an h ∈ C c ( G ) + such that k a / − h k < ǫ and k h k ≤ k h ∗ h ∗ − a k < ǫ and k h ∗ ∗ h − a k < ǫ . Then for h ∗ ∗ h ∈ C c ( G ) + ,Lemma 4.2.5 in [Sim17] shows that there is a function f ∈ C ( G (0) ) + such that k f k = 1, f ∗ h ∗ ∗ h ∗ f = f E ( h ∗ ∗ h ) f and k f ∗ h ∗ ∗ h ∗ f k ≥ k E ( h ∗ ∗ h ) k − ǫ . Thenone has k f ∗ h ∗ ∗ h ∗ f k > k E ( h ∗ ∗ h ) k − ǫ ≥ k E ( a ) k − ǫ ≥ ǫ. Then define g = ( f ∗ h ∗ ∗ h ∗ f − ǫ ) + = ( f E ( h ∗ ∗ h ) f − ǫ ) + ∈ C ( G (0) ) + \ { } . Thenusing Lemma 1.6 and 1.7 in [Phi14], one has g ∼ ( h ∗ f ∗ h ∗ − ǫ ) + - ( h ∗ h ∗ − ǫ ) + - a as desired. If G is ample then choose a compact open set V ⊂ supp( g ) and then1 V - g - a . Then 1 V is what we want. (cid:3) Lemma 9.3.
Let G be a locally compact Hausdorff ´etale effective groupoid on acompact space. Suppose G has the GSBP, and for any ǫ > , n ∈ N , non-zeropositive element g ∈ C ( G (0) ) + and finite collection F ⊂ C c ( G ) in which the support supp( f ) for any f ∈ F is a compact bisection contained in an open bisection V f andsatisfying µ ( ∂s (supp( f ))) = 0 for any µ ∈ M ( G ) , there is an order zero c.p.c. map ψ : M n ( C ) → C ∗ r ( G ) such that the following hold:(1) C ∗ r ( G ) − ψ (1 n ) - g .(2) for any x ∈ M n ( C ) with k x k = 1 and any f ∈ F one has k [ ψ ( x ) , f ] k r < ǫ. Then the C ∗ -algebra C ∗ r ( G ) is tracially Z -stable.If G is assumed ample then one may further require that function g above issupported on a compact open set and all function f ∈ F are supported on compactopen bisections.Proof. Let a ∈ C ∗ r ( G ) + \{ } , ǫ > F a finite set in C ∗ r ( G ) and n ∈ N . we aim to finda c.p.c. map ψ satisfies Definition 9.1. Since C c ( G ) is dense in C ∗ r ( G ), without loss ofany generality, we may assume F ⊂ C c ( G ). Then Proposition 2.7 implies that onemay assume further that each support supp( f ) of f ∈ F is a compact set contained in an open bisection V . Let f ∈ F and write K = supp( f ) ⊂ V . Now since G hasthe GSBP, there is an open set O such that s ( K ) ⊂ O ⊂ O ⊂ s ( V ) with µ ( ∂O ) = 0for any µ ∈ M ( G ). Then choose a function g ∈ C ( G (0) ) such that g = 1 on s ( K ),0 < g ≤ O and g = 0 on G (0) \ O . Now define h = g ( s ( f ) + ǫ/ ∈ C ( G (0) ).Observe that supp o ( h ) = O and k h − s ( f ) k ∞ ≤ ǫ/
3. Now define f ′ ∈ C c ( G ) by f ′ ( x ) = h ( s ( x )) for x ∈ V and f ′ = 0 on G \ V . Note that supp o ( f ′ ) = ( s | V ) − ( O )and thus supp( f ′ ) = ( s | V ) − ( O ) . In addition, since f − f ′ is supported on ( s | V ) − ( O ),which is a bisection. Then Proposition 2.9 implies that k f − f ′ k r = k f − f ′ k ∞ = sup x ∈ ( s | V ) − ( O ) | f ( x ) − f ′ ( x ) | = sup u ∈ O | s ( f )( u ) − h ( u ) | ≤ ǫ/ . Denote by F ′ = { f ′ : f ∈ F } obtained by the argument above. Now choose anon-zero positive function f ∈ C ( G (0) ) + such that f - a by Lemma 9.2. Then for ǫ >
0, finite set F ′ , n ∈ N and f , by assumption, there is an order zero c.p.c. map ψ : M n ( C ) → C ∗ r ( G ) such that the following hold:(i) 1 C ∗ r ( G ) − ψ (1 n ) - f .(ii) for any x ∈ M n ( C ) with k x k = 1 and any f ′ ∈ F ′ one has k [ ψ ( x ) , f ′ ] k r < ǫ/ . Then first, one has 1 C ∗ r ( G ) − ψ (1 n ) - f - a by our choice of f . In addition, foreach f ∈ F , one has k [ ψ ( x ) , f ] k r ≤ k [ ψ ( x ) , f ′ ] k + 2 k ψ ( x ) k r · k f − f ′ k r ≤ ǫ. This shows that C ∗ r ( G ) is tracially Z -stable.Now if G is ample, in the proof above, for any f ∈ F , one can choose O to becompact open and the corresponding f ′ is supported on O . In addition, the f canbe chosen to be supported on a compact open set by Lemma 9.2. (cid:3) Then we generically show how to construct c.p.c. order zero map from M n ( C ) to C ∗ r ( G ) to establish the tracial Z -stability. Remark 9.4.
Let G be a locally compact Hausdorff ´etale groupoid on a compactspace. Let n ∈ N and ǫ >
0. In addition, let N ∈ N such that N > /ǫ and K be acompact set in G . Suppose A , B , C and D are open castles such that(i) A is K -extendable to B and C is K -extendable to D .(ii) B is nested in D with multiplicity at least nN .(iii) A is nested in C with multiplicity at least nN .(iv) µ ( S A (0) ) > − ǫ/ µ ∈ M ( G ).If we write A and B explicitly, say, by A = { A li,j : i, j ∈ F l , l ∈ I } and B = { B li,j : i, j ∈ E l , l ∈ I } where F l ⊂ E l for each l ∈ I then Proposition 6.12 implies that thereare compact castle A ′ = { A ′ li,j : i, j ∈ F l , l ∈ I } and B ′ = { B ′ li,j : i, j ∈ E l , l ∈ I } such that(i) all A ′ li,j and B ′ li,j are compact sets;(ii) for each l ∈ I , one has A ′ li,j ⊂ A li,j for any i, j ∈ F l and B ′ li,j ⊂ B li,j for any i, j ∈ E l ;(iii) A ′ is K -extendable to B ′ and(iv) µ ( S A ′ (0) ) > − ǫ/ µ ∈ M ( G ).Now for each l ∈ I fix an i ∈ E l . We first define a function h B li,i by choosing acontinuous function in C ( G (0) ) + such that supp( h B li,i ) ⊂ B li,i and h B li,i ≡ B ′ li,i . Then for j, k ∈ E l , we define h B lj,i be the function such that LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 59 s ( h B lj,i ) = h B li,i and define h B lj,j = r ( h B lj,i ). By this process, we have a collection { h B : B ∈ B} of functions in C c ( G ) such that(i) s ( h B ) = h s ( B ) and r ( h B ) = h r ( B ) for each B ∈ B .(ii) h r ( B ) ∗ B = h B and 1 B ∗ h s ( B ) = h B for each B ∈ B .(iii) supp( h B ) ⊂ B for each B ∈ B .(iv) h B ≡ B ′ for any B ∈ B where B ′ ⊂ B is the compact bisection containedin B ′ .In this case we say the collection { h B : B ∈ B} above is B - compatible .Now we write C and D explicitly by C = { C pt,s : t, s ∈ T p , p ∈ J } and D = { D pt,s , t, s ∈ S p , p ∈ J } . Let H (0) ⊂ D (0) be a subset containing C (0) . Now, since C (0) ⊂ H (0) , one has that H (0) contains some D -levels from multisection ( D p ) (0) forany p ∈ J . Denote by ( H p ) (0) = ( D p ) (0) ∩ H (0) . Now, for each p ∈ J and let l ∈ I such that B l is nested in D p with multiplicity at least nN . Fix a level D pt,t where t ∈ S p and define P p,t,l = { B ∈ ( B l ) (0) : B ⊂ D pt,t } . Note that | P p,t,l | ≥ nN . Thenfor m = 1 , . . . , n choose a subset P p,t,l,m ⊂ P p,t,l such that | P p,t,l,m | = ⌊| P p,t,l | /n ⌋ . Inaddition, choose a bijection Λ p,t,l,m : P p,t,l, → P p,t,l,m . Then for any D ∈ ( H p ) (0) there is a bisection D ′ ∈ D p such that s ( D ′ ) = D pt,t and r ( D ′ ) = D . Now define P D,l = { r ( D ′ B ) : B ∈ P p,t,l } and P D,l,m = { r ( D ′ B ) : B ∈ P p,t,l,m } for all 1 ≤ m ≤ n . In addition, define mapsΘ D,l,m : P D,l, → P D,l,m byΘ
D,l,m ( r ( D ′ B )) = r ( D ′ Λ p,t,l,m ( B ))for any B ∈ P p,t,l, and define Θ D,l,k,m = Θ
D,l,k ◦ Θ − D,l,m . From this construction,for any p ∈ J and l ∈ I such that B l is nested in D p , we actually have the followingconfiguration.(i) P D,l = { B ∈ ( B l ) (0) : B ⊂ D } has the cardinality | P D,l | > nN for any D ∈ ( H p ) (0) .(ii) There are collections P D,l,m ⊂ P D,l such that | P D,l,m | = ⌊| P D,l | /n ⌋ for any D ∈ ( H p ) (0) and 1 ≤ m ≤ n .(iii) There are bijective maps Θ D,l,k,m : P D,l,m → P D,l,k for any D ∈ ( H p ) (0) and1 ≤ m, k ≤ n . For any 1 ≤ k, m, p ≤ n , these functions also satisfy(a) Θ D,l,m,m is the identity map;(b) Θ − D,l,k,m = Θ
D,l,m,k ;(c) Θ
D,l,k,m Θ D,l,m,p = Θ
D,l,k,p ..(iv) For any D ∈ D such that s ( D ) , r ( D ) ∈ ( H p ) (0) one has r ( D Θ s ( D ) ,l,k,m ( B )) = Θ r ( D ) ,l,k,m ( r ( DB ))for any B ∈ P s ( D ) ,l,k,m and 1 ≤ , k, m ≤ n .In this case, we call such a collection of all sets P D,l,m together with all maps Θ
D,l,k,m for any p ∈ J , l ∈ I such that B l is nested in D p , D ∈ ( H p ) (0) , 1 ≤ k, m ≤ n , a H (0) - B (0) - nesting system .Now for D ∈ ( H p ) (0) , l ∈ I such that B l is nested in D p and 1 ≤ k, m ≤ n , wedenote by R D,l,k,m = { B ∈ B l : s ( B ) ∈ P D,l,m and r ( B ) = Θ D,l,k,m ( s ( B )) } . For each p ∈ J write I p = { l ∈ I : B l is nested in D p with multiplicity at least nN. } and for each D ∈ ( H p ) (0) define Q k,m,D = G l ∈ I p R D,l,k,m . In addition, we fix an arbitrary function κ : H (0) → [0 , e km the matrixin M n ( C ) whose ( k, m )-entry is 1 while other entries are zero. Now we define a map ψ : M n ( C ) → C ∗ r ( G ) by ψ ( e km ) = X D ∈H (0) X B ∈ Q k,m,D κ ( D ) h B and is linearly extended to define on the whole M n ( C ). We will show in the followinglemma that the map ψ is a c.p.c. order zero map.On the other hand, note that for each p ∈ J the index set I p consists exactly all l ∈ I such that A l is nested in C p with multiplicity at least nN . Then for any C p -level C pt,t and l ∈ I p , there are at most n − A ∈ ( A l ) (0) with A ⊂ C pt,t so that ψ (1 n ) is not supported on. Then choose one such level, denoted by A p,l , in ( A l ) (0) .Now, for any µ ∈ M ( G ), first the fact that A l is nested in C p with multiplicity atleast nN implies that X p ∈ J X l ∈ I p nN | T p | µ ( A p,l ) ≤ µ ( [ A (0) ) ≤ . Then one has µ ( [ { A ∈ A (0) : ψ (1 n ) ≡ A } ) ≤ X p ∈ J X l ∈ I p ( n − | T p | µ ( A p,l ) ≤ /N. and thus µ ( [ { A ′ ∈ A ′ (0) : ψ (1 n ) ≡ A ′ } ) ≥ − ǫ/ − /N ≥ − ǫ. Write f = 1 C ∗ r ( G ) − ψ (1 n ). Then one has µ (supp( f )) < ǫ for any µ ∈ M ( G ). Lemma 9.5.
Let G be a locally compact Hausdorff ´etale groupoid on a compactspace. The map ψ defined in Remark 9.4 is a c.p.c. order zero map.Proof. Let A , B , C and D be open castles defined above. Let H (0) , P D and Q k,m,D bespecific sets defined in Remark 9.4 above as well. Now define ϕ : M n ( C ) → C ∗ r ( G ) ∗∗ by ϕ ( e km ) = X D ∈H (0) X B ∈ Q k,m,D B and extending linearly where 1 B is the indicator function on the open set B . It isstraightforward to see ϕ above is a homomorphism by using (a), (b) and (c) of (iii)in the configuration of H (0) - B (0) -nesting system. Define h ∈ C c ( G ) by h = ψ (1 n ) = n X i =1 X D ∈H (0) X B ∈ Q i,i,D κ ( D ) h B where ψ is the map defined in Remark 9.4. Then consider h ∗ ϕ ( e km ) = ( n X i =1 X D ′ ∈H (0) X B ′ ∈ Q i,i,D ′ κ ( D ′ ) h B ′ ) ∗ ( X D ∈H (0) X B ∈ Q k,m,D B ) LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 61 = n X i =1 X D ′ ∈H (0) X B ′ ∈ Q i,i,D X D ∈H (0) X B ∈ Q k,m,D κ ( D ′ ) h B ′ ∗ B . Let B ′ ∈ Q i,i,D ′ and B ∈ Q k,m,D . Note that κ ( D ′ ) h B ′ ∗ B = κ ( D ) h B if B ′ = r ( B ), D = D ′ and i = k . Otherwise, κ ( D ′ ) h B ′ ∗ B = 0. Thus one has h ∗ ϕ ( e km ) = X D ∈H (0) X B ∈ Q k,m,D κ ( D ) h B = ψ ( e km ) . Similarly, one has ϕ ( e km ) ∗ h = ( X D ∈H (0) X B ∈ Q k,m,D B )( n X i =1 X D ′ ∈H (0) X B ′ ∈ Q i,i,D ′ κ ( D ′ ) h B ′ )= X D ∈H (0) X B ∈ Q k,m,D n X i =1 X D ′ ∈H (0) X B ′ ∈ Q i,i,D κ ( D ′ )1 B ∗ h B ′ . Let B ′ ∈ Q i,i,D ′ and B ∈ Q k,m,D . Note that κ ( D ′ )1 B ∗ h B ′ = κ ( D ) h B if B ′ = s ( B ), D = D ′ and i = m . Otherwise, κ ( D ′ ) h B ′ ∗ B = 0. This implies that ϕ ( e km ) ∗ h = X D ∈H (0) X B ∈ Q k,m,D κ ( D ) h B = ψ ( e km ) . This shows that the homomorphism ϕ in fact maps M n ( C ) into C ∗ r ( G ) ∗∗ ∩ { h } ′ and ϕ ( a ) ψ (1 n ) = ψ ( a ) for any a ∈ M n ( C ). Then Theorem 3.3 in [WZ09] shows that ψ is a c.p.c. order zero map. (cid:3) Lemma 9.6.
Let G be a locally compact Hausdorff ´etale groupoid on a compactspace. Let U , U O and O be precompact open bisections such that U i ⊂ O i for i = 1 , and µ ( ∂s ( U i )) = 0 for any µ ∈ M ( G ) . Then µ ( ∂s ( U U )) = µ ( ∂r ( U U )) =0 for any µ ∈ M ( G ) .Proof. For any µ ∈ M ( G ), one first has µ ( ∂r ( U i )) = µ ( ∂s ( U i )) = 0 for i = 1 , µ ( ∂ ( r ( U ) ∩ s ( U )) ≤ µ ( ∂r ( U ))+ µ ( ∂s ( U )) = 0. Thus one has µ ( ∂s ( U U )) = µ ( r ( O − ∂ ( r ( U ) ∩ s ( U ))) = 0 and in the similar way one also has µ ( ∂r ( U U )) = µ ( r ( O ∂ ( r ( U ) ∩ s ( U ))) = 0. (cid:3) Now, we are ready to establish the following theorem.
Theorem 9.7.
Let G be a locally compact Hausdorff σ -compact ´etale minimalgroupoid on a compact metrizable space. Suppose G is almost elementary. Then C ∗ r ( G ) is simple and tracially Z -stable.Proof. First fix a metric d on G (0) and an integer n ∈ N . Since G is almost elementary,Proposition 6.10 and Theorem 8.5 imply that G is effective and has the GSBP. Thuswe prove this theorem by using Lemma 9.3. Now let ǫ > , n ∈ N , g be a non-zeropositive function in C ( G (0) ) + and F a finite collection of functions in C c ( G ) suchthat for any f ∈ F the support, supp( f ), of f is contained in an open bisection V f and satisfies µ ( ∂s (supp( f ))) = 0 for any µ ∈ M ( G ). Write m = | F | . Since G is minimal, there is an η > µ (supp o ( g )) > η for any µ ∈ M ( G )Note that η < / N ∈ N such that N ≥ max { / n ǫ, /η } . Denote by O f = supp o ( f ). Since r ( V f ∂s ( O f )) = ∂r ( O f ),one also has µ ( ∂r ( O f )) = 0 for any µ ∈ M ( G ). Define S = ( [ f ∈ F ∂s ( O f )) ∪ ( [ f ∈ F ∂r ( O f )) . Now Lemma 6.13 implies that there is a δ > µ ( ¯ B d ( S, δ )) < η (2 m ) N +1 forany µ ∈ M ( G ). Define a compact set K = N +1 [ i =1 (( [ f ,...,f i ∈ F U f · U f · · · · U f i ) ∪ G (0) where U f = O f or O − f for any f ∈ F . Choose an open cover V of G (0) in which anymember V ∈ V has diameter less than δ and for any u, v ∈ V and f ∈ F one has( ⋆⋆⋆ ) | s ( f )( u ) − s ( f )( v ) | < ǫ/ n and | r ( f )( u ) − r ( f )( v ) | < ǫ/ n . Since G is almost elementary, For the compact set K , the cover V , and the integer n , Theorem 8.9, Remark 8.10 and Lemma 9.6 imply that there are open castles A , B , C and D such that(1) both ¯ C = { C : C ∈ C} and ¯ D = { D : D ∈ D} are compact castles;(2) A is K -extendable to B and ¯ C is K -extendable to ¯ D ;(3) For any i ≤ N +1, f , . . . , f i ∈ F and C -level C , either C ⊂ s ( U f · U f · · ·· U f i )or C ∩ s ( U f · U f · · · · U f i ) = ∅ , where U f k = O f k or U f k = O − f k for any1 ≤ k ≤ i .(4) For any i ≤ N + 1 and f , . . . , f i ∈ F if a C -level C ⊂ s ( U f · U f · · · · U f i )where U f k = O f k or U f k = O − f k for any 1 ≤ k ≤ i then U f · U f · · · · U f i C = D for some D ∈ D .(5) B is nested in D with multiplicity at least nN (6) A is nested in C with multiplicity at least nN ;(7) any D -level is contained in a member of V ;(8) µ ( G (0) \ S A (0) ) < η for any µ ∈ M ( G ).Now for each f ∈ F define T f = { u ∈ G (0) : d ( u, G (0) \ s ( O f )) ≥ δ } ∪ { u ∈ G (0) : d ( u, s ( O f )) ≥ δ } and T f = { u ∈ G (0) : d ( u, G (0) \ r ( O f )) ≥ δ } ∪ { u ∈ G (0) : d ( u, r ( O f )) ≥ δ } . In addition, define R = T f ∈ F ( T f ∩ T f ). Note that G (0) \ R ⊂ ¯ B ( S, δ ) and thus µ ( R ) ≥ − η (2 m ) N +1 for any µ ∈ M ( G ). Then for any D -level D ∈ D (0) , sincediam d ( D ) < δ , if D ∩ R = ∅ then D ∩ S = ∅ and thus either D ⊂ s ( O f ) or D ∩ s ( O f ) = ∅ and either D ⊂ r ( O f ) or D ∩ r ( O f ) = ∅ for any f ∈ F .Now define C (0)0 = { C ∈ C (0) : C ∩ S = ∅} . Observe that for any µ ∈ M ( G ) onehas µ ( [ { A ∈ A (0) : A ⊂ C, C ∈ C (0)0 } ) ≥ − η − η/ (2 m ) N +1 and for each f ∈ F and C ∈ C (0)0 one has either O f C ∈ D or O f C = ∅ and either O − f C ∈ D or O − f C = ∅ . Define C ′ (0)0 = { C ∈ C (0)0 : there exists D ∈ D (0) such that D = r ( O f C ) or D = r ( O − f C ) LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 63 for some f ∈ F and D ∩ S = ∅ . } and C (0)1 = C (0)0 \ C ′ (0)0 . Observe that for any D ∈ D (0) with D ∩ S = ∅ one has D ⊂ ¯ B ( S, δ ) and there are at most 2 m levels C ∈ C ′ (0)0 such that r ( O f C ) = D or r ( O − f C ) = D for some f ∈ F . This implies that µ ( S C ′ (0)0 ) < η (2 m ) N for any µ ∈ M ( G ). Thus one has µ ( [ C (0)1 ) ≥ µ ( [ { A ∈ A (0) : A ⊂ C, C ∈ C (0)1 } ) ≥ − η − η/ (2 m ) N +1 − η/ (2 m ) N . In addition, for any C ∈ C (0)1 and f, g ∈ F , the condition (3) for C above implieseither U f U g C = ∅ or U f U g C ∈ D where U f = O f or O − f and U g = O g or O − g .Then by induction, suppose we have C (0) k for k < N such that(i) either U f i . . . U f C ∈ D or U f i . . . U f C = ∅ for any f , . . . , f i ∈ F and i ≤ k ,where U f i = O f i or U f i = O − f i for any i ≤ k .(ii) if D = U f i . . . U f C ∈ D for some f , . . . , f i ∈ F and i ≤ k then r ( D ) ∩ S = ∅ .(iii) U f k +1 . . . U f C ∈ D for any f , . . . , f k +1 ∈ F if it is not empty by (ii) justabove and the condition (3), (4) in the setting of C before.(iv) for any µ ∈ M ( G ) one has µ ( S C (0) k ) ≥ µ ( S { A ∈ A (0) : A ⊂ C, C ∈ C (0) k } ) ≥ − η − η (2 m ) N +1 − P ki =1 η (2 m ) N +1 − i .Define C ′ (0) k = { C ∈ C (0) k : there exists D ∈ D (0) such that D = r ( U f k +1 . . . U f C ) ,U f i = O f i or U f i = O − f i , f , . . . , f k +1 ∈ F and D ∩ S = ∅} and C (0) k +1 = C (0) k \ C ′ (0) k . Then, similarly, for any D ∈ D (0) and D ∩ S = ∅ , there are atmost (2 m ) k +1 levels C ∈ C ′ (0) k such that D = r ( U f . . . U f k +1 C ). Then µ ( S C ′ (0) k ) < (2 m ) k +1 · η (2 m ) N +1 < η (2 m ) N − k holds and thus one has µ ( C (0) k +1 ) ≥ µ ( [ { A ∈ A (0) : A ⊂ C, C ∈ C (0) k +1 } ) ≥ − η − η (2 m ) N +1 − k +1 X i =1 η (2 m ) N +1 − i . for any µ ∈ M ( G ). In addition, by definition of C (0) k +1 , it is straightforward to verifythe corresponding properties (i)-(iii) above for k + 1. This finishes our inductivedefinition for k = 0 , . . . , N . Now we look at C (0) N , which satisfies correspondingproperties (i)-(iv) for k = N . In particular, one has µ ( C (0) N ) ≥ µ ( [ { A ∈ A (0) : A ⊂ C, C ∈ C (0) N } ) ≥ − η − η (2 m ) N +1 − N X i =1 η (2 m ) N +1 − i > − η > . and thus in particular C (0) N is not empty. Now define D (0)0 = C (0) N and inductivelydefine D (0) k = { D ∈ D (0) : D = r ( U f k . . . U f C ) , U f i = O f i or O − f i , for i = 1 , . . . , k, f , . . . , f k ∈ F and C ∈ C (0) N } \ k − G i =0 D (0) i for k = 1 , . . . , N + 1 (some D (0) k may be empty). Define H (0) = F Nk =0 D (0) k , which isa subset of D (0) and contains C (0) N = D (0)0 . Now we define a c.p.c. order zero mapby using Remark 9.4 via choosing a B -compatible functions { h B ∈ C c ( G ) : B ∈ B} and a H (0) - B (0) -nesting system. Note that C (0) N here plays the role as C (0) in Remark9.4. Then we define a function κ : H (0) → [0 ,
1] by κ ( D ) = 1 − k/N if D ∈ D (0) k for k = 0 , . . . , N . Finally, we define ψ : M n ( C ) → C ∗ r ( G ) by ψ ( e ij ) = X D ∈H (0) X B ∈ Q i,j,D κ ( D ) h B and extending linearly. Lemma 9.5 implies that ψ is a c.p.c. order zero map. Inaddition, by Remark 9.4, for function h = 1 C ∗ r ( G ) − ψ (1 n ) one has µ (supp( h )) < η < µ (supp o ( g ))for any µ ∈ M ( G ). Then since G has groupoid strict comparison by Theorem 6.19,one has supp o ( h ) - G supp o ( g ), which implies that1 C ∗ r ( G ) − ψ (1 n ) = h - g by Proposition 6.1 in [Ma20].Now, for any f ∈ F , e ij ∈ M n ( C ), define sets S f = { D ∈ H (0) : D ⊂ s ( O f ) , r ( O f D ) ∈ H (0) } and R f = { D ∈ H (0) : D ⊂ r ( O f ) , r ( O − f D ) ∈ H (0) } . Observe that the map σ f : S f → R f defined by σ f ( D ) = r ( O f D ) is bijective. Notethat for any D ∈ H (0) , there is a C ∈ C (0) N and U f , · · · , U f k for some k ≤ N suchthat D = r ( U f · · · U f k C ), where each U f i = O f i or O − f i . Then one has O f D = ∅ or r ( O f D ) ∈ H (0) ⊔ D N +1 , which entails that r ( O f D ), if not empty, is in the samemultisection of D with the given level D .Define a map π f : S f → D by s ( π f ( D )) = D and r ( π f ( D )) = σ f ( D ). Thendefine another bijective map θ i,j,f,D : Q i,j,D → Q i,j,σ f ( D ) in the following way. Forany B ∈ Q i,j,D , define θ i,j,f,D ( B ) ∈ B such that s ( θ i,j,f,D ( B )) = r ( π f ( D ) s ( B )) and r ( θ i,j,f,D ( B )) = r ( π f ( D ) r ( B )) The map θ i,j,f,D is well-defined because the property(iv) of the definition of H (0) - B (0) -nesting system in Remark 9.4. Note that π f ( D ) B = θ i,j,f,D ( B ) π f ( D ) s ( B ) ∈ B . Now one has[ ψ ( e ij ) , f ] = X D ∈H (0) X B ∈ Q i,j,D κ ( D ) f ∗ h B − X D ∈H (0) X B ∈ Q i,j,D κ ( D ) h B ∗ f = X D ∈H (0) D ⊂ s ( O f ) X B ∈ Q i,j,D κ ( D ) f ∗ h B − X D ∈H (0) D ⊂ r ( O f ) X B ∈ Q i,j,D κ ( D ) h B ∗ f = X D ∈ S f X B ∈ Q i,j,D κ ( D ) f ∗ h B − X D ∈ R f X B ∈ Q i,j,D κ ( D ) h B ∗ f LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 65 = X D ∈ S f X B ∈ Q i,j,D ( κ ( D ) f ∗ h B − κ ( σ f ( D )) h θ i,j,f,D ( B ) ∗ f ) . The second equality above is due to the fact that either D ⊂ s ( O f ) or D ∩ s ( O f ) = ∅ and either D ⊂ r ( O f ) or D ∩ r ( O f ) = ∅ for any D ∈ H (0) . On the other hand,if ∅ 6 = D ∈ H (0) with D ⊂ s ( O f ) but r ( O f D ) / ∈ H (0) (note that in this case r ( O f D ) ∈ D N +1 ), then D ∈ D N necessarily. In this case, observe that κ ( D ) = 0. Inthe same way, if ∅ 6 = D ∈ H (0) with D ⊂ r ( O f ) but r ( O − f D ) / ∈ H (0) then κ ( D ) = 0.This establishes the third equality above. Finally, the fourth equality above is touse bijections σ f and θ i,j,f,D defined above. Now for fixed i, j, f write a D,B = κ ( D ) f ∗ h B − κ ( σ f ( D )) h θ i,j,f,D ( B ) ∗ f for simplicity. Note that a D,B ∈ C c ( G ) and supported on the bisection π f ( D ) B ∈B and thus k a D,B k r = k a D,B k ∞ by Proposition 2.9. Now for any D ∈ S f with D ∈ D (0) k , if k = 0, then σ f ( D ) ∈ D (0)0 ⊔ D (0)1 . If 1 ≤ k ≤ N − σ f ( D ) ∈D (0) k − ⊔ D (0) k ⊔ D (0) k +1 . Finally, if k = N then necessarily one has σ f ( D ) ∈ D (0) N − ⊔ D (0) N .Therefore, in any case, for D ∈ S f , one has | κ ( D ) − κ ( σ f ( D )) | < /N < ǫ/ n . On the other hand, for any γ ∈ B ′ = π f ( D ) B = θ i,j,f,D ( B ) π f ( D ) s ( B ), there isa unique decomposition of γ by γ = α β = β α , where α ∈ π f ( D ) r ( B ), α ∈ π f ( D ) s ( B ), β ∈ B and β ∈ θ i,j,f,D ( B ). In addition, by B -compatibility of h B , onehas h B ( β ) = h s ( B ) ( s ( β )) = h s ( B ′ ) ( s ( γ )) = h B ′ ( γ )= h r ( B ′ ) ( γ ) = h r ( θ i,j,f,D ( B )) ( r ( β )) = h θ i,j,f,D ( B ) ( β ) . Finally, since D ⊂ V for some V ∈ V , then ( ⋆⋆⋆ ) implies that | f ( α ) − f ( α ) | = | s ( f )( s ( α )) − s ( f )( s ( α )) | < ǫ/ n . This implies that for any γ = α β = β α ∈ B ′ as decomposed above one has | ( f ∗ h B )( γ ) − ( h θ i,j,f,D ( B ) ∗ f )( γ ) | = | f ( α ) h B ( β ) − h θ i,j,f,D ( B ) ( β ) f ( α ) | = | h B ( β ) || f ( α ) − f ( α ) | ≤ ǫ/ n . This implies that k f ∗ h B − h θ i,j,f,D ( B ) ∗ f k ∞ = sup γ ∈ B ′ | ( f ∗ h B )( γ ) − ( h θ i,j,f,D ( B ) ∗ f )( γ ) | ≤ ǫ/ n and thus one has k a D,B k r = k a D,B k ∞ = k κ ( D ) f ∗ h B − κ ( σ f ( D )) f ∗ h B + κ ( σ f ( D )) f ∗ h B − κ ( σ f ( D )) h θ i,j,f,D ( B ) ∗ f k ∞ ≤ | κ ( D ) − κ ( σ f ( D )) |k f ∗ h B k ∞ + κ ( σ f ( D )) k f ∗ h B − h θ i,j,f,D ( B ) ∗ f k ∞ ≤ ǫ/ n = ǫ/n . Finally, observe that all such a D,B for D ∈ S f and B ∈ Q i,j,D are pairwise disjointin the sense that a D ,B a ∗ D ,B = a ∗ D ,B a D ,B = 0 whenever ( D , B ) = ( D , B ).This implies that for any f ∈ F and e ij ∈ M n ( C ) one has k [ ψ ( e ij ) , f ] k r = k X D ∈ S f X B ∈ Q i,j,D a D,B k r = max D ∈ S f B ∈ Q i,j,D k a D,B k r ≤ ǫ/n . Then for any x = P ≤ i,j ≤ n x ij e ij ∈ M n ( C ) with k x k = 1 one has k [ ψ ( x ) , f ] k r ≤ X ≤ i,j ≤ n k [ ψ ( e ij ) , f ] k r ≤ ǫ This thus establishes that C ∗ r ( G ) is tracially Z -stable by Lemma 9.3. Finally, since G is minimal and effective, C ∗ r ( G ) is simple by Proposition 4.3.7 in [Sim17]. Thesefinishes the proof. (cid:3) Using Theorem 9.7 directly, we have the following natural corollaries.
Corollary 9.8.
Let G be a locally compact Hausdorff σ -compact ´etale minimalgroupoid on a compact metrizable space. Suppose G is almost elementary. Then C ∗ r ( G ) has strict comparison for positive elements.Proof. Theorem 9.7 implies that C ∗ r ( G ) is simple unital tracially Z -stable C ∗ -algebras.Then Theorem 3.3 in [HO13] shows that C ∗ r ( G ) has the strict comparison for positiveelements. (cid:3) In the ample case, one can drop the condition of metrizability on the unit space,based on the same reason in Remark 8.11. In fact, using Theorem 8.12, Lemma 9.2,9.3 for ample groupoids and Proposition 7.2, the same argument of Theorem 9.7 inthe compact open setting show the following result.
Corollary 9.9.
Let G be a locally compact ample Hausdorff σ -compact ´etale minimalgroupoid on a compact space. Suppose G is almost elementary. Then C ∗ r ( G ) istracially Z -stable and thus has strict comparison for positive elements. We remark that Corollaries 9.8 and 9.9 do not assume nuclearity of C ∗ r ( G ). How-ever, given the current technologies and interests in the classification and structuretheory of C ∗ -algebras, the most interesting case appears to be where G is amenableand second countable. Corollary 9.10.
Let G be a locally compact Hausdorff amenable second countable´etale minimal groupoid on a compact space. Suppose G is almost elementary. Then C ∗ r ( G ) is unital simple separable nuclear and Z -stable and thus has nuclear dimen-sion one. In addition, in this case C ∗ r ( G ) is classified by its Elliott invariant. Finally,If M ( G ) = ∅ , then C ∗ r ( G ) is quasidiagonal and if M ( G ) = ∅ then C ∗ r ( G ) is a unitalKirchberg algebra.Proof. Since G is assumed to be amenable, Theorem 9.7 implies C ∗ r ( G ) is unitalsimple separable nuclear and tracially Z -stable and thus Z -stable by Theorem 4.1in [HO13]. In this case, the nuclear dimension dim nuc ( C ∗ r ( G )) = 1 by Theorem Aand Corollary C in [CET + C ∗ r ( G ) is classified by Elliott invariantby the recent progress of classification theorem for unital simple nuclear separable C ∗ -algebras having finite nuclear dimendion and satisfying the UCT via combining[EGLN15, GLN15, TWW17, Phi00]. Finally, if M ( G ) = ∅ then there is a non-zerotracial state on C ∗ r ( G ). This implies that C ∗ r ( G ) is stably finite and thus quasidiag-onal by Corollary 6.1 in [TWW17]. On the other hand, if M ( G ) = ∅ , then C ∗ r ( G )is traceless and thus C ∗ r ( G ) is purely infinite by Corollary 5.1 in [Rør04]. Therefore,in this case C ∗ r ( G ) is a unital Kirchberg algebra. (cid:3) Now we apply our results to almost finite grouoids in Matui’s sense and obtainthe following result.
LMOST ELEMENTARINESS AND FIBERWISE AMENABILITY 67
Corollary 9.11.
Let G be a locally compact Hausdorff σ -compact ample ´etale mini-mal groupoid on a compact space. Suppose G is almost finite in Matui’s sense. Then C ∗ r ( G ) is tracially Z -stable and thus has the strict comparison for positive elements.If we assume G is also amenable and second countable then C ∗ r ( G ) is Z -stable andquasidiagonal.Proof. Proposition 7.3 and Theorem 7.4 shows that G is fiberwise amenable andalmost elementary. Then Proposition 5.9 implies that M ( G ) = ∅ . Now Corollary9.8 and 9.10 shows the result. (cid:3) Then we may recover the following result due to Kerr in [Ker20].
Corollary 9.12.
Let α : Γ y X be a minimal free action of a countable discreteamenable group Γ on a compact metrizable space X . Suppose α is almost finite inKerr’s sense. Then the crossed product C ( X ) ⋊ r Γ is Z -stable and quasidiagonal.Proof. Theorem 7.8 shows that the transformation groupoid X ⋊ α Γ is fiberwiseamenble and almost elementary. In addition, amenability of Γ implies that X ⋊ α Γis amenable. Then Corollary 9.10 shows the result. (cid:3)
We finally provide several applications of our result on Z -stability of almost finitegroupoids. Example 9.13.
Recently, in [IWZ19], Ito, Whittaker and Zacharias established Z -stability of Kellendonk’s C ∗ -algebra of an aperiodic and repetitive tiling with finitelocal complexity through generalizing the approach for group actions in [Ker20] togroupoid actions. In addition, they showed that such a C ∗ -algebra is a reduced C ∗ -algebra of a locally compact Hausdorff ´etale second countable minimal princi-ple almost finite tiling groupoid. Thus, their result is a direct application of ourCorollary 9.11 .Recall a geometric groupoid G in Example 7.9, constructed by Elek is a locallycompact Hausdorff second countable ´etale minimal principal almost finite amplegroupoid G , which is not amenable. Therefore, Corollary 9.11 implies that C ∗ r ( G ) isnot nuclear but tracially Z -stable.Finally, it was shown in [OS20, Theorem 2.10] that all minimal actions of theinfinite dihedral group Z ⋊ Z on the Cantor set is almost finite in Matui’s sense(and thus their transformation groupoids are almost elementary). Such an actioncould be non-free (see Remark 6.11). But the crossed product of any such action is Z -stable by Corollary 9.11. References [ABBL20] Pere Ara, Christian B¨onicke, Joan Bosa, and Kang Li. Strict comparison for C ∗ -algebrasarising from almost finite groupoids. preprint, arXiv:2002.12221, 2020.[ALLW18] Pere Ara, Kang Li, Fernando Lled´o, and Jianchao Wu. Amenability of coarse spacesand K -algebras. Bull. Math. Sci. , 8(2):257–306, 2018.[BW92] Jonathan Block and Shmuel Weinberger. Aperiodic tilings, positive scalar curvature andamenability of spaces.
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X. Ma: 305 Mathematics Building, UB North Campus, Buffalo NY, 14260-2900,United States
Email address : [email protected] J. Wu: Department of Mathematics, Mailstop 3368, Texas A&M University, CollegeStation, TX 77843-3368, United States
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