Equicontinuous foliated spaces
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J. A. ´ALVAREZ L ´OPEZ* AND A. CANDEL † C ontents Introduction 11. Foliated spaces 22. Pseudogroups of local transformations 63. Coarse quasi-isometries 84. Coarse quasi-isometry type of orbits 95. F¨olner orbits 156. An example of non-recurrent compact generation 187. Quasi-local metric spaces 198. Equicontinuous pseudogroups 209. Quasi-e ff ective pseudogroups 2610. Coarse quasi-isometry type of orbits with trivial groups of germs 2911. Minimality of the orbit closures 3112. The closure of a strongly equicontinuous pseudogroup 3213. Local metric spaces 3414. Pseudogroups of local isometries 3715. Isometrization of strongly equicontinuous pseudogroups 3816. A non-standard description of weak equicontinuity 4217. Strongly equicontinuous foliated spaces 43References 46I ntroduction The theme of this paper originates in the study of the generic geometry of leaves of afoliated space. In [3], we analyze the problem of when all (or almost all) the leaves of afoliated space are quasi-isometric. In this paper, a dynamical condition on a foliated spaceguaranteeing that all the leaves without holonomy are quasi-isometric will be discussed.The condition is on the structure of the action of the holonomy pseudogroup, and is calledequicontinuity. That such condition may imply that all leaves without holonomy are quasi-isometric comes from the structure theorem of Riemannian foliations. These foliationsare the models of equicontinuous foliated spaces, and P. Molino’s work [17] describingtheir structure has as a consequence that the holonomy covers of all the leaves are quasi-isometric via a di ff eomorphism. Indeed, given a compact, connected manifold M endowedwith a Riemannian foliation, Molino shows that there is a fiber bundle over M with compactfiber (the transverse orthonormal frame bundle), and with a foliation transverse to the fibers *Research of the first author supported by DGICYT Grant PB95-0850. † Research of the second author supported by NSF Grants. whose leaves are the holonomy covers of the leaves of M . Furthermore, there is a group ofautomorphisms of this bundle which permutes the leaves.The concept of Riemannian foliation is easily formulated by saying that the holonomypseudogroup is a pseudogroup of local isometries of a Riemannian manifold. This gener-alizes to equicontinuous pseudogroups of local transformations of topological spaces. Assuch, the concept of equicontinuity appears in R. Sacksteder’s paper [24]. The parallelismbetween Riemannian foliations and equicontinuous pseudogroups is developed by E. Ghysin [17, Appendix E], see also M. Kellum’s paper [16] for this connection.Thus, in certain measure, what is done in this paper may be seen as a generalizationof that aspect of Molino’s theory pointed out above. First we show that all leaves withoutholonomy have the same coarse quasi-isometry type. For general equicontinuous foliatedspaces, it seems not to be possible to obtain this result, the main obstruction being thevery general structure of the transverse models. This obstruction can be overcome byimposing certain regularity to the transverse structure. The proof of this result requirescertain amount of work on pseudogroups and on the geometric structure of their orbits,and on how to pass from coarse quasi-isometries between orbits to leaves.When the foliated space is smooth, then it is possible to introduce a metric tensor onthe leaves that varies continuously from leaf to leaf. In this case the above results can beimproved by using quasi-isometries via di ff eomorphisms between leaves. Moreover, forgeneral equicontinuous foliated spaces, it can be shown that the universal covers of allleaves are quasi-isometric to each other via di ff eomorphisms. These results are obtainedwith the help of normal bundle theory.From our study of pseudogroups, it also follows that equicontinuous foliated spaces(with some mild conditions) satisfy two other typical properties of Riemannian foliations.First, the leaf closures are homogeneous spaces and form a partition. Second, the holo-nomy pseudogroup is indeed given by local isometries with respect to some “local metric,”and has a closure in certain sense. The existence of this closure of the holonomy pseu-dogroup is an important ingredient of our topological description of Riemannian foliationswith dense leaves given in [2].The main results of this paper and [2] were conjectured and greatly justified by E. Ghys[17, Appendix E]. The concept of equicontinuity for foliations was also studied by M. Kel-lum [16] and C. Tarquini [27]. Kellum dealt with the more restrictive setting of transverselyquasi-isometric foliations, and Tarquini showed that equicontinuous transversely confor-mal foliations are Riemannian, which also follows from our main result of [2] in the caseof dense leaves. Acknowledgments.
The work of the first author was supported by DGICYT GrantPB95-0850, and that of the second by NSF Grants No. DMS-9973086 and DMS-0049077.The authors were at the Universidade de Santiago de Compostela and at CSUN while thispaper was being written, and thank these institutions for their support. We also thankF. Alcalde Cuesta for helpful conversations.1. F oliated spaces
This section collects and develops some information on foliated spaces which will beused later. General references on foliated spaces are [11], [18], [5].The definition of the concept of foliated space (or lamination) requires that of smoothfunction. Let Z be a Polish space ( i.e. , a completely metrizable separable space) and let U be an open set in R n × Z . A map f : U → R p is smooth (of class C k ) at the point ( x , z ) ifthere is a neighborhood of ( x , z ) in U of the form D × Y (where D is an open ball in R n QUICONTINUOUS FOLIATED SPACES 3 and Y is an open subset of Z ) such that f is continuous on D × Y and the partial derivativesup to order ≤ k of all coordinate functions f i of f exist and are continuous at all points( x , z ) ∈ D × Y .A locally compact Polish space X is said to have the structure of a foliated space (ofclass C k ) if there is a collection of charts (flow boxes) ( U i , φ i ), where { U i } is a covering of X by open sets, the maps φ i are homeomorphisms φ i : U i → B i × Z i , for Polish spaces Z i and open balls B i of finite radius in R n , such that the coordinate changes are of the form φ i ◦ φ − j ( x j , z j ) = ( x i ( x j , z j ) , z i ( z j )) , where x i : φ j ( U i ∩ U j ) → R n is of class C k and z i is continuous. The sets φ − i ( B i × { z } ) arecalled plaques.Since the foliated space X is locally compact, given an open cover U = { U i , φ i } by flowboxes of X , it is always possible to find a locally finite covering { V α , ϕ α } by flow boxeswhich is a refinement of U in the sense that each V α has compact closure on some U i ( α ) andthe corresponding chart φ i ( α ) extends ϕ α . Such cover { V α } is called regular [11], [5].A synonym of the term foliated space is lamination, which is sometimes reserved fora foliated space which is embedded as a closed subset of a manifold. It may also beconvenient to denote the foliated space by ( X , F ), with F referring to the particular foliatedstructure on X .The foliated structure of a space X induces a locally euclidean topology on X , the basicopen sets being the open subsets of the plaques, which is finer than the original. The con-nected components of X in this topology are called leaves. The smooth structure impliesthat each leaf is a connected manifold of dimension n and of class C k . If x is a point of X ,the leaf which contains x will usually be denoted by L x .Concepts of manifold theory readily extend to foliated spaces. In particular, if F is atleast of class C , there is a continuous vector bundle T X over X whose fiber at each point x of X is the tangent space of the manifold L x at x .Some very basic properties of foliated spaces will now be listed. They easily followfrom the definition and standard techniques of manifold theory extended to “manifoldswith parameters.” A basic observation is the following [18], which is immediate from theparacompactness of Polish spaces and the local structure of foliated spaces. Proposition 1.1.
Every open cover of a foliated space of class C k admits a subordinatepartition of unity of class C k . The important consequence is that a foliated space of class C k , with k ≥
1, alwaysadmits a metric tensor; i.e. , and inner product on
T X inducing a Riemannian metric oneach leaf, and varying continuously from leaf to leaf. The induced distance on a leaf L willbe denoted by d L . If X is compact, then each leaf is a complete Riemannian manifold ofbounded geometry. (This may not be the case in general.)Another consequence of the existence of partitions of unity is the following. The proofis like that for manifolds [5]. Proposition 1.2.
If X is a compact foliated space of class C k , k ≥ , then there is a C k embedding ϕ of X in the separable real Hilbert space E . Moreover, a given metric tensoralong the leaves can be extended to a metric tensor on E . A given metric tensor along the leaves of X admits an extension to a metric tensor on E . This is so because a metric tensor may be viewed as a section of some bundle over E with contractible fibers, and X being closed in E implies that a section over X extends toone over E . It may only be continuous, of course, but that is su ffi cient to define length. JES ´US A. ´ALVAREZ L ´OPEZ AND ALBERTO CANDEL If X is compact, then any two metric tensors on the leaves of X are quasi-isometric, andso, for the purposes of this paper, the metric tensor induced from the standard metric of E will be taken by default.This embedding of a foliated space X in E also gives rise to a normal vector bundle,the fibers of which are isomorphic to E . This structure permits to formalize concepts andresults like “local projection of leaves onto leaves” or “Reeb’s stability.” This structure willbe described presently.Let X be a compact foliated space, of class C k with k ≥
1, embedded in the Hilbertspace E . The restriction of the embedding to each leaf is not an embedding, but only aninjective immersion. The smoothness of X being at least C implies that the map whichassigns to a point x ∈ X the subspace T x X of E is continuous (as a map of X into the spaceof n -dimensional subspaces of E ). It follows that if F is a subspace complementary to one T x X in E , then it is also complementary to T y X for y close to x .The key point is that each tangent space T x X is a finite dimensional subspace of E ,hence is closed and has an orthogonal complement. If i : L → E is the inclusion of aleaf of X in E , then there are charts about x in L and E such that the corresponding localrepresentation of i is of the form y ( y , T y X ⊥ = { y } × E for y in this plaque containing x ,and by continuity, the a ffi ne subspaces y + T y X ⊥ meet nearby leaves transversely. Since X is compact, it follows that this holds for all y in a neighborhood of radius r about x in X ,the radius r being independent of the point x . Theorem 1.3.
Let X be a compact foliated space embedded in E as above, of class C .Let i : L → E denote the inclusion of a leaf L in X ⊂ E . Then there is a vector bundlep : N → L and a neighborhood W of the zero section of N such that the following hold: (1)
The map i : L → E extends to a local di ff eomorphism ϕ : W → E ; (2) there is a foliated space Y ⊂ W, of the same dimension as X, having L as a leafand transverse to the fibers of N; and (3) as foliated spaces, Y = ϕ − ( X ∩ ϕ ( W )) and the restriction of p to each leaf of Y isa local di ff eomorphism into L. Let N ( X ) = { ( x , v ) | v ⊥ T x X } ⊂ X × E . Then the exponential map λ : N ( X ) → E is λ ( x , v ) = x + v . Let N ( X , ε ) denote the set of pairs ( x , v ) ∈ N ( X ) with k v k < ε . The normalbundle N ( L ) to each leaf L is contained in N ( X ).The di ff erential λ ∗ at each point ( x , ∈ N ( X ) is the identity (under the canonical iden-tification T ( x , N ( X ) = T x X × T x X ⊥ ). Therefore, by continuity of λ ∗ on N ( X ) and compact-ness of X , there exists ε > λ ∗ is an isomorphism at ( x , v ) for x ∈ X and k v k < ε .This does not mean that λ is locally a homeomorphism. What it means is the following: Lemma 1.4.
For each point x ∈ X there exists a neighborhood U x in the leaf through xsuch that λ is a di ff eomorphism of N ( U x , ε ) into E . It will be shown that there exists ε > r > λ is a di ff eomorphismof N ( B ( x , r ) , ε ) into E , for every x ∈ X . Otherwise, by compactness of X , there existsequences ( x n , v n ) , ( y n , w n ) in N ( X ), with k v n k , k w n k → x n , y n → x , and such that λ ( x n , v n ) , λ ( y n , w n ) for each n . Working on a local flow box around x in X , choosing flatcoordinates around the plaque through x , and taking into account that the normal subspaceto points of X varies continuously, it is then obvious that it is possible to find new sequences( x ′ n , v ′ n ) and ( y ′ n , w ′ n ) as above with the added property that all the points x ′ n and y ′ n are in thesame plaque as x . This contradicts Lemma 1.4. QUICONTINUOUS FOLIATED SPACES 5
The vector bundle N ( X ) has a metric on the fibers so that the continuous map φ ( x , v ) = λ ( x , v ) − x is a linear isometry on each fiber. By compactness of X , there exists ε > L of X the restricted map λ : N ( L , ε ) → E is a local di ff eomorphism.Moreover, there exists r > L and every point x ∈ L , the map λ : N ( B ( x , r ) , ε ) → E is a di ff eomorphism.It follows that if L is a leaf of X , then the neighborhood N ( L , ε ) contains a foliatedspace Y , which is lifted from X via the local di ff eomorphism λ . This completes the proofof Theorem 1.3. Remark.
The hypothesis that X be of class C is required so that the normal bundle andexponential map be of class C . On the other hand, a manifold of class C admits a com-patible structure of class C r , for any 1 ≤ r ≤ ∞ . The proof (see [12, Chapter 2]) is basedon approximation results that can be adapted to ‘manifolds with parameters’, that is, tofoliated spaces. Since it is out of place to do this here, the C hypothesis is kept here andin Theorems 17.2, 17.3. Remark.
Let x ∈ L , and let T be a transversal through x , which may be taken to lie in thea ffi ne subspace x + T x X ⊥ of E . If G ( x ) denotes the group of germs of local homeomorphismsof T which fix x , then there is the germinal holonomy representation π ( L , x ) → G ( x ) . The construction above shows that this representation is equivalent to that of L as a leaf ofthe foliated space Y of N ( L , ε ).The following two propositions complement Theorem 1.3. Proposition 1.5.
Let L be a leaf of the compact foliated space of Theorem 1.3. Then thereexists ε > and an ε -disc bundle W → L which carries a foliated space Y and there existsa constant K > such that the projection p : S → L of each leaf S of Y into L is a localdi ff eomorphism ( of class C ) of distortion in the interval [1 / K , K ] . The disk bundle W → L and foliated space Y ⊂ W with the stated properties have beenconstructed in the previous proof. Because the space X is compact, it admits a finite regularcover by flow boxes. The projection Y → L amounts to finitely many projections betweenplaques of these flow boxes. As the cover is finite and regular, there are global bounds forthe distortion of these projections within flow boxes. The claim made in the propositionfollows from these observations.A similar useful fact is the following. Proposition 1.6.
Let L be as in Proposition 1.5. Given R > , there exists δ > such that,if x ∈ L and y is a point of W with p ( y ) = x, and such that the distance in the fiber of Wbetween x and y is < δ , then the ball B ( y , R ) in the leaf of Y through y is contained in W.Proof. The proof of this fact uses again a finite regular cover by flow boxes. The localcoordinate changes in the transverse direction are uniformly continuous maps, and only afinite number of coordinate changes are required to run through a ball of radius R , becauseplaques have a definite size. (cid:3) Remark.
The construction of Theorem 1.3 and the discussion that follows it (especiallyProposition 1.5), extend to a covering π : L ′ → L of a leaf L , simply by consideringthe normal bundle of the immersion L ′ → L ֒ → E . The output is a disc bundle W → L ′ containing a foliated space Y transverse to the fibers and having L ′ as a leaf. The projectionof leaves of Y into L is a local di ff eomorphism of bounded distortion. JES ´US A. ´ALVAREZ L ´OPEZ AND ALBERTO CANDEL
Continuing with the notation so far introduced, let L ′ → L be a covering of a leaf L of X , let x ∈ L ′ and let D be a compact manifold with boundary. Let y ∈ D and let f : D → L ′ be a continuous map with f ( y ) = x . The following is a Reeb stability type of result. Proposition 1.7.
Suppose that the induced homomorphism π ( D , y ) → π ( L ′ , x ) takes π ( D , x ) into the kernel of the holonomy representation of L ′ ( as a leaf of the foliated spaceY ) . Then there is a transversal Z ⊂ Y through x and a smooth map F from the productfoliation D × Z into X such that F | D × { x } = f and F ( D × { y } ) is contained in the leafthrough y, for all y ∈ Z. If f : D ֒ → L is an embedding, Z can be chosen so that F embeds D × Z in Y . In partic-ular, if π ( L ′ , x ) → π ( L ) has image contained in the kernel of the holonomy representationof L as a leaf of X , then, given x ∈ L ′ and R > δ > y is a point in the fiber p − ( x ) at distance < δ from x , then the ball B ( y , KR ) contains p − ( B ( x , R )) ∩ Y , and the component of this last set which contains y , contains the ball B ( y , R / K ).Moreover, the absence of holonomy permits to choose δ so that there is a transversal Z in the fiber through x such that the union of the leaves of p − ( B ( x , R )) ∩ Y through pointsof Z is parametrized as a product B ( x , R ) × Z , and the induced metric on the leaves is atbounded distance from that of B ( x , R ).2. P seudogroups of local transformations A pseudogroup of local transformations of a topological space Z is a collection H ofhomeomorphisms between open subsets of Z that contains the identity on Z and is closedunder composition (wherever defined), inversion, restriction and combination of maps.Such pseudogroup H is generated by a set E ⊂ H if every element of H can be obtainedfrom E by using the above pseudogroup operations; to simplify arguments, the sets ofgenerators to be considered will be assumed to be symmetric ( h − ∈ E if h ∈ E ).The orbit of an element x ∈ Z is the set H ( x ) of elements h ( x ), for all h ∈ H whosedomain contains x . These orbits are the equivalence classes of an equivalence relation on Z . Note that an arbitrary equivalence relation R ⊂ Z × Z is defined by a pseudogroup on Z if and only if R is a union of sets R i , i ∈ I , such that the restriction to each R i of bothfactor projections Z × Z → Z are homeomorphisms onto open subsets. Indeed, take thesets R i to be the graphs of all local transformations in the pseudogroup. Moreover R isdefined by a countably generated pseudogroup on Z if and only if R is a countable unionof sets R i satisfying the above condition. This follows because a countable set of localtransformations of Z gives rise to a countable family of composites with maximal domain.The set of germs of all transformations in the pseudogroup H at all points of theirdomains, endowed with the ´etale topology, is a topological groupoid, product and inversionbeing induced by composition and inversion of maps, respectively. Thus, for each x ∈ Z ,the set of germs at x of transformations h ∈ H with x ∈ dom h and h ( x ) = x is a groupcalled the group of germs at x . If x , y ∈ Z are in the same H -orbit, then the groups of germsat x and y are isomorphic: an isomorphism is given by conjugation with the germ at x ofany transformation g ∈ H with x ∈ dom g and g ( x ) = y . The group of germs of an orbitis therefore well defined, up to isomorphisms, as the group of germs at any point of thatorbit. In particular, a distinguished type of orbits are those with trivial group of germs.Pseudogroups of local transformations must be thought of as natural generalizations ofgroup actions on topological spaces (each group action generates a pseudogroup). But themain example to keep in mind is the holonomy pseudogroup of a foliated space ( X , F ) QUICONTINUOUS FOLIATED SPACES 7 associated to a regular covering by flow boxes ( U i , φ i ), whose construction is now recalled.If φ i : U i → B i × Z i for Polish spaces Z i and open balls B i of finite radius in R n , let p i : U i → Z i denote the composite of φ i with the factor projection R n × Z i → Z i ; thefibers of these p i are the plaques. If U i meets U j , let Z i , j = p i ( U i ∩ U j ), and regularity ofthe cover permits to define a homeomorphism h i , j : Z i , j → Z j , i such that p j = h i , j ◦ p i on U i ∩ U j [5, 11]. (If the covering by flow boxes is not regular, one can define generatorsof a pseudogroup via local sections of the projections p i .) Such a collection ( U i , p i , h i , j )is called a defining cocycle for F [8, 9]. These h i , j generate a pseudogroup H of localtransformations of Z = F i Z i , which is called the holonomy pseudogroup of ( X , F ) (withrespect to the covering ( U i , φ i )).There is a canonical bijection between the set of leaves and the set of H -orbits, whichis given by L H ( p i ( x )) if x ∈ L ∩ U i . Each Z i can be considered as a local transversalof F via φ i and the identification Z i ≡ { } × Z i ⊂ B i × Z i . It may be assumed that all ofthese local transversals are disjoint from each other, and thus that Z is embedded in X as acomplete transversal. Each H -orbit injects into the corresponding leaf in this way.The holonomy groups of the leaves can be defined as the groups of germs of the corre-sponding orbits. Thus leaves with trivial holonomy group correspond to orbits with trivialgroup of germs. Moreover, with the same arguments of [11], it follows that, for a generalpseudogroup H of local transformations of a topological space Z , if H has a countable setof generators, then the union of orbits with trivial group of germs is a residual subset of Z ;in particular, this union is dense in Z if Z is a Polish space.It is well known that all defining cocycles of a foliated space induce holonomy pseu-dogroups that are equivalent in the sense given by the following definition; thus the relevantproperties of pseudogroups of local transformations of a topological space are those thatare invariant by these equivalences. Definition 2.1 (Haefliger [8, 9]) . Let H , H ′ be pseudogroups of local transformations oftopological spaces Z , Z ′ , respectively. An ´etale morphism Φ : H → H ′ is a maximalcollection Φ of homeomorphisms of open subsets of Z to open subsets of Z ′ such that: • If φ ∈ Φ , h ∈ H and h ′ ∈ H ′ , then h ′ ◦ φ ◦ h ∈ Φ ; • the sources of elements of Φ form a covering of Z ; and • if φ, ψ ∈ Φ , then ψ ◦ φ − ∈ H ′ .An ´etale morphism Φ : H → H ′ is called an equivalence if the collection Φ − = { φ − | φ ∈ Φ } is also an ´etale morphism H ′ → H , which is called the inverse of Φ . The composite oftwo ´etale morphisms Φ : H → H ′ and Ψ : H ′ → H ′′ is the collection Ψ ◦ Φ = { ψ ◦ φ | φ ∈ Φ , ψ ∈ Ψ } , which is an ´etale morphism H → H ′′ . Finally, an ´etale morphism Φ : H → H ′ is generated by a subset Φ ⊂ Φ if all the elements of Φ can be obtained by restriction andcombination of composites h ′ ◦ φ ◦ h with h ∈ H , φ ∈ Φ and h ′ ∈ H ′ .An ´etale morphism Φ : H → H ′ clearly induces a continuous map between the cor-responding spaces of orbits, ¯ Φ : Z / H → Z ′ / H ′ , which is a homeomorphism if Φ is anequivalence.A basic example of a pseudogroup equivalence is the following. Let H be a pseu-dogroup of local transformations of a space Z , let U ⊂ Z be an open subset that meets every H -orbit, and let G denote the restriction of H to U . Then the inclusion map U ֒ → Z gener-ates an equivalence G → H . In fact, this example can be used to describe any pseudogroupequivalence in the following way. Let H , H ′ be pseudogroups of local transformations ofspaces Z , Z ′ , and Φ : H → H ′ an equivalence. Then let H ′′ be the pseudogroup of local JES ´US A. ´ALVAREZ L ´OPEZ AND ALBERTO CANDEL transformations of Z ′′ = Z ⊔ Z ′ generated by H ∪ H ′ ∪ Φ . Then the inclusions of Z , Z ′ in Z ′′ generate equivalences Ψ : H → H ′′ and Ψ : H ′ → H ′′ so that Φ = Ψ − ◦ Ψ .For pseudogroups of local transformations of locally compact spaces, the followingresult characterizes the existence of relatively compact open subsets that meet all orbits. Lemma 2.2.
Let H be a pseudogroup of local transformations of a locally compact Z. Theorbit space Z / H is compact if and only if there exists a relatively compact open subset thatmeets every H -orbit.Proof. If an open subset U ⊂ Z meets every H -orbit, then the restriction U → Z / H of thequotient map Z → Z / H is onto. Thus Z / H is compact because it is a continuous image ofthe compact space U .Assume that Z / H is compact. Since Z is locally compact, each x ∈ Z has relativelycompact open neighborhood U x . Let Q x denote the image of U x by the quotient map Z → Z / H . Since Z / H is compact, its open covering { Q x | x ∈ Z } has a finite sub-covering Q x , . . . , Q x m . Then the open set U = U x ∪ · · · ∪ U x n of Z is relatively compact and meetsall orbits of H . (cid:3) Examples of pseudogroups with compact space of orbits are the holonomy pseudogroupsof compact foliated spaces, as orbit and leaf spaces can be identified. But such pseu-dogroups satisfy a stronger compactness condition that is defined as follows.
Definition 2.3 (Haefliger [8]) . Let H be a pseudogroup of local transformations of a locallycompact space Z . Then H is compactly generated if there is a relatively compact open set U in Z meeting each orbit of H , and such that the restriction G of H to U is generated bya finite symmetric collection E ⊂ G so that each g ∈ E is the restriction of an element ¯ g of H defined on some neighborhood of the closure of the source of g .It was observed in [8] that this notion is invariant by equivalences and that the rela-tively compact open set U meeting each orbit can be chosen arbitrarily. If E satisfies theconditions of Definition 2.3, it will be called a system of compact generation of H on U .3. C oarse quasi - isometries The concept of coarse quasi-isometry was introduced by M. Gromov in [7]; it is alsocalled rough isometry in the context of potential theory [15]. A net in a metric space M ,with metric d , is a subset A ⊂ M such that d ( x , A ) < C for some C > x ∈ M ;the term C -net is also used. A coarse quasi-isometry between M and another metric space M ′ is the choice of a bi-Lipschitz bijection between nets of M and M ′ ; in this case, M and M ′ are said to have the same coarse quasi-isometry type or to be coarsely quasi-isometric .This definition involves two constants that will be called coarse distortions : one for the netsand another one for the bi-Lipschitz equivalence. A collection of coarse quasi-isometriesis said to be uniform when the same coarse distortions are valid for all of them.Recall that the Hausdor ff distance between subspaces X , Y of some metric space withmetric d , is defined as d H ( X , Y ) = max ( sup x ∈ X d ( x , Y ) , sup y ∈ Y d ( y , X ) ) . Now let M , M ′ be arbitrary metric spaces with metrics d , d ′ . The Gromov-Hausdor ff dis-tance (also called abstract Hausdor ff distance ) between M , M ′ , denoted by d GH ( M , M ′ ),is the infimum of the Hausdor ff distances d H ( M , M ′ ) over all metrics on M ⊔ M ′ that re-strict to d , d ′ on M , M ′ . Note that d GH ( M , M ′ ) may be equal to ∞ . If d GH ( M , M ′ ) < ∞ , QUICONTINUOUS FOLIATED SPACES 9 then M , M ′ are called Hausdor ff equivalent . On the other hand, the metric spaces M , M ′ are called Lipschitz equivalent when there exists a bi-Lipschitz bijection M → M ′ . Then M , M ′ are coarsely quasi-isometric if and only if there are some metric spaces N , N ′ suchthat the pairs M , N and M ′ , N ′ are Hausdor ff equivalent, and N , N ′ are Lipschitz equivalent.There is also a categorical description of coarse quasi-isometries. Two maps f , g : M → M ′ are called parallel [7], or bornotopic [23] or uniformly close [4] if there is some R > d ′ ( f ( x ) , g ( x )) < R for all x ∈ M . A map f : M → M ′ is said to be large scaleLipschitz [7] if there are constants λ, c > d ′ ( f ( x ) , f ( y )) ≤ λ d ( x , y ) + c for all x , y ∈ M ; note that f need not be continuous. Then coarse quasi-isometries can beconsidered as isomorphisms in the category of metric spaces and parallel classes of largescale Lipschitz maps.The above description of coarse quasi-isometry is similar to the definition of anothertype of “coarse” equivalence. A map f : M → M ′ is called e ff ectively proper [4] if for all r > s > d ′ ( f ( x ) , f ( y )) < r = ⇒ d ( x , y ) < s for all x , y ∈ M . The map f is called uniformly bornologous [23] or ( coarsely ) Lipschitz [4] if for all r > s > d ( x , y ) < r = ⇒ d ′ ( f ( x ) , f ( y )) < s for all x , y ∈ M . Then two metric spaces are called uniformly close [4] if there is anisomorphism between them in the category of metric spaces and uniformly close classesof e ff ectively proper coarsely Lipschitz maps. Note that every large scale Lipschitz map iscoarsely Lipschitz, and it is also e ff ectively proper if it has a large scale Lipschitz inverseup to the uniform closeness of maps. Therefore coarsely quasi-isometric metric spaces areuniformly close.Uniform closeness of metric spaces is a slight modification of the concept of bornotopyequivalence introduced in [23], which is an isomorphism in the category of proper met-ric spaces and bornotopy classes of e ff ectively proper uniformly bornologous Borel maps.Here, a metric space is called proper when its closed bounded subsets are compact. Thusbornotopy equivalence is the same as uniform closeness for all spaces that will be consid-ered in this paper. 4. C oarse quasi - isometry type of orbits Let H be a pseudogroup of local transformations of a space Z , and E a symmetric setof generators of H . For each h ∈ H and each x ∈ dom h , let | h | E , x be defined as follows.If h is the identity around x , set | h | E , x =
0. Otherwise, | h | E , x is the minimal positive integer k such that h = h k ◦ · · · ◦ h around x for some h , . . . , h k ∈ E . Let R ⊂ Z × Z denote theequivalence relation induced by H (whose equivalence classes are the orbits). Then, for( x , y ) ∈ R , let d E ( x , y ) = min {| h | E , x | h ∈ H , x ∈ dom h , h ( x ) = y } . In this way, E induces a map d E : R → N whose restriction to each orbit is a metric. Thisis a well known construction of a metric on the orbits; especially, for group actions.Unlike the case of group actions, for a pseudogroup H of local transformations of aspace Z with a symmetric set E of generators, the coarse quasi-isometry type of the in-duced metric d E on the orbits may depend on the choice of E , even if E is finite. This isdue to the fact that not only composition of maps is used to generate a group action, but restriction and combination of maps are also used to generate H . Moreover, an equiva-lence of pseudogroups may not preserve the coarse quasi-isometry type of the orbits forany choice of generators, as can be seen in the following simple example. Example 4.1.
Let H be the pseudogroup on R generated by the action of Z by translations,and let G be the restriction of H to some open interval U ⊂ R . If U is of length >
1, thenit meets every H -orbit, and thus H is equivalent to G . But the H -orbits are infinite, whileeach G -orbit is finite if U is of bounded length. So, for the metrics induced by any choiceof symmetric families of generators of H , G , the H -orbits have infinite diameter and the G -orbits finite diameter, and thus cannot be coarsely quasi-isometric.In the measure theoretic setting, this problem is solved by considering Kakutani equiv-alences [14], which are kind of measure theoretic counterparts of ´etale equivalences withthe additional requirement that the coarse quasi-isometry type of the orbits is preserved. Inthe present topological context, the above problem will be addressed without adding moreconditions to ´etale equivalences. Instead, appropriate representatives of pseudogroups andsets of generators will be chosen to determine a coarse quasi-isometry type on the orbits.The choice of appropriate pseudogroup representatives is easy, while the choice of appro-priate generators is rather delicate.Let H be a pseudogroup of local transformations of a locally compact space Z withcompact orbit space. By Lemma 2.2, there is a relatively compact open subset U of Z thatmeets all H orbits. If H is indeed compactly generated, the restriction G of H to U is arepresentative of H whose orbits will be shown to have a canonical coarse quasi-isometrytype, which is determined by any symmetric set E of generators of G satisfying certainconditions. The first condition on E is that it must be a system of compact generation of H on U . But this is not enough because there are systems of compact generation on the sameopen set inducing di ff erent coarse quasi-isometry types in the same orbit; such an examplewill be given in Section 6. So a second new condition is introduced as follows. Definition 4.2.
A finite symmetric family E of generators of a pseudogroup H of localtransformations of a locally compact space Z is said to be recurrent if there exists a rela-tively compact open subset U ⊂ Z and some R > d E -ball of radius R inany H -orbit meets U ; i.e. , for any x ∈ Z there exists h ∈ H with x ∈ dom h , | h | E , x < R and h ( x ) ∈ U .The role played by U in Definition 4.2 can actually be played by any relatively compactopen subset that meets all orbits, as shown by the following result. Lemma 4.3.
Let H be a pseudogroup of local transformations of a locally compact spaceZ, and let E be a recurrent finite symmetric family of generators of H . If V ⊂ Z is an openset that meets every orbit, then there exists S > such that any d E -ball of radius S in any H -orbit meets V.Proof. By definition, there exist a relatively compact open subset U ⊂ Z and a positivenumber R such that any d E -ball of radius R in any H -orbit meets U . Since V also meetsevery orbit, there exists a finite family F ⊂ H such that: • the sources of elements of F cover the compact closure U ; • the targets of elements of F are contained in V ; and • each element of F is a composite of elements of E .Let r > g ∈ F can be written as a composition of at most r elements of E . QUICONTINUOUS FOLIATED SPACES 11
Fix any x ∈ Z . On the one hand, there is some h ∈ H with x ∈ dom h , | h | E , x < R and h ( x ) ∈ U . On the other hand, there is some g ∈ F whose domain contains h ( x ). So x ∈ dom( gh ), gh ( x ) ∈ V , and | gh | E , x ≤ | g | E , h ( x ) + | h | E , x ≤ r + R . Thus the result follows with S = r + R . (cid:3) Let H be a compactly generated pseudogroup of local transformations of a locally com-pact space Z . A system of compact generation of H on a relatively compact open subset U ⊂ Z that meets every orbit is called recurrent if it is recurrent when considered as finitesymmetric set of generators of the restriction of H to U . An example of a non-recurrentsystem of compact generation will be given in Section 6. On the other hand, the existenceof recurrent systems of compact generation will be a consequence of the following result. Lemma 4.4.
With the above notation, let E be a system of compact generation of H on U.For each g ∈ E, fix an extension ¯ g ∈ H of g with dom g ⊂ dom ¯ g. Suppose that every x ∈ Uhas an open neighborhood V x in Z such thatV x ⊂ dom ¯ g x , V x ∩ U ⊂ dom g x , ¯ g x ( V x ) ⊂ Ufor some g x ∈ E. Then E is recurrent.Proof.
For each x ∈ U , let W x = ¯ g x ( V x ); its closure can be assumed to be containedin U . Compactness of U implies that U ⊂ V x ∪ · · · ∪ V x n , for some finite set of points x , . . . , x n ∈ U . Let V k = V x k , W k = W x k and g k = g x k for k = , . . . , n , so W = W ∪· · ·∪ W n is a relatively compact open set in U . Moreover, each y ∈ U belongs to some V k ; thus y ∈ V k ∩ U ⊂ dom g k and g k ( y ) ∈ W , yielding d E ( y , W ∩ H ( y )) ≤ (cid:3) Corollary 4.5.
Let H be a compactly generated pseudogroup of local transformations ofa locally compact space Z, and let U be a relatively compact open subset of Z that meetsall H -orbits. Then there exists a recurrent system E of compact generation of H on Usatisfying the following property. The extension ¯ g ∈ H of each g ∈ E with dom g ⊂ dom ¯ gcan be chosen so that E = { ¯ g | g ∈ E } is also a recurrent system of compact generation onsome relatively compact open subset U ′ ⊂ Z with U ⊂ U ′ .Proof. Since U meets every H -orbit and U is compact, there exists a finite family F ⊂ H satisfying the following properties: • each f ∈ F is the restriction of some ¯ f ∈ H whose domain is relatively compactand contains dom f ; • each ¯ f is the restriction of some ˜ f ∈ H with dom ¯ f ⊂ dom ˜ f ; • the sources of elements of F cover U ; and • im ¯ f ⊂ U for every f ∈ F .For each f ∈ F , let f ′ denote its restriction U ∩ dom f → f ( U ∩ dom f ) , let F ′ = { f ′ | f ∈ F } , and set F ′− = { f ′− | f ′ ∈ F ′ } .There exists a system G of compact generation of H on U , and E = G ∪ F ′ ∪ F ′− isalso a system of compact generation of H on U . Moreover, E satisfies the condition ofLemma 4.4 because U ⊂ [ f ∈ F dom f , and im ¯ f ⊂ U for every f ∈ F . Thus E is recurrent. Now let U ′ = [ f ∈ F dom ¯ f , which is a relatively compact open subset of Z containing U . Let F = n ¯ f | f ∈ F o and F − = n ¯ f − | f ∈ F o , which are subsets of the restriction G ′ of H to U ′ . The extensions ¯ g of the maps g ∈ G satisfying dom g ⊂ dom ¯ g can obviously be chosen so that: • each ¯ g has source and range in U ′ ; • the set G = { ¯ g | g ∈ G } is symmetric; and • each ¯ g is the restriction of some ˜ g ∈ H with dom ¯ g ⊂ dom ˜ g .Then E = G ∪ F ∪ F − is a finite symmetric subset of G ′ which generates G ′ because G generates G and im ¯ f ⊂ U for all f ∈ F . The above properties guarantee that E is a systemof compact generation of H on U ′ . Finally, E is recurrent by Lemma 4.4 since im ¯ f ⊂ U for all f ∈ F . (cid:3) The following is the promised result that shows the invariance of the coarse quasi-isometry type of the orbits by equivalences when appropriate representatives of pseu-dogroups and generators are chosen.
Theorem 4.6.
Let H , H ′ be compactly generated pseudogroups of local transformationsof locally compact spaces Z , Z ′ , and let U , U ′ be relatively compact open subsets of Z , Z ′ that meet all orbits of H , H ′ , respectively. Let G , G ′ denote the restrictions of H , H ′ toU , U ′ , and let E , E ′ be recurrent symmetric systems of compact generation of H , H ′ onU , U ′ , respectively. Suppose that there exists an equivalence H → H ′ , and consider theinduced equivalence G → G ′ and homeomorphism U / G → U ′ / G ′ . Then the G -orbits withd E are uniformly coarsely quasi-isometric to the corresponding G ′ -orbits with d E ′ . In other words, Theorem 4.6 asserts that, for pseudogroups H of local transformationsof locally compact spaces Z with given sets of generators E , the coarse quasi-isometrytype of the orbits is uniformly invariant by equivalences when the following conditions aresatisfied. First, H must be the restriction of a pseudogroup H ′ acting on a larger locallycompact space where Z is open, relatively compact and meets all H ′ -orbits. Second, E must be a recurrent system of compact generation of H ′ on Z .In order to prove Theorem 4.6, the following preliminary results will be required. Lemma 4.7.
Let H be a compactly generated pseudogroup of local transformations ofa locally compact space Z, let U , U ′ be relatively compact open subsets of Z such thatU ∩ U ′ meets all H -orbits, and let E , E ′ be recurrent systems of compact generation of H over U , U ′ , respectively. Then, for any open set V that meets all H -orbits and withV ⊂ U ∩ U ′ , there exists some C > such that C d E ′ ( x , y ) ≤ d E ( x , y ) ≤ C d E ′ ( x , y ) for all x , y ∈ V lying in the same H -orbit.Proof. Let G denote the restriction of H to U . By Lemma 4.3, there exists some R > d E -ball of radius R in any G -orbit meets V . Let Φ ⊂ G denote the finite set ofrestrictions of the form g : V ∩ dom g → g ( V ∩ dom g ) , QUICONTINUOUS FOLIATED SPACES 13 where g runs over the composites of at most R elements of E , wherever defined. It is notedthat the images of elements of Φ cover U . Moreover, it may be assumed that R ≥
2, andthus that the identity map of V ∩ dom g belongs to Φ for all g ∈ E with V ∩ dom g , ∅ .Let F denote the finite set of composites ψ − ◦ g ◦ φ , wherever defined, where φ, ψ ∈ Φ and g ∈ E . Observe that each f ∈ F is the restriction of some ¯ f ∈ G with dom f ⊂ dom ¯ f . Furthermore, for each x ∈ dom ¯ f , it holds that (cid:12)(cid:12)(cid:12) ¯ f (cid:12)(cid:12)(cid:12) E ′ , y ≤ (cid:12)(cid:12)(cid:12) ¯ f (cid:12)(cid:12)(cid:12) E ′ , x , for all y in someneighborhood of x in dom ¯ f . Hence, since F is finite and the domain of each f ∈ F isrelatively compact in U ′ , there exists an integer S > | f | E ′ , x ≤ S for all f ∈ F and all x ∈ dom f .Let x , y ∈ V be points in the same H -orbit. If x = y , then d E ( x , y ) = d E ′ ( x , y ) =
0, andthe statement holds trivially with any C >
0. If x , y , then d E ( x , y ) = k ≥
1. Let g ∈ G be such that x ∈ dom g , g ( x ) = y and | g | E , x = k . This element g may be assumed to be ofthe form g = g k ◦ · · · ◦ g for some g , . . . , g k ∈ E . Then, for i = , . . . , k −
1, there exists φ i ∈ Φ whose image contains x i = g i ◦ · · · ◦ g ( x ) and such that z i = φ − i ( x i ) ∈ V . Such g can be written as g = f k ◦ · · · ◦ f around x , where f , . . . , f k are the elements of F given by f = φ − ◦ g , f k = g k ◦ φ k − , f i = φ − i ◦ g i ◦ φ i − , for i = , . . . , k −
1. Observe that x ∈ dom f and z i ∈ dom f i + for all i = , . . . , k − | g | E ′ , x ≤ | f k | E ′ , z k − + · · · + | f | E ′ , z + | f | E ′ , x ≤ kS , yielding d E ′ ( x , y ) ≤ S d E ( x , y ) . Similarly, d E ( x , y ) ≤ S ′ d E ′ ( x , y )for some integer S ′ >
0, and result follows with C = max { S , S ′ } . (cid:3) Corollary 4.8.
Let H be a compactly generated pseudogroup of local transformations ofa locally compact space Z and let U , U ′ be relatively compact open subsets of Z such thatU ∩ U ′ meets all H -orbits. Let G , G ′ denote the restrictions of H to U , U ′ , and let E , E ′ berecurrent systems of compact generation of H over U , U ′ , respectively. Then the G -orbitswith d E are uniformly coarsely quasi-isometric to the corresponding G ′ -orbits with d E ′ .Proof. There exists an open set V meeting every H -orbit and such that V ⊂ U ∩ U ′ . ByLemma 4.3, there also exist R , R ′ > d E -ball of radius R in any G -orbit meets V , and any d E ′ –ball of radius R ′ in any G ′ -orbit meets V . That is, for every G -orbit O andfor every G ′ -orbit O ′ , the intersection O ∩ V is an R -net in O and O ′ ∩ V is an R ′ -net in O ′ .So the result follows from Lemma 4.7. (cid:3) Proof of Theorem 4.6.
Let Φ : H → H ′ be an equivalence, and let H ′′ be the pseudogroupof local transformations of Z ′′ = Z ⊔ Z ′ generated by H ∪ H ′ ∪ Φ , considered as sets oflocal transformations of Z ′′ in the obvious way. Then U , U ′ and U ′′ = U ⊔ U ′ are relativelycompact open subsets of Z ′′ that meet all H ′′ -orbits. The restrictions of H ′′ to U and U ′ are G and G ′ , thus E and E ′ are recurrent systems of compact generation of H ′′ on U and U ′ , respectively. Let G ′′ denote the restriction of H ′′ to U ′′ , and select a recurrent system E ′′ of compact generation of H ′′ on U ′′ . By Corollary 4.8, the G ′′ -orbits with d E ′′ areuniformly coarsely quasi-isometric to the corresponding G -orbits with d E , and also to thecorresponding G ′ -orbits with d E ′ . (cid:3) Let H be a compactly generated pseudogroup of local transformations of a locally com-pact space Z , U a relatively compact open subset of Z that meets all H -orbits, and G therestriction of H to U . By considering the identity map on the G -orbits and inclusions ofsystems of compact generation on U , we get an inductive system of metric spaces. Notealso that distances between points in the same G -orbit do not increase by considering largersystems of compact generation. By Theorem 4.6, the corresponding “inductive system ofcoarse quasi-isometry types” has a limit, which is uniformly reached just when a recurrentsystem of compact generation is considered. The following consequence of Lemma 4.7shows that it happens so with the corresponding “inductive system of Lipschitz types.” Corollary 4.9.
With the above notation, let E be a recurrent symmetric system of compactgeneration of H on U. Any other symmetric system E ′ of compact generation of H on Uis recurrent if and only if there exists some C > such that (4.1) 1 C d E ′ ( x , y ) ≤ d E ( x , y ) ≤ C d E ′ ( x , y ) for all x , y ∈ U lying in the same G -orbit.Proof. Fix any open set V that meets all G -orbits and with V ⊂ U .Suppose that E ′ is recurrent. Then, by Lemma 4.3, there is some R > O ∩ V is an R -net in ( O , d E ′ ) for every G -orbit O . To show that (4.1) holds for some C >
0, assumefirst that E ′ ⊂ E . Hence the first inequality of (4.1) holds for any C ≥
1. Take arbitrarypoints x , y ∈ U lying in the same G -orbit. We can assume that x , y , otherwise (4.1)holds trivially for any C >
0. There are points x ′ , y ′ ∈ V with d E ′ ( x , x ′ ) , d E ′ ( y , y ′ ) ≤ R .So d E ( x , x ′ ) , d E ( y , y ′ ) ≤ R as well because E ′ ⊂ E . By Lemma 4.7, there is some C ′ > x ′ , y ′ , such that1 C ′ d E ′ ( x ′ , y ′ ) ≤ d E ( x ′ , y ′ ) ≤ C ′ d E ′ ( x ′ , y ′ ) . Therefore d E ( x , y ) ≤ d E ( x , x ′ ) + d E ( x ′ , y ′ ) + d E ( y , y ′ ) ≤ d E ( x ′ , y ′ ) + R ≤ C ′ d E ′ ( x ′ , y ′ ) + R ≤ C ′ ( d E ′ ( x ′ , x ) + d E ′ ( x , y ) + d E ′ ( y , y ′ )) + R ≤ C ′ ( d E ′ ( x , y ) + R ) + R ≤ ( C ′ + C ′ R + R ) d E ′ ( x , y )since d E ′ ( x , y ) ≥
1, yielding the second inequality of (4.1) with C = C ′ + C ′ R + R .When E ′ E , the union E ′′ = E ∪ E ′ is a recurrent symmetric system of compactgeneration of H on U . We have shown that there are some C , C > C d E ′′ ( x , y ) ≤ d E ( x , y ) ≤ C d E ′′ ( x , y ) , C d E ′′ ( x , y ) ≤ d E ′ ( x , y ) ≤ C d E ′′ ( x , y )for all x , y ∈ U lying in the same G -orbit, and therefore (4.1) holds with C = C C .Now assume that (4.1) holds for some C > x , y ∈ U lying in the same G -orbit.By Lemma 4.3, there is some R > O ∩ V is an R -net in ( O , d E ) for every G -orbit QUICONTINUOUS FOLIATED SPACES 15 O . Then it easily follows that O ∩ V is an CR -net in ( O , d E ′ ) for every G -orbit O , and thus E ′ is recurrent by Lemma 4.3. (cid:3)
5. F¨ olner orbits
The F¨olner condition will be used in the next section to distinguish coarse quasi-isometrytypes of orbits. The property that F¨olner orbits give rise to invariant measures will beneeded also. This was shown by S. Goodman and J. Plante [6] for pseudogroups acting oncompact metric spaces, and is partially improved in this section by using recurrent compactgeneration instead of a compact space. For compact foliated spaces, it is well known thatF¨olner leaves induce invariant transverse probability measures. So recurrence can be use-ful to show that compactly generated pseudogroups behave like compact foliated spaces,which is in the spirit of a famous project of A. Haefliger [10].Let M be a metric space with metric d . A quasi-lattice Γ of M is a net of M such thatfor every r ≥ K r ≥ Γ ∩ B ( x , r )) ≤ K r for every x ∈ M .Not every metric space has a quasi-lattice, but metric spaces with bounded complexity in areasonable sense do; see e.g. [4]) for examples. The metric space M is said to be of coarsebounded geometry if it has a quasi-lattice.For any r >
0, the r-boundary of each subset S ⊂ M is the set ∂ r S = { x ∈ S | d ( x , S ) < r and d ( x , M \ S ) < r } . The notation ∂ Mr S will be also used for ∂ r S . Then M is called amenable [4] if it has aquasi-lattice Γ and a sequence of finite subsets S n ⊂ Γ such that(5.1) lim n →∞ ∂ Γ r S n S n = r >
0. Such a sequence S n will be called a F¨olner sequence in Γ . Since ∂ Γ r S \ S ⊂ [ x ∈ S ∩ ∂ Γ r S ( Γ ∩ B ( x , r ))for every S ⊂ Γ , it follows that(5.2) ∂ r S \ S ) ≤ K r · S ∩ ∂ r S )if Γ ∩ B ( x , r )) ≤ K r for any x ∈ Γ . Therefore the amenability condition (5.1) is equivalentto lim n →∞ S n ∩ ∂ Γ r S n ) S n = r > Theorem 5.1 (Block-Weinberger [4]) . Let M , M ′ be uniformly close metric spaces ofcoarse bounded geometry. Then M is amenable if and only if so is M ′ . This result was proved in [4] in the following way. First, the uniformly finite homology H uf • ( M ) is introduced for any metric space M . Second, it is shown that, if two metric spaces M , M ′ are uniformly close, then H uf • ( M ) (cid:27) H uf • ( M ′ ). Finally, it is shown that a metric space M of coarse bounded geometry is amenable if and only if H uf0 ( M ) ,
0, and Theorem 5.1follows. In particular, amenability is a coarse quasi-isometry invariant for metric spaces ofcoarse bounded geometry, which can be also proved directly without too much di ffi culty.The following lemma will be useful in the in the proof of the main result of this section. Lemma 5.2.
Let Γ be a quasi-lattice in some metric space, and let A be a C-net in Γ forsome C > . Fix any K > such that every ball of radius C in Γ has at most K points.Then S ≤ (cid:16) S ∩ ∂ Γ C S (cid:17) + K · A ∩ S ) for any S ⊂ Γ .Proof. For every x ∈ S , there is some a ∈ A so that d ( x , a ) < C , and thus x ∈ S ∩ ∂ Γ C S if a < S . Therefore S ⊂ (cid:16) S ∩ ∂ Γ C S (cid:17) ∪ [ a ∈ A ∩ S ( Γ ∩ B ( a , C )) , yielding the inequality of the statement. (cid:3) We are interested in the case of a metric space M whose points are the vertices of someconnected graph, and where the distance between two points is the minimum number ofcontiguous edges needed to join them. In this case, besides the amenability condition of[4], M may be also F¨olner in the usual graph sense, which is defined as follows. The boundary ∂ S of any S ⊂ M is the set of points x ∈ S such that there is some edge joining x with some point in M \ S ; i.e. , ∂ S = S ∩ ∂ S with the notation of [4]. Then M is F¨olner (as a graph) when there is a sequence of finite subsets S n ⊂ M such thatlim n →∞ ∂ S n S n = . Note that M is a quasi-lattice in itself just when there is a uniform upper bound K on thenumber of edges that meet at every vertex; indeed, B ( x , r ) ≤ K r for all r ≥ K . In this case, since ∂ r S ⊂ [ x ∈ ∂ S B ( x , r ) , it follows that ∂ r S ≤ K r · ∂ S for any r >
0. Hence, when there is such a uniform upper bound K , M is amenable (asmetric space) if and only if it is F¨olner (in the graph sense).Consider again a pseudogroup H of local transformations of a space Z with the metric d E on the orbits induced by a finite symmetric set E of generators of H . Then we get agraph by introducing an edge between two points x , y ∈ Z whenever there is some g ∈ E with g ( x ) = y . Thus E is an upper bound for the number of edges that meet at everyvertex. Observe that each orbit of H is given by the vertices of a connected component ofthis graph, and d E is the metric induced by this graph on its connected components. Thefollowing notation and terminology will be used in this setting: • Let ∂ E S denote the boundary of any finite subset S of an orbit with respect to thegraph structure induced by E ; • for r >
0, let ∂ Er S denote the r -boundary of any finite subset S of an orbit withrespect to the metric d E ; • a F¨olner sequence of an orbit with the metric d E (or with the graph structure in-duced by E ) will be called an E-F¨olner sequence ; and • an orbit with an E -F¨olner sequence will be called E-F¨olner or E-amenable . Theorem 5.3.
Let H be a compactly generated pseudogroup of local transformations of alocally compact metric space Z, let U be a relatively compact open subset of Z that meetsall H -orbits, and let G be the restriction of H to U. Consider the metric on the G -orbitsinduced by a recurrent symmetric system E of compact generation of H on U. If some QUICONTINUOUS FOLIATED SPACES 17 G -orbit is E-F¨olner, then there is a non-trivial non-negative G -invariant Borel measure onU of finite mass.Proof. Let S n be an E -F¨olner sequence of some orbit O of G . As in [6], a measure µ isconstructed on U as a limit of averaging measures on the finite sets S n . Let C ( U ) be theBanach space of continuous functions f : U → R that vanish at infinity, endowed with thesupremum norm k k given by k f k = sup x ∈ U | f ( x ) | For each n ∈ N , let µ n : C ( U ) → R be defined by µ n ( f ) = S n X x ∈ S n f ( x )for f ∈ C ( U ). Each µ n is obviously linear and continuous; i.e. , it is an element of the(algebraic-topological) dual space C ( U ) ′ . Moreover it is easy to check that | µ n ( f ) | ≤ k f k for all n ∈ N and f ∈ C ( U ). Therefore, by the Banach-Alaoglu theorem, the set { µ n | n ∈ N } is relatively compact in C ( U ) ′ with the weak ∗ topology; i.e. , the topology of pointwiseconvergence. Then, by passing to a subsequence if necessary, we can assume that thesequence µ n converges pointwise to some µ in C ( U ) ′ , which can be considered as a Borelmeasure of finite mass on U by the Riesz representation theorem. This µ is non-negativesince all the µ n are probability measures. The G -invariance of µ follows from the E -F¨olnercondition of the sequence S n since, as shown in [6], | µ ( f ◦ g ) − µ ( f ) | ≤ k f k lim n ∂ E S n S n for all g ∈ E and f ∈ C ( U ) with supp f ⊂ im g .Finally, we show that µ is not trivial. Take any open set U ′ that meets all G -orbits andwith U ′ ⊂ U , and consider any non-negative function f ∈ C ( U ) with f ( x ) = x ∈ U ′ . By Lemma 4.3, there is some C > O ∩ U ′ is a C -net in O with d E . Then µ ( f ) ≥ lim n S n ∩ U ′ ) S n ≥ ( E ) − C lim n S n − (cid:16) S n ∩ ∂ EC S n (cid:17) S n = ( E ) − C > S n is E -F¨olner. (cid:3) Remarks. (i) In the proof of Theorem 5.3, the measure µ could be trivial if E were notrecurrent. This is di ff erent from the arguments of [6] because U is not compact. Forinstance, for the pseudogroup on R generated by the translation g ( x ) = x +
1, the sets { n , n + , . . . , n } ( n ∈ N ) form a { g , g − } -F¨olner sequence in an orbit, and the limit ofcorresponding averaging measures is trivial.(ii) The statement of Theorem 5.3 could be more general. As in [6], the definitions andarguments could be modified to remove the condition that all sets of the F¨olner sequencelie in the same orbit: it would be enough to have what is called an averaging sequence in[6]. But our result is simpler and general enough for our purposes in the next section.
6. A n example of non - recurrent compact generation In this section, we give an example showing that there exist non-recurrent systems ofcompact generation, and that the coarse quasi-isometry type of the orbits may depend onthe system of compact generation if recurrence is not considered.Fix real numbers a < a ′ < a ′′ < b ′′ < b ′ < b , and choose homeomorphisms φ : R → ( a , b ) and ˜ g : R → R satisfying the followingproperties: • φ ( x ) = x for all x ∈ [ a ′′ , b ′′ ]; • φ ( x ) > x for all x ∈ ( −∞ , a ′′ ); • φ ( x ) < x for all x ∈ ( b ′′ , ∞ ); • φ ( a ) = a ′ and φ ( b ) = b ′ ; • ˜ g ( x ) = x for all x ∈ ( −∞ , a ] ∪ [ b , ∞ ); • ˜ g ( x ) > x for all x ∈ ( a , b ); and • ˜ g ( a ′′ ) < b ′′ .Let H be the pseudogroup of local transformations of R generated by φ and ˜ g . Thebounded open interval U = ( a , b ) meets all H -orbits because φ ( R ) = U , and let G denotethe restriction of H to U .Now define a map ˜ g : R → R by setting˜ g ( x ) = φ ◦ ˜ g ◦ φ − ( x ) if a < x < b , x otherwise.Such a ˜ g is a homeomorphism and satisfies the following properties: • ˜ g ( x ) = x for all x ∈ ( −∞ , a ′ ] ∪ [ b ′ , ∞ ); • ˜ g ( x ) > x for all x ∈ ( a ′ , b ′ ); and • φ ◦ ˜ g ( x ) = ˜ g ◦ φ ( x ) for all x ∈ U .We now prove that G is generated by the restrictions ˜ g , ˜ g : U → U , which will bedenoted by g , g . It is enough to prove that the restriction φ : U → φ ( U ) is in the pseu-dogroup G ′ generated by g , g . Since ˜ g ( a ′′ ) < b ′′ , the collection of sets U n = g n ( a ′′ , b ′′ ), n ∈ Z , is an open covering of U , and thus it su ffi ces to prove that each restriction φ : U n → φ ( U n ) is in G ′ . But φ is the identity on ( a ′′ , b ′′ ) = g − n ( U n ), yielding g − n = φ ◦ g − n = g − n ◦ φ on U n . So φ = g n ◦ g − n on U n , which belongs to G ′ , as desired.Since dom g i ⊂ dom ˜ g i for i = ,
2, it follows that E = { g , g , g − , g − } is a system ofcompact generation of H on U . For the open subset V = ( a ′ , b ) ⊂ U , it will be shown that(6.1) lim x → a d E ( x , V ∩ G ( x )) → ∞ . Since V meets every G -orbit, it follows from Lemma 4.3 and (6.1) that E is not recurrent.To prove (6.1), let ν ( x ) = min { n ∈ N | g n ( x ) ∈ V } for each x ∈ U . Clearly, x ∈ V ⇐⇒ ν ( x ) = , x < y = ⇒ ν ( x ) ≥ ν ( y ) , lim x → a ν ( x ) = ∞ . Take any x ∈ U and some h ∈ G with x ∈ dom h , h ( x ) ∈ V , | h | E , x = d E ( x , V ∩ G ( x )) . QUICONTINUOUS FOLIATED SPACES 19
For n = | h | E , x , we have h = g ε n i n ◦· · ·◦ g ε i around x for some i , . . . , i n ∈ { , } and ε , . . . , ε n ∈{± } . Let x k = g ε k i k ◦ · · · ◦ g ε i ( x ) for every k = , . . . , n . Since x k < V for each k < n , either x k + ≤ x k (yielding ν ( x k + ) ≥ ν ( x k )), or g ε k + i k + = g (yielding ν ( x k + ) = ν ( x k ) − ν ( x ) ≤ n = d E ( x , V ∩ G ( x ))because ν ( x n ) =
0, and (6.1) follows.Finally, let F be a recurrent symmetric system of compact generation of H on U . Wewill show that no G -orbit with the metric d E is coarsely quasi-isometric to itself with themetric d F . Suppose that this is not true for some G -orbit O . Since the open interval I = ( a , a ′ ) meets every orbit and since g is the identity on I , there is some point x ∈ O ∩ I such that the set X = { g − m ( x ) | m ∈ N } ⊂ O ∩ I satisfies ∂ E X = { x } . Hence O is E -F¨olner; for instance, an E -F¨olner sequence for O isgiven by the sets S n = { g − m ( x ) | ≤ m ≤ n } . Then O is also F -F¨olner by Theorem 5.1 since we are assuming that the metrics d E , d F on O are coarsely quasi-isometric. So, by Theorem 5.3 and because F is recurrent, there is anon-trivial non-negative G -invariant Borel measure µ on U of finite mass. Fix any t ∈ U ,and let I n = ( a , g n ( t )) for each n ∈ Z . Since µ is G -invariant and g ( I n ) = I n + , all sets I n have the same µ -measure, which is finite since µ has finite mass. Then µ ( I n + \ I n ) = n , whence µ ( U ) = U = S n ( I n + \ I n ). This is a contradiction because µ isnon-trivial and non-negative. 7. Q uasi - local metric spaces The concept of equicontinuity can be defined for pseudogroups of local transformationsof uniform spaces, but we are mainly concerned with the case of metric spaces in thispaper. Nevertheless, it is enough to consider only certain part of the local geometry ofmetric spaces, which is extracted in the following definition. Moreover it is easier to workwith pseudogroups and their equivalences when all other geometric information is removedfrom metric spaces.
Definition 7.1.
Let { ( Z i , d i ) } i ∈ I be a family of metric spaces such that { Z i } i ∈ I is a coveringof a set Z , each intersection Z i ∩ Z j is open in ( Z i , d i ) and ( Z j , d j ), and for all ε > δ ( ε ) > i , j ∈ I and z ∈ Z i ∩ Z j , there issome open neighborhood U i , j , z of z in Z i ∩ Z j (with respect to the topology induced by d i and d j ) such that(7.1) d i ( x , y ) < δ ( ε ) = ⇒ d j ( x , y ) < ε for all ε > x , y ∈ U i , j , z . Such a family will be called a cover of Z by quasi-locallyequal metric spaces . Two such families are called quasi-locally equal when their unionalso is a cover of Z by quasi-locally equal metric spaces. This is an equivalence relationwhose equivalence classes are called quasi-local metrics on Z . For each quasi-local metric Q on Z , the pair ( Z , Q ) is called a quasi-local metric space .Any quasi-local metric Q on Z induces a uniformity so that, for any { ( Z i , d i ) } i ∈ I ∈ Q , thefamilies U r = [ i ∈ I , x ∈ Z i B i ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∈ Z , r > , form a base of uniform covers, where B i ( x , r ) denotes the open ball in ( Z i , d i ) of center x and radius r . The open balls of all metric spaces ( Z i , d i ) form a base of the correspondingtopology on Z . Any topological concept or property of ( Z , Q ) refers to this underlyingtopology. Any quasi-local metric space ( Z , Q ) is locally metrizable, and thus first countableand completely regular. If ( Z , Q ) is Hausdor ff and paracompact, then it is metrizable [25]and normal [30, Theorem 20.10], and every point finite open cover of ( Z , Q ) is shrinkable[30, Theorem 15.10]. Moreover ( Z , Q ) is a locally compact Polish space if and only if itis Hausdor ff , paracompact, separable and locally compact; this is the type of quasi-localmetric spaces that will be mainly considered in this paper. Remark.
A quasi-local metric is a “local structure” in the sense that it is determined by its“restriction” to the sets of any open covering. This property is specially useful to deal withpseudogroup equivalences, and is not satisfied by general uniformities. This is anotherreason to consider quasi-local metric spaces instead of general uniform spaces.If a quasi-local metric space ( Z , Q ) is paracompact, then there is some { ( Z i , d i ) } i ∈ I ∈ Q so that the covering { Z i } i ∈ I is locally finite. In this case, { ( Z i , d i ) } i ∈ I satisfies the followingslightly stronger condition. Lemma 7.2.
Let ( Z , Q ) be a quasi-local metric space. If { Z i } i ∈ I is locally finite for some { ( Z i , d i ) } i ∈ I ∈ Q , then there is some open neighborhood U x of each x ∈ Z and some assign-ment ε δ ( ε ) such that (7.1) holds for all ε > , i , j ∈ I and y ∈ U x ∩ Z i ∩ Z j .Proof. With the notation of Definition 7.1, the set U z = \ i , j ∈ I , z ∈ Z i ∩ Z j U i , j , z is an open neighborhood of each z ∈ Z and satisfies the stated property. (cid:3) Example 7.3.
Any metric d on a set Z induces a unique quasi-local metric Q on Z sothat { ( Z , d ) } ∈ Q . It will be shown in Section 14 that every Hausdor ff paracompact quasi-local metric space ( Z , Q ) is indeed induced by some metric d on Z (Theorem 15.1), but theinformation of ( Z , d ) contained in Q is just what is relevant for our study of equicontinuouspseudogroups. Example 7.4.
For each t >
0, consider the metric d t on R defined by d t ( x , y ) = q t ( x − y ) + ( x − y ) for x = ( x , x ) and y = ( y , y ). Then, for T ⊂ R + , the family { ( R , d t ) | t ∈ T } is a cover of R by quasi-locally equal metric spaces if and only if T is relatively compact in R + . Hencethere are no maximal covers by quasi-locally equal metric spaces in general.8. E quicontinuous pseudogroups This section develops the concept of equicontinuity for pseudogroups, as suggested byE. Ghys in [17, Appendix E]. To motivate our definitions, consider a group G of home-omorphisms of a space Z . On the one hand, some uniformity is needed on Z for theusual definition of equicontinuity of G (see, e.g. , A. Weil [29]). But, on the other hand,equicontinuity of G does not imply that each map in G is uniformly continuous; i.e. , thesehomeomorphisms may not preserve the uniformity of Z . This gives a di ffi culty when try-ing to generalize equicontinuity to pseudogroups in a way compatible with pseudogroupequivalences. More precisely, let H , H ′ be pseudogroups on spaces Z , Z ′ , and Φ : H → H ′ QUICONTINUOUS FOLIATED SPACES 21 an equivalence. Suppose that H is equicontinuous in some reasonable way, which shoulduse some uniformity on Z . Then one has to use Φ to construct a uniformity on Z ′ so that H ′ is equicontinuous too. The following is a standard way to do this kind of construction:the uniformity of Z must be restricted to domains of homeomorphisms in Φ , which areused to define uniformities on the sets of some open covering of Z ′ , and then these localuniformities must be combined to yield a uniformity on the whole of Z ′ . Some conditionsmust be satisfied to achieve such a combination. First, we need some type of uniformitythat is determined by its restriction to the sets of any open covering, which holds for quasi-local metrics as indicated in the remark of Definition 7.1. Secondly, these uniformities onopen sets of Z ′ must be compatible on the overlaps, which means that the local transfor-mations of H must preserve the uniformity of Z ; i.e. , they must be uniformly continuous!Therefore the type of equicontinuity needed for pseudogroups seems to be “equi-uniformcontinuity;” i.e. , the transformations of a pseudogroup are not only required to be “simul-taneously” continuous at every point, but also required to be “simultaneously” uniformlycontinuous. Moreover, surprisingly, there are some unsolved di ffi culties to show that rea-sonable definitions of “equicontinuity at every point” and “equi-uniform continuity” areequivalent for compactly generated pseudogroups. So we define equicontinuity for pseu-dogroups by requiring that the “transformations with small domain” are “simultaneously”uniformly continuous. Indeed, what may be understood as “transformations with smalldomain” gives rise to two versions of equicontinuity. The first one is weaker and looksmore natural, but the second one fits our needs. Definition 8.1.
Let ( Z , Q ) be a quasi-local metric space. A pseudogroup H of local homeo-morphisms of ( Z , Q ) is called weakly equicontinuous if, for some { ( Z i , d i ) } i ∈ I ∈ Q and every ε >
0, there is some δ ( ε ) > h ∈ H , i , j ∈ I and z ∈ Z i ∩ h − ( Z j ∩ im h ), there is some neighborhood U h , i , j , z of z in Z i ∩ h − ( Z j ∩ im h )such that(8.1) d i ( x , y ) < δ ( ε ) = ⇒ d j ( h ( x ) , h ( y )) < ε for all ε > x , y ∈ U h , i , j , z .A pseudogroup H acting on a space Z will be called weakly equicontinuous when it isweakly equicontinuous with respect to some quasi-local metric inducing the topology of Z . Remarks. (i) Note that weak equicontinuity is a local property on ( Z , Q ) to a large extent;the only global aspect is the assignment ε δ ( ε ), which is valid for all possible h , i , j , z .(ii) The condition (7.1) of Definition 7.1 is the particular case of (8.1) for h equal to theidentity map on Z . So the whole structure of quasi-local metrics is needed to define weaklyequicontinuous pseudogroups.(iii) The condition of weak equicontinuity given in Definition 8.1 can be described ascertain compatibility of H with Q : H is equicontinuous on ( Z , Q ) if and only if Q can berealized as a combination of h ∗ ( Q | im h ) for h running through H , where the restrictions,pull-backs and combinations of quasi-local metrics are defined in an obvious way (whenappropriate conditions are satisfied).(iv) In Definition 8.1, the assignment ε δ ( ε ) depends on { ( Z i , d i ) } i ∈ I ∈ Q , but thisdefinition is of course independent of the choice of { ( Z i , d i ) } i ∈ I ∈ Q ; i.e. , any other choiceof { ( Z i , d i ) } i ∈ I satisfies the definition with some other assignment ε δ ( ε ). The following result shows that weak equicontinuity is a property of equivalence classesof pseudogroups. The definition was worded in precisely such way for this property to holdtrue; in fact, this is rather evident by the above remarks (i) and (iii).
Lemma 8.2.
Let H , H ′ be equivalent pseudogroups. Then H is weakly equicontinuous ifand only if H ′ is equicontinuous.Proof. Let Z , Z ′ be the acted on by H , H ′ . Assuming that H is weakly equicontinuouswith respect to some quasi-local metric Q inducing the topology of Z , we will show thatso is H ′ . Thus there is some { ( Z i , d i ) } i ∈ I ∈ Q and some assignment ε δ ( ε ) such that,for all h ∈ H and i , j ∈ I , the condition (8.1) holds on some neighborhood U h , i , j , z of each z ∈ Z i ∩ h − ( Z j ∩ im h ).Let Φ : H → H ′ be a pseudogroup equivalence. There is an open covering { Z ′ a } a ∈ A of Z ′ such that, for each a ∈ A , there is some φ a ∈ Φ and some i a ∈ I with Z ′ a ⊂ im φ a anddom φ a ⊂ Z i a . Let d ′ a denote the restriction to Z ′ a of the metric on im φ a that correspondsvia φ a to the restriction of d i a to dom φ a . For h ′ ∈ H ′ , a , b ∈ A , z ′ ∈ Z ′ a ∩ h ′− ( Z ′ b ∩ im h ′ ) , let U ′ h ′ , a , b , z ′ = φ a ( U h , i a , i b , z ), where h = φ − b ◦ h ′ ◦ φ a ∈ H , z = φ − a ( z ′ ) ∈ Z i a ∩ h − ( Z i b ∩ im h ) . Now, given any ε >
0, suppose d ′ a ( x ′ , y ′ ) < δ ( ε ) for x ′ , y ′ ∈ U ′ h ′ , a , b , z ′ . Then thepoints x = φ − a ( x ′ ) and y = φ − a ( y ′ ) lie in U h , i a , i b , z and satisfy d i a ( x , y ) < δ , yielding d i b ( h ( x ) , h ( y )) < ε by (8.1), and thus d ′ b ( h ′ ( x ′ ) , h ′ ( y ′ )) < ε . Therefore (8.1) is satisfiedby H ′ , { ( Z ′ a , d ′ a ) } a ∈ A , the same assignment ε δ ( ε ), and the above choice of neighbor-hoods U ′ h ′ , a , b , z ′ . It follows that { ( Z ′ a , d ′ a ) } a ∈ A is a cover of Z ′ by quasi-locally equal metricspaces (remark (ii) of Definition 8.1), and H ′ is weakly equicontinuous with respect to thecorresponding quasi-local metric, which obviously induces the given topology of Z ′ . (cid:3) On paracompact spaces, the following slightly di ff erent description of weak equiconti-nuity will be useful to understand the stronger version. Lemma 8.3.
Let H a pseudogroups acting on a paracompact quasi-local metric space ( Z , Q ) . Then H is weakly equicontinuous if and only if there is some { ( Z i , d i ) } i ∈ I ∈ Q andsome symmetric subset S ⊂ H such that any germ of any map in H is the germ of somemap in S and, for every ε > , there is some δ ( ε ) > so that (8.1) holds for all h ∈ S ,i , j ∈ I and x , y ∈ Z i ∩ h − ( Z j ∩ im h ) .Proof. The “only if” part follows because, with the notation of Lemma 7.2 and Defini-tion 8.1, if U h , z = U z ∩ \ i , j ∈ I , z ∈ Z i ∩ Z j U h , i , j , z for h ∈ H , i , j ∈ I and z ∈ dom h , then (8.1) is satisfied by the family S of all possiblerestrictions h : U h , z → h ( U h , z ).Reciprocally, for all h , i , j , z as in Definition 8.1, take U h , i , j , z equal to some open neigh-borhood of z in Z i ∩ h − ( Z j ∩ im h ) where h is equal some map in S . Then (8.1) obviouslyholds for all x , y ∈ U h , i , j , z . (cid:3) The stronger version of equicontinuity is defined by requiring that there is a set S asin Lemma 8.3 that is also closed under compositions, which is some kind of a non-localcondition . QUICONTINUOUS FOLIATED SPACES 23
Definition 8.4.
Let H be a pseudogroup of local homeomorphisms of a quasi-local metricspace ( Z , Q ). Then H is called strongly equicontinuous if there exists some { ( Z i , d i ) } i ∈ I ∈ Q and some symmetric set S of generators of H that is closed under compositions such that,for every ε >
0, there is some δ ( ε ) > h ∈ S , i , j ∈ I and x , y ∈ Z i ∩ h − ( Z j ∩ im h ).A pseudogroup H acting on a space Z will be called strongly equicontinuous when it isstrongly equicontinuous with respect to some quasi-local metric inducing the topology of Z . Remarks. (i) A typical choice of S in Definition 8.4 is the set of all possible compositesof some symmetric set of generators. In fact, given any S satisfying the condition of strongequicontinuity, it is obviously possible to find a symmetric set of generators E given byrestrictions of elements of S , and then the set of all composites of elements of E alsosatisfies the condition of strong equicontinuity.(ii) If h ∈ H satisfies (8.1) for all i , j ∈ I and x , y ∈ Z i ∩ h − ( Z j ∩ im h ), then so does itsrestriction to any open set. Hence S can be assumed to be also closed under restrictionsto open sets. Nevertheless, Definition 8.4 is not satisfactory with S = H because then thebasic test of Example 8.6 below is not satisfied (see Example 8.7).(iii) The definition of strong equicontinuity is independent of the choice of { ( Z i , d i ) } i ∈ I ∈ Q .Hence it is possible to assume that { Z i } i ∈ I locally finite in Definition 8.4 when ( Z , Q ) isparacompact. Example 8.5.
The pseudogroup H generated by the identity map on any quasi-local metricspace ( Z , Q ) is obviously weakly equicontinuous by the remark (ii) of Definition 8.1. If( Z , Q ) is paracompact, then H is also strongly equicontinuous by Lemma 7.2; in fact, thedefinition of strong equicontinuity is satisfied with S equal to the family of the identitymaps on all finite intersections of the sets U z given by Lemma 7.2. Example 8.6.
Recall that a group G of homeomorphisms of a metric space ( Z , d ) isequicontinuous, or better “equi-uniformly continuous,” if for every ε > δ ( ε ) > d ( x , y ) < δ ( ε ) = ⇒ d ( g ( x ) , g ( y )) < ε for all g ∈ G and x , y ∈ Z . The pseudogroup H generated by such a G is strongly equicon-tinuous because Definition 8.4 is satisfied with S = G and { ( Z i , d i ) } i ∈ I = { ( Z , d ) } . Of course,if ( Z , d ) is compact, G is “equicontinuous at every point” if and only if it is “equi-uniformlycontinuous.” Example 8.7.
The group of translations on R is strongly equicontinuous with respect tothe euclidean metric, and thus generates a strongly equicontinuous pseudogroup H withrespect to the corresponding quasi-local metric. The following simple argument shows thatsome proper subset S ⊂ H must be taken to verify the definition of strong equicontinuous.Let { Z i } i ∈ I be any open covering of R , and d i a metric on each Z i inducing its topology.For any fixed index i , take real numbers a < b such that [ a , b ] ⊂ Z i . Let r > < r < b − a . Let U = ( a , a + r ) ∪ ( a + r , a + r ) , V = ( a , a + r ) ∪ ( b − r , b ) , which are contained in Z i , and let h : U → V be the map in H defined by h ( x ) = x if a < x < a + r , x + b − a − r if a + r < x < a + r . The points x n = a + n − n r , y n = a + n + n r satisfy the following properties: • x n , y n ∈ U ; • d i ( x n , y n ) → x n , y n → a in Z i ; and • d i ( h ( x n ) , h ( y n )) > C for some C > h ( x n ) → a and h ( y n ) → b in Z i .Then h does not satisfy (8.1) for any { ( Z i , d i ) } i ∈ I as above.Even though an apparently non-local condition was added to define strong equiconti-nuity, the following result shows that this property is invariant by equivalences of pseu-dogroups acting on locally compact Polish spaces. Lemma 8.8. If H , H ′ are equivalent pseudogroups acting on locally compact Polishspaces. Then H is strongly equicontinuous if and only if H ′ strongly equicontinuous.Proof. Let Φ : H → H ′ be a pseudogroup equivalence. Let Z , Z ′ be the locally compactPolish spaces acted on by H , H ′ . Assuming that H is strongly equicontinuous with respectto some quasi-local metric Q inducing the topology of Z , we will show that so is H ′ . Thus H satisfies the condition of strong equicontinuity for some { ( Z i , d i ) } i ∈ I ∈ Q and somesymmetric set S of generators of H that is closed under compositions. By the remark (ii)of Definition 8.4, we can assume that S is also closed under restrictions to open sets; soevery transformation of H is a combination of maps in S . Claim 1.
There is a subset Φ ⊂ Φ such that: • For each φ ∈ Φ , there is some i ∈ I so that dom φ ⊂ Z i ; • Z ′ = S φ ∈ Φ im φ ; and • φ − ◦ ψ ∈ S for all φ, ψ ∈ Φ . To prove Claim 1, first note that, since Z ′ is a locally compact Polish space and Φ anequivalence, there is a sequence φ , φ , . . . in Φ such that: • The domain of each φ n is contained in some Z i ; • Z ′ = S n im φ n ; • the domain of each φ n is relatively compact in Z ; and • every φ n is a restriction of some ˜ φ n ∈ Φ with dom φ n ⊂ dom ˜ φ n .Then an increasing sequence of finite subsets Φ , n ⊂ Φ is defined by induction on n so that φ − ◦ ψ ∈ S for all φ, ψ ∈ Φ , n andim φ ∪ · · · ∪ im φ n ⊂ [ φ ∈ Φ , n im φ . Let Φ , = ∅ to begin with, and assume that Φ , n is defined for some n ≥ Φ , n + , first set Φ ′ , n = Φ , n ∪ n ˜ φ n + o . There is an open covering U n + of dom ˜ φ n + such that the restriction of φ − ◦ ˜ φ n + to every U ∈ U n + is in S for all φ ∈ Φ ′ , n , which follows since Φ is an equivalence, Φ ′ , n is finite and every map in H is acombination of maps in S . Then the compact set dom φ n + is covered by a finite subfamily U ′ n + ⊂ U n + , and let Φ , n + be the union of Φ , n and the set of restrictions of ˜ φ n + to all setsof U ′ n + . We get by induction that φ − ◦ ψ ∈ S for all φ, ψ ∈ Φ , n + because S is symmetric,and im φ ∪ · · · ∪ im φ n + ⊂ [ φ ∈ Φ , n im φ ∪ im φ n + ⊂ [ φ ∈ Φ , n + im φ . QUICONTINUOUS FOLIATED SPACES 25
Therefore Claim 1 follows with Φ = S n Φ , n .Now, let S ′ be the family of all possible composites φ ◦ h ◦ ψ − for h ∈ S and φ, ψ ∈ Φ ,which is symmetric and generates H ′ . Moreover the following argument shows that S ′ isclosed under compositions. Take arbitrary elements h ′ , h ′ ∈ S ′ ; thus h ′ = φ ◦ h ◦ ψ − and h ′ = φ ◦ h ◦ ψ − for h , h ∈ S and φ , ψ , φ , ψ ∈ Φ . Since ψ − ◦ φ ∈ S by Claim 1, itfollows that h ◦ ψ − ◦ φ ◦ h ∈ S because S is closed under compositions. So h ′ ◦ h ′ = φ ◦ h ◦ ψ − ◦ φ ◦ h ◦ ψ − ∈ S ′ . According to Claim 1, take any open covering { Z ′ a } a ∈ A of Z ′ such that, for each a ∈ A ,there is some φ a ∈ Φ and some i a ∈ I with Z ′ a ⊂ im φ a and dom φ a ⊂ Z i a . Let d ′ a denotethe restriction to Z ′ a of the metric on im φ a that corresponds via φ a to the restriction of d i a to dom φ a . Choose an assignment ε δ ( ε ) > S and { ( Z i , d i ) } i ∈ I satisfy thecondition of strong equicontinuity. Let h ′ be an arbitrary element of S ′ , which is equal tosome composite φ ◦ h ◦ ψ − for h ∈ S and φ, ψ ∈ Φ . Let a , b ∈ A and x ′ , y ′ ∈ Z ′ a ∩ dom h ′ with h ′ ( x ′ ) , h ′ ( y ′ ) ∈ Z ′ b , and let g = φ − b ◦ h ′ ◦ φ a . Since ψ − ◦ φ a and φ b ◦ φ − are in S byClaim 1, it follows that g = φ − b ◦ φ ◦ h ◦ ψ − ◦ φ a ∈ S because S is closed under compositions. The points x = φ − a ( x ′ ) and y = φ − a ( y ′ ) liein Z i a ∩ dom g , and g ( x ) , g ( y ) ∈ Z i b . Moreover, if d ′ a ( x ′ , y ′ ) < δ ( ε ), then d i a ( x , y ) <δ ( ε ), yielding d i b ( g ( x ) , g ( y )) < ε by (8.1), and thus d ′ b ( h ′ ( x ′ ) , h ′ ( y ′ )) < ε . It follows that { ( Z ′ a , d ′ a ) } a ∈ A is a cover of Z ′ by quasi-locally equal metric spaces (remark (ii) of Defini-tion 8.1), and H ′ satisfies the condition of strong equicontinuity with S ′ and { ( Z ′ a , d ′ a ) } a ∈ A .Hence H ′ is strongly equicontinuity with respect to the quasi-local metric Q ′ representedby { ( Z ′ a , d ′ a ) } a ∈ A , which obviously induces the given topology of Z ′ . (cid:3) Problem 1.
Give mild conditions so that weak and strong equicontinuity are equivalent.
Problem 2.
It is possible to give pseudogroup versions of weak and strong “equicontinu-ity at every point” as in Definitions 8.1 and 8.4. Are they equivalent to our versions ofequicontinuity ( in the uniform sense ) for compactly generated pseudogroups? For the purposes of this paper, the key property of strong equicontinuity is the following.
Proposition 8.9.
Let H be a compactly generated and strongly equicontinuous pseu-dogroup acting on a locally compact Polish quasi-local metric space ( Z , Q ) , and let Ube any relatively compact open subset of ( Z , Q ) that meets every H -orbit. Suppose that { ( Z i , d i ) } i ∈ I ∈ Q satisfies the condition of Definition 8.4, E is any system of compact gener-ation of H on U, and ¯ g satisfies the condition of Definition 2.3 for each g ∈ E. Let { Z ′ i } i ∈ I be any shrinking of { Z i } i ∈ I . Then there is a finite family V of open subsets of ( Z , Q ) whoseunion contains U and such that, for any V ∈ V , x ∈ U ∩ V, and h ∈ H with x ∈ dom h andh ( x ) ∈ U, the domain of ˜ h = ¯ g n ◦ · · · ◦ ¯ g contains V for any expression h = g n ◦ · · · ◦ g around x with g , . . . , g n ∈ E, and moreover V ⊂ Z ′ i and ˜ h ( V ) ⊂ Z ′ i for some i , i ∈ I.Proof.
We can assume that { Z i } i ∈ I is locally finite. Let S be a symmetric set of generatorsof H that is closed under compositions and restrictions to open subsets so that the conditionof strong equicontinuity is satisfied by S and { ( Z i , d i ) } i ∈ I (Definition 8.4).Observe that any system E of compact generation of H on U satisfies the statement ofthis result if so does some other system of compact generation of H on U whose elementsare restrictions of elements of E . Therefore it can be assumed that, for all g , g ∈ E , wehave(8.2) dom g ∩ im g , ∅ = ⇒ dom ¯ g ∪ im ¯ g ⊂ Z ′ i for some i , j ∈ I ; in particular, since E is symmetric, for each g ∈ E there exists some i , j ∈ I such that(8.3) dom ¯ g ⊂ Z ′ i , im ¯ g ⊂ Z ′ j . By the same reason, we can also suppose that ¯ g ∈ S for all g ∈ E .Since U is relatively compact and { Z i } i ∈ I is locally finite, U meets only a finite numberof the sets Z i . Thus there exists ε > Z ′ i ∩ dom g , ∅ = ⇒ d i (cid:0) Z ′ i ∩ dom g , Z i \ dom ¯ g (cid:1) > ε , for all i ∈ I and all g ∈ E , because Z ′ i ∩ dom g is a relatively compact subset of Z i . Let δ = δ ( ε ) > ε ; it is no restrictionto assume that δ < ε .Let V be a finite family of open subsets of Z whose union contains U and such thateach V ∈ V is contained in some Z ′ i and has d i -diameter smaller than δ . Fix any V ∈ V , x ∈ U ∩ V and h ∈ H with x ∈ dom h and h ( x ) ∈ U . Since E generates the restriction of H to U , there exist g , . . . , g n ∈ E so that h = g n ◦ · · · ◦ g in some neighborhood of x , and let˜ h = ¯ g n ◦ · · · ◦ ¯ g . If V ⊂ dom ˜ h , then V ⊂ Z ′ i and ˜ h ( V ) ⊂ Z ′ i for some i , i ∈ I by (8.3). Soit only remains to show that V ⊂ dom ˜ h , which will be done by induction on n .The result is true for n =
1. Indeed, V ⊂ Z ′ i for some i , and d i ( x , y ) < δ < ε for all y ∈ V .Thus V ⊂ dom ¯ g by (8.4).For n >
1, let f = g n − ◦ · · · ◦ g , which is defined in some neighborhood of x . By theinduction hypothesis, the domain of ˜ f = ¯ g n − ◦ · · · ◦ ¯ g contains V . Thendom g n ∩ im g n − , ∅ , and thus dom ¯ g n ∪ im ¯ g n − ⊂ Z ′ j for some j ∈ I by (8.2). In particular,im ˜ f ⊂ im ¯ g n − ⊂ Z ′ j , yielding d j (cid:16) ˜ f ( x ) , ˜ f ( y ) (cid:17) < ε for all y ∈ V by strong equicontinuity since d i ( x , y ) < δ and˜ f ∈ S . Therefore ˜ f ( y ) ∈ dom ¯ g n by (8.4) because ˜ f ( x ) = f ( x ) ∈ Z ′ j ∩ dom g n ; i.e. , thedomain of ˜ h = ¯ g n ◦ ˜ f contains V as desired. (cid:3) Remarks. (i) With the notation of Proposition 8.9, given any symmetric set S of generatorsof H that is closed under compositions, we can choose E with the extensions ¯ g in S . So S contains all maps ˜ h of the statement of Proposition 8.9.(ii) Note that, with the conditions of Proposition 8.9, the pseudogroup H is complete asdefined by Haefliger in [9].It makes sense to consider the generalization to complete strongly equicontinuous pseu-dogroups of known results for complete pseudogroups of local isometries of Riemannianmanifolds [8], [9]. But, for simplicity, only compactly generated strongly equicontinuouspseudogroups will be considered in this paper.9. Q uasi - effective pseudogroups Recall the following property that is invariant by equivalences of pseudogroups; it isinteresting for our results on strongly equicontinuous pseudogroups.
QUICONTINUOUS FOLIATED SPACES 27
Definition 9.1 (Haefliger [8]) . A pseudogroup H of local transformations of a space Z is called quasi-analytic when the following holds for every h ∈ H : if x ∈ dom h and h is the identity on some open set whose closure contains x , then h is the identity on aneighborhood of x . Example 9.2.
Pseudogroups of local isometries of Riemannian manifolds are quasi-analyticbecause every such local isometry with connected domain is determined by its di ff erentialat any point. Example 9.3.
Let Z be the compact subspace of R that is the union of a horizontal 2-dimensional euclidean disk centered at the origin with a compact segment of the verticalaxis containing the origin. Consider the quasi-local metric on Z induced by the restrictionof the euclidean metric of R . The space Z is invariant by rotations around the vertical axis.The pseudogroup H generated by any such non-trivial rotation is strongly equicontinuousbut not quasi-analytic.If H is a quasi-analytic pseudogroup on a space Z , then every h ∈ H with connecteddomain is the identity on dom h if it is the identity on some non-trivial open set. Becauseof this, quasi-analyticity is interesting for our purposes when Z is locally connected, whichis a very strong condition. To remove it, a slightly di ff erent property is defined inspired bythe terminology of group actions. Definition 9.4.
A pseudogroup H of local transformations of a space Z is said to be quasi-e ff ective if it is generated by some symmetric set S that is closed under compositions, andsuch that any transformation in S is the identity on its domain if it is the identity on somenon-empty open subset of its domain. Remarks. (i) In Definition 9.4, the family S can be assumed to be also closed under re-strictions to open sets. So every map in H is a combination of maps in S in this case.(ii) If the pseudogroup H is strongly equicontinuous and quasi-e ff ective, then H is gener-ated by a symmetric subset S that is closed under compositions and satisfies the conditionsof both Definitions 8.4 and 9.4.The following result can be proved with arguments similar to those in the proof ofLemma 8.8. Lemma 9.5. If H , H ′ are equivalent pseudogroups acting on locally compact Polishspaces Z , Z ′ , then H is quasi-e ff ective if and only if so is H ′ . Lemma 9.6.
Any quasi-e ff ective pseudogroup is quasi-analytic.Proof. Let H be a quasi-e ff ective pseudogroup of local transformations of a space Z . So H satisfies the condition of Definition 9.4 with some symmetric set S that generates H andis closed under compositions and restrictions to open sets (remark (i) of Definition 9.4).Then H is obviously quasi-analytic because any h ∈ H is a combination of elements of S . (cid:3) Example 9.7.
Let r n , s n be two sequences of real numbers satisfying 0 < r n < s n and r n , s n ↓
0. For each n ∈ Z + , let U n denote the (multiplicative) group of n th roots of 1 in C , and fix a generator α n of each U n . Then let Z be the compact subspace of R × C that isthe union of the origin and the subspaces { s n } × r n U n , n ∈ Z + . Let H be the pseudogroupon Z generated by the homeomorphism h : Z → Z that fixes the origin and satisfies h ( s n , z ) = ( s n , α n z ) for any z ∈ r n U n and n ∈ Z + . Note that h is an isometry with respect tothe restriction of the metric on R × C induced by the norm defined by k ( t , z ) k = max {| t | , | z |} . So H is strongly equicontinuous with respect to the corresponding quasi-local metric on Z .Moreover H is quasi-analytic because, on the one hand, the origin is the only non-isolatedpoint of Z and, on the other hand, if some power h m is the identity on some open set whoseclosure contains the origin, then m =
0. But H is not quasi-e ff ective, as follows easily byusing that, for any neighborhood U of the origin in Z , there is some m ∈ Z + such that h m fixes some point in U di ff erent from the origin, which is an open subset.The above example shows that being quasi-e ff ective is a strictly stronger property thanquasi-analyticity, even for equicontinuous pseudogroups. Nevertheless, the following re-sult shows that both properties are equivalent when quasi-analyticity fits our needs. Lemma 9.8.
Let H be a compactly generated strongly equicontinuous pseudogroup on alocally connected and locally compact Polish space Z. Then H is quasi-e ff ective if andonly if it is quasi-analytic.Proof. The “only if” part was shown in Lemma 9.6.Now assume that H is quasi-analytic. Let U be any relatively compact open subset of Z that meets every H -orbit, and let G denote the restriction of H to U . By Lemma 9.5, itis enough to show that G is quasi-e ff ective. Let E be any system of compact generation of H on U , and let ¯ g be an extension of each g ∈ E satisfying the condition of Definition 2.3.Take a family V of open subsets of Z satisfying the statement of Proposition 8.9 for theabove U , E and extensions ¯ g . Since Z is locally connected, we can assume that all sets in V are connected. Let S be the set of maps h ∈ G such that: • h is a restriction of some composite of elements of E ; and • the domain and range of h are contained in elements of V .Such an S generates G , is symmetric, and is closed under compositions and restrictions toopen sets. Suppose that some h ∈ S is the identity on some non-trivial open subset of itsdomain. We have h = g n ◦ · · · ◦ g : O → P , where g , . . . , g n ∈ E and O , P are open subsets of U that are contained in elements of V ;say O ⊂ V ∈ V . Then the domain of ˜ h = ¯ g n ◦ · · · ◦ ¯ g contains V by Proposition 8.9. Since h is the identity on some non-trivial open subset of O , the germ of ˜ h at some point of V isequal to the germ of the identity. So ˜ h is the identity on V because H is quasi-analytic and V is connected. Thus h is the identity on O and the result follows. (cid:3) The following result combines strong equicontinuity and quasi-e ff ectiveness. Proposition 9.9.
Let H be a compactly generated, strongly equicontinuous and quasi-e ff ective pseudogroup of local homeomorphisms of a locally compact Polish space Z. Sup-pose that the conditions of strong equicontinuity and quasi-e ff ectiveness are satisfied witha symmetric set S of generators of H that is closed under compositions ( Definitions 8.4and 9.4 ) . Let A , B be open subsets of Z such that A is compact and contained in B. If x andy are close enough points in Z, thenf ( x ) ∈ A = ⇒ f ( y ) ∈ Bfor all f ∈ S whose domain contains x and y.Proof.
Suppose that the condition of strong equicontinuity is satisfied with S , some quasi-local metric Q , some { ( Z i , d i ) } i ∈ I ∈ Q such that { Z i } i ∈ I is locally finite, and some assignment ε δ ( ε ). Let { Z ′ i } i ∈ I be a shrinking of the open covering { Z i } i ∈ I . Since A is relatively QUICONTINUOUS FOLIATED SPACES 29 compact and { Z i } i ∈ I locally finite, A only meets finitely many of the sets Z i . Thus, becauseeach Z ′ i ∩ A is relatively compact in Z i , there exists some ε > Z ′ i ∩ A , ∅ , Z i \ B = ⇒ d i ( Z ′ i ∩ A , Z i \ B ) > ε for all i ∈ I .Fix x , y ∈ Z . Since H is compactly generated, A and x are contained in some relativelycompact open U that meets all orbits. Let V be a finite family of open sets that covers U and satisfies the conditions of Proposition 8.9. If x and y are close enough, then bothof these points lie in some set V ∈ V . Furthermore V ⊂ Z i for some i ∈ I , and we have d i ( x , y ) < δ ( ε ) if x and y are close enough.Now take any f ∈ S with x , y ∈ dom f and f ( x ) ∈ A . According to Proposition 8.9,there is some f ′ ∈ S whose domain contains V and so that f , f ′ have the same germ at x ;furthermore there is some j ∈ I such that f ′ ( V ) ⊂ Z j . In particular, f ( x ) = f ′ ( x ) ∈ Z j .Since y ∈ dom f ∩ dom f ′ , we also get f ( y ) = f ′ ( y ) ∈ Z j by quasi-e ff ectiveness. Therefore d j ( f ( x ) , f ( y )) < ε by strong equicontinuity, and the result follows from (9.1). (cid:3)
10. C oarse quasi - isometry type of orbits with trivial groups of germs To compare di ff erent orbits of pseudogroups, some connection between them is needed;so the following terminology will be used. A pseudogroup H acting on a space Z is calledtransitive when some orbit is dense in Z , and it is called minimal if every orbit is dense.A non-trivial subset Y ⊂ Z is called minimal if it is closed, H -invariant, and every orbitin Y is dense in Y ; equivalently, if Y is a minimal element of the family of all non-trivial H -invariant closed subsets of Z . Theorem 10.1.
Let H be a compactly generated, strongly equicontinuous and quasi-e ff ective pseudogroup of local transformations of a locally compact Polish space Z. As-sume the space of orbits Z / H is connected ( for example, if H is transitive ) . Let G denotethe restriction of H to some relatively compact open subset U ⊂ Z that meets every orbit.Then, with respect to any recurrent system of compact generation of H on U, all G -orbitswith trivial group of germs are uniformly coarsely quasi-isometric to each other.Proof. Let E be a recurrent system of compact generation of H on U , and for each g ∈ E let ¯ g denote its extension satisfying the conditions of Definition 2.3. According to Corol-lary 4.5 and Theorem 4.6, it may be assumed that E = { ¯ g | g ∈ E } is also a recurrent systemof compact generation on some relatively compact open subset U ′ ⊂ Z with U ⊂ U ′ . Let G ′ denote the restriction of H to U ′ . By considering restrictions of elements of E to opensubsets of their domains, we can assume that E ⊂ S for some subset S ⊂ H satisfying theconditions of Definition 9.4. Take a family V of open subsets of Z satisfying the statementof Proposition 8.9 for the above U , E and extensions ¯ g .Since Z is a Polish space, the union of orbits with trivial group of germs is a densesubset of Z . Hence, because Z / H is connected, it is enough to establish coarse quasi-isometries between the G -orbits of points x , y ∈ U that are close enough to each otherand have trivial group of germs; moreover the corresponding coarse distortions must beindependent of x and y . Thus it can be assumed that x and y are in the same element V ∈ V . Consider the map φ x , y : G ( x ) → G ′ ( y ) given by h ( x ) ˜ h ( y ), where h ∈ G , ˜ h ∈ S , x ∈ dom h , V ⊂ dom ˜ h , and both h , ˜ h have the same germ at x . Here, the germ of h at x isdetermined by the value h ( x ) because the group of germs at x is trivial. There exists suchan ˜ h for any h by Proposition 8.9 and since E ⊂ S . Moreover ˜ h is unique on V because H satisfies the condition of Definition 9.4 with S . Note also that φ x , y takes values in G ′ ( y ) byProposition 8.9. Therefore φ x , y is well defined. Claim 2. φ x , y : G ( x ) → G ′ ( y ) is injective. To prove this claim, take f , f ∈ S whose domains contain V . So φ x , y ( f ( x )) = f ( y )and φ x , y ( f ( x )) = f ( y ). If f ( y ) = f ( y ), then f , f have the same germ at y because thegroup of germs of H at y is trivial. It follows that f = f on V because both of these mapsare in S and their domains contain V . Hence h ( x ) = h ( y ) as desired. Claim 3.
We have d E ( φ x , y ( z ) , φ x , y ( z )) ≤ d E ( z , z ) for all z , z ∈ G ( x ) . We now show this assertion. We have z = f ( x ), z = f ( x ), φ x , y ( z ) = f ( y ) and φ x , y ( z ) = f ( y ) for some f , f ∈ S whose domains contain V . Suppose d E ( z , z ) = k ≥ G ( x ). This means that there is a minimal decomposition h ◦ h − = g k ◦ · · · ◦ g about z with g , . . . , g k ∈ E . Hence f ◦ f − and ¯ g k ◦· · ·◦ ¯ g are equal on f ( V ) because bothof these maps are in S and their domains contain f ( V ). This yields d E ( f ( y ) , f ( y )) ≤ k in G ′ ( y ), which finishes the proof of Claim 3.Let A be an open subset of Z intersecting every H -orbit and such that A ⊂ U . Claim 4.
There is some C > , independent of x , y, such thatd E ( z , z ) ≤ C d E (cid:16) φ x , y ( z ) , φ x , y ( z ) (cid:17) for all z , z ∈ G ( x ) ∩ A. To prove this estimate, take any z , z ∈ G ( x ). Again, there are f , f ∈ S , whosedomains contain V , such that z = f ( x ), z = f ( x ), φ x , y ( z ) = f ( y ) and φ x , y ( z ) = f ( y ).Suppose d E ( f ( y ) , f ( y )) = k . Then there is a decomposition f ◦ f − = ¯ g k ◦ · · · ◦ ¯ g around f ( y ) for some g , . . . , g k ∈ E . So f ◦ f − = ¯ g k ◦ · · · ◦ ¯ g on f ( V ) because both of thesemaps are in S and their domains contain f ( V ). It follows that d E ( z , z ) ≤ k . Hence d E ( z , z ) ≤ Ck for some C > x and y by Lemma 4.7 since z , z ∈ A and E , E are recurrent systems of compact generation on U , U ′ . Claim 5.
If x , y are close enough, then G ′ ( y ) ∩ A ⊂ φ x , y ( G ( x )) . By Proposition 9.9, if x , y are close enough in V , then(10.1) f ( y ) ∈ A = ⇒ f ( x ) ∈ U for all f ∈ S whose domain contains V . Then G ′ ( y ) ∩ A = G ( y ) ∩ A and thus everypoint in G ′ ( y ) ∩ A can be written as f ( y ), for some f ∈ G whose domain contains y . ByProposition 8.9, there exists ˜ f ∈ H whose domain contains V , and such that f and ˜ f havethe same germ at y . By (10.1), from ˜ f ( y ) = f ( y ) ∈ A , it follows that ˜ f ( x ) ∈ U . Thus therestriction h of ˜ f to some neighborhood of x is in G , and so f ( y ) = ˜ f ( y ) = φ x , y ( h ( x )) ∈ φ x , y ( G ( x )) , which shows Claim 5.Since E is recurrent, Lemma 4.3 implies that there exist R , R ′ > d E -ball of radius R in any G -orbit meets A , as well as every d E -ball of radius R ′ in any G ′ -orbit.Therefore φ x , y ( G ( x ) ∩ A ) is an ( R + R ′ )-net in (cid:16) G ′ ( y ) , d E (cid:17) by Claims 3 and 5. Moreover φ x , y : ( G ( x ) ∩ A , d E ) → (cid:16) φ x , y ( G ( x ) ∩ A ) , d E (cid:17) QUICONTINUOUS FOLIATED SPACES 31 is a bi-Lipschitz map whose distortion is independent of x , y by Claims 2, 3 and 4. Hence( G ( x ) , d E ) is coarsely quasi-isometric to (cid:16) G ′ ( y ) , d E (cid:17) , where the coarse distortions are inde-pendent of the choices of x , y . The result now follows from Theorem 4.6 since E and E arerecurrent systems of compact generation of H on U , U ′ . (cid:3) Example 10.2.
In the pseudogroup H on Z of Example 9.3, all orbits have trivial groups ofgerms, except the origin. Moreover Z is locally compact and locally connected, and Z / H isconnected because so is Z . But, if H is generated by an irrational rotation, the statement ofTheorem 10.1 does not hold because H is not quasi-analytic. Indeed, there are two coarsequasi-isometry types of orbits with trivial group of germs: the orbits of the points in thevertical segment are trivial, and all other orbits are quasi-isometric to the integers.An action of a group Γ on a space will be called quasi-e ff ective when it generates aquasi-e ff ective pseudogroup. A quasi-e ff ective action may not be e ff ective, as shown bythe following example. Example 10.3.
Let Z be a finite discrete space with more than two elements, and let Γ bethe group of all permutations of Z . Then the canonical action of Γ on Z is not e ff ectivebut it is quasi-e ff ective: the condition of Definition 9.4 is satisfied with the set S of maps { x } → { y } with x , y ∈ Z . Corollary 10.4.
Let Γ be a finitely generated group acting quasi-e ff ectively and equicon-tinuously on a compact separable metric space Z with connected space of orbits ( for ex-ample, if some orbit is dense ) . Then all orbits with trivial group of germs are uniformlycoarsely quasi-isometric to each other.
11. M inimality of the orbit closures
For compactly generated pseudogroups of local isometries of a Riemannian manifold,the closures of the orbits are manifolds, and the restriction of the pseudogroup to each orbitclosure is a minimal pseudogroup [17, Appendix D]—this is just a pseudogroup version ofMolino’s theory for Riemannian foliations. In the more general situation considered here,at least the minimality of the orbit closures holds true.
Theorem 11.1.
Let H be a compactly generated strongly equicontinuous pseudogroup oflocal transformations of a locally compact Polish space Z. Then the closure of each orbitis a minimal set. In particular, such a pseudogroup is minimal if it is transitive.Proof. It is required to show that if the orbit of a point x ∈ Z approaches another point y ,then the orbit of y also approaches x . If U is a relatively compact open subset of Z thatmeets every H -orbit, then it can be assumed that x , y ∈ U .Let V be a finite family of open subsets of Z whose union covers U as in Proposition 8.9.Let V , W ∈ V be such that x ∈ V and y ∈ W . If h n ∈ H is a sequence such that h n ( x ) → y ,then it can be assumed that dom h n = V and h n ( x ) ∈ W for all n . Moreover, there are maps f n ∈ H with dom f n = W , and such that f n and h − n have the same germ at h n ( x ) for all n . Strong equicontinuity of H then implies that f n ( y ) → x as follows. Suppose that thecondition of strong equicontinuity of H is satisfied for a locally finite covering { ( Z i , d i ) } i ∈ I of Z by quasi-locally equal metric spaces, some symmetric set S of generators of H thatis closed under compositions, and some assignment ε δ ( ε ) (Definition 8.4). It may beassumed that V ⊂ Z i and W ⊂ Z j for some i , j ∈ I , and that h n , f n ∈ S for all n . Given ε >
0, there exists an integer N > d j ( h n ( x ) , y ) < δ ( ε ) for all n ≥ N . Hence d i ( x , f n ( y )) = d i ( f n ◦ h n ( x ) , f n ( y )) < ε for all n ≥ N by strong equicontinuity. (cid:3) Corollary 11.2.
Let H be a compactly generated and strongly equicontinuous pseudogroupof local transformations of a locally compact Polish space Z. Then the orbit closures of H define a partition of Z.
12. T he closure of a strongly equicontinuous pseudogroup
In the study of pseudogroups of local isometries of Riemannian manifolds, an importantrole is played by the closure of such a pseudogroup [9]. It is defined by using the spaceof 1-jets, which is not available in our more general setting. But the closure of our typeof pseudogroups can be also defined by using the compact-open topology on the spaces oflocal transformations defined on small enough open subsets.As usual, for spaces Y , Z , let C ( Y , Z ) denote the set of continuous maps Y → Z , whichwill be denoted by C c-o ( Y , Z ) when it is endowed with the compact-open topology. For opensubspaces O , P of a space Z , the space C c-o ( O , P ) will be considered as an open subspaceof C c-o ( O , Z ) in the canonical way. Theorem 12.1.
Let H be a quasi-e ff ective, compactly generated and strongly equicontin-uous pseudogroup of local transformations of a locally compact Polish space Z. Let S bea symmetric set of generators of H that is closed under compositions and restrictions toopen subsets, and satisfies the conditions of strong equicontinuity and quasi-e ff ectiveness ( Definitions 8.4 and 9.4 ) . Let e H be the set of maps h between open subsets of Z that satisfythe following property: for every x ∈ dom h, there exists a neighborhood O x of x in dom hso that h | O x is in the closure of C ( O x , Z ) ∩ S in C c-o ( O x , Z ) . Then: • e H is closed under composition, combination and restriction to open sets; • every map in e H is a homeomorphism around every point of its domain; • the maps of e H that are homeomorphisms form a pseudogroup H that contains H ; • H is strongly equicontinuous; • the orbits of H are equal to the closures of the orbits of H ; and • e H and H are independent of the choice of S .Proof. The family e H is obviously closed under combination of maps and restrictions toopen sets. Moreover e H is closed under composition of maps because Z is locally compactHausdor ff (see e.g. [19, p. 289, Exercise 4]).Given any relatively compact open subset U that meets all H -orbits, by Proposition 8.9,its remark (i) and Proposition 9.9, there is some finite family V of open subsets of Z andanother relatively compact open set U such that: • U is covered by the family V ; • any germ of any map in the restriction of H to U is a germ of some map in S whose domain belongs to V ; and • f ( V ) ⊂ U for any V ∈ V and f ∈ S with V ⊂ dom f and f ( V ) ∩ U , ∅ . Inparticular, any V ∈ V is contained in U .For any map h : V → W in e H and any x ∈ V , we will show that there is someneighborhood O of x in V such that the restriction h : O → h ( O ) is a homeomorphismwhose inverse is also in e H . It can be assumed that there are some U , U and V as aboveso that V , W ∈ V . Furthermore we can suppose that h is the limit in C c-o ( V , Z ) of somesequence of maps h n ∈ C ( V , Z ) ∩ S . Take any open neighborhood V ′ of x with V ′ ⊂ V .Since V ′ is compact, it follows that h n (cid:16) V ′ (cid:17) ⊂ W for n large enough. QUICONTINUOUS FOLIATED SPACES 33
The germ of each h − n at h n ( x ) is equal to the germ of some f n ∈ S whose domain is W .By quasi-e ff ectiveness, each f n is equal to h − n on h n ( V ) ∩ W , which contains h n ( V ′ ). Hence V ′ ⊂ im f n , and f − n is equal to h n on V ′ .By strong equicontinuity, the set C ( W , U ) ∩ S is equicontinuous in the usual sense. So,by the Ascoli theorem and the compactness of U , we can assume that f n is convergent tosome map f in C c-o ( W , Z ), which is in e H .We have that h n ( x ) → y = h ( x ), yielding f n ( y ) → x by strong equicontinuity as in theproof of Theorem 11.1. Therefore f ( y ) = x ∈ V ′ , and there is some open neighborhood W ′ of y with W ′ ⊂ W and f (cid:16) W ′ (cid:17) ⊂ V ′ . Since W ′ is compact, we get f n (cid:16) W ′ (cid:17) ⊂ V ′ for n large enough. So f − n is equal to h n on f n ( W ′ ), yielding that the composite h n ◦ f n is theidentity on W ′ for n large enough. It follows that h ◦ f is the identity on W ′ because Z islocally compact Hausdor ff . Similarly, for any open neighborhood O of x with O ⊂ V ′ and h (cid:16) O (cid:17) ⊂ W ′ , we get that f ◦ h is the identity on O . So h : O → h ( O ) is a homeomorphismwhose inverse is f : h ( O ) → O , which is in e H as desired.Now, from what was proved for e H , it follows directly that H is a pseudogroup. More-over H contains H because e H contains S by definition.We now show that H is strongly equicontinuous. Suppose that H satisfies the conditionof strong equicontinuity (Definition 8.4) with the above S , some quasi-local metric Q ,some { ( Z i , d i ) } i ∈ I ∈ Q with { Z i } i ∈ I locally finite, and some assignment ε δ ( ε ). Let S be the set of homeomorphisms that are in the union of the closures of C ( O , Z ) ∩ S in C c-o ( O , Z ) with O running on the open sets of Z . By definition, every element of H isa combination of maps in S . Since S is closed under restrictions to open sets, it easilyfollows that so is S . The set S is also closed under compositions because so is S and Z islocally compact Hausdor ff . Moreover S is symmetric since it is closed under restrictionsto open sets and because H is a pseudogroup whose elements are combinations of maps in S . A typical “ ε/ H satisfies the strong equicontinuity conditionwith S and the above family { ( Z i , d i ) } i ∈ I . Take any h : O → P in S , i , j ∈ I and x , y ∈ Z i ∩ h − ( Z j ∩ im h ). Suppose that d i ( x , y ) < δ ( ε/
3) for some ε >
0. Such an h is the limit ofsome sequence of maps h n ∈ C ( O , Z ) ∩ S in C c-o ( O , Z ). On the one hand, since the compact-open topology is equal to the topology of uniform convergence on compact sets, it followsthat d j ( h n ( x ) , h ( x )) < ε/ d j ( h n ( x ) , h ( x )) < ε/ n large enough. On the other hand,we have d j ( h n ( x ) , h n ( y )) < ε/ n since h n ∈ S . Therefore d j ( h ( x ) , h ( y )) < ε asdesired by the triangle inequality.We now show that the orbits of H are equal to the orbit closures of H . Given two points x , y in the same orbit closure of H , it has to be shown that x , y are in the same orbit of H .There is a sequence h n ∈ H with h n ( x ) → y . It can be assumed that x , y ∈ U for somerelatively compact open set U that meets all H -orbits. As above, by Proposition 8.9, itsremark (i) and Proposition 9.9, we can suppose that h n ∈ S , dom h n = V and h n ( V ) ⊂ U for some fixed open set V and some relatively compact open set U . Thus h n is a sequencein C ( V , U ) ∩ S , which is an equicontinuous family of maps. Therefore we can assume that h n is convergent in C c-o ( V , Z ) by the Ascoli theorem, and let h be its limit. Then h ( x ) = y and h ∈ e H by definition. Thus x , y are in the same orbit of H because the restriction of h to some open neighborhood of x is in H .Finally H is independent of the choice of S because it is the pseudogroup generated bythe local transformations of Z lying in the union of closures of C ( O , Z ) ∩ H in C c-o ( O , Z ) with O running on the open sets of Z . Obviously, H is independent of S if and only if e H is also. (cid:3) Definition 12.2.
Let H be a quasi-e ff ective, compactly generated and strongly equicontin-uous pseudogroup of local transformations of a locally compact Polish space Z . With thenotation of Theorem 12.1, the pseudogroup H is called the closure of H .As a direct consequence of Theorem 12.1, in the present general setting, the orbit clo-sures satisfy the following property of manifolds. Definition 12.3.
A topological space is homogeneous if the pseudogroup of all local home-omorphisms has exactly one orbit.
Corollary 12.4.
Let H be a quasi-e ff ective, compactly generated and strongly equicontin-uous pseudogroup of local transformations of a locally compact Polish space Z. Then theclosure of each orbit is homogeneous.
13. L ocal metric spaces
Pseudogroups of local isometries make sense on metric spaces but, with more general-ity, this type of pseudogroup can be defined on local metric spaces, which are introducedas follows.
Definition 13.1.
Two metrics on the same set are said to be locally equal when they inducethe same topology and each point has a neighborhood where both metrics are equal. Let { ( Z i , d i ) } i ∈ I be a family of metric spaces such that { Z i } i ∈ I is a covering of a set Z , eachintersection Z i ∩ Z j is open in ( Z i , d i ) and ( Z j , d j ), and the metrics d i , d j are locally equalon Z i ∩ Z j whenever this is a non-empty set. Such a family will be called a cover of Z bylocally equal metric spaces . Two such families are called locally equal when their unionalso is a cover of Z by locally equal metric spaces. This is an equivalence relation whoseequivalence classes are called local metrics on Z . For each local metric D on Z , the pair( Z , D ) is called a local metric space . Remarks. (i) Observe the analogy between the definitions of local metrics and quasi-localmetrics: for every local metric D , there is a unique quasi-local metric Q so that D ⊂ Q .In particular, all topological properties of quasi-local metric spaces hold for local metricspaces.(ii) In contrast with quasi-local metrics, local metrics can be also characterized as maximalcovers of Z by locally equal metric spaces; there always exist such maximal families.(iii) The concept of local metric has the following sheaf theoretic description, which showsits naturality. Suppose that the set Z is endowed with a topology a priori , even though thistopology will be later determined by the local metric. Then, for each open subset U ⊂ Z , let M ( U ) denote the set of all metrics on U that induce its topology. Such an M is a presheafon Z with the usual restriction of metrics, and a local metric on Z is just a global section ofthe sheaf e M determined by M . By Example 13.2 below, the presheaf M is a sheaf only inthe uninteresting case where the only metrizable open sets contain just one point. Example 13.2. If Z is metrizable and contains at least two points x , y , then there are in-finitely many metrics that are locally equal to any given metric d inducing the topology of Z ; for instance, all the metrics d r , 0 < r < d ( x , y ), given by d r ( z , z ′ ) = min { d ( z , z ′ ) , r } for z , z ′ ∈ U . QUICONTINUOUS FOLIATED SPACES 35
Example 13.3.
Let B be any open disc in R , Z = R \ B . Let d denote the restriction ofthe euclidean distance of R to Z , and d ′ the distance map on Z induced by the restrictionof the Riemannian metric of R . Also, let [ x , y ] denote the segment that joins each pair ofpoints x , y ∈ Z . We have d ( x , y ) = d ′ ( x , y ) if [ x , y ] ∩ B = ∅ , and d ( x , y ) < d ′ ( x , y ) otherwise.So both metrics d , d ′ are locally equal, and thus define the same local metric space ( Z , D ).The proof of the following result is analogous to the proof of Lemma 7.2. Lemma 13.4.
Let ( Z , D ) be a local metric space. If { Z i } i ∈ I is locally finite for some { ( Z i , d i ) } i ∈ I ∈ D , then there is some open neighborhood U z of each z ∈ Z such that themetrics d i , d j are equal on U z ∩ Z i ∩ Z j for all i , j ∈ I. Each metric d on a set Z induces a unique local metric D so that { ( Z , d ) } ∈ D . Thefollowing shows that the reciprocal holds when ( Z , D ) is Hausdor ff and paracompact. Theorem 13.5.
A local metric space ( Z , D ) is induced by some metric on Z if and only if ( Z , D ) is Hausdor ff and paracompact.Proof. The “only if” part holds by the Stone theorem (see e.g. [30, Theorem 20.9]). Nowsuppose that ( Z , D ) is Hausdor ff and paracompact. Claim 6.
There is some { ( U a , D a ) } a ∈ A ∈ D such that { U a } a ∈ A is locally finite, and D a , D b are equal on U a ∩ U b for all a , b ∈ A with U a ∩ U b , ∅ . Indeed, since ( Z , D ) is paracompact, there is some { ( Z i , d i ) } i ∈ I ∈ D such that { Z i } i ∈ I islocally finite. Let { Z ′ i } i ∈ I be a shrinking of { Z i } i ∈ I . For each i ∈ I and x ∈ Z ′ i , let V i , x be anopen neighborhood of x that is contained in Z ′ i and meets only a finite number of sets Z j for j ∈ I . Therefore, for any y ∈ V i , x , there is some open neighborhood W i , x , y of y in V i , x such that: • If y < Z ′ j for some j ∈ I , then W i , x , y ∩ Z ′ j = ∅ ; • if y ∈ Z ′ j for some j ∈ I , then W i , x , y ⊂ Z j and the metrics d i , d j are equal on W i , x , y .Again, because ( Z , D ) is paracompact, there is an open locally finite refinement { U a } a ∈ A ofthe open cover given by all possible sets W i , x , y as above. For each a ∈ A , choose any W i , x , y containing U a , and let D a denote the restriction of d i to U a . Then Claim 6 follows easilywith such a family { ( U a , D a ) } a ∈ A .A metric D on Z is now defined as follows. With the notation of Claim 6, let { U ′ a } a ∈ A be a shrinking of the open covering { U a } a ∈ A . A pair ( z , z ) ∈ Z × Z will be said to beadmissible if there is some a ∈ A such that z , z ∈ U ′ a , and moreover { z , z } ∩ U ′ b , ∅ = ⇒ { z , z } ⊂ U b for all b ∈ A . For each ( x , y ) ∈ Z × Z , let S x , y denote the set of all finite sequences ( z , . . . , z n )in Z , with arbitrary length n ∈ Z + , such that z = x , z n = y , and ( z k − , z k ) is an admissiblepair for every k = , . . . , n . Then set D ( x , y ) = S x , y = ∅ , and let D ( x , y ) = inf ( z ,..., z n ) ∈ S x , y n X k = D a k ( z k − , z k )if S x , y , ∅ , where z k − , z k ∈ U ′ a k with a k ∈ A for each k = , . . . , n . This definition isindependent of the choices of the indices a k by Claim 6. Claim 7.
Let a ∈ A, x ∈ U ′ a and y ∈ Z with S x , y , ∅ . ThenD ( x , y ) ≥ min { D a ( x , y ) , D a ( x , U a \ U ′ a ) } if y ∈ U ′ a ,D a ( x , U a \ U ′ a ) if y < U ′ a . To prove this assertion, let ( z , . . . , z n ) ∈ S x , y and a , . . . , a k ∈ A with z k − , z k ∈ U ′ a k for k = , . . . , n . On the one hand, if z , . . . , z n ∈ U ′ a , we have n X k = D a k ( z k − , z k ) = n X k = D a ( z k − , z k ) ≥ D a ( z , z n ) = D a ( x , y )by Claim 6. On the other hand, suppose { z , . . . , z n } U ′ a . Then n ≥
1, and let n = min { k ∈ { , . . . , n } | z k < U ′ a } . Since z n − ∈ U ′ a , we get z n ∈ U a because ( z n − , z n ) is an admissible pair. So n X k = D a k ( z k − , z k ) ≥ n X k = D a k ( z k − , z k ) ≥ D a ( z , z n ) ≥ D a ( x , U a \ U ′ a )by Claim 6, which completes the proof of Claim 7.The above D is a pseudometric on Z because the following holds for all x , y , z ∈ Z :( x , x ) ∈ S x , x , ( z , . . . , z n ) ∈ S x , y = ⇒ ( z n , . . . , z ) ∈ S y , x , ( z , . . . , z m ) ∈ S x , y ( z m , . . . , z m + n ) ∈ S y , z ) = ⇒ ( z , . . . , z m + n ) ∈ S x , z . To show that D is indeed a metric, suppose D ( x , y ) = x , y ∈ Z ; thus S x , y , ∅ .Take any a ∈ A with x ∈ U ′ a . Since D a ( x , U a \ U ′ a ) >
0, it follows from Claim 7 that y ∈ U ′ a and D a ( x , y ) ≤ D ( x , y ) =
0. So x = y as desired because D a is a metric.It remains to check that { ( Z , D ) } ∈ D . Fix any z ∈ Z and any a ∈ A with z ∈ U ′ a .The following assertion follows easily because { U a } a ∈ A is locally finite and { U ′ a } a ∈ A is ashrinking of { U a } a ∈ A . Claim 8.
There is some open neighborhood P z of z in U ′ a such thatP z ∩ U ′ a , ∅ = ⇒ P z ⊂ U a for all a , b ∈ A. Since { U a } a ∈ A is locally finite and ( Z , D ) is Hausdor ff , the set O x = \ a ∈ A , x ∈ U ′ a U a \ [ b ∈ A , x < U b U ′ b is an open neighborhood of every x in Z . If x ∈ U ′ a , it is easy to see that ( x , y ) ∈ S x , y forany y ∈ U ′ a ∩ O x , and thus D ( x , y ) ≤ D a ( x , y ). Since x ∈ P z = ⇒ P z ⊂ O x for all x ∈ Z by Claim 8, it follows that D ( x , y ) ≤ D a ( x , y ) for all x , y ∈ P z .On the other hand, we get from Claim 7 that D ( x , y ) ≥ D a ( x , y ) for all x ∈ U ′ a and all y in the open ball in ( U a , D a ) of center x and radius ρ ( x ) = D a ( x , U a \ U ′ a ). So D ( x , y ) ≥ D a ( x , y ) for all x , y in the open ball in ( U a , D a ) of center z and radius ρ ( z ). Thereforethe metrics D , D a are equal on some neighborhood of x , and the result follows. (cid:3) Remarks. (i) Theorem 13.5 is very similar to the Smirnov metrization theorem [25], [19,pp. 260–261] (see also J. Nagata [20, Chapter VI.3] for a stronger result), which showsthat a topological space is metrizable if and only if it is Hausdor ff , paracompact and lo-cally metrizable: in Theorem 13.5, the existence of a local metric is slightly stronger thanlocal metrizability, and the existence of a metric that induces a given local metric is slightlystronger than metrizability. QUICONTINUOUS FOLIATED SPACES 37 (ii) By the proof of Theorem 13.5, any paracompact Hausdor ff local metric D can be con-sidered as the germ of some metric on Z around the diagonal of Z × Z . But even in thiscase, the introduction of local metrics makes sense to emphasize the fact that we are onlyconsidering distances between “very close” points.(iii) With the sheaf theoretic point of view given in the remark (iii) of Definition 13.1, eventhough M is never a sheaf for interesting spaces, it is closer to be so for Hausdor ff para-compact spaces: in this case, Theorem 13.5 asserts that the canonical homomorphism ofpresheaves, M → e M , is surjective on all open sets. Example 13.6.
Let P be the open upper half-plane { ( x , y ) ∈ R | y > } , and L the real axis { ( x , | x ∈ R } . Consider the half-disk topology on Z = P ∪ L [26, pp. 96–97], which hasa base given by the euclidean open sets in P and the sets of the form { z } ∪ ( P ∩ U ), where z ∈ L and U is any euclidean open neighborhood of z in R . This space is not metrizablebecause it is not paracompact. But this topology is induced by a local metric D on Z , whichis determined by the family { ( P , d P ) } ∪ { ( U z , d z ) | z ∈ L } , where d P is the restriction of the euclidean metric to P , U z = { z }∪ P , and d z is the restrictionof the euclidean metric to U z . Example 13.7.
With more generality, let ( Z , d ) be a metric space, let { Z i } i ∈ I be a coveringof Z , and let d i be the restriction of d to Z i for each i ∈ I . Then the metrics d i , d j have equalrestriction to the overlap Z i ∩ Z j for all i , j ∈ I , and thus the family { ( Z i , d i ) } i ∈ I defines alocal metric D on Z . If the sets Z i are open in ( Z , d ), then D is induced by the metric d ,otherwise the topology induced by D is strictly finer than the topology induced by d , and D may not be induced by any metric, as in Example 13.6.Even though we are only interested on paracompact Hausdor ff spaces, the followingproblem is interesting. Problem 3.
Is any locally metrizable topology induced by some local metric? In particu-lar, is there a compatible local metric on every non-paracompact manifold? For instance,is there a compatible local metric on the Long Line [26, pp. 71–72] ?
14. P seudogroups of local isometries
The idea of a local metric as measuring distances between “very close” points is spe-cially appropriate to define local isometries.
Definition 14.1.
Let ( Z , D ) be a local metric space, and let h be a homeomorphism be-tween open subsets of ( Z , D ). Then h is called a local isometry of ( Z , D ) if there is some { ( Z i , d i ) } i ∈ I ∈ D such that, for i , j ∈ I and z ∈ Z i ∩ h − ( Z j ∩ im h ), there is some neighborhood U h , i , j , z of z in Z i ∩ h − ( Z j ∩ im h ) so that d i ( x , y ) = d j ( h ( x ) , h ( y )) for all x , y ∈ U h , i , j , z . Remarks. (i) For a map h between open subsets of a local metric space ( Z , D ), the prop-erty of being a local isometry is completely local, and h may not be isometric for a givenmetric inducing D (Examples 8.7 and 14.2).(ii) About the condition that the metrics d i , d j are locally equal on Z i ∩ Z j for any { ( Z i , d i ) } i ∈ I ∈ D , it just means that the identity map on any open subset of ( Z , D ) is a local isometry.(iii) A homeomorphism h between open subsets of a local metric space ( Z , D ) is a localisometry when it preserves the local metric in the obvious sense: h ∗ ( D | im h ) = D | dom h ,where the restrictions and pull-backs of local metrics are defined in an obvious way. Withthe sheaf theoretic description of local metrics (remark (iii) of Definition 13.1), this means that h induces an isomorphism between the restrictions of e M to its domain and image.(iv) The definition of local isometry is completely independent of the choice of the family { ( Z i , d i ) } i ∈ I ∈ D . So the same { ( Z i , d i ) } i ∈ I can be chosen to verify Definition 14.1 for anyfamily of local isometries. Therefore the concept of pseudogroup of local isometries iscompletely analogous to the concept of weakly equicontinuous pseudogroup. Example 14.2.
On the local metric space ( Z , D ) of Example 13.3, let H be the restrictionof the pseudogroup generated by all translations of R . Then H is a pseudogroup of localisometries of ( Z , D ). The maps in H with connected domain are isometries with respectto d , but many of them are not isometries with respect to d ′ . For instance, let U be anyrelatively compact and connected open subset of Z containing points x , y with [ x , y ] ∩ B , ∅ .Then there is a translation h of R such that h ( U ) ⊂ Z and [ h ( x ) , h ( y )] ∩ B = ∅ . So d ′ ( h ( x ) , h ( y )) = d ( h ( x ) , h ( y )) = d ( x , y ) < d ′ ( x , y ) , and thus the restriction h : U → h ( U ) is an element of H with connected domain that doesnot preserve d ′ .Arguments similar to those used in the proof of Lemma 8.2 prove the following lemma. Lemma 14.3.
Let H , H ′ be equivalent pseudogroups on spaces Z , Z ′ . Then H is a pseu-dogroup of local isometries for some local metric inducing the topology of Z if and only if H ′ is a pseudogroup of local isometries for some local metric inducing the topology of Z ′ . Unlike the concept of equicontinuity, it is not necessary to introduce weak and strongversions of the concept of pseudogroup of local isometries by the following result.
Lemma 14.4.
Let H be a pseudogroup of local transformations of a paracompact localmetric space ( Z , D ) . Then H is a pseudogroup of local isometries of ( Z , D ) if and onlyif there is some { ( Z i , d i ) } i ∈ I ∈ D and some symmetric set S of generators of H that isclosed under compositions and such that d i ( x , y ) = d j ( h ( x ) , h ( y )) for all h ∈ S , i , j ∈ I andx , y ∈ Z i ∩ h − ( Z j ∩ im h ) .Proof. Take any { ( Z i , d i ) } i ∈ I ∈ D such that { Z i } i ∈ I is locally finite. With the notation ofLemma 13.4 and Definition 14.1, for each h ∈ H and z ∈ dom h , let U h , z = U z ∩ \ i , j ∈ I , z ∈ Z i ∩ Z j U h , i , j , z , which is an open neighborhood of z . Then the result holds with S equal to the set ofcompositions of all restrictions of the form h : U h , z → h ( U h , z ) and their inverses. Wehave used that composition of isometries is an isometry, which fails for the equicontinuouscondition (8.1) with a fixed assignment ε δ ( ε ). (cid:3)
15. I sometrization of strongly equicontinuous pseudogroups
On the type of spaces we are considering, it will be shown that compactly generatedquasi-e ff ective strongly equicontinuous pseudogroups are pseudogroups of local isometriesfor some local metric. We begin with the following version of Theorem 13.5 for quasi-localmetric spaces. Most of its proof is also similar to the proof of Theorem 13.5, but there aresome new di ffi culties. Theorem 15.1.
A quasi-local metric space ( Z , Q ) is induced by some metric on Z if andonly if ( Z , Q ) is Hausdor ff and paracompact. QUICONTINUOUS FOLIATED SPACES 39
Proof.
As in the proof of Theorem 13.5, the “only if” part holds by the Stone theorem.Now suppose that ( Z , Q ) is Hausdor ff and paracompact. The following assertion can beproved in the same way as Claim 6. Claim 9.
There is some { ( U a , D a ) } a ∈ A ∈ Q and some δ ( ε ) > for each ε > such that { U a } a ∈ A is locally finite, and D a ( x , y ) < δ ( ε ) = ⇒ D b ( x , y ) < ε for all ε > , a , b ∈ A and x , y ∈ U a ∩ U b . We can also assume that the family { ( U a , D a ) } a ∈ A given by Claim 6 satisfies that the D a -diameter of each U a is smaller than 1. If x , y ∈ U a for some a ∈ A , let D ( x , y ) = sup a ∈ A , x , y ∈ U a D a ( x , y ) . Note that D ( x , y ) ≤ D a -diameter of each U a , and that D ( x , y ) isindependent of a . Moreover D is obviously symmetric, we have D ( x , y ) = x = y , and the following assertion follows directly from Claim 9. Claim 10.
We have D a ( x , y ) < δ ( ε ) = ⇒ D ( x , y ) < ε for all a ∈ A and x , y ∈ U a . But D may not satisfy the triangle inequality on any open set because there may bepoints x , y , z ∈ U a so that x , y ∈ U a and z < U a for some a ∈ A . So D may not be a metricon the sets of any open covering of Z ; otherwise, Theorem 13.5 could be used to conclude.Yet D is used to define a metric on Z with the idea of the proof of Theorem 13.5.Let { U ′ a } a ∈ A be a shrinking of { U a } a ∈ A . A pair ( z , z ) ∈ Z × Z will be said to be admissibleif there is some a ∈ A such that z , z ∈ U ′ a , and moreover { z , z } ∩ U ′ b , ∅ = ⇒ { z , z } ⊂ U b for any b ∈ A . For each ( x , y ) ∈ Z × Z , let S x , y denote the set of all finite sequences( z , . . . , z n ) in Z , with arbitrary length n ∈ Z + , such that z = x , z n = y , and ( z k − , z k ) is anadmissible pair for every k = , . . . , n . Now set D ( x , y ) = S x , y = ∅ , and D ( x , y ) = inf ( z ,..., z n ) ∈ S x , y n X k = D ( z k − , z k )if S x , y , ∅ . Claim 11.
Let a ∈ A, x ∈ U ′ a and y ∈ Z with S x , y , ∅ . ThenD ( x , y ) ≥ min { D a ( x , y ) , D a ( x , U a \ U ′ a ) } if y ∈ U ′ a ,D a ( x , U a \ U ′ a ) if y < U ′ a . To prove this assertion, let ( z , . . . , z n ) ∈ S x , y and a , . . . , a k ∈ A with z k − , z k ∈ U ′ a k for k = , . . . , n . On the one hand, if z , . . . , z n ∈ U ′ a , we have n X k = D ( z k − , z k ) ≥ n X k = D a ( z k − , z k ) ≥ D a ( z , z n ) = D a ( x , y ) . On the other hand, suppose { z , . . . , z n } U ′ a . Then n ≥
1, and let n = min { k ∈ { , . . . , n } | z k < U ′ a } . Since z n − ∈ U ′ a , we get z n ∈ U a because ( z n − , z n ) is an admissible pair. So n X k = D ( z k − , z k ) ≥ n X k = D a ( z k − , z k ) ≥ D a ( z , z n ) ≥ D a ( x , U a \ U ′ a ) , which completes the proof of Claim 7.With the same arguments as in the proof of Theorem 13.5, it follows that D is a metricon Z by using Claim 11.It remains to check that { ( Z , D ) } ∈ Q . Fix any z ∈ Z and any a ∈ A with z ∈ U ′ a . Weget the following assertion as in the proof of Theorem 13.5. Claim 12.
There is some open neighborhood P z of z in U ′ a such thatP z ∩ U ′ a , ∅ = ⇒ P z ⊂ U a for all a , b ∈ A. Also, as in the proof of Theorem 13.5, the set O x = \ x ∈ U ′ a , a ∈ A U a \ [ x < U b , b ∈ A U ′ b is an open neighborhood of every x in Z , and we have ( x , y ) ∈ S x , y for any x ∈ U ′ a and y ∈ U ′ a ∩ O x . So D ( x , y ) ≤ D ( x , y ) for all y ∈ U ′ a ∩ O x , yielding(15.1) D a ( x , y ) < δ ( ε ) = ⇒ D ( x , y ) < ε by Claim 10. Since x ∈ P z = ⇒ P z ⊂ O x for all x ∈ Z by Claim 8, it follows that (15.1) holds for all x , y ∈ P z .On the other hand, as in the proof of Theorem 13.5, we get from Claim 11 that D ( x , y ) ≥ D a ( x , y ) for all x , y in the open ball in ( U a , D a ) of center z and radius D a ( z , U a \ U ′ a ).Therefore the families of metric spaces { ( Z , D ) } and { ( U a , D a ) } a ∈ A are quasi-locally equal; i.e. , Q is induced by D . (cid:3) Remarks. (i) Theorem 15.1 can be also compared with the Smirnov metrization theorem.(ii) By Theorem 15.1, in the paracompact Hausdor ff case, a quasi-local metric is almostthe same concept as a local metric; the only di ff erent being that di ff erent local metrics mayinduce the same quasi-local metric (Example 7.4).Our “isometrization” result for pseudogroups can be stated as follows. Theorem 15.2.
Let H be a compactly generated, quasi-e ff ective and strongly equicontin-uous pseudogroup of local transformations of a locally compact Polish space Z. Then H isa pseudogroup of local isometries with respect to some local metric inducing the topologyof Z.Proof. The pseudogroup H is strongly equicontinuous with respect to some quasi-localmetric Q that induces the topology of Z . Such a Q is induced by some metric d on Z according to Theorem 15.1. So, by remark (iv) of Definition 8.4, the condition of strongequicontinuity is satisfied by the family { ( Z , d ) } with some assignment ε δ ( ε ) and somesymmetric set S of generators of H that is closed under compositions. We can also supposethat S is closed under restrictions to open sets by remark (iii) of Definition 8.4. Furthermorewe can assume that the condition of quasi-e ff ectiveness is also satisfied with S (remarks ofDefinition 9.4). This means that any element of S is equal to the identity on its domain if QUICONTINUOUS FOLIATED SPACES 41 it is equal to the identity on some non-trivial open subset; so two elements of S are equalon the intersection of their domains if they have the same germ at some point.Let U be any relatively compact open subset of Z that meets every H -orbit, and E anysymmetric system of compact generation of H on U . For each g ∈ E , let ¯ g be its extensionsatisfying the conditions of Definition 2.3, and let E = { ¯ g | g ∈ E } . We can choose S , E and the extensions ¯ g so that E ⊂ S .Let V be a finite family of open subsets of Z given by Proposition 8.9 for the above d , S , U , E and extensions ¯ g . We can suppose that the d -diameter of every V ∈ V is smallerthan δ (1). Let R ⊂ H be the set of all compositions of elements in E , and R ⊂ H the set ofall compositions of elements in E ; so R , R ⊂ S . For each V ∈ V , and x , y ∈ V , let d V ( x , y ) = sup h ∈ R , V ⊂ dom h d ( h ( x ) , h ( y )) , Such d V is well defined by Proposition 8.9, and we have d V ( x , y ) ≤ V and because R ⊂ S . It is easy to check that d V is a metric on V . Moreoverwe have the following fact. Claim 13.
The metrics d V , d W are equal on V ∩ W for all V , W ∈ V . Take sets V , W ∈ W with V ∩ W , ∅ to verify this assertion. It su ffi ces to show that, forall h ∈ R whose domain contains V , there is some h ′ ∈ R whose domain contains W and sothat h , h ′ are equal on V ∩ W : this clearly yields d V ( x , y ) ≤ d W ( x , y ) for all x , y ∈ V ∩ W ,and the reverse inequality is similarly obtained. Thus let h ∈ R with V ⊂ dom h . The germof h at any x ∈ V ∩ W is equal to the germ of some f ∈ R at x . By Proposition 8.9, there issome h ′ ∈ R whose domain contains W and equal to f around x . Since h , h ′ ∈ S and havethe same germ at x , these maps are equal on V ∩ W by quasi-e ff ectiveness, and the claimfollows.Therefore the collection { ( V , d V ) | V ∈ V } defines a local metric D on the union U ofthe sets V ∈ V . Moreover, on the one hand, we obviously have d V ( x , y ) ≥ d ( x , y ) for all V ∈ V and x , y ∈ V . On the other hand, d ( x , y ) < δ ( ε ) = ⇒ d V ( x , y ) < ε for all ε > V ∈ V and x , y ∈ V by strong equicontinuity since R ⊂ S . Thus D inducesthe restriction Q of Q to U . Claim 14.
We have d W ( f ( x ) , f ( y )) = d V ( x , y ) for all V , W ∈ V , f ∈ R and x , y ∈ V ∩ f − ( W ∩ im h ) . To prove this equality, let V , W , f , x , y be as in the statement of this claim. Then wehave f = g m ◦ · · · ◦ g for g , . . . , g m ∈ E . Let ˜ f = ¯ g m ◦ · · · ◦ ¯ g ∈ R . Then V ⊂ dom ˜ f by Proposition 8.9. For any h ∈ R with W ⊂ dom h , the germ of h at any fixed point z ∈ W ∩ im f is equal to the germ at z of some element of R ; say g m + n ◦ · · · ◦ g m + forsome g m + n , . . . , g m + ∈ E . Hence ¯ g m + n ◦ · · · ◦ ¯ g m + ∈ R has the same germ at z as h and itsdomain contains W again by Proposition 8.9. It follows that h = ¯ g m + n ◦ · · · ◦ ¯ g m + on W by quasi-e ff ectiveness. Since g m + n ◦ · · · ◦ g is defined around f − ( z ) ∈ V , the domain of¯ g m + n ◦ · · · ◦ ¯ g contains V by Proposition 8.9 once more, and we have h ◦ ˜ f = ¯ g m + n ◦ · · · ◦ ¯ g on V ∩ ˜ f − ( W ∩ im ˜ f ) by quasi-e ff ectiveness. So h ◦ f is equal to some element of R on V ∩ f − ( W ∩ im f ), which yields d W ( f ( x ) , f ( y )) = sup h ∈ R , W ⊂ dom h d ( h ◦ f ( x ) , h ◦ f ( y )) ≤ sup h ′ ∈ R , V ⊂ dom h ′ d ( h ′ ( x ) , h ′ ( y )) = d V ( x , y ) . We also get d W ( f ( x ) , f ( y )) ≥ d V ( x , y ) by applying the above argument to f − instead of f ,and Claim 14 follows.Claim 14 shows that the restriction of H to U is a pseudogroup of local isometries withrespect to the restriction of D to U , and therefore the theorem follows by Lemma 14.3. (cid:3) According to Theorem 15.2, the following problem may be di ffi cult and interesting. Problem 4.
Find an example of a strongly equicontinuous pseudogroup that is not a pseu-dogroup of local isometries for any local metric.
16. A non - standard description of weak equicontinuity The following simple non-standard description of weak equicontinuity shows the nat-urality of this condition, even though strong equicontinuity is what is mainly used in ourstudy. The reference for non-standard analysis is Robinson [21]; we do not use any tech-nique particular to non-standard analysis, only the concept of monad, which is now defined.Fix a non-principal ultrafilter F on the set N of positive integers; i.e. , F defines a pointin the corona of the Stone- ˇCech compactification of N . Let ( Z , d ) be any metric space. Forany x ∈ Z , the monad of x in ( Z , d ), denoted by M ( x , Z , d ) or simply M ( x ), is the quotientset of the set of sequences y n in Z such that { n ∈ N | d ( x , y n ) < r } ∈ F for all r >
0, where two such sequences y n , z n are identified when { n ∈ N | y n = z n } ∈ F . If ( Z ′ , d ′ ) is another metric space, any continuous map f : ( Z , d ) → ( Z ′ , d ′ ) induces amap f ∗ : M ( x , Z , d ) → M ( f ( x ) , Z ′ , d ′ ) for every x ∈ Z , which is defined as follows: if y ∈ M ( x , Z , d ) is represented by the sequence y n , then f ∗ ( y ) is represented by the sequence f ( y n ).The monad of 0 in R with the euclidean metric is the set I of infinitesimal numbers. Theinfinitesimal number represented by the zero constant sequence will be denoted by . For ε , δ ∈ I , represented by sequences ε n , δ n , the inequality ε < δ means that { n ∈ N | ε n < δ n } ∈ F . Moreover the metric d on Z defines a map d ∗ : M ( x ) → I for every x ∈ Z in the followingway: if y ∈ M ( x ) is represented by the sequence y n , then d ∗ ( y ) is represented by thesequence d ( x , y n ).Now suppose that Q is a quasi-local metric on Z and { ( Z i , d i ) } i ∈ I ∈ Q . If x ∈ Z i ∩ Z j for i , j ∈ I , then M ( x , Z i , d i ) ≡ M ( x , Z j , d j ) canonically. Thus M ( x , Z i , d i ) can be called themonad of x in ( Z , Q ), and denoted by M ( x , Z , Q ) or simply M ( x ). It also follows that anycontinuous map between quasi-local metric spaces, f : ( Z , Q ) → ( Z ′ , Q ′ ), induces a map f ∗ : M ( x , Z , Q ) → M ( f ( x ) , Z ′ , Q ′ ) for each x ∈ Z . QUICONTINUOUS FOLIATED SPACES 43
Theorem 16.1.
Let H be a pseudogroup of local homeomorphisms of a quasi-local metricspace ( Z , Q ) , and let { ( Z i , d i ) } i ∈ I ∈ Q . Then H is weakly equicontinuous if and only if forevery ε ∈ I , ε > , there is some δ ( ε ) ∈ I , δ ( ε ) > , such that (16.1) d i ∗ ( y ) < δ ( ε ) = ⇒ d j ∗ ( h ∗ ( y )) < ε for all h ∈ H , i , j ∈ I, x ∈ Z i ∩ h − ( Z j ∩ im h ) and y ∈ M ( x ) .Proof. Suppose first that H is weakly equicontinuous. So the condition of weak equicon-tinuity is satisfied with { ( Z i , d i ) } i ∈ I , some assignment ε δ ( ε ) and neighborhoods U h , i , j , z (Definition 8.1). We can assume that δ ( ε ) < ε for all ε >
0. Given any ε ∈ I , ε > ,take some sequence ε n representing ε . We can assume that ε n > n . Then thesequence δ ( ε n ) also represents some infinitesimal number, which is denoted by δ ( ε ). Nowtake h ∈ H , i , j ∈ I , x ∈ Z i ∩ h − ( Z j ∩ im h ) and y ∈ M ( x ) with d i ∗ ( y ) < δ ( ε ). So { n ∈ N | d i ( x , y n ) < δ ( ε n ) } ∈ F . Moreover { n ∈ N | y n ∈ U h , i , j , x } ∈ F because y ∈ M ( x ). Therefore { n ∈ N | y n ∈ Z i ∩ h − ( Z j ∩ im h ) , d j ( h ( x ) , h ( y n )) < δ ( ε n ) } ∈ F by weak equicontinuity, yielding d j ∗ ( h ∗ ( y )) < ε , and (16.1) follows.Now suppose that (16.1) holds for some assignment ε δ ( ε ) and all h , i , j , x , y as inthe statement. According to Definition 8.1, if H is not weakly equicontinuous, then thereexists some ε > h ∈ H , i , j ∈ I and z ∈ Z i ∩ h − ( Z j ∩ im h ) so that, in every neighborhood U of z in Z i ∩ h − ( Z j ∩ im h ), there are points x U , y U with d j ( h ( x U ) , h ( y U )) ≥ ε . So, forevery n ∈ N , there are points x n , y n ∈ Z i ∩ h − ( Z j ∩ im h ) with d i ( x n , z ) , d i ( y n , z ) < / n , d j ( x n , y n ) ≥ ε . On the other hand, there is some N ∈ N so that d j ( h ( z ) , h ( x n )) < ε/ n ≥ N . Thus(16.2) d j ( h ( z ) , h ( y n )) ≥ d j ( h ( x n ) , h ( y n )) − d j ( h ( z ) , h ( x n )) > ε/ n ≥ N . Given any ε ∈ I , ε > , let δ ( ε ) be represented by a sequence δ n . We cansuppose that δ n > n . Then there is some k n ≥ N with 1 / k n < δ n for every n ,the sequence y ′ n = y k n represents an element y ′ ∈ M ( z ), and we have d i ∗ ( y ′ ) < δ ( ε ). So d j ∗ ( h ∗ ( y ′ )) < ε , which contradicts (16.2). (cid:3)
17. S trongly equicontinuous foliated spaces
Let ( X , F ) be a compact foliated space. Compact generation and recurrence are prop-erties satisfied by its holonomy pseudogroup with the generators given by a finite definingcocycle. To see this, fix a finite defining cocycle ( U i , p i , h i , j ) of F with p i : U i → Z i and h i , j : Z i , j → Z j , i , where Z i , j = p i ( U i ∩ U j ). Let H be the representative of the holonomypseudogroup of F induced by ( U i , p i , h i , j ) on Z = F i Z i . Suppose that ( U i , p i , h i , j ) is ashrinking of another defining cocycle (cid:16) e U i , ˜ p i , ˜ h i , j (cid:17) with ˜ p i : e U i → e Z i and ˜ h i , j : e Z i , j → e Z j , i ,where e Z i , j = ˜ p i (cid:16) e U i ∩ e U j (cid:17) . This means that, for each i , U i ⊂ e U i , Z i = ˜ p i ( U i ), and p i is therestriction of ˜ p i . Thus Z i ⊂ e Z i , Z i , j ⊂ e Z i , j , and h i , j is the restriction of ˜ h i , j . Let e H be therepresentative of the holonomy pseudogroup of F induced by (cid:16) e U i , ˜ p i , ˜ h i , j (cid:17) on e Z = F i e Z i .It easily follows that Z is relatively compact in e Z , H is the restriction of e H to Z , Z meetsall e H -orbits, and the transformations h i , j form a system of compact generation of e H on Z .This system is proved to be recurrent as follows. Fix any point x in the closure of some Z i in e Z i . Then ˜ p − i ( x ) cuts some e U j . Thus V i , j = ˜ p i (cid:16) e U i ∩ U j (cid:17) is a neighborhood of x in e Z i such that V i , j ⊂ e Z i , j , V i , j ∩ Z i = Z i , j and ˜ h i , j ( V i , j ) ⊂ Z j . Hence the transformations h i , j form a recurrent system of compact generation by Lemma 4.4. Therefore, according toTheorem 4.6, the coarse quasi-isometry type of the H -orbits with the metric induced bythe generators h i , j is kept uniformly fixed when varying the defining cocycle. So each leafof F determines a coarse quasi-isometry type of metric spaces. See e.g. [11] for a moredetailed description of the coarse quasi-isometry between the leaves and the orbits of theholonomy pseudogroup, which is already implicit in [22]. The bornotopy type of leaves isstudied in [13], and most of its discussion also applies to the coarse quasi-isometry type.If F is at least of class C , then there exists a metric tensor on the leaves that is contin-uous on X . In this case, it is well known that the already explained embedding of Z into X , as a complete transversal of F , defines a uniform collection of coarse quasi-isometriesbetween the H -orbits and the corresponding leaves. This is just a consequence of havinga uniform upper bound of the diameter of the plaques for the given finite defining cocycle.Therefore the coarse quasi-isometry type determined by each leaf is given by just itselfwith such a metric tensor.The foliated space ( X , F ) will be called weakly equicontinuous , strongly equicontinu-ous , quasi-analytic , or quasi-e ff ective , respectively, if any representative of its holonomypseudogroup is such. This is well defined because all of these properties on pseudogroupsare invariant by equivalences (Lemmas 8.2, 8.8 and 9.5). Then Theorem 10.1 has the fol-lowing consequence. Theorem 17.1.
Let ( X , F ) be an equicontinuous, compact and quasi-e ff ective foliatedspace. Assume that the space of leaves is connected ( for example, if F is transitive, orif X is connected ) . Then all leaves with trivial holonomy group determine the same coarsequasi-isometry type. When ( X , F ) is at least of class C , the leaves can be endowed with a metric tensorwhich is continuous on X , and the above result can be improved to obtain quasi-isometriesvia di ff eomorphisms between covers of leaves.This will be done with the help of the normal quasi-foliated bundles of Section 1. As theprevious results make evident, the problem with the holonomy and topological structure ofthe transversal makes it di ffi cult to e ff ectively use the equicontinuity to push whole leavesonto others. In any case, the following weaker solution to this problem can be provided. Theorem 17.2.
Let ( X , F ) be a strongly equicontinuous, compact foliated space of class C with connected space of leaves ( for example, if F is transitive, or if X is connected ) . Thenthe universal covers of all the leaves are uniformly quasi-isometric via di ff eomorphisms.Proof. Let L ′ be the universal cover of a leaf L , and let N ( L ′ ) be the normal bundle de-scribed in Section 1. With respect to some metric on N ( L ′ ), which is boundedly distortedvia the embedding, there is a product neighborhood N ( L ′ , ε ) of the zero section in N ( L ′ )carrying a lamination Y , as described in Theorem 1.3. The projection of the leaves in thisneighborhood onto the zero section L ′ is locally a di ff eomorphism with bounded distor-tion, by Proposition 1.5. On the other hand, the strong equicontinuity of the pseudogroupreadily implies that these projections are actually covering maps. More precisely, with thenotation of Section 1, strong equicontinuity implies that given ε > δ > S is a leaf of Y and meets the fiber p − ( x ) of some point x ∈ L ′ at a point at distance < δ from x , then it meets every fiber p − ( y ) at points at distance < ε from the base y . Thisimplies that the projection p : S → L ′ is a covering map, for it is a local homeomorphismwhich has the path covering property, the only obstruction to lifting a path being that such QUICONTINUOUS FOLIATED SPACES 45 leaf S runs o ff the neighborhood N ( L ′ , ε ). Since L ′ is simply connected, it follows that p isa di ff eomorphism. That p has bounded distortion was already discussed in Proposition 1.5.The above paragraph shows that universal covers of pairs of leaves are uniformly quasi-isometric if both leaves are close enough. Then the result follows since the space of leavesis compact and connected. (cid:3) Theorem 17.3.
Let ( X , F ) be a strongly equicontinuous, compact and quasi-e ff ective foli-ated space of class C with connected space of leaves ( for example, if F is transitive, or ifX is connected ) . Then the holonomy covers of all the leaves are uniformly quasi-isometricvia di ff eomorphisms.Proof. The proof is similar to the proof of Theorem 17.2, but using the holonomy cover L ′′ of a given leaf L instead of using the universal cover L ′ . As N ( L ′ , ε ) in the above proof,the neighborhood N ( L ′′ , ε ) of the zero section in the normal bundle p : N ( L ′′ ) → L ′′ ,carrying a lamination Y , satisfies the following property: given 0 < ε < ε , there exists δ > S is a leaf of Y meeting some fiber p − ( x ) of a point x ∈ L ′′ at a point atdistance < δ from x , then S meets every fiber p − ( y ) at a distance < ε from the base point y ∈ L ′′ , and it follows that p : S → L ′′ is a covering map, whose triviality has to be proved.This would finish the proof because the distortion of p : S → L ′′ is uniformly bounded forthe leaves S of Y , as in the proof of Theorem 17.2.Observe that Z ( x , r ) = Y ∩ p − ( x ) ∩ N ( L ′′ , δ ) is a transversal of Y through x for any r ≤ ε . Then the key property of the above statement can be stated as follows: if h is any holonomy map of Y defined on some neighborhood of x in Z ( x , δ ), and whoseimage is contained in Z ( x , ε ), then h can be extended to a holonomy transformation withdomain Z ( x , δ ) and image contained in Z ( x , ε ). Moreover, under the present hypothesis,this extension can be assumed to be unique. The only such holonomy transformation isthe identity on Z ( x , δ ) because L ′′ has trivial holonomy group in Y since it is the holonomycover of L . Therefore any such a leaf S meets every fiber of p at just one point; i.e. , p : S → L ′′ is a di ff eomorphism, as desired. (cid:3) Remark.
There are versions of Theorems 17.2 and 17.3 for the coarse quasi-isometry typeof the universal coverings or the holonomy covers of all leaves when the foliated space isnot of class C ; in particular, this generalizes Theorem 17.1. The coarse quasi-isometrytypes of these covers can be defined again via the generators of a representative of theholonomy pseudogroup induced by a finite defining cocycle: the orbits can be thoughtas graphs in an obvious way, and thus the corresponding covers can be constructed. Thecoarse quasi-isometry types of such covers can be proved to be invariant by equivalenceswhen the metrics are induced by recurrent systems of compact generation. Hence, versionsof Theorems 4.6 and 10.1 for covers of the orbits need to be proved first. These generaliza-tions are easy to make, but the required notation becomes complicated; thus they are leftto the reader.One of the fundamental results of Molino’s theory of Riemannian foliations is the fol-lowing. The closure of the leaves partition the manifold into the leaves of a larger singu-lar foliation [17]. In the general situation considered here, the following weaker resultsare available; they follow directly by applying Theorem 11.1, Corollary 11.2 and Corol-lary 12.4 to the holonomy pseudogroup. Theorem 17.4.
Let ( X , F ) be a strongly equicontinuous, compact foliated space. Then theclosure of each leaf is a minimal set. In particular, F is minimal if it is transitive. Corollary 17.5.
The leaf closures define a partition of any strongly equicontinuous, com-pact foliated space.
Theorem 17.6.
Let ( X , F ) be a strongly equicontinuous, compact and quasi-e ff ective foli-ated space. Then the closure of each leaf is a homogeneous space The next result shows another geometric aspect of the structure of a strongly equicontin-uous foliated space. It was shown by H. Winkelnkemper [31] for the holonomy groupoid orgraph, and by F. Alcalde Cuesta [1] for the homotopy groupoid of a Riemannian foliation.
Corollary 17.7.
Let ( X , F ) be a strongly equicontinuous, compact foliated space of classC with connected leaf space. Then the homotopy groupoid of ( X , F ) is, canonically, afiber bundle whose structural group can be reduced to the group of di ff erentiable quasi-isometries of the typical fiber, this being the universal cover of a leaf. If F is quasi-e ff ective,then the same is true for the holonomy groupoid, the fiber now being the holonomy coverof a leaf.Proof. The homotopy groupoid G of ( X , F ) consists of equivalence classes of paths onleaves, two paths α and β being equivalent if they have the same endpoints and the closedloop αβ − is homotopically trivial in the leaf which contains it. If [ α ] is a point of G , thenlet s [ α ] = α (0) denote the source map s : G → X . The fiber of s over a point x ∈ X is G x , and is canonically identified with the universal cover of the leaf L x . The foliatedspace being of class C means that its leaves can be endowed with a continuous metrictensor. Let U be a flow box for X with leaf space Z and such that its plaques are convexsubsets of the leaves with respect to the chosen metric tensor. Let x ∈ U , and let P be theplaque containing x , so that U is of the form P × Z . Then, if G U denotes the restrictionof G to U , G U = s − U , there is a map G U → Z × G P which sends a point [ α ] ∈ U to( q ( α (0)) , [ p ø α (0)]), where q : U → Z is the projection into the space of leaves of U . Sincethe plaque P is convex, there is a unique geodesic path in P joining x to any given point of P , so there is a well defined map G P → P × G x obtaining by precomposing a path startingat some y ∈ P with the unique geodesic in P from x to y . Let φ : G U → U × G x denotethe composition of this two maps. This map is a homeomorphism, and from the previouswork (Theorem 17.2), it follows that this map is a quasi-isometry on the fibers, that is, itsends the fiber s − ( y ) = G y quasi-isometrically onto G x . The quasi-isometry distortion isbounded, and there is a commutative diagram G U φ −−−−−−→ U × G x y y U U from which the result readily follows.If the foliated space is quasi-e ff ective, then the same property for the holonomy groupoidis proved similarly by using Theorem 17.3. (cid:3) To conclude, it is quite reasonable to expect that the theory presented in this paper can beextended to include larger classes of foliated spaces which have some sort of transverse uni-formity, paralleling certain well-known structures of classical topological dynamics [28], e.g. , distal actions. R eferences [1] F. Alcalde Cuesta,
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The graph of a foliation , Ann. Global Anal. Geom. (1983), 51–75. ∗ D epartamento de X eometr ´ ıa e T opolox ´ ıa , F acultade de M atem ´ aticas , U niversidade de S antiago de C om - postela , C ampus U niversitario S ur , 15706 S antiago de C ompostela , S pain E-mail address : [email protected] † D epartment of M athematics , CSUN, N orthridge , CA 91330, U.S.A. E-mail address ::