Topological description of Riemannian foliations with dense leaves
aa r X i v : . [ m a t h . G T ] N ov TOPOLOGICAL DESCRIPTION OF RIEMANNIAN FOLIATIONS WITHDENSE LEAVES
J. A. ´ALVAREZ L ´OPEZ* AND A. CANDEL † C ONTENTS
Introduction 11. Local groups and local actions 22. Equicontinuous pseudogroups 53. Riemannian pseudogroups 94. Equicontinuous pseudogroups and Hilbert’s 5th problem 105. A description of transitive, compactly generated, strongly equicontinuous andquasi-effective pseudogroups 146. Quasi-analyticity of pseudogroups 14References 15I
NTRODUCTION
Riemannian foliations occupy an important place in geometry. An excellent survey isA. Haefliger’s Bourbaki seminar [6], and the book of P. Molino [13] is the standard refer-ence for Riemannian foliations. In one of the appendices to this book, E. Ghys proposesthe problem of developing a theory of equicontinuous foliated spaces paralleling that ofRiemannian foliations; he uses the suggestive term “qualitative Riemannian foliations” forsuch foliated spaces.In our previous paper [1], we discussed the structure of equicontinuous foliated spacesand, more generally, of equicontinuous pseudogroups of local homeomorphisms of topo-logical spaces. This concept was difficult to develop because of the nature of pseudogroupsand the failure of having an infinitesimal characterization of local isometries, as one doeshave in the Riemannian case. These difficulties give rise to two versions of equicontinuity:a weaker version seems to be more natural, but a stronger version is more useful to gen-eralize topological properties of Riemannian foliations. Another relevant property for thispurpose is quasi-effectiveness, which is a generalization to pseudogroups of effectivenessfor group actions. In the case of locally connected foliated spaces, quasi-effectiveness isequivalent to the quasi-analyticity introduced by Haefliger [4]. For instance, the followingwell-known topological properties of Riemannian foliations were generalized to stronglyequicontinuous quasi-effective compact foliated spaces [1]; let us remark that we also as-sume that all foliated spaces are locally compact and Polish:
Date : September 25, 2018.*Research of the first author supported by DGICYT Grant PB95-0850. † Research of the second author partially supported by NSF Grant DMS-0049077. • Leaves without holonomy are quasi-isometric to one another (our original motiva-tion). • Leaf closures define a partition of the space. So the foliated space is transitive(there is a dense leaf) if and only if it is minimal (all leaves are dense). • The holonomy pseudogroup has a closure defined by using the compact-opentopology on small enough open subsets.In this paper, we show, in fact, that there are few ways of constructing nice equicontinu-ous foliated spaces beyond Riemannian foliations. The definition of Riemannian foliationused here is slightly more general than usual: a foliation is called Riemannian when itsholonomy pseudogroup is given by local isometries of some Riemannian manifold (a Rie-mannian pseudogroup); thus leafwise smoothness is not required. Our main result is thefollowing purely topological characterization of Riemannian foliations with dense leaveson compact manifolds.
Theorem.
Let ( X, F ) be a transitive compact foliated space. Then F is a Riemannianfoliation if and only if X is locally connected and finite dimensional, F is strongly equicon-tinuous and quasi-analytic, and the closure of its holonomy pseudogroup is quasi-analytic. This theorem is a direct consequence of the corresponding result for pseudogroups,whose proof uses the material developed in [1] as well as the local version of the solutionof Hilbert 5th problem due to R. Jacoby [9].An earlier result in this direction was that of M. Kellum [10, 11] who proved this prop-erty for certain pseudogroups of uniformly Lipschitz diffeomorphisms of Riemannian man-ifolds. Also, R. Sacksteder work [17] can be interpreted as giving a characterization ofRiemannian pseudogroups of one-dimensional manifolds. Another similar result, provedby C. Tarquini [19], states that equicontinuous transversely conformal foliations are Rie-mannian; note that, in the case of dense leaves, this result of Tarquini follows easily fromour main theorem. 1. L
OCAL GROUPS AND LOCAL ACTIONS
The concept of local group and allied notions is developed in Jacoby [9]. Some of thesenotions are recalled in this section, for ease of reference.
Definition 1.1.
A local group is a quintuple ( G, e, · , ′ , D ) satisfying the following condi-tions:(1) ( G, D ) is a topological space;(2) · is a function from a subset of G × G to G ;(3) ′ is a function from a subset of G to G ;(4) there is a subset O of G such that(a) O is an open neighborhood of e in G ,(b) O × O is a subset of the domain of · .(c) O is a subset of the domain of ′ ,(d) for all a, b, c ∈ O , if a · b and b · c ∈ O , then ( a · b ) · c = ( a · b ) · c .(e) for all a ∈ O , a ′ ∈ O , a · e = e · a = a and a ′ · a = a · a ′ = e ,(f) the map · : O × O → G is continuous,(g) the map ′ : O → G is continuous;(5) the set { e } is closed in G .Jacoby employs the notation G for the quintuple ( G, e, · , ′ , D ) , but here it will be simplydenoted by G . IEMANNIAN FOLIATIONS 3
The collection of all sets O satisfying condition (4) will be denoted by Ψ G . This is aneighborhood base of e ∈ G ; all of these neighborhoods are symmetric with respect to theinverse operation (3) . Let Φ( G, n ) denote the collection of subsets A of G such that theproduct of any collection of ≤ n elements of A is defined, and the set A n of such productsis contained in some O ∈ Ψ G .If G is a local group, then H is a subgroup of G if H ∈ Φ( G, , e ∈ H , H ′ = H and H · H = H .If G is a local group, then H ⊂ G is a sub-local group of G in case H is itself a localgroup with respect to the induced operations and topology.If G is a local group, then Υ G denotes the set of all pairs ( H, U ) of subsets of G so that(1) e ∈ H ; (2) U ∈ Ψ G ; (3) for all a, b ∈ U ∩ H , a · b ∈ H ; (4) for all c ∈ U ∩ H , c ′ ∈ H .Jacoby [9, Theorem 26] proves that H ⊂ G is a sub-local group if and only if thereexists U such that ( H, U ) ∈ Υ G .Let G be a local group and let Π G denote the pairs ( H, U ) so that (1) e ∈ H ; (2) U ∈ Ψ G ∩ Φ( G, ; (3) for all a, b ∈ U ∩ H , a · b ∈ H ; (4) for all c ∈ U ∩ H , c ′ ∈ H ;(5) U r H is open. Given such a pair ( H, U ) ∈ Π G , there is a (completely regular,hausdorff) topological space G/ ( U, H ) and a continuous open surjection T : U → G/ ( U, H ) such that T ( a ) = T ( b ) if and only if a ′ · b ∈ H ( cf. [9, Theorem 29]).If ( H, V ) is another pair in Π G , then the spaces G/ ( H, U ) and G/ ( H, V ) are locallyhomeomorphic in an obvious way. Thus the concept of coset space of H is well defined inthis sense, as a germ of a topological space. The notation G/H will be used in this sense;and to say that
G/H has certain topological property will mean that some G/ ( H, U ) hassuch property.Let ∆ G be the set of pairs ( H, U ) such that ( H, U ) ∈ Π G and, for all a ∈ H ∩ U and b ∈ U , b ′ · ( a · b ) ∈ H . A subset H ⊂ G is called a normal sub-local group of G if thereexists U such that ( H, U ) ∈ ∆ G . If ( H, U ) ∈ ∆ G then the quotient space G/ ( H, U ) admits the structure of a local group (see [9, Theorem 35] for the pertinent details) and thenatural projection T : U → G/ ( H, U ) is a local homomorphism. As before, another suchpair ( H, V ) produces a locally isomorphic quotient local group.Let us recall the main results of Jacoby [9] on the structure of locally compact localgroups because they will be needed in the sequel. Theorem 1.2 (Jacoby [9, Theorem 96]) . Any locally compact local group without smallsubgroups is a local Lie group.
In the above result, a local group without small subgroups is a local group where someneighborhood of the identity element contains no nontrivial subgroup.
Theorem 1.3 (Jacoby [9, Theorems 97–103]) . Any locally compact second countable localgroup G can be approximated by local Lie groups. More precisely, given V ∈ Ψ G ∩ Φ( G, , there exists U ∈ Ψ G with U ⊂ V and there exists a sequence of compact normalsubgroups F n ⊂ U such that ( ) F n +1 ⊂ F n , ( ) T n F n = { e } , ( ) ( F n , U ) ∈ ∆ G , and ( ) G/ ( F n , U ) is a local lie group. Theorem 1.4 (Jacoby [9, Theorem 107]) . Any finite dimensional metrizable locally com-pact local group is locally isomorphic to the direct product of a Lie group and a compactzero-dimensional topological group.
An immediate consequence of Theorem 1.4 is that any locally euclidean local group isa local Lie group, which is the local version of Hilbert 5th problem obtained by Jacoby.
JES ´US A. ´ALVAREZ L ´OPEZ AND ALBERTO CANDEL
All local groups appearing in this paper will be assumed, or proved, to be locally com-pact and second countable.
Definition 1.5.
A local group G is a local transformation group on a subspace X ⊂ Y ifthere is given a continuous map G × X → Y , written ( g, x ) gx , such that • ex = x for all x ∈ X ; and • g ( g x ) = ( g g ) x , provided both sides are defined.This map G × X → Y is called a local action of G on X ⊂ Y .The typical example of local action is the following. Let H be a sub-local group of G .If ( H, U ) ∈ Π G and T : U → G/ ( H, U ) is the natural projection, then U is a sub-localgroup of G and the map ( u, T ( g )) T ( u · g ) defines a local action of U on the opensubspace T ( U ) of G/ ( H, U ) .If G is a local group acting on X ⊂ Y and the action is locally transitive at x ∈ X in the sense that there is a neighborhood V ∈ Ψ G such that V x includes a neighborhoodof x in X , then there is a sub-local group H of G and an open subset U ⊂ G such that ( H, U ) ∈ Π G and the orbit map g ∈ G gx ∈ X induces a local homeomorphism G/ ( H, U ) → X at x , which is equivariant with respect to the action of U . Theorem 1.6.
Let G be a locally compact, separable and metrizable local group. Supposethat there is a local action of G on a finite dimensional subspace X ⊂ Y and that theaction is locally transitive at some x ∈ X . Fix some ( H, U ) ∈ Π G so that the orbitmap g gx induces a local homeomorphism G/ ( H, U ) → X at x . Then there exists aconnected normal subgroup K of G such that K ⊂ H , ( K, U ) ∈ Π G and G/ ( K, U ) isfinite dimensional.Proof. This is a local version of [15, Theorem 6.2.2], whose proof shows the followingassertion that will be used now.
Claim 1.
Let A be a locally compact, separable and metrizable topological group, and let B be a closed subgroup of A such that A/B is of finite dimension and connected. Let N n be a sequence of compact normal subgroups so that T n N n = { e } and every A/N n is aLie group. Then there is some index n such that the connected component of the identityof N n is contained in B . The following observation is also needed.
Claim 2.
Let A be a local group, let ( B, V ) ∈ Π A , let T : A → A/ ( B, V ) denote thenatural projection, and let C be a compact subgroup of A contained in V ∩ V . Then B ∩ C is a compact subgroup of C , a map C/ ( B ∩ C ) → A/ ( B, V ) is well defined by theassignment a ( B ∩ C ) T ( a ) , and this map is an embedding. This assertion can be proved as follows. On the one hand, B ∩ C is compact because B is closed and C compact. On the other hand, B ∩ C is a subgroup of C because C isa subgroup, C ⊂ V , and a · b ∈ B and a ′ ∈ B for all a, b ∈ V since ( B, V ) ∈ Π A .The map C/ ( B ∩ C ) → A/ ( B, V ) is well defined and injective because C ⊂ V and T ( a ) = T ( b ) if and only if a · b ′ ∈ B for a, b ∈ V . This injection is continuous becauseit is induced by the inclusion C ֒ → V . Thus this map is an embedding since C/ ( B ∩ C ) is compact and A/ ( B, V ) is Hausdorff.Now, with the notation of the statement of this theorem, let F n be a sequence of compactnormal subgroups of G as provided by Jacoby’s theorem [9] (quoted as Theorem 1.3). Itmay be assumed that ( F n , U ) ∈ ∆ G and F n ⊂ U ∩ U for all n . If K n is the identity IEMANNIAN FOLIATIONS 5 component of each F n , then the natural quotient map G/ ( K n , U ) → G/ ( F n , U ) has zerodimensional fibers, because they are locally homeomorphic to the zero-dimensional group F n /K n . Because each G/ ( F n , U ) is a local Lie group, it is finite dimensional, and thus G/ ( K n , U ) is also finite dimensional (see [8, Ch. VII, § K ∩ H is a compact subgroup of K , and there is a canonical embedding K / ( K ∩ H ) → G/ ( H, U ) . Moreover K / ( K ∩ H ) is connected since so is K .Then the dimension of K / ( K ∩ H ) is less or equal than the dimension of G/ ( H, U ) by [8, Theorem III 1], and thus K / ( K ∩ H ) is of finite dimension. On the other hand,each canonical embedding K / ( K ∩ F n ) → G/ ( F n , U ) , given by Claim 2, realizes K / ( K ∩ F n ) as a compact subgroup of the local Lie group G/ ( F n , U ) because K ∩ F n is a normal subgroup of K . So every K / ( K ∩ F n ) is a Lie group. Then, by Claim 1 with A = K , B = K ∩ H and N n = K ∩ F n , there is some index n such that the identitycomponent K of F = K ∩ F n is contained in K ∩ H . This F is a normal subgroup of G , and thus K is a connected normal subgroup of G . Furthermore ( K, U ) , ( F, U ) ∈ ∆ G ,and dim G/ ( K, U ) = dim G/ ( F, U ) ≤ dim G/ ( K , U ) + dim K / ( K ∩ F n ) by [8, Theorem III 4]. So G/ ( K, U ) is of finite dimension as desired. (cid:3)
2. E
QUICONTINUOUS PSEUDOGROUPS 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 a pseudogroup H is generated by a set E ⊂ H if every element of H can be obtainedfrom E by using the above pseudogroup operations; the sets of generators will be assumedto be symmetric for simplicity ( h − ∈ E if h ∈ E ). The orbit of an element x ∈ Z is theset H ( x ) of elements h ( x ) , for all h ∈ H whose domain contains x . These orbits are theequivalence classes of an equivalence relation on Z .Pseudogroups of local transformations are natural generalizations of group actions ontopological spaces (each group action generates a pseudogroup). Another important ex-ample of a different nature is the holonomy pseudogroup of a foliated space defined by aregular covering by flow boxes [2, 4, 5, 7].The study of pseudogroups can be simplified by using certain equivalence relation in-troduced by Haefliger [4, 5]. For instance, any pseudogroup of local transformations isequivalent to its restriction to any open subset that meets all orbits; indeed, the whole ofthis equivalence relation is generated by this very basic type of examples. This conceptof pseudogroup equivalence is very important in the study of foliated spaces because theequivalence class of the holonomy pseudogroup depends only on each foliated space; it isindependent of the choice of a regular covering by flow boxes.For a pseudogroup H of local transformations of a locally compact space Z , Haefligerintroduced also the concept of compact generation: H is compactly generated if there is arelatively compact open set U in Z meeting each orbit of H , and such that the restriction G of H to U is generated by a finite symmetric collection E ⊂ G so that each g ∈ E isthe restriction of an element ¯ g of H defined on some neighborhood of the closure of thesource of g . This notion is invariant by equivalences and the relatively compact open set U meeting each orbit can be chosen arbitrarily. If E satisfies the above conditions, it is calleda system of compact generation of H on U . JES ´US A. ´ALVAREZ L ´OPEZ AND ALBERTO CANDEL
The concept of strong and weak equicontinuity was introduced in [1] for pseudogroupsof local transformations of spaces whose topology is induced by the following type ofstructure. Let { ( Z i , d i ) } i ∈ I be a family of metric spaces such that { Z i } i ∈ I is a covering ofa set Z , each intersection Z i ∩ Z j is open in ( Z i , d i ) and ( Z j , d j ) , and for all ε > there issome δ ( ε ) > so that the following property holds: for all i, j ∈ I and z ∈ Z i ∩ Z j , thereis some 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 d i ( x, y ) < δ ( ε ) = ⇒ d j ( x, y ) < ε for all ε > and all x, y ∈ U i,j,z . Such a family is 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 . Such a Q induces a topologyon Z so that, for each { ( Z i , d i ) } i ∈ I ∈ Q , the family of open balls of all metric spaces ( Z i , d i ) form a base of open sets. Any topological concept or property of ( Z, Q ) refersto this underlying topology. It was also observed in [1] that ( Z, Q ) is a locally compactPolish space if and only if it is hausdorff, paracompact, separable and locally compact.The strongest version of equicontinuity was defined in [1] as follows. Let H be a pseu-dogroup of local homeomorphisms of a quasi-local metric space ( 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 > , there issome δ ( ε ) > so that d i ( x, y ) < δ ( ε ) = ⇒ d j ( h ( x ) , h ( y )) < ε for all h ∈ S , i, j ∈ I and x, y ∈ Z i ∩ h − ( Z j ∩ im h ) .The condition on S to be closed under compositions is precisely what distinguishesstrong and weak equicontinuity [1, Lemma 8.3]. A typical choice of S is the set of allpossible composites of some symmetric set of generators. In fact, given any S satisfyingthe condition of strong equicontinuity, it is obviously possible to find a symmetric set ofgenerators E given by restrictions of elements of S , and then the set of all composites ofelements of E also satisfies the condition of strong equicontinuity.A pseudogroup H acting on a space Z will be called strongly equicontinuous when itis strongly equicontinuous with respect to some quasi-local metric inducing the topologyof Z . This notion is invariant by equivalences of pseudogroups acting on locally compactPolish spaces [1, Lemma 8.8].A key property of strong equicontinuity is the following. Proposition 2.1 ([1, Proposition 8.9]) . Let H be a compactly generated and stronglyequicontinuous pseudogroup acting on a locally compact Polish quasi-local metric space ( Z, Q ) , and let U be 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 strong equicontinuity. Let E be any system of compact generation of H on U , and let ¯ g be an extension of each g ∈ E with dom g ⊂ dom ¯ g . Also, let { Z ′ i } i ∈ I be any shrinking of { Z i } i ∈ I . Then there is a finitefamily V of open subsets of ( Z, Q ) whose union contains U and such that, for any V ∈ V , x ∈ U ∩ V , and h ∈ H with x ∈ dom h and h ( x ) ∈ U , the domain of ˜ h = ¯ g n ◦ · · · ◦ ¯ g contains V for any composite h = g n ◦ · · · ◦ g defined around x with g , . . . , g n ∈ E , andmoreover V ⊂ Z ′ i and ˜ h ( V ) ⊂ Z ′ i for some i , i ∈ I . IEMANNIAN FOLIATIONS 7
The following terminology was introduced in [1] to study strongly equicontinuous pseu-dogroups. A pseudogroup H of local transformations of a space Z is said to be quasi-effective 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. The family S can be assumed to be also closedunder restrictions to open sets, and thus every map in H is a combination of maps in S in this case. Moreover, if H is strongly equicontinuous and quasi-effective, then S canbe chosen to satisfy the conditions of both strong equicontinuity and quasi-effectiveness.The notion of quasi-effectiveness is invariant by equivalences of pseudogroups acting onlocally compact Polish spaces [1, Lemma 9.5]. Moreover this property is equivalent toquasi-analyticity for pseudogroups acting on locally connected and locally compact Polishspaces [1, Lemma 9.6]; recall that a pseudogroup H is called quasi-analytic if every h ∈ H is the identity around some x ∈ dom h whenever h is the identity on some open set whoseclosure contains x [4]. Proposition 2.2 ([1, Proposition 9.9]) . Let H be a compactly generated, strongly equicon-tinuous and quasi-effective pseudogroup of local homeomorphisms of a locally compactPolish space Z . Suppose that the conditions of strong equicontinuity and quasi-effectivenessare satisfied with a symmetric set S of generators of H that is closed under compositions.Let A, B be open subsets of Z such that A is compact and contained in B . If x and y areclose enough points in Z , then f ( x ) ∈ A = ⇒ f ( y ) ∈ B for all f ∈ S whose domain contains x and y . Recall that a pseudogroup is called transitive when it has a dense orbit, and is called minimal when all of its orbits are dense.
Theorem 2.3 ([1, Theorem 11.1]) . Let H be a compactly generated and strongly equicon-tinuous pseudogroup of local transformations of a locally compact Polish space Z . If H istransitive, then H is minimal. For spaces
Y, Z , let C ( Y, Z ) denote the family of continuous maps Y → Z , which willbe 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 2.4 ([1, Theorem 12.1]) . Let H be a quasi-effective, compactly generated andstrongly equicontinuous pseudogroup of local transformations of a locally compact Polishspace Z . Let S be a symmetric set of generators of H that is closed under compositionsand restrictions to open subsets, and satisfies the conditions of strong equicontinuity andquasi-effectiveness. Let e H be the set of maps h between open subsets of Z that satisfy thefollowing property: for every x ∈ dom h , there exists a neighborhood O x of x in dom h so that the restriction 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 . JES ´US A. ´ALVAREZ L ´OPEZ AND ALBERTO CANDEL
If a pseudogroup H satisfies the conditions of Theorem 2.4, then the pseudogroup H iscalled the closure of H .Note that a pseudogroup H of local transformations of a locally compact space Z isquasi-effective just when there is a symmetric set S of generators of H that is closedunder compositions and restrictions to open subsets, and such that the restriction map ρ VW : S ∩ C ( V, Z ) → S ∩ C ( W, Z ) is injective for all open subsets V, W of Z with W ⊂ V . Ifmoreover Z is a locally compact Polish space, and H is compactly generated and stronglyequicontinuous, then any such ρ VW is bijective for V, W small enough by Proposition 2.1.Moreover ρ VW is continuous with respect to the compact-open topology [14, p. 289], but itmay not be a homeomorphism as shown by the following example. Example 2.5.
Let Z be the union of two tangent spheres in R , and let h : Z → Z bethe combination of two rotations, one on each sphere, around the common axis and withrationally independent angles. Then h generates a compactly generated, strongly equicon-tinuous and quasi-effective pseudogroup H of local transformations of Z ; indeed, h is anisometry for the path metric space structure on Z induced from that of R . Nevertheless,it is easy to see that the closure H is not quasi-effective. Lemma 2.6.
Let H be a compactly generated, strongly equicontinuous and quasi-effectivepseudogroup of local transformations of a locally compact Polish space Z . Then H isquasi-effective if and only if there is a symmetric set S of generators of H that is closedunder compositions and restrictions to open subsets, and such that ρ VW : S ∩ C ( V, Z ) → S ∩ C ( W, Z ) is a homeomorphism with respect to the compact-open topologies for smallenough open subsets V, W of Z with W ⊂ V .Proof. The result follows directly by observing that, according to Theorem 2.4, H is quasi-effective just when there is some symmetric set S of generators of H that is closed undercompositions and satisfies the following condition: for any sequence h n in S and opennon-empty subsets V, W of Z , with W ⊂ V ⊂ dom h n for all n , if h n | W → id W in C c-o ( W, Z ) , then h n | V → id V in C c-o ( V, Z ) . (cid:3) Corollary 2.7.
Let H be a compactly generated, strongly equicontinuous and quasi-analytic pseudogroup of local transformations of a locally connected and locally compactPolish space Z . Then H is quasi-analytic if and only if there is a symmetric set S of gener-ators of H that is closed under compositions and restrictions to open subsets, and such that ρ VW : S ∩ C ( V, Z ) → S ∩ C ( W, Z ) is a homeomorphism with respect to the compact-opentopologies for small enough open subsets V, W of Z with W ⊂ V . Finally, let us recall from [1] certain isometrization theorem, which states that equicon-tinuous quasi-effective pseudogroups are indeed pseudogroups of local isometries in somesense. First, 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 locallyequal on Z i ∩ Z j whenever this is a non-empty set. Such a family will be called a coverof Z by locally equal metric spaces . Two such families are called locally equal when theirunion also is a cover of Z by locally equal metric spaces. This is an equivalence relationwhose equivalence classes are called local metrics on Z . For each local metric D on Z ,the pair ( Z, D ) is called a local metric space . Observe that every metric induces a uniquelocal metric in a canonical way. In turn, every local metric canonically determines a uniquequasi-local metric. Note also that local metrics induced by metrics can be considered as IEMANNIAN FOLIATIONS 9 germs of metrics around the diagonal. Moreover a local or quasi-local metric is inducedby some metric if and only if it is hausdorff and paracompact [1, Theorems 13.5 and 15.1].Now, a local homeomorphism h of a local metric space ( Z, D ) is called a local isometry 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 . This definition is independent of the choice of thefamily { ( Z i , d i ) } i ∈ I ∈ D . Then the isometrization theorem is the following. Theorem 2.8 ([1, Theorem 15.1]) . Let H be a compactly generated, quasi-effective andstrongly equicontinuous pseudogroup of local transformations of a locally compact Polishspace Z . Then H is a pseudogroup of local isometries with respect to some local metricinducing the topology of Z .
3. R
IEMANNIAN PSEUDOGROUPS
Definition 3.1.
A pseudogroup H of local transformations of a space Z is called a Rie-mannian pseudogroup if Z is a Hausdorff paracompact C ∞ -manifold and all maps in H are local isometries with respect to some Riemannian metric on Z . Example 3.2.
Let G be a local Lie group, G ⊂ G a compact subgroup. Then the canon-ical local action of some neighborhood of the identity in G on some neighborhood of theidentity class in G/G generates a transitive Riemannian pseudogroup. In fact, since G is compact, there is a G -left invariant and G -right invariant Riemannian metric on someneighborhood of the identity in G , which induces a G -invariant Riemannian metric onsome neighborhood of the identity class in G/G . With more generality, if Γ ⊂ G a densesub-local group, then the canonical local action of some neighborhood of the identity in Γ on some neighborhood of the identity class in G/G generates a transitive Riemannianpseudogroup. Moreover this Riemannian pseudogroup is complete in the sense of [4]. It iswell known that any transitive complete Riemannian pseudogroup is equivalent to a pseu-dogroup of this type, which follows from the pseudogroup version of Molino descriptionof Riemannian foliations.The pseudogroup version of the main result of this paper is the following topologicalcharacterization of transitive compactly generated Riemannian pseudogroups. Theorem 3.3.
Let H be a transitive, compactly generated pseudogroup of local transfor-mations of a locally compact Polish space Z . Then H is a Riemannian pseudogroup if andonly if Z is locally connected and finite dimensional, H is strongly equicontinuous andquasi-analytic, and H is quasi-analytic. Remark.
The closure H of H exists by virtue of Theorem 2.4, because the space Z islocally connected, hence the pseudogroup H is quasi-effective because it is quasi-analytic[1, Lemma 9.6].The following is a direct consequence of the above theorem. Corollary 3.4.
Let H be a compactly generated, strongly equicontinuous and quasi-analytic pseudogroup of local transformations of a locally compact Polish space Z . Thenthe H -orbit closures are C ∞ manifolds if and only if they are locally connected and finitedimensional, and the induced pseudogroup H is quasi-analytic on them.Proof. This follows from Theorem 3.3 because the closure of H acting on the closure ofan orbit is equivalent to a pseudogroup like Example 3.2. (cid:3) The proof of Theorem 3.3 will be given in the next section; in the interim, some exam-ples illustrating the necessity of several hypotheses are described.
Example 3.5.
Let Z be the product of countably infinitely many circles. This is a compact,locally connected Polish group which acts on itself by translations in an equicontinuousway. Let Z → Z be an injective homomorphism with dense image. Then the action of Z on Z induced by this homomorphism is minimal and equicontinuous, and so it gener-ates a minimal, quasi-analytic and equicontinuous pseudogroup, which is not Riemannianbecause Z is of infinite dimension. Example 3.6.
Let Z be the set of p -adic numbers x ∈ Q p with p -adic norm | x | p ≤ . Thenthe operation x x + 1 defines an action of Z on Z which is minimal and equicontinuous(it preserves the p -adic metric on Z ). Thus it generates a minimal, quasi-analytic andequicontinuous pseudogroup, which is not Riemannian because Z is zero-dimensional. Example 3.7.
A related example is as follows. Let Z be the standard cantor set in [0 , ⊂ R together with all integer translates. Then there is a pseudogroup H acting on Z which isgenerated by translations of the line which locally preserve Z . In fact, H is a pseudogroupof local isometries for two geometrically distinct metrics, the euclidean and the dyadic. Example 3.8.
The previous example can be generalized, replacing Z by the universalMenger curve [3, Ch. 15]. This space Z (to be precise, a modification of it) is constructedas an invariant set of the pseudogroup of local homeomorphisms of R generated by themap f ( x ) = 3 x and the three unit translations parallel to the coordinate axes. There is apseudogroup acting on Z generated by euclidean isometries which locally preserve Z . It isfairly easy to see that such a pseudogroup is minimal, quasi-analytic and equicontinuous.Moreover Z is locally connected and of dimension one. However, this pseudogroup is notcompactly generated.4. E QUICONTINUOUS PSEUDOGROUPS AND H ILBERT ’ S TH PROBLEM
This section is devoted to the proof of Theorem 3.3. The “only if” part is obvious,so it is enough to show the “if” part, which has essentially two steps. In the first one,a local group action on Z is obtained as the closure of the set of elements of H whichare sufficiently close to the identity map on an appropriate subset of Z . This constructionfollows Kellum [10]. The second step invokes the theory behind the solution to the localversion of Hilbert’s 5th problem in order to show that the local group is a local Lie group,and thus this local action is isometric for some Riemannian metric if its isotropy subgroupsare compact. So H is proved to be Riemannian by showing that it is of the type describedin Example 3.2.By Theorem 2.8, there is a local metric structure D on Z with respect to which theelements of H are local isometries. Take any { ( Z i , d i ) } i ∈ I ∈ D satisfying the conditionof strong equicontinuity. Let U be a relatively compact non-trivial open subset of Z , and V a family of open subsets which cover U as in Proposition 2.1. Let V be an elementof V which meets U , which is assumed to be contained in Z i for some i ∈ I , and let D ⊂ V be an open connected subset with compact closure also contained in V . Accordingto Proposition 2.1, if h ∈ H is such that dom h ⊂ D and h ( D ) ∩ U = ∅ , then there existsan element ˜ h ∈ H which extends h and whose domain contains V . Moreover, as H isquasi-analytic and D connected, such extension ˜ h is unique on D . In particular, such h admits a unique extension to a homeomorphism of D onto its image.Under the current hypothesis, the completion H of H is a quasi-analytic pseudogroupof transformations of Z whose action on Z has a single orbit. Let H D be the collection of IEMANNIAN FOLIATIONS 11 homeomorphisms h | D with h ∈ H an element whose domain contains D . Let D ′ ⊂ D bea connected, compact set with non-empty interior, and let H DD ′ = (cid:8) h ∈ H D | h ( D ′ ) ∩ D ′ = ∅ (cid:9) . By the strong equicontinuity of H and Proposition 2.2, the set D ′ can be chosen so that allthe translates h ( D ′ ) , h ∈ H DD ′ , are contained in a fixed compact subset K of D . Oncethis choice of D ′ is made, let G = H DD ′ be the resulting space.The space G is endowed with the compact open topology as a subset of C ( D, Z ) . Everyelement of G is actually defined on V , hence on D , and so the compact open topology canbe described by the supremum metric given by d ( g , g ) = sup x ∈ D d i ( g ( x ) , g ( x )) , where d i is the distance function on Z i ⊂ Z as above. Lemma 4.1.
Endowed this topology, G is a compact space.Proof. It has to be shown that any sequence g n of elements of G has a convergent subse-quence. By equicontinuity, g n may be assumed to be made of elements of H . By Propo-sition 2.1 and the definition of G , each g n can be extended to a homeomorphism whosedomain contains V . According to Theorem 2.4, the sequence g n converges uniformly on D to a map g ∈ e H . It needs to be shown that g : D → g ( D ) is a homeomorphism and thatit satisfies g ( D ′ ) ∩ D ′ = ∅ .To verify this last condition, note that, for each n , there exists x n ∈ D ′ such that g n ( x n ) ∈ D ′ , by the definition of G . Since D ′ is compact, it may be assumed that x n → x ∈ D ′ , yielding g n ( x n ) → g ( x ) ∈ D ′ since g n → g uniformly on D . Thus, g ( D ′ ) ∩ D ′ = ∅ . Each g n : D → g n ( D ) ⊂ V is a homeomorphism. Thus, by Proposition 2.1, thereare maps h n ∈ H defined on V such that h n ◦ g n = id on g n ( D ) . If g fails to be ahomeomorphism on D , then there are points x, y ∈ D with d i ( x, y ) > and g ( x ) = g ( y ) = z . The map g is a homeomorphism around each point of D , as Theorem 2.4 shows.Thus there are disjoint neighborhoods O x and O y of x and y , respectively, such that g maps each of them homeomorphically onto a neighborhood W of z . Since the sequences g n ( x ) , g n ( y ) both converge to z , they may be assumed to be contained in W . Furthermore,perhaps further shrinking W , the restrictions h n | W form an equicontinuous family, whichtherefore converges to a map h which inverts g on W . This situation contradicts the factthat h n ( g n ( x )) and h n ( g n ( y )) do not have the same limit. (cid:3) The following lemma is similar to the corresponding one in Kellum [10].
Lemma 4.2.
The space G , endowed with the compact-open topology and the operationsjust described, is a locally compact local group.Proof. Let g , g be two elements of G . Then the composition g ◦ g is defined on D ′ because g ( D ′ ) ⊂ D . Therefore there exists h ∈ H D which extends g ◦ g . By quasi-analyticity of H , this extension is unique and thus it defines a map ( g , g ) g · g from G × G into H . If g , g are sufficiently close to the identity of D in the compact opentopology of C ( D, Z ) , then also g · g ∈ G .The existence of a unique identity element e for G , as well as the existence of an inverseoperation on G , is proved analogously. Finally, by Corollary 2.7 and the quasi-analyticity of H , it is easy to see that the localgroup multiplication and inverse map are continuous with the compact open topology on G . (cid:3) Remarks.
The quasi-analyticity for H was used in this proof. Note that it would beneeded even to prove, in a similar way, that Γ is a local group. The final section of thepaper discusses the necessity of this condition in some more detail.By Theorem 2.8, we can assume that all elements of G are isometries with respect to d i . Then it easily follows that the above distance d on G is left invariant.The following lemma follows easily; cf. [10]. Lemma 4.3.
The map G × D ′ → D defined by ( g, x ) g ( x ) makes G into a local groupof transformations on D ′ ⊂ D . Let
Γ = H ∩ G , which is a finitely generated dense sub-local group of G . The followingis a direct consequence of the minimality of H . Lemma 4.4. H is equivalent to the pseudogroup on any non-empty open subset of D ′ generated by the local action of Γ on D ′ ⊂ Z . Let x be a point in the interior of D ′ , which will remain fixed from now on. Note that,by construction, all elements of G are defined at x . Let φ : G → D be the orbit mapgiven by φ ( g ) = g ( x ) . This map is continuous because the action is continuous. Lemma 4.5.
The image of the orbit map φ contains a neighborhood of x .Proof. H is minimal by Theorem 2.3, and thus the space Z is locally homogeneous withrespect to the pseudogroup H by Theorem 2.4. More precisely, Proposition 2.1 and The-orem 2.4 show that, given x ∈ D ′ , there exists h ∈ H with domain dom h = D suchthat h ( x ) = x . Since both x, x ∈ D ′ , it follows that h ∈ G . The statement followsimmediately from this. (cid:3) Let G denote the collection of elements g ∈ G such that g ( x ) = x . Lemma 4.6.
The set G is a compact subgroup of G .Proof. First, G is compact because, being the stabilizer of a point, it is a closed subset of G and G is a compact hausdorff space.Second, it follows from the definitions of G and of its group multiplication that theproduct of two elements of G is defined and belongs to G , and likewise the inverse ofevery element. More precisely, if g , g ∈ G , then g ◦ g is an element of H which fixes x , hence g ◦ g ( D ′ ) ∩ D ′ = ∅ . (cid:3) In the special case of the group G which stabilizes x considered here, the equivalencerelation ∼ on G used to define a representative coset space of G can also be defined as h ∼ g if and only if h ( x ) = g ( x ) . Lemma 4.7.
The orbit map φ : G → Z induces a map ψ : G/G → Z which is ahomeomorphism of a neighborhood of the identity class in G/G onto a neighborhood of x in Z .Proof. This follows directly from the preceding discussion on coset spaces and the finitedimensionality of Z . (cid:3) IEMANNIAN FOLIATIONS 13
Corollary 4.8. H is equivalent to the pseudogroup induced by the canonical local actionof some neighborhood of the identity in Γ on some neighborhood of the identity class in G/G .Proof. This follows from Lemmas 4.4 and 4.7. (cid:3)
Corollary 4.9.
G/G is finite dimensional.Proof. This follows directly from Lemma 4.7 and the finite dimensionality of Z . (cid:3) Lemma 4.10.
The group G contains no non-trivial normal sub-local group of G .Proof. If N ⊂ G is a normal sub-local group contained in G and n ∈ N , then for each g in a suitable neighborhood of e in G there is some n ′ ∈ N so that n φ ( g ) = n g ( x ) = g n ′ ( x ) = g ( x ) = φ ( g ) . Thus n acts trivially on a neighborhood of x in D ′ . This is possible only if n = e , because H is quasi-analytic. (cid:3) Lemma 4.11.
The local group G is finite dimensional.Proof. By Corollary 4.9,
G/G is finite dimensional. By Theorem 1.6, there exists acompact normal subgroup ( K, U ) in ∆ G such that K ⊂ G and G/ ( K, U ) is finite di-mensional. Lemma 4.10 implies that K is trivial; thus G is finite dimensional because it islocally isomorphic to G/K . (cid:3) Finally, since G is a compact subgroup of G , the following finishes the proof of Theo-rem 3.3 according to Example 3.2 and Corollary 4.8. Lemma 4.12.
The group G is a local Lie group.Proof. This is a local version of [16, Theorem 73]. By Theorem 1.2, it is enough to showthat G has no small subgroups. The local group G is finite dimensional and metrizable, soTheorem 1.4 implies that there is a neighborhood U of e in G which decomposes as thedirect product of a local Lie group L and a compact zero-dimensional normal subgroup N .Then P = N ∩ G is a normal subgroup of G , and G /P is a Lie group because it is agroup which is locally isomorphic to the local Lie group G/ ( N, U ) ( cf . [9, Theorem 36]).Furthermore, since N is zero-dimensional, so is P . Thus there exists a neighborhood V of e in G which is the direct product of a connected local lie group M and the normalsubgroup P . It may be assumed that V ⊂ U . Since M is connected and N is zero-dimensional, it follows that M ⊂ L . Summarizing, there is a local isomorphism between G and the direct product L × N , which restricts to a local isomorphism of G to M × P .Therefore, there exists a neighborhood of the class G in G/G which is homeomorphicto a neighborhood of the class of the identity in the product L/M × N/P . It followsthat a neighborhood of x in Z is homeomorphic to the product of an euclidean ball andan open subspace T ⊂ N/P . Since Z is by assumption locally connected and N/P zero-dimensional, it follows that T is finite, and hence that N/P is a discrete space. So P is an open subset of N and thus there exists a neighborhood W of e in G such that W ∩ P = W ∩ N . By the local approximation of Jacoby (Theorem 1.3), there exists acompact normal subgroup K ⊂ W such that G/K is a local Lie group. Then G contains P ∩ K , which is equal to the normal subgroup N ∩ K of G because K ⊂ W . Thus, byLemma 4.10, N ∩ K is trivial. On the other hand, N/ ( N ∩ K ) is a zero-dimensional liegroup, hence N ∩ K is open in N . It follows that N is finite, and thus that G is a local liegroup. (cid:3)
5. A
DESCRIPTION OF TRANSITIVE , COMPACTLY GENERATED , STRONGLYEQUICONTINUOUS AND QUASI - EFFECTIVE PSEUDOGROUPS
The following example is slightly more general than Example 3.2.
Example 5.1.
Let G be a locally compact, metrizable and separable local group, G ⊂ G a compact subgroup, and Γ ⊂ G a dense sub-local group. Suppose that there is a leftinvariant metric on G inducing its topology. This metric can be assumed to be also G -right invariant by the compactness of G . Then the canonical local action of Γ on someneighborhood of the identity class in G/G induces a transitive strongly equicontinuousand quasi-effective pseudogroup of local transformations of a locally compact Polish space.In fact, this is a pseudogroup of local isometries in the sense of [1].The proof of the following theorem is a straightforward adaptation of the first part ofthe proof of Theorem 3.3, by using quasi-effectiveness instead of quasi-analyticity. Theorem 5.2.
Let H be a transitive, compactly generated, strongly equicontinuous andquasi-effective pseudogroup of local transformations of a locally compact Polish space,and suppose that H is also quasi-effective. Then H is equivalent to a pseudogroup of thetype described in Example 5.1. The study of compact generation for the pseudogroups of Example 5.1 is very deli-cate [12]. But those pseudogroups are obviously complete, and it seems that compactgeneration could be replaced by completeness in Theorem 5.2, obtaining a better result.This would require the generalization of our work [1] to complete strongly equicontinuouspseudogroups. 6. Q
UASI - ANALYTICITY OF PSEUDOGROUPS
The most elusive of the hypotheses of Theorem 3.3 is that concerning the quasi-analyticityof the closure of a quasi-analytic pseudogroup. We do not have an example of a pseu-dogroup H as in the main theorem whose closure fails to be quasi-analytic. Thus thissection offers some examples and observations relevant to this problem.It is quite easy to see that the definition of length space has a local version as describedin [1, Section 13], and that the two theorems of Section 15 of such paper are also availablefor local length spaces.There are many examples of metric spaces which admit actions of pseudogroups ofisometries which are not quasi-analytic. The following two are examples of real trees (seeShalen [18]). Example 6.1.
Let X = R be endowed with the metric given by d (( x , y ) , ( x , y )) = ( | y | + | x − x | + | y | if x = x | y − y | if x = x Given a subset F of the real axis there is an isometry f of X given by f ( x, y ) = ( ( x, y ) if x ∈ F ( x, − y ) if x / ∈ F This family of isometries f forms a normal subgroup of the group of isometries of X .Thus this group is not quasi-analytic, although X is not locally compact. IEMANNIAN FOLIATIONS 15
Example 6.2.
Let X = R ∗ R be the free product of two copies of R . Then X has thestructure of a real tree, and is a homogeneous space with respect to its group of isometries.It is not quasi-analytic. Definition 6.3.
A local length space X is analytic at a point x ∈ X if the following holds:if γ, γ ′ are geodesic arcs (parametrized by arc-length) defined on an interval about ∈ R ,such that γ (0) = γ ′ (0) = x and that γ = γ ′ on some interval ( − a, , then they have thesame germ at . The space X is analytic if it is analytic at every point. Example 6.4.
A Riemannian manifold is an analytic length space. Real trees with manybranches, as in the above examples, are not analytic.In relation to Theorem 3.3, if a local length space is known to be analytic at one pointand admits a transitive action of a pseudogroup of local isometries, then it is analytic.
Proposition 6.5.
Let H be a pseudogroup of local isometries of an analytic local lengthspace X . Then H is quasi-analytic.Proof. If H is not quasi-analytic, then, by definition, there exists an element f of H , anopen subset U in dom( f ) such that f | U = id , and a point x in the closure of U such that f is not the identity in any neighborhood of x . Therefore, there is a sequence of points x n converging to x such that f ( x n ) = x n . If n is sufficiently large, then there is a geodesicarc contained in the domain of f and joining x n and a point y ∈ U . This geodesic arc ismapped by f to a distinct geodesic arc having the same germ at one of its endpoints. (cid:3) The following example shows that the converse is false.
Example 6.6.
Let X be the euclidean space R endowed with the metric induced by thesupremum norm k ( x, y ) } = max {| x | , | y |} . Then X is a length space which is not locallyanalytic. Indeed, if f : I → R is a function such that | f ( s ) − f ( t ) | ≤ | s − t | for all s, t ∈ I ,then t ∈ I ( t, f ( t )) ∈ X is a geodesic. However, every local isometry is locally equalto a linear isometry, hence the pseudogroup of local isometries is quasi-analytic.R EFERENCES[1] J. A. ´Alvarez L´opez and A. Candel,
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