Minimality of the action on the universal circle of uniform foliations

Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.


Introduction
Consider a Reebless foliation F on a closed 3-manifold M without spherical or projective plane leaves. This implies that the universal cover Ă M of M is homeomorphic 1 to R 3 [Pal] and that every leaf of F is a properly embedded plane [Nov]. We denote by r F to the lift of F to Ă M . A specific class of foliations are those called uniform which means that in the universal cover, any two leaves are at finite Hausdorff distance from each other. See section 2.2 for the several variations of the definition of uniform foliations. Fibrations over the circle are one obvious example. Much more generally, slitherings, introduced by Thurston in [Th] (see also [Ca 4  This result has no restriction on the intrinsic metric in the leaves. Theorem 1.1 implies that all Reebless uniform foliations are obtained from either slithering foliations or blow ups of slithering foliations as explained in [Ca 4 , Construction 9.14 and Theorem 9.15] using a result of Thurston [Th,Theorem 2.7]. The proof implicitly uses the fact that the foliation is R-covered. We provide a proof of the R-covered property here. The requirement of Reebless in Theorem 1.1 is not superfluous: any foliation in the 3-sphere S 3 (or any closed M 3 with finite fundamental group) is uniform, however none are R-covered, because they have Reeb components. To prove Theorem 1.1 it is enough to show that the leaf space is 1 We note that we will always work with Ă M as a Riemannian manifold where the metric is induced by lifting the metric of M to the universal cover. As such, the manifold Ă M may be very different from R 3 even if homeomorphic.
Hausdorff (see e.g. [Ca 4 ,CC], a quick account of the most relevant material is presented in §2.1).
Some uniform foliations are quite special, for example linear foliations in T 3 or in nilmanifolds. These foliations have leaves that are parabolic. But for most uniform foliations, one can apply a beautiful result of Candel [Can] to see that there is a metric on M making each leaf negatively curved (see e.g. [FP 1 , §5.1] for a specific statement).
For our next result we will consider the following setting: F will be a uniform foliation on a closed Riemannian 3-manifold M such that the metric restricted to each leaf of F is Gromov hyperbolic (in particular, it has to be Reebless since the torus does not admit a Gromov hyperbolic metric). We will call such foliations uniform hyperbolic foliations.
For such foliations, one can consider, for each leaf L P r F the circle at infinity S 1 pLq defined as the set of geodesic rays up to being a finite Hausdorff distance apart (see §2.4). The fact that the foliation is R-covered is very useful to define a universal circle S 1 univ which is essentially a canonical way to identify all the S 1 pLq as one varies L P r F. The precise definition will be given in §2.5. See also [Th,Ca 4 , Ca 1 , Fen 3 , CD, Fra] among other places where universal circles are defined in even more general situations.
Our main result is the following: Theorem 1.2. Let F be a uniform hyperbolic foliation on a 3-manifold M . Then, the fundamental group π 1 pM q acts minimally on the universal circle S 1 univ . Moreover, the diagonal action on pairs of different points of S 1 univ has dense orbits.
This result extends a very well known result about actions of hyperbolic groups on their Gromov boundary (see [Gr,§8.2]) and complements well with [Ca 1 , Lemma 5.2.2] which is stated for non-uniform R-covered foliations. Note that in the case where the foliation is a fibration this follows from the corresponding result for fundamental group actions of surfaces in their boundary. For Anosov foliations the result is also easily proved using the following: the flow is R-covered and since the foliation is uniform, the flow is skewed [Fen 2 ]. The structure of skewed Anosov flows is very rich and well understood [Fen 2 , Th]: the second statement of theorem 1.2 follows from the existence of a dense orbit of the flow. The first statement follows from the minimality of the Anosov foliation [Fen 2 ] and the structure of the flow. Theorem 1.2 was motivated by some applications to partially hyperbolic dynamics (it will be used in [FP 2 ]). We hope this result may have independent interest or find other applications.
Some proofs of intermediate steps are simpler if one restricts to the case of atoroidal 3-manifolds where one has transverse pseudo-Anosov flows that helps understanding the action on the universal circle ([Th, Ca 1 , Fen 3 ]).
When the manifold has a non-trivial JSJ decomposition, the proof includes a careful study of the intersection between leaves of the foliation and the pieces of the JSJ decomposition. This results in a new way to look at the universal circle that may be of independent interest and holds for general (both uniform and non uniform) R-covered foliations. See Proposition 4.9.
Because of our applications, at the end of the paper we explain how the results hold also for branching foliations, which are a technical object featuring often in partially hyperbolic dynamics.

Preliminaries
2.1. Reebless foliations. We will be mainly concerned with Reebless foliations in this article. See [Ca 4 , §4] for a broad introduction.
A Reeb component is a foliation of the solid torus, such that the boundary is a leaf. In addition all the leaves in the interior are planes and spiral or limit towards the boundary. There is a circle worth of leaves in the interior. By an abuse of terminology we also consider Reeb component a quotient of this, which may be a foliation of a solid Klein bottle. If a foliation by surfaces F in a closed 3-manifold M does not have Reeb components it follows from a celebrated result of Novikov [Nov] that when lifted to the universal cover, the foliation is made of simply connected leaves and the leaf space F is a simply connected (possibly non-Hausdorff) one-dimensional manifold. If there is a leaf of r F which is a sphere or a projective plane, it follows that the foliation r F is equivalent to the trivial foliation by spheres in S 2ˆR . If there are no projective space or spherical leaves of F then a result of Palmeira [Pal] implies that Ă M is homeomorphic to R 3 . We refer the reader to [CC,Ca 4 ] for a broad treatment, we will assume some familiarity with the theory of foliations.
We will not be too precise about regularity of our foliations. Everything works for foliations of class C 1,0`a s defined in [CC] (i.e. continuous with C 1 leaves tangent to a continuous distribution). Thanks to [Ca 2 ] in view of the nature of our result, this is a quite general assumption.
To show that a foliation is R-covered, it is enough to show that its leaf space is Hausdorff (see e.g. [Fen 3 , Lemma 2.2]).
A taut foliation is a foliation such that every leaf intersects a closed transversal. Notice that taut foliations must be Reebless 2 Another relevant result about foliations in 3-manifolds is the following (see [Gab] or [CC,Theorem II.9.5.5]): Theorem 2.1 (Roussarie-Gabai). Let F be a taut foliation in a 3-manifold M and let T Ă M be an embedded incompressible torus or Klein bottle. Then, T can be isotoped to be either a leaf of F or in general position with respect to F. In particular in the second case the induced foliation by F in T does not have singularities. If F is taut one can isotope T to be either a leaf of F or transverse to F.

2.2.
Uniform foliations. In this paper we will mainly concentrate in the following class of foliations.
Definition 2.2. Let F be a foliation in a manifold M . We say that F is uniform, if for any two leaves L, F of the lifted foliation r F to Ă M , then the Hausdorff distance between L and F is finite.
There have been several forms of the definition of uniform foliations, which we review here. Our definition is the weakest or most general possible. In his seminal article [Th,Definition 2.1], Thurston originally defined uniform foliation as a codimension one foliation in any dimension satisfying Definition 2.2 and such that in addition any closed transversal is not null homotopic. Calegari [Ca 1 , Definition 2.1.5] or [Ca 4 , Definition 9.13], defined uniform for codimension one foliations in 3-manifolds M satisfying Definition 2.2 and so that the foliation is also taut. The first author [Fen 3 , Definition 2.4] defined uniform for codimension foliations in 3-manifolds satisfying Definition 2.2.
Note that Definition 2.2 does not require M to be 3-dimensional or F codimension one, but we will restrict to this case in this paper.
Thurston [Th] remarks on the connection of the uniform property with the Reebless condition for codimension one foliations in 3-manifolds. After [Th,Definition 2.1] it is stated that if a foliation verifies that every closed transversal is not-nullhomotopic then there are no Reeb components. This is true if one additionally assumes that the foliation is uniform, and we prove this in § 3.1.
can be invariant under γ. Otherwise one gets a π 1 -injective annulus in M i with boundary in boundary of M i which is not homotopic rel boundary to boundary of M i . Using this annulus and annuli in boundary components of M i one can piece together a π 1 -injective torus or Klein bottle in M i which is not homotopic to the boundary, contradicting that M i is atoroidal. Now, if M i is a Seifert piece, then we claim that if Ă M j i is fixed by γ then by a similar argument we see that if more than one wall is fixed, then γ must belong to the center of π 1 pM i q (i.e. the element generated by the fibers of the Seifert fibering) in which case, γ cannot belong to the center of the Seifert pieces that are adjacent to Ă M j i . This shows that any connected component of the fixed point set of γ has diameter at most two. But since T is a tree and γ acts by isometries, the fixed point set is connected. This concludes.
2.4. Boundaries at infinity. Let X be a negatively curved complete space with curvature bounded from below and above. See [Gr, Led] for general references.
For such a space we define a boundary at infinity B 8 X defined as the equivalence relation of geodesic rays up to being at a bounded distance (see [Led, §I]). When X is a surface, the negative curvature implies that if X is simply connected then it is homeomorphic to D 2 and one can identify the boundary B 8 X with the circle of directions T 1 x X at any point x P X. So, for simply connected surfaces of negative curvature, we denote the boundary at infinity as S 1 pXq " B 8 X.
The metric in S 1 pXq is only well defined up to Hölder equivalence since it is intended to be an invariant under quasi-isometries. For our purposes, it will be convenient to choose a special metric on S 1 pXq called the visual metric. For this, we fix a point x 0 P X and we measure the length of an interval I Ă S 1 pXq by looking at the angle formed by the interval in T 1 x 0 X of vectors whose geodesic ray starting at x 0 lands in a point of I. The visual measure is the Lebesgue measure induced by this metric. This is clearly dependent on the point, but we will always explicit the point we are considering.
2.5. Universal circles. In this section we will review the construction of the universal circle for an R-covered foliation F on a closed 3-manifold M so that it admits a metric which restricts in each leaf to a negatively curved surface with curvature 5 close to´1. Denote by r F the lift of F to Ă M , the universal cover of M . For each L P r F we define S 1 pLq to be the boundary at infinity of L, which is well defined thanks to the fact that L is negatively curved. First there is the cylinder at infinity A which is the union of the S 1 pLq for L leaf in r F. The topology in A is given by: given x in Ă M , let τ be a small transversal to r F through x. For every point y of τ , y is in L P r F . For every v in the unit tangent bundle of L at y, let γ v be the geodesic ray starting at y with direction v. The ideal point z v of γ v is a point in S 1 pLq. It is well known that since L has negative curvature, the map v Ñ z v is a homeomorphism. In the same way one defines a map which is the map v Ñ z v for any y in τ . Put a topology in B τ so that this map is a homeomorphism. Do this for a π 1 pM q invariant collection of transversals with union intersecting every L P r F. In [Fen 3 ] it is proved that the topology in the intersection of subsets of A is well defined. This makes A into an open annulus, and π 1 pM q acts by homeomorphisms on this. In addition there is a topology on Ă M Y A making it homeomorphic to D 2ˆR and so that each L Y S 1 pLq corresponds to D 2ˆt tu for some t. Again π 1 pM q acts by homeomorphisms on this topology. We now describe the universal circle of F.
2.5.1. Case of F uniform. We denote, for L, F P r F a map τ L,F : LYS 1 pLq Ñ F Y S 1 pF q which has the following properties: ‚ τ L,F | L is a quasi-isometry with constant c ą 1 depending only on the Hausdorff distance between L and F , ‚ τ L,F | S 1 pLq is a homeomorphism, ‚ τ F,G˝τL,F | S 1 pLq " τ L,G | S 1 pLq . See [Th,§5] or [Ca 1 , Corollary 5.3.16] or [Fen 3 , Proposition 3.4]. Roughly the construction of such a collection of maps τ L,F is as follows: Recall that a quasi-isometry of constant c ą 1 is a map φ : L Ñ F so that c´1d L px, yq´c ă d F pφpxq, φpyqq ă cd L px, yq`c. Given L, F , the Hausdorff distance between them is a 0 ą 0. Given any x in F there is y in L with dpx, yq ă a 0`1 . Let 5 For foliations, [Can] provides a metric of curvature exactly´1, but since we want to apply this result in a slightly more general case (that is of branching foliations), we only use that the curvature is uniformly close to´1. Notice also that the metric constructed by [Can] may be only C 0 transversally to leaves, and since we are concerned only with quasi-isometric properties of leaves, it is more than fine to have just negative curvature or CAT(-1) leaves. See [BFP,§A.3] for a more complete account. τ L,F pxq " y. This map is well defined up to an error a 1 , with a 1 depending only on a 0 , see [Fen 3 ,§3].
The map τ L,F is a quasi-isometry, so it extends to L Y S 1 pLq, and it is a homemorphism into its image restricted to S 1 pLq.
Recall that a quasigeodesic is a quasi-isometry from Z or R into Ă M . Since the map τ L,F | L is a quasi-isometry it takes quasigeodesics in F to quasigeodesics in L. It is easy to see that x P S 1 pLq " y P S 1 pF q if and only if a quasigeodesic α in L with ideal point x is a finite Hausdorff distance from a quasigeodesic in F with ideal point y.
The universal circle of F is then defined as the circle S 1 univ which is A{ " where x P S 1 pLq " y P S 1 pF q if y " τ L,F pxq. Notice that it is easy to see that S 1 univ can be identified with S 1 pLq for every L P r F, so one can think of S 1 univ as a cannonical way to identify all boundaries at infinity of leaves. The fundamental group π 1 pM q acts on S 1 univ by homeomorphisms. This is because any γ in π 1 pM q sends pairs of quasigeodesics in leaves which are a finite Hausdorff distance apart to like pairs in γpLq, γpF q.
Remark 2.5. Let γ in π 1 pM q and L a leaf of r F. The action of γ P π 1 pM q on S 1 univ can be represented by an action on S 1 pLq identifying S 1 pLq -S 1 univ and so the action is obtained by the deck transformation composed with τ γL,L . We denote the action of γ on S 1 pLq obtained as τ γL,L˝γ by ρpγq. Hence one can assume that F is minimal. There is also a canonical collapsing between the cylinders at infinity. So assume that F is minimal. In [Fen 3 , §3] it is proved that for any L, F in r F there is a dense set of directions between them which is a contracting direction between them. This means the following: Fix x in L. There is a dense set of points B in S 1 pLq so that for any y in B if γ is a geodesic ray in L starting in L and with ideal point y, then γ is asymptotic to F (and hence to any leaf in between L, F ). Asymptotic means that distance between γ and F goes to 0 as points escape in γ. For any E between L, F there is a geodesic ray in E asymptotic to γ. This defines an ideal point in S 1 pEq. The union of these ideal points over such E is a continuous curve in A. The union of these for all y in B is a dense set in the subset D of A between S 1 pLq and S 1 pLq. This extends uniquely to a foliation in D by intervals, each interval intersects a circle at infinity once and only once. One iterates this procedure making L, F escape compact sets of the leaf space in opposite directions. This defines a foliation in A by vertical lines, each intersecting a circle at infinity once and only once.
The universal circle of F is the quotient A{ " where is the equivalence relation of being in the same leaf of the vertical foliation. The group of deck transformations π 1 pM q acts by homeomorphisms preserving the vertical foliation in A. This is because it sends the contracting directions as above to contracting directions. Hence π 1 pM q acts by homeomorphisms on the universal circle S 1 univ . Both the vertical foliation and the universal circle pull back to the original foliation before collapsing complementary regions of the minimal set L.
The existence of a the universal circle is much more general. It exists for every foliation with Gromov hyperbolic leaves [CD]. In addition in [CD] a universal circle is constructed for every tight essential lamination. See [Ca 4 ] for more on this theory.
3. Uniform foliations: proof of Theorem 1.1 In this section we prove Theorem 1.1. We first discuss in §3.1 the Reebless assumption. This subsection is independent of the proof of Theorem 1.1 and can be safely skipped. The results in §3.2 hold in more generality than the case of uniform foliations and could be of independent interest.
As explained in §2.1, Theorem 1.1 is immediate if the foliation has spherical or projective plane leaves by the Reeb stability theorem which implies in that case that up to finite cover the foliation is the trivial foliation by spheres in S 2ˆS1 . So in this section we will assume throughout that leaves of F are not spheres or projective planes.
3.1. Some remarks on the Reebless assumption. It can certainly be the case that a foliation with Reeb components is uniform yet not R-covered. Indeed, if M has finite fundamental group, any foliation in M has Reeb components by Novikov's theorem [Nov] while the universal cover is compact, so the foliation is uniform. Notice that a Reeb component has non-Hausdorff leaf space: every neighborhood of the boundary leaf contains all the leaves of the interior of the solid torus as these all accumulate the boundary. Foliations of closed 3-manifolds with finite fundamental group are all examples of uniform non-R-covered foliations: We don't know how to prove this in all generality, however we can prove the following intermediate fact.
Lemma 3.1. If F a foliation in M has a Reeb component and it is uniform then every leaf in the universal cover has compact closure. In particular, any non-torsion element γ P π 1 pM q acts freely on the leaf space Let us first assume that the fundamental group of the boundary tori of the Reeb component does not map to 0 in the fundamental group π 1 pM q of M . This implies that the Reeb torus lifts to a Reeb cylinder where leaves accumulate on one end of the cylinder. Let γ represent the deck transformation associated with the core of the Reeb component. Assume that the basepoint p is in a liftγ. Then in Ă M , dpγ n p, pq Ñ 8. One can see this in the Cayley graph of π 1 pM q with an edge metric. The Cayley graph is quasi-isometric to the universal cover Ă M . In the Cayley graph there are finitely many elements in the ball of any radius and this implies that dpγ n p, pq Ñ 8. In particular this implies that the leaves inside of the cylinder are not a bounded distance away from the cylinder, so the foliation is not uniform. Now, assume that the Reeb component lifts to Ă M , so there are compact leaves of r F. It follows that every leaf L P r F is a finite Hausdorff distance from a compact leaf. In particular L is bounded, therefore its closure is compact.
In particular this implies that if F is uniform and every closed transversal is not null homotopic then F is Reebless, as announced in subsection 2.2.

3.2.
Lifts of leaves at a bounded distance. In this section we prove some general results about Reebless foliations. We will use them in the next subsection to prove Theorem 1.1.
Recall that the leaf space L F " Ă M { r F in this case is a simply connected 1-dimensional manifold which is possibly non-Hausdorff. This is because for every leaf L P r F if t is a transversal (i.e. a curve transverse to r F homeomorphic to an open interval and intersecting L) it holds that t intersects each leaf of r F at most once (cf. §2.1). We note here that a natural idea would be to use the Hausdorff distance between leaves in the universal cover to show that the leaf space is Hausdorff. For a Reebless, uniform foliation, leaves in Ă M separate and hence the Hausdorff distance between leaves induces a metric in the leaf space, which we can call the Hausdorff metric. However in general this metric induces a topology which is completely different from the quotient topology in the leaf space due to lack of compactness. Consider for example an Anosov flow which is R-covered but not topologically conjugate to a suspension. It follows that if F is the weak stable foliation of this flow, then F is also uniform [Th]. Transversely to this foliation there is a strong unstable foliation and using that it is very easy to see that there is a 0 ą 0 such that for any two distinct leaves L, E of r F, then the Hausdorff distance between L and E is finite but bigger than a 0 . In other words the Hausdorff metric induces the discrete topology in the leaf space. This is completely different from the quotient topology making it homeomorphic to the reals. Notice that it is easy to see that the topology induced by the Hausdorff metric is always bigger than the quotient topology.
As explained in §2.1, when F is not R-covered, there are non-separated leaves of r F: that is, leaves L, F P r F so that for every transversals t L , t F to respectively L and F one has a leaf E P r F which intersects both t L and t F . Notice that if L, F P r F are distinct non-separated leaves, then they cannot intersect a common foliation chart, so the distance between points in one leaf to the other leaf is bounded from below.
We give some more definitions. We refer the reader to [BFFP, Section 3 and Appendix B] for a broader introduction with similar notation. We can assume that the foliation is transversally oriented by going to a double cover and this makes no problem in our results since we are working in Ă M . Given two leaves L, F P r F, the region between L and F is the intersection of the connected component of Ă M zL containing F and the connected component of Ă M zF containing L.
Remark 3.2. If L and F are non-separated and distinct, then no transversal to L can intersect F . Otherwise any leaf intersecting the transversal between L, F would separate F from L.
The following general result holds.
Proposition 3.3. Let F be a Reebless foliation of a closed 3-manifold M . Assume that there are distinct leaves L, F P r F a finite Hausdorff distance apart, and which are non separated from each other (i.e. there is a sequence of leaves L n P r F which converges to both in the leaf space). Then, both L and F project into compact leaves in M .
Proof. Let A and B be the projections of L and F respectively to M . Assume that A is not compact. Hence there is a sequence of points p i in A such that p i Ñ p so that p i are not in the same plaques of a local chart around p. Without loss of generality we can assume that the sequence p i is strictly monotone in the plaques of the chart.
One can lift the points p i to points x i P L and consider γ i P π 1 pM q so that γ i x i Ñ x 0 a lift of p. Let L 0 be the leaf of r F through x 0 . The fact that the points p i converge to p in different local leaves implies that γ i L are pairwise distinct leaves of r F´as transversals intersect a leaf only once in Ă M . It is exactly this property that we will show produces a contradiction. Denote by R ą 0 a bound of the Hausdorff distance between L and F . One can choose points y i P F so that dpy i , x i q ă R`1. Up to a subsequence, we can assume that γ i y i Ñ y 0 P F 0 P r F. Notice that L 0 ‰ F 0 for otherwise one could fix a curve in L 0 from x 0 to y 0 and that would lift to nearby leaves, giving that γ i L and γ i F intersect the same transversal for large i which is impossible since L is non-separated from F and L " F (cf. Remark 3.2). Now, pick transversals t x 0 and t y 0 to the leaves L 0 and F 0 through x 0 and y 0 respectively. For large i it follows that the plaques through γ i x i and γ i y i intersect t x 0 and t y 0 respectively. and so t x 0 is a transversal to γ i L and t y 0 a transversal to γ i F . Since γ i L and γ i F are non-separated it follows that there are leaves intersecting both t x 0 and t y 0 which implies that L 0 and F 0 are non-separated from each other.
Assume first that γ i L does not belong to the region between L 0 and F 0 . In this case L 0 separates γ i L from γ i F which is a contradiction since they are non-separated. In fact, if one considers a transversal t to γ i L which is contained in the component of Ă M zL 0 not containing F 0 it follows that every leaf intersecting t must remain in this component while γ i F must intersect a small transversal to F 0 so belong to a different connected component of Ă M zL 0 showing that γ i L and γ i F cannot be non-separated. The same works for γ i F . Suppose now that both γ i L and γ i F belong to the region between L 0 and F 0 . Recall that pγ i Lq converges to L 0 and now γ i L is in the complementary component of L 0 containing F 0 . In particular the sequence pγ i Lq also converges to F 0 . Hence for i big γ i L intersects t F 0 . Since for i big the leaf γ i F also intersects t F 0 this would show that γ i F, γ i L intersect a common transversal contradicting the fact that F, L not separated from each other.
In other words, what these arguments really show is that the assumption that γ i L are all distinct leads to a contradiction.
This finishes the proof of the proposition.
The following result also holds in great generality. Notice that even if we assumed that F is uniform, the result is not immediate since a priori we don't know if the region between two leaves has to be contained in a bounded neighborhood of one of the leaves. This is indeed what we show here for leaves which project into compact surfaces. Given a leaf L of r F, let Γ L be the subgroup of deck transformations fixing L, in other words, the stabilizer of L in π 1 pM q. Notice that πpLq " L{Γ L .
Proposition 3.4. Let F be a transversely oriented, Reebless foliation of a closed 3-manifold. Let L, F P r F leaves at bounded Hausdorff distance whose projection to M are compact surfaces. Let r N be the region between L and F . Then r N projects to a compact r0, 1s-bundle in Ă M {Γ L .
Proof. Notice that γF is at bounded Hausdorff distance from L for every γ P Γ L since deck transformations are isometries and γL " L. As F projects onto a compact surface, it follows that the orbit of F by π 1 pM q is a closed subset of Ă M . Let R ą 0 be the Hausdorff distance between L and F and consider a closed ball B of radius R`1 centred at a point x 0 P L. After covering B with finitely many foliation charts, by compactness one sees that only finitely many translates of F can intersect B. Since every translate of F by some element of Γ L must intersect B, this implies that the action of Γ L in F has finitely many translates of F . One deduces that the stabilizer of F in Γ L is a finite index subgroup of Γ L .
The symmetric argument says that Γ F has a finite index subgroup fixing L. We deduce that Γ " Γ F X Γ L is finite index in both Γ F and Γ L .
Consider the quotient Ă M { Γ of Ă M by the group Γ. It follows that both L and F project to compact leaves in Ă M { Γ . The region r N between L, F projects to a compact 3-manifold with boundary N Γ in Ă M { Γ , whose boundaries are the quotients A and B of L and F . Since F is Reebless and transversely oriented then leaves of F are π 1 injective in π 1 pM q, so π 1 pAq, π 1 pBq inject into π 1 pN Γ q. Clearly π 1 pAq, π 1 pBq also surject into π 1 pN Γ q. In addition N Γ is irreducible.
It follows (see [Hem,Theorem 10.2]) that N Γ is homeomorphic to Aˆr0, 1s and Aˆt1u corresponds to B. Projecting to Ă M {Γ L one gets that r N also projects to an r0, 1s-bundle with a boundary a leaf homeomorphic to C " L{ Γ L . This uses that F is transversely oriented.
We need one additional result.
Proposition 3.5. Suppose that F is a Reebless foliation in N " T 2ˆr 0, 1s, so that each boundary component is a leaf of F. Suppose that in r N , the boundary leaves E " r T 2ˆt 0u and G " r T 2ˆt 1u are not separated from each other in the leaf space of r F. Then F is not a uniform foliation.
Proof. Given a leaf L in the interior of r N , we will show it cannot be a finite Hausdorff distance from either one of the boundary leaves. Since N is a product there is b 0 ą 0 so that r N is contained in the neighborhood of size b 0 of E, and likewise for G. We will show that E cannot be in a bounded neighborhood of any such L as above.
Lifting to a double cover if necessary we can assume that F is transversely orientable.
Since F is Reebless the fundamental group of leaves injects in π 1 pN q, so the leaves are either planes, annuli or tori. If there is a compact leaf in the interior of N , then its fundamental group injects in Z 2 " π 1 pN q, so it is a torus, and hence it is isotopic to T 2ˆt 0u. It lifts to a leaf Z in r N which separates E from G, contradiction. So the leaves in the interior of N are only planes and annuli.
Let A " T 2ˆt 0u, B " T 2ˆt 1u. We look at the holonomy of F along a boundary leaf, say A. We want to find an element of π 1 pAq with contracting holonomy. Fix x a basepoint in A, let τ be a small transversal to F at x. Let α represent a simple closed curve in A not null homotopic. If either α or α´1 has contracting holonomy, that is the element we want. Otherwise there are p i in τ converging to x so that α holonomy fixes p i . Fix i, let C be the leaf through p i . Then C is an annulus. Let now β another simple closed curve which generates π 1 pT 2 q together with α. If holonomy of β fixes p i also then C is in fact a compact leaf, but in the interior of N , which we showed it is not possible. So replacing β by its inverse, the holonomy image of p i under β is closer to x. If the iterates converge to x, then β is the desired element. Otherwise the iterates converge to y not x, and the leaf through y is compact, again a contradiction.
Let then α be a simple closed curve in A with contracting holonomy. We think of α also as a deck transformation. Then α fixes E.
Fix a point y in E and a transversal τ . Since holonomy of the foliation F is contracting in the α direction this means that α´1pLq intersects τ and in a point closer to E. The contracting holonomy means that the sequence pα n pLqq converges to E as n Ñ´8. In fact this is an if and only if property: if there is L intersecting τ so that pα n pLqq converges to E as n Ñ´8, then α has contracting holonomy.
But α also preserves G. Since E, G non separated from each other, and pα n pLqq converges to E, it follows that pα n pLqq also converges to G when n converges to minus infinity. By the if and only if characterization above, this implies the following: If β is a simple closed curve in B freely homotopic to α then the holonomy of F along β is contracting as well.
We proceed with the proof of the proposition. We consider a model of N as T 2ˆr 0, 1s so that r N is homeomorphic to R 2ˆr 0, 1s with coordinates pa, b, cq and any deck transformation acts as pθpa, bq, cq, where θ is a translation of R 2 . In that way we can choose coordinates so that αpa, b, cq " ppa, bq`p1, 0q, cq.
Suppose now that E is in a neighborhood of size a 0 of L. For any n there is a point p n in L which is ă a 0 distant from p´np1, 0q, 0q.
Claim 3.6. Given ą 0, there is a 1 ą 0 so that any point in a leaf U of r F, it is less than a 1 along U from a point distant from E or G.
Proof. Suppose not. Project to N , we get bigger and bigger sets in leaves which avoid an neighborhood of the boundary. Taking a limit we find a leaf V of F avoiding an neighborhood of the boundary. The closure of V is a lamination in N disjoint from the boundary. It is an essential lamination W . Double N to get a Seifert fibered space, W is still an essential lamination. By Brittenham's result [Brit], W has a sublamination that is either vertical or horizontal in the double of T 2ˆI . If W is vertical it would have to intersect a boundary component of N . This is a horizontal T 2 in the double manifold. This is a contradiction. Suppose that W is horizontal. It is also contained in T 2ˆI , hence a "topmost" leaf would have to be compact, hence a torus. This is contained in the interior of N , again a contradiction. This proves the claim.
We fix ą 0 so that the foliation F restricted to the neighborhood of the boundary of N is entirely described by the holonomy maps. Let a 1 ą 0 given by the claim. So given n, there is q n in L, which is less than a 1 along L from p n and q n is away from the boundary. Hence q n " pp´n, 0q`v n , t n q where v n is bounded under n and |t n | ă ε or |t n | ą 1´ε. Up to subsequence we assume that all v n i are very close to v 0 (projection to N all in a fixed foliated chart). Now apply the holonomy of α n to q n . Since q n is close to the boundary and the holonomy of α is contracting in the neighborhood of size of both A and B it follows that the holonomy image of q n i is pv n i , t n i q where t n i is either arbitrarily close to 0 or to 1. None is either 0 or 1 as L is in the interior of N . They are all points in L, and this contradicts that L cannot intersect a transversal more than once.
This contradiction proves that the assumption that E is at a bounded distance from L is impossible. Hence the foliation F is not uniform.
3.3. Proof of Theorem 1.1. Now we are ready to prove Theorem 1.1. Let F be a uniform Reebless foliation on M . We want to show that F is R-covered, so we assume by contradiction that there are leaves L and F of r F which are non-separated in the leaf space L F " Ă M { r F of F. Up to a double cover we may assume that F is transversely oriented. Proposition 3.3 implies that both L and F project to compact surfaces in M . Let Γ L be the stabilizer of L in π 1 pM q. Proposition 3.4 shows that the region r N between L and F projects to a compact r0, 1s-bundle W in Ă M {Γ L , with one boundary L{Γ L .
Suppose that there is a deck translate βpLq of L or F inside r N . It projects to a surface in Ă M {Γ L contained in the r0, 1s-bundle W . Since πpLq is compact in M , then H " βpLq{ Γ L is also compact. Since H is π 1 -injective in W it follows that H is isotopic in W to a boundary component. Lifting to Ă M this implies that βpLq separates F from L, contradicting that they are non separated.
Let A " πpLq. Suppose that there is a closed transversal to F through A. Lift to Ă M , with the transversal intersecting L and entering r N . It cannot exit r N as F, L do not intersect a closed transversal. Hence this produces a deck translate of L inside r N which we just proved cannot happen. Hence there are no closed transversal through either A or B " πpF q.
On the other hand suppose there are E i converging to F Y L so that πpE i q is compact. For i big enough πpE i q is isotopic to A, and hence E i separates F from L, contradiction. Hence πpE i q is non compact and there are transversals through πpE i q for i big enough. It follows that the region between A and B is a dead end component, see [Ca 4 ,Definition 4.27]. By [Ca 4 , Lemma 4.28], A, B are two sided tori or Klein bottles. Lifting to a double cover we can assume that both A, B are tori.
It can be that A " B, but in any case r N projects in Ă M {Γ L to a compact submanifold homeomorphic to T 2ˆr 0, 1s.
We can now apply Proposition 3.5. Let G be a leaf in r N . By Proposition 3.5 it follows that L is not a bounded distance from G in r N . Suppose that this does not happen in Ă M . Then there are points p i in L which are ą i distant from G along path distance in r N , but a bounded distance in Notice that q 1 i is a bounded distance in r N from q i in L´just follow along the lift of the I-bundle structure to r N . If one uses the parametrization pa, b, cq as in Proposition 3.5 one can assume up to moving them boundedly in L, that p i , q i have all coordinates integers and the last coordinate 0. Consider a generating set of π 1 pM q which includes 2 generators of the torus A. Then p i , q i are vertices of the Cayley graph. Modulo deck transformations sending p i back to a base point, it follows that q i is a bounded neighborhood of the origin. So only finitely many elements of π 1 pM q are allowed. It follows that q i is a bounded distance from p i along L. This is a contradiction.
This completes the proof of Theorem 1.1.

Universal circles and JSJ trees
In this section we will show that for R-covered foliations (uniform or not) one can recover the universal circle from the JSJ decomposition of the manifold (cf. Proposition 4.9), if the manifold has a non trivial JSJ decomposition. This will allow us to prove Proposition 4.11 that we will need in the proof of Theorem 1.2. Proposition 4.11 states that the action of the fundamental group on the universal circle does not have fixed points which is certainly a fact that needs to be established if one desires to obtain minimality of the action.
Consider an R-covered foliation F by leaves with curvature uniformly close to´1 on a closed 3-manifold M , so that M has non trivial JSJ decomposition. In particular the leaves are Gromov hyperbolic. If F is not taut, then there are dead end components, see [Ca 4 , Definition 4.27]. In particular there are either tori or Klein bottle leaves. This is disallowed by F having Gromov hyperbolic leaves. Hence F is taut.
We will consider that M is orientable and F transversely orientable. The only difference in the non-orientable case is that in the JSJ decomposition we also have to consider Klein bottles. These Klein bottles lift to embedded tori in some cover of M . Then all the results follow with the same proofs.
4.1. The trace of JSJ tori in the universal circle. Let M 1 , . . . M k be the pieces of its JSJ decomposition. Let T be a torus of the JSJ decomposition. In this section we show Proposition 4.4 which states that one can associate to each lift of a torus of the JSJ decomposition some points in the universal circle.
We first need the following lemma that puts (after isotopy) the JSJ tori in general position.
Lemma 4.1. Any lift r T to Ă M intersects every leaf of r F. In addition one can isotope T so that r T intersects every leaf of r F in a single component, and so that the foliation induced by F in T has no Reeb components.
Proof. Let G " Z 2 be the isotropy group of r T . The set of r F leaves intersected by r T is connected. If this set is not the whole leaf space, it is a non trivial interval in the leaf space. Let F be an endpoint. Since the leaf space is homeomorphic to R, it follows that G preserves F . So π 1 pπpF qq has a Z 2 subgroup and the projection πpF q is therefore a torus or Klein bottle. This contradicts that the leaves of F are Gromov hyperbolic.
Since F is taut, by Theorem 2.1 we can isotope T to be either a leaf of F or transverse to F. The first option is disallowed because of Gromov hyperbolic leaves. Hence assume that T is transverse to F, let G be the induced foliation in T .
Claim 4.2. It is possible to isotope T so that G has no Reeb annuli.

Proof.
A Reeb annulus is a foliation of the annulus so that boundaries are leaves, all other leaves spiral toward the boundary leaves, and there is no transversal arc intersecting both boundary leaves. Suppose that G has a Reeb annulus A. The two boundary leaves of A lift to curves in Ă M , contained in leaves of r F which are non separated from each other. This is because of the Reeb annulus, so in r A the boundary infinite lines are non separated from each other. Since the foliation is R-covered, the two leaves of r F containing these infinite lines α, β are the same leaf L. Since πpαq, πpβq are freely homotopic in T , then α, β are a bounded distance from each other in Ă M . We now use a fact of R-covered foliations: for any a 0 ą 0, there is a 1 ą 0, so that if two points x, y in a leaf F of r F are less that a 0 in Ă M , then they are less than a 1 in L (see [Fen 1 , Proposition 2.1]). This holds only for R-covered foliations. Hence α, β are a bounded distance from each other in L. It now follows that πpαq, πpβq are isotopic closed curves in πpLq and bound an annulus B in πpLq. The interior of B cannot intersect A, because any interior leaf of G in A limits to the boundary of A, and A, B are transverse to each other. Hence A Y B is a torus. This torus is not π 1 injective because one can produce an essential arc across A together with one across B to yield a closed curve which is null homotopic. One can easily see this as r B is contained in the fixed leaf L, and r A has both boundaries in L. Hence A Y B is compressible and there is a compressing disk D intersecting A Y B only in the boundary. Cutting A Y B along D, produces a sphere. Since M is irreducible, this sphere bounds a ball. Gluing back together one sees that A Y B bounds a solid torus.
What we proved is that B is isotopic to A in M . So then one can isotope A across the solid torus to the other side of B and eliminate this Reeb annulus in G. Doing this finitely many times eliminates all Reeb annuli in G. This proves the claim. See also [Ca 1 , Theorem 5.3.13] for a similar statement.
Since there are no Reeb annuli in G, it follows that F intersects T in a foliation uniformly equivalent 6 to a linear foliation of the two dimensional torus. In particular any two leaves of r G are connected by a tranversal to r G, hence a transversal to r F as well. It follows that any leaf F of r F intersects r T in a single component.
This finishes the proof of the lemma.
Remark 4.3. The reason we choose the definition of non-trivial JSJ decomposition is to exclude Sol and N il geometries for which some of the arguments do not work. These cases are not problematic to us and can be dealt with separately, and in a different way. A good thing about manifolds with non-trivial JSJ decomposition under our definition is that the tori of the decompositons are quasi-isometrically embedded: the map between the universal covers is a quasi-isometric embedding. This follows from [KL, Theorem 1.1] (see also [Ng,Section 3.1]). In particular when lifted to Ă M , every quasigeodesic in the lift of the torus lifts maps to a quasigeodesic in Ă M .
Let T be a torus of the JSJ decomposition, put in good position as in Lemma 4.1. Let G be the induced foliation by F in T . Given L leaf of r F, and a lift r T of T , then by Remark 4.3, the curve LX r T is a quasigeodesic of Ă M . It is also a leaf of r G. Since it is a quasigeodesic in Ă M , then it is necessarily also a quasigeodesic in L, with ideal points a L p r T q, b L p r T q in S 1 pLq. Orient the foliation G so that b L p r T q corresponds to the forward direction in G. Varying the leaf, produces corresponding ideal points a F p r T q, b F p r T q in S 1 pF q for any F leaf of r F.
Proposition 4.4. The collection tb F p r T qu as F varies over leaves of r F is a leaf of the vertical foliation in the cylinder at infinity A. Equivalently, the point tb F p r T qu is well defined in S 1 univ and independent of the leaf F . Proof. We will fix a lift r T of some torus T of the JSJ decomposition. So, we will not include the reference to r T in the notation. Suppose first that F is uniform. Let α L be the intersection of L and r T , that is a leaf of r G. For any L, F leaves of r F, the curves α L , α F are a bounded distance from each other in r T´since there are no Reeb annuli in G. It follows that α L , α F are a bounded distance from each other in Ă M . By the remark above, α L is a quasigeodesic in L, hence, the ray β L defining b L is a bounded distance in L from a geodesic ray in L. Since F is uniform, this ray in L is a bounded distance from a geodesic ray in F defining τ L,F pb L q. But β L is a bounded distance from a corresponding ray β F of α F (same direction given by the foliation G). This is bounded distance in Ă M . Hence β F is a bounded distance in Ă M from the geodesic ray defining τ L,F pb L q. Since F is R-covered, this again implies that β F is a bounded distance from this geodesic ray in F . In particular the ideal point of β F is τ L,F pb L q. But by definition the ideal point of β F is b F . Hence b F " τ L,F pb L q. This proves the proposition in this case.
6 By this we mean that in the universal cover each leaf is bounded Hausdorff distance from a leaf of the linear foliation. See [Th,Definition 2.1] for a general definition of being uniformly equivalent.
Suppose now that F is not uniform. By the description in §2.5.2 we can assume that F is minimal. Hence for any L, F in r F there is a dense set of directions in S 1 pLq which are asymptotic to F . Fix a transversal τ to G in T . Lift this to a transversalτ in r T . For any L intersectingT , let x L "τ X L. Let r L be the geodesic ray in L starting at x L and with ideal point b L . As L varies the corresponding rays β L in r G are boundedly close to each other in r T and hence in Ă M . Hence the same happens for the geodesic rays r L as L varies. It follows that the ideal points of β L vary continuously with L. Hence the functions a L , b L from the leaf space into A are continuous.
Suppose that for some L, F , then τ L,F pb L q " b F . Since the set of contracting directions between L and F is dense in S 1 pLq and b E varies continuously with E, it follows that there is some E between L, F so that b E corresponds to a direction in E which is contracting with both L and F . Hence the ray β E in E X r T is asymptotic to a curve in L. This implies that in r T , the curve β E is asymptotic to a curve in r T XL. But this can only be β L´a s r T XL is a single curve and has a ray β L corresponding to that direction. In particular this implies that b E " τ L,E pb L q. The same holds for the pair E, F . By the composition property of the maps τ L,F , it now follows that τ L,F pb L q " b F . This finishes the proof of the proposition.

JSJ universal circles.
Our setup has an R-covered foliation F by leaves with curvature very close to´1 in M with non trivial JSJ decomposition. If T is a torus in the JSJ decomposition we use Lemma 4.1 and isotope T to be transverse to F and so that the induced foliation in T does not have any Reeb annuli. Recall that in Proposition 2.4 we introduced the JSJ tree T of M . Let T 1 , ..., T k be the tori in the JSJ decomposition. The fundamental group of M naturally acts on the tree T. The tree T is infinite and in general not locally compact: there are infinitely many edges adjoining any given vertex. We observe that if M has a trivial JSJ decomposition, that is, M is either Seifert or atoroidal, then the object constructed above would be a single point. We now consider the case that M has an R-covered foliation.
Let W " π´1pT 1 Y . . . Y T k q. In other words a component of W is an arbitrary lift r T of one of the JSJ tori.
Lemma 4.5. Suppose that M has a non-trivial JSJ decomposition and F is an R-covered foliation by leaves with curvature very close to´1. Then the JSJ tree T has an embedding into the plane well defined up to isotopy. This determines a well defined circular ordering on the set of ends of T. A deck transformation either preserves the circular ordering, or reverses the circular ordering on the set of ends.
Proof. The curvature condition implies that F is Reebless.
Hence the leaves of r F are properly embedded planes in Ă M . First fix a leaf F of r F. Lemma 4.1 shows that any lift r T of a JSJ torus intersects F in a single component. This component is a quasigeodesic in F . For each vertex y of T, associated to a component V of Ă M´W , it has at least two edges adjoining it, let r T be one of them. Since r T intersects F transversely, then V also intersects F . In addition since any lift r T 1 of a JSJ torus separates Ă M , and each such lift intersects F in a single component, it also follows that V also intersects F in a single component. Choose a point p V in V X F representing the vertex y of T. It r T is an edge of T adjoining components V , Z of Ă M´W , choose an embedded arc in F connecting p V to p Z , and intersecting r T in a single point. This represents an embedding of the edge r T of T into F . In this way we construct an embedding of T into F . The choices of the points p V are well defined up to isotopy in V X F . The choices of the embedded arcs are also well defined up to isotopy. Therefore the embedding of T into F is well defined up to isotopy. Fix one such embedding and call T F the image tree in F . Now if L is another leaf of r F, then the same reasoning applies. Notice that if V , Z components of Ă M´W define and edge r T , then V X L, Z X L are adjoining in L along r T X L just as in F . In addition the circular ordering around a vertex is also the same whether considering it wrt to F or to L. It follows that the embeddings of T in F and L are isomorphic, preserving the circular ordering at the corresponding vertices.
It follows that the embedding in the plane is well defined up to isotopy. This induces a circular ordering in the set of ends of T.
If γ is a deck transformation, and F a leaf of r F, then γ also induces a homeomorphism of the embedding of T in F : given V components of Ă M´W , then γpV q also intersects F in a single component, and likewise for r T component of W . This produces the required homeomorphism of the tree T. In addition this homeomorphism is induced by a homeomorphism between F and γpF q, which can be either orientation preserving or reversing. It follows that this homeomorphism either preserves the circular ordering of the ends of T or reverses it.
Remark 4.6. We emphasize some facts proved in this lemma: if V is a component of Ă M´W , and F is a leaf of r F, then V intersects F and in a single component (cf. Lemma 4.1). Similarly if r T is a component of W then r T intersects F in a single component. Therefore the trees T and T F are canonically isomorphic. In particular if F, L are leaves of r F, then T F , T L are canonically isomorphic, with the circular order of the edges at any vertex preserved by the isomorphism (see also Proposition 4.4).
We produced a set with a circular order and a group action so that each group element either preserves the circular order or reverses it. Given these properties, a circle with an induced action can be created. This procedure from set with circular order and group action to action on a circle was developed by Calegari and Dunfield in [CD]. We refer to [CD,Theorem 3.2] for specific details. Here we will only briefly describe the construction of the circle with the induced action.
Since the set of ends is cyclically ordered there is an embedding of the set of ends into a circle preserving the circular order. First take the closure of the image of the set of ends. If the tree were locally finite (finitely many edges at any vertex), then the set of ends would be order complete, and the image is a closed subset of the circle.. The fundamental group still acts on the closure. There may be gaps in the image. Now collapse every closure of a complementary interval (that is a gap) to a point, producing a circle S 1 JSJ , called the JSJ universal circle of F. Deck transformations either preserve or reverse the circular ordering so induce homeomorphisms of the circle that either preverse or reverse orientation.
Remark 4.7. The JSJ universal circle depends on the foliation F: given a different R-covered foliation F 1 , it may induce a different circular ordering of the edges at a given vertex of the tree T. This will produce a different circular order on the set of ends of T and hence a different JSJ universal circle. The tree T is the same and so are its ends. But the the set of edges around a vertex in T does not come with a natural circular order. This is the information that the R-covered foliation is providing, because it gives an embedding of the tree into the plane. Different R-covered foliations may give different such circular orders.
Let T be a π 1 -injective torus in M , put in good position as in Lemma 4.1. Given F leaf of r F, we define the lamination G F whose leaves are the intersections of lifts r T of T with F . In fact G F also depends on T , but for notational simplicity we omit this dependence.
Lemma 4.8. For each π 1 -injective torus T of M and for each F leaf of r F, then the set of ideal points of leaves of G F is dense in S 1 pF q. In addition for any non degenerate interval J of S 1 pF q there are leaves of G F with both ideal points in J.
Proof. Suppose the first property is not true, let T be a π 1 -injective torus and F a leaf of r F so that the set of ideal points of leaves of G F is not dense in S 1 pF q.
Then there is a non trivial interval I in S 1 pF q which is disjoint from the ideal points of of G F . Since the curves in G F are uniform quasigeodesics in F they are a uniform bounded distance from geodesics in F . Hence up to considering a subinterval, it follows that I bounds a half plane P in F which is disjoint from G F . Therefore there are disks D i with radius converging to infinity disjoint from G F . Up to taking subsequences and deck transformations g i , then g i pD i q converges to a full leaf L which is disjoint from G L . But this is impossible since any lift r T of T intersects every leaf of r F. This proves the first property of the lemma. Now suppose that J is a non degenerate interval so that no leaf of G F has both ideal points in J. Let x be an interior point of J. Let x i a sequence of distinct points in J converging monotonically to x. There are leaves c i of G F with an ideal point arbitrarily close to x i . Since the x i are distinct in J we can choose the c i to be distinct as well. The other endpoints of c i are not J, hence at least a 1 ą 0 from the first endpoint of c i which is arbitrarily close to x. Since the c i are uniform quasigeodesics, then up to subsequence we may assume that c i converges to a quasigeodesic c. But then different c i , c j have points that are arbitrarily close to each other. This is a contradiction: different tori in the JSJ decomposition are compact and disjoint. This implies that there is a constant a 2 ą 0, so that if C, C 1 are different lifts of JSJ tori, then points p P C, p 1 P C 1 satisfy that distance from p to p 1 is at least a 2 .
This finishes the proof of the lemma.
We can now prove the following proposition that gives a different way to think about the universal circle of a foliation in terms of the JSJ universal circle.
Proposition 4.9. Suppose that M has a non-trivial JSJ decomposition and F is an R-covered foliation with Gromov hyperbolic leaves. Then there is a canonical homeomorphism between the universal circle S 1 univ of F and the JSJ universal circle S 1 JSJ of F. This homeomorphisms is equivariant under deck transformations.
Proof. For simplicity fix a leaf F of r F. The universal circle of F is canonically identified with S 1 pF q. The JSJ universal circle can be obtained from the intersections with F . What we will prove is that considering F , both of these are canonically homeomorphic.
Let T F be the embedded tree in F which is the homeomorphic image of T. Fix a basepoint p in T F . Let B be the set of ends of T F . Since T F is a tree it is easy to see that each end is uniquely associated to an embedded ray in T F starting at p. Let e be an end in B associated to a ray α in T F , which is also an embedded ray in F . Then α keeps intersecting lifts C i of one of the JSJ tori, let c i " C i X F . Recall that c i is a quasigeodesic with uniform constants, so globally a 0 distant from a geodesic in F . Any two lifts C, C 1 of JSJ tori have a minimum separation between them. Hence the corresponding points C XF, C 1 XF also have a minimum separation between them. Therefore the geodesics associated to c i also escape in F and they define a unique ideal point in S 1 pF q which we call f peq. This defines a map f from the set of ends B to S 1 pF q.
Given appropriate orientations on S 1 pF q and the circular order on the set of ends of T F , it follows that the map f preserves this circular order. In particular as one goes around once in the circular order of the ends of T F , then one also goes around once in S 1 pF q. By Lemma 4.8, for each non degenerate interval J in S 1 pF q there is a leaf c of L F with both ideal points in J. Hence any end e of T F which is associated with a path in the tree T F which crosses c will have f peq in J. It follows that the image of f is dense in S 1 pF q.
Recall the construction of the JSJ universal circle S 1 JSJ of F: we map the set of ends B to a circle S 1 preserving the circular order, take the closure and then collapse the gaps.
By the first step we can think of B as a subset of S 1 . Let H be the closure in S 1 of the image of f . Since f preserves circular order it induces a map f 1 from H into S 1 pF q. This map is weakly monotone. Since the image of B under f is dense in S 1 pF q it follows that given the endpoints of a gap of H they have the same image in S 1 pF q under f 1 . This implies that f 1 induces a map f˚from the JSJ universal circle S 1 JSJ of F to S 1 pF q. Finally by the same reasoning if two points have the same image under f 1 then they have to be boundary points of a gap of H in S 1 . This implies that f˚is a homeomorphism.
Any deck transformation γ permutes the lifts of JSJ tori and components of Ă M´W . It sends infinite embedded paths in the tree T F to infinite paths in the tree T γpF q . The tree T γpF q is canonically homeomorphic to the tree T F and this identification is compatible with the identifications of S 1 pγpF qq and S 1 pF q. It follows that the homeomorphisms f˚are equivariant. This finishes the proof of the proposition. Proposition 4.11. If F is a uniform R-covered foliation by hyperbolic leaves and ξ P S 1 univ then there is γ P π 1 pM q such that γpξq ‰ ξ. Proof. We first treat the case where the JSJ decomposition of M is trivial. If M is Seifert with hyperbolic base, the universal circle is identified with the boundary of the universal cover of the base. The base is a hyperbolic surface S, maybe with finitely many orbifold singular points. If δ is a generator of the center of π 1 pM q then π 1 pM q{ ă δ ą is isomorphic to a closed surface group π 1 pSq where S may have finitely many orbifold (or cone) points and acts on the boundary B r S. The stabilizer of each point in B r S is at most infinite cyclic. The deck transformation δ acts by the identity on the universal circle of the foliation. It now follows that the stabilizer of a point of the universal circle is at most a Z ' Z subgroup. By homological reasons Z ' Z cannot be the fundamental group of an irreducible closed 3-manifold [Hem]. This finishes the proof in the Seifert case.
If M is atoroidal then it is hyperbolic 7 and we then assume Ă M " H 3 . In this case we show that the stabilizer of ξ is at most infinite cyclic. Suppose that γ is in the stabilizer of ξ. Let F be a leaf of r F and ξ F be the ideal point of S 1 pF q associated to ξ. Thurston [Th] proved that the embedding F Ñ Ă M extends to a continuous map F Y S 1 pF q Ñ Ă M Y S 2 8 where S 2 8 is the boundary B 8 Ă M " B 8 H 3 (cf. § 2.4). Let p be the image of ξ F under this extended map. Let β be a geodesic ray in F with ideal point ξ F . Then γpβq is a geodesic ray in γpF q. Since γpξq " ξ, and F is uniform, it follows that γpβq has a subray which is a bounded distance from β. In Ă M Y S 2 8 the image of β limits to p. Since γpβq has a subray a bounded distance from a ray of β, it follows that γppq is equal to p. Hence γ is in the stabilizer of p. But it is well known that the stabilizer of a point in S 2 8 is at most cyclic. This finishes the proof in the atoroidal case.
Foliations in manifolds with (virtually) solvable fundamental group are classified and cannot be uniform R-covered with hyperbolic leaves (see [Pla] or [HP, Appendix B] for the C 0 -case). In fact the result does not work for manifolds with (virtually) solvable fundamental group. So the remaining case to be analyzed in the proof is is when M has a non-trivial JSJ decomposition in our sense (which excludes being a torus bundle up to a finite cover).
Now we consider the case that the JSJ decomposition of M is not trivial. Let T be the tree of lifts of the pieces of the JSJ decomposition as in Proposition 2.4. Fix F a leaf of r F. Recall from the proof of Lemma 4.8 that the following holds: for any lift Ă M j 0 i 0 of a piece M i 0 of the JSJ decomposition of M , it intersects F in a single component. Let ξ F be the point of S 1 pF q corresponding to ξ. We consider 2 distinct lifts Ă M j i as follows. First take an arbitrary Ă M j i so that ξ F is not an ideal point of i is greater than 3. Let now γ be a non trivial deck transformation that fixes Ă M k i . By Proposition 2.4 the diameter of the fixed point set of γ acting on the JSJ tree is less than or equal to 2. In particular γp Ă M j i q " Ă M j i . Since A 1 separates A 2 from ξ F , it now follows that γpξq is not equal to ξ.
This concludes the proof of the proposition.
Remark 4.12. One can give a different proof of Proposition 4.11 using different machinery that we chose not to present in detail. Indeed, if there is a global fixed point ξ in the universal circle of a uniform R-covered foliation by hyperbolic leaves, then the one-dimensional foliation by geodesics in each leaf landing as a geodesic fan on ξ is equivariant and therefore descends to a one-dimensional foliation (which if chosen to be tangent to a unit vector field defines a flow) in M . By an argument in [Ca 3 ] (see the proof of [Ca 3 , Theorem 5.5.8]) this flow is (topologically) Anosov 8 for which F is the weak stable foliation. This is impossible since the flow would be R-covered and not a suspension (because the center stable foliation is uniform). This flow also does not have periodic orbits freely homotopic to their inverses, because the orbits always point in the direction of ξ. This contradicts what is proved in [Fen 2 , Bar].

Proof of Theorem 1.2
We fix in π 1 pM q a finite symmetric set of generators S and denote by |γ| the word length of γ with respect to S. We will be concerned with sequences going to infinity, so the choice of S is irrelevant.
Theorem 1.2 concerns the action of π 1 pM q on the universal circle S 1 univ . The universal circle is canonically homeomorphic to S 1 pLq for any L leaf of r F. By Remark 2.5, in order to prove Theorem 1.2, it is equivalent to consider the action of π 1 pM q on S 1 pLq. So fix a leaf L P r F and denote by This induces an action of π 1 pM q on S 1 pLq (but in general does not induce an action on L). Again via the identification with the universal circle S 1 univ this is exactly the action defined on S 1 univ in Remark 2.5. In this way ρ is a group homomorphism from π 1 pM q into Homeo`pS 1 pLqq.
Fix a point x 0 P L. The point x 0 allows us to define a visual measure (cf. §2.4) in S 1 pLq that we will also fix.
The first important property is the following: Lemma 5.1. Given a compact interval I Ă L F " Ă M { r F containing L we have that if γ n P π 1 pM q satisfies γ n L P I and |γ n | Ñ 8, then for every x P L we have d L px, ρpγ n qxq Ñ 8. In particular, given C Ă L compact, there is K ą 0 such that if |γ| ą K and γL P I then ρpγqC X C " H.
Proof. Fix a compact fundamental domain Y of M in Ă M . For a given R ą 0 there is a bounded set G Ă Ă M which consists of the points z in leaves F P I such that τ F,L pzq P B R pxq where B R pxq denotes the ball of radius R in L. The set G is bounded because the quasi-isometry constants of τ F,L | L depend only on the Hausdorff distance between F and L. Since F P I the Hausdorff distance is bounded. Now, one can cover G by finitely many fundamental domains, implying that if γ verifies that γL P I and |γ| is sufficiently large, then ρpγqx cannot be in B R pxq. This completes the first part of the Lemma.
For the second statement, notice that estimates are uniform, so by compactness one gets the statement.
This allows us to show the following: Lemma 5.2. For every finite interval I Ă L F containing L and ε ą 0 there is K ą 0 such that if |γ| ą K and γL P I we have that there are (not necessarily disjoint) intervals I γ , J γ of length (for the visual distance in S 1 pLq) smaller than ε and such that ρpγqpS 1 zI γ q Ă J γ .
Proof. Given the finite interval I there exists a uniform constant c ą 1 so that for every F P I the map τ F,L : F Ñ L is a quasi-isometry with constant c. It follows that the image by ρpγq of a geodesic in L is a c-quasigeodesic in L whenever γL P I. Notice that τ F,L | L is not necessarily continuous, so τ F,L | L pcq not necessarily a continuous curve. But the quasi-isometry inequalities still hold. Fix x 0 in L. Let C Ă L be a compact set containing x 0 with the property that every quasigeodesic in L with constants bounded by c which does not intersect C verifies that its visual measure is smaller than ε{2. Now, we can apply Lemma 5.1 to find K such that if γ verifies that γL P I and |γ| ą K then one has that ρpγ´1qCXC " H. By choosing K a bit larger, one can assume that there is a geodesic in L which separates ρpγ´1qC from C (see figure 1). This uses that the diameter of ρpγ´1qC X C is uniformly bounded. This allows us to define I γ as the (shortest) interval determined by the endpoints of (i.e. the one so that I γ Y l Ă L Y S 1 pLq leaves C on the outside) and J γ to the (shortest) interval joining the endpoints of the quasigeodesic ρpγqp q. Now we are in condition to prove minimality of the action: Proposition 5.3. The action of π 1 pM q on S 1 univ is minimal. In particular, given ξ P S 1 pLq and an open interval U Ă S 1 pLq there exists γ P π 1 pM q such that ρpγqξ P U .
Proof. We first fix an open set U Ă S 1 pLq.
Fix T a compact fundamental domain of M in Ă M . Every other fundamental domain will be a translate of T by a deck transformation. Let D " diampT q, which is also the diameter of any translate of T . Let I Ă L F be a compact interval around L such that the union Ť F PI F contains the neighborhood of size 2D of the leaf L. Notice that this interval can be chosen thanks to the fact that F is R-covered and uniform.
Choose points ξ 1 ‰ ξ 2 in the interior of U and take fundamental domains T n 1 , T n 2 of M in Ă M such that they intersect L in points very close to ξ 1 and ξ 2 respectively, more precisely, such that the intersection T n i X L is non empty and T n i X L Ñ ξ i in L Y S 1 pLq. See figure 2. Now, we can choose γ n so that γ n pT n 1 q " T n 2 . Since the diameter of T n i is fixed, T n i X L Ñ ξ i and ξ 1 " ξ 2 then as n Ñ 8, for any y n 1 P T n 1 , y n 2 P T n 2 , dpy n 1 , y n 2 q Ñ 8. It follows that |γ n | Ñ 8. Also γ n L P I so that Lemma 5.2 applies. This also uses that F is R-covered. Let I γn , J γn be the intervals provided by Lemma 5.2. We choose ε ą 0 small so that the 2ε-neighborhood of both ξ 1 and ξ 2 in S 1 pLq is contained in U .
By contradiction, we assume that there are arbitrarily large n so that neither I γn nor J γn are contained in U . TakeÛ the subinterval of U obtained by removing from U the neighborhoods of the endpoints (i.e. if U " pa, bq we considerÛ " pa`ε, b´εq). Notice that ξ 1 , ξ 2 are inÛ . The choice of I γn and J γn implies that they are both disjoint fromÛ and therefore γ nÛ XÛ " H. This will be a contradiction as follows: take c n a geodesic in L intersecting T n 1 X L whose endpoints are close to ξ 1 and contained in U . Then the image by ρpγ n q of c n is a uniform quasigeodesic, because γ n L is in a compact interval I in the leaf space. This uniform quasigeodesic intersects T n 2 and therefore has at least one endpoint in a neighborhood of ξ 2 if n is large enough (note that we cannot ensure that both endpoints of ρpγ n qpc n q are contained inÛ ). This implies that γ nÛ XÛ ‰ H which is a contradiction.
Therefore up to a subsquence and replacing U by a slightly smaller open set, it follows that either I γn or J γn is contained in U . Up to taking γ´1 n we can assume that J γn Ă U . We now choose an arbitrary point ξ in S 1 pLq. Pick η P π 1 pM q so that ρpηqξ ‰ ξ (cf. Proposition 4.11). If necessary choose ε smaller so that the distance in S 1 pLq from ξ to ρpηqξ is bigger than 10ε.
Assume first that ξ R I γn for arbitrarily large n. In this case, one concludes since ρpγ n qξ P J γn Ă U as desired. If ξ P I γn for all large n, then by the choice of ε it follows that ρpηqξ R I γn for large enough n. This implies that ρpγ n ηqξ P J γn Ă U completing the proof of the proposition.
We devote the rest of the section to the proof of transitivity of the action on pairs of points. First, we show that we can find attractor/repeller configurations in any pair of open sets.
Lemma 5.4. For every U, V open intervals in S 1 pLq there is γ P π 1 pM q such that ρpγqpS 1 pLqzU q Ă V .
Proof. Consider a sufficiently large compact interval I Ă L F as in the proof of Proposition 5.3 so that the union of its leaves contains a neighborhood of size larger than the diameter of a fundamental domain around L.
As in the proof of Proposition 5.3, it is possible to construct a sequence γ n P π 1 pM q such that |γ n | Ñ 8 and such that the neighborhoods I γn and J γn verify (up to taking a subsequence) that I γn Ñ ξ 1 and J γn Ñ ξ 2 where it could be that ξ 1 " ξ 2 . This is just taking very large elements that move a fundamental domain intersecting L into other fundamental domain intersecting L and applying Lemma 5.2. Now, using Proposition 5.3 we choose η 1 and η 2 in π 1 pM q satisfying ρpη 1 qpξ 1 q P U and ρpη 2 qpξ 2 q P V . It follows that for sufficiently large n the deck transformation β n " η 2˝γn˝η´1 1 verifies that ρpβ n qpS 1 zU q Ă V .
To see this, notice that ρpη´1 1 qpU q contains I γn for suficiently large n because ρpη 1 qpξ 1 q P U . Similarly, if n is large enough, then ρpη 2 qpJ γn q is contained in V . Since ρpγ n qpS 1 pLqzI γn q Ă J γn , this completes the proof.
To complete the proof of Theorem 1.2 it is enough to show: Proposition 5.5. Given open intervals U 1 , V 1 Ă S 1 pLq and U 2 , V 2 Ă S 1 pLq there existsγ P π 1 pM q such that ρpγqU 1 X U 2 ‰ H and ρpγqV 1 X V 2 ‰ H. In particular, there exists a pair ξ 1 ‰ ξ 2 P S 1 univ whose π 1 pM q-orbit is dense in S 1 univˆS 1 univ ztdiagonalu. Proof. By reducing the intervals we can assume without loss of generality that the four intervals U 1 , U 2 , V 1 , V 2 are disjoint.
Apply Lemma 5.4 to find deck transformations γ and η which verify that ρpγqpS 1 pLqzU 1 q Ă V 2 and ρpηqpS 1 pLqzρpγqV 1 q Ă U 2 . Now, the transformationγ " ηγ is the desired one. Indeed, ρpγqU 1 X ρpγqV 1 " H, or ρpγqU 1 Ă S 1 pLqzρpγqV 1 , which implies that ρpηγqU 1 Ă U 2 . In addition ρpηγqV 1 " ρpηqρpγqV 1 Ą S 1 pLqzU 2 Ą V 2 . The existence of dense orbits is now standard. Indeed, pick a countable basis tU n u of intervals generating the topology of S 1 pLq. The set A n,m of pairs of different points ξ 1 , ξ 2 such that there exists γ P π 1 pM q such that ρpγqξ 1 P U n and ρpγqξ 2 P U m is clearly open and it is dense because of what we just proved. Then, the intersection Ş n,m A n,m is a residual subset by Baire's category theorem and the orbit of points in A n,m is always dense in S 1 univˆS 1 univ .

Branching foliations
In this section we just point out that all our results work in the setting of branching foliations as they appear in the study of partially hyperbolic dynamics. These objects were introduced by Burago-Ivanov [BI]. We give here a definition that excludes a priori the existence of Reeb component like objects.
A branching foliation F bran in a 3-manifold M is a collection of immersed surfaces (tangent to a continuous distribution) called leaves with the following properties. If r F bran is the lift of the collection to Ă M then: ‚ Each leaf L of r F bran is a properly embedded plane in Ă M and separates Ă M in two open regions L ' and L a . Denote L`" L Y L ' and L´" L Y L a . ‚ Every point in Ă M belongs to at least one leaf L P r F bran . ‚ The leaves do not topologically cross. That is, given two leaves L and F of r F bran we have that F Ă L`or F Ă L´. ‚ Given a sequence of points x n Ñ x P Ă M and leaves L n with x n P L n it follows that through x there is a leaf L P r F bran which is the uniform limit in compact parts of L n .
In [BFFP,§3] (see also [BFP,§3]) a careful study of the properties of these objects is performed, including a study of the leaf space associated to such a branching foliation. In particular, it makes perfect sense to talk about uniform branching foliations and R-covered ones. Moreover, in the partially hyperbolic setting there exists foliations in M that approach the center stable and center unstable branching foliations. In this setting this can be used to have in general situations a metric in M which gives curvature arbitrarily close to´1 to all leaves of F. In this setting, one can define a universal circle as one does for general foliations.
All arguments performed in this note thus hold for branching foliations. We state the result in this context for future use and explain how it can be deduced from the results proved in this paper. (Note that we could have performed our arguments directly in the branching foliation setting, but we decided to work out the foliation case first since we believe that many people may only be interested by the true foliation case.) Theorem 6.1. Let F be a uniform branching foliation. Then, it is Rcovered. Moreover, if M admits a metric making every leaf negatively curved, then the action of π 1 pM q is minimal in the universal circle S 1 univ and moreover it acts transitively in pairs of points of S 1 univ .
Proof. Note that [BI,Theorem 7.2] (see also [BFP,Theorem 3.3]) shows that a transversely oriented branching foliation F can be approximated by a true foliation F ε together with a continuous and surjective map h ε : M Ñ M , which is ε close to the identity, and is a local diffeomorphism from L P F ε to h ε pLq P F. In particular the ε-close property shows that the approximating foliation is also uniform. Moreover, the map h ε lifted to Ă M induces a homeomorphism between the leaf spaces (see [BFP,Theorem 3.3 (ii)]). In this way, we can associate to a uniform branching foliation F another uniform (non branching) foliation F ε and thus apply Theorem 1.1 to F ε to obtain the same statement for F. To show the minimality, we can either use the same arguments as in the previous section (which work without modifications), or alternatively, we can also use F ε and note that h ε induces a conjugacy between the actions in the universal circle.
There is always a double cover of M so that F lifts to a transversely oriented branching foliation, so the result follows.