Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology

We consider a minimal compact lamination by hyperbolic surfaces. We prove that if it admits a leaf whose holonomy covering is not topologically trivial, then the horocycle flow on its unitary tangent bundle is minimal.


Introduction
The geodesic and horocycle flows on hyperbolic surfaces are two classical examples of flows in homogeneous spaces. Their dynamical and ergodic properties were studied in the 1930s by E.Hopf and G.A.Hedlund. A compact hyperbolic surface S is the quotient of the Poincaré upper-half plane H by a cocompact Fuchsian group Γ. Its unit tangent bundle T 1 S, that can be seen as the quotient of P SL(2, R) by Γ, is the phase space of these flows. The geodesic flow is uniformly hyperbolic on T 1 S, and its stable manifolds are precisely the horocyclic orbits. Alternatively, these flows can be seen as coming from the action of the diagonal and unipotent one-parameter subgroups of P SL(2, R) on Γ\P SL(2, R).
In this context, Hedlund proved in 1936 that the horocycle flow is minimal -that is, all its orbits are dense [7].
M.Ratner completed in the 1990s the ergodic-theoretical and topological descriptions of the dynamics of unipotent groups on homogeneous spaces, with a conclusive theorem in a subject that had seen important contirbutions by S.G.Dani, H.Furstenberg, G.A.Margulis, J.Smillie, among others.
Hedlund's theorem has a converse (see for example [4]): If the horocycle flow on a hyperbolic surface S is minimal, then S must be compact. In fact, the dynamics of the horocycle flow on non-compact surfaces is quite complex. The finite-volume case has been studied by S.G.Dani and J.Smillie. In the infinite-volume setting, A.N.Starkov conjectured that this flow it is ergodic (with respect to Liouville measure) if and only if the action of Γ on the boundary ∂H is ergodic (with respect to Lebesgue measure). By duality, this is equivalent to the ergodicity of the joint action of the geodesic and horocycle flows on T 1 S, which is the action of the affine group B. This result was proved in a special case by M.Babillot and F.Ledrappier and independently by M.Pollicott, and in general by V. Kaimanovich. For these and other related results see for example [17] and [10]. See also [15] for a complete survey in a more general setting. As for the topological dynamics, the behavior of horocycles poses many questions in the non-compact case. M.Kulikov has constructed in [11] a non-compact hyperbolic surface with infinitely generated fundamental group and without minimal sets for the horocycle flow and S. Matsumoto has announced other examples in [14]. The dynamics of the recurrent horocycle orbits under different assumptions has been extensively studied by Y.Coudène, F.Ledrappier, F.Maucourant and B.Schapira, among others.
In the present paper we consider a compact lamination (M, F) by hyperbolic surfaces. Its unit tangent bundle is a lamination obtained by taking the unit tangent bundles of the leaves of F, and it is defined by a continuous P SL(2, R)-action.
We study the horocycle flow -that is, the action of the unipotent subgroup U of P SL(2, R)-on this space when the foliation is minimal. The actions of the diagonal group D and the affine group B play an important role in this study. The idea of studying the geodesic flow for compact laminations is not new. Its dynamics and its relation with the dynamics of the foliation F have been studied in several contexts, see for example [9], [2].
As the lamination is minimal, one might expect that the compactness of the ambient space forces the minimality of the horocycle flow, like for compact hyperbolic surfaces. Nevertheless, this is not the case, since there are examples where not even the B-action is minimal (see [1] and [13]). In [13], the last two authors have posed the following question: Is the minimality of the B action equivalent to the minimality of the horocycle flow? This question has been answered in the case of homogeneous Riemannian foliations by the first two authors in [1].
The main result of this paper gives a positive answer to the question above for a large family of laminations by hyperbolic surfaces: Theorem 1. Let (M, F) be a minimal compact lamination by hyperbolic surfaces. If F admits a leaf whose holonomy covering is not homeomorphic to the plane, then the horocycle flow on the unitary tangent bundle T 1 F is minimal.
A foliation by hyperbolic surfaces that satisfies the hypotheses of this theorem is the Hirsch foliation, described below. It was this foliation that motivated many of the ideas present in this paper. Furthermore, we believe that in this example, because of the simplicity of its construction, many of the arguments we use become particularly transparent.
Example 1 (The Hirsch foliation). Let P be the closed unit disk minus two open disks of radius 1/4 centered at -1/2 and 1/2 in the complex plane; namely, a pair of pants. P is then invariant under the involution T : z ∈ P → −z ∈ P . Consider the suspension S of T . S is a non-trivial fiber bundle over S 1 , whose holonomy interchanges the two interior boundary components of the pair of pants. Therefore, it can be obtained from a solid torus removing from its interior a thinner solid torus wrapped two times around the generator of its fundamental group. Let p : S → S 1 be the fibration. The boundary of S consists of two tori T 1 and T 2 , and the restriction of p to each torus T i is itself a fibration p i with fiber S 1 . (See Figure 1).

Figure 1. The Hirsch foliation
Let M be the 3-manifold obtained from S by glueing smoothly T 1 and T 2 via a diffeomorphism h that sends p −1 1 (θ) to p −1 2 (θ) for every θ in S 1 . The fibration in S projects onto a foliation F having two types of leaves: there are countably many leaves of genus one and a Cantor set of ends, and all other leaves have no genus and a Cantor set of ends (they are so-called Cantor trees). Therefore, all leaves are hyperbolic surfaces. (See Figure 2.) All leaves are dense and those of genus one are exactly the ones with non-trivial holonomy.  The non-homogeneous Lie foliations constructed by G. Hector, S. Matsumoto and G. Meigniez in [6] are other examples of minimal foliations satisfying the hypotheses of Theorem 1 for which the horocycle flow is always minimal.

Preliminaries and notation
Hyperbolic surfaces. A hyperbolic surface S is the quotient of the hyperbolic plane H under the left action of a discrete subgroup Γ of the group P SL(2, R) of orientation preserving isometries of H. This group acts freely and transitively on the unit tangent bundle T 1 H of H, which means that we can make the identification T 1 S = Γ\P SL(2, R).
Let D and U be the diagonal and unipotent subgroups of P SL(2, R) Their right actions define the geodesic flow g t and the horocycle flow h s on T 1 S, respectively. Therefore, the joint action of g t and h s is the action of the affine group If I ⊂ R, the subset {g t (u); t ∈ I} of the orbit of a point u ∈ T 1 S under the geodesic flow is g I (u), and a similar notation will be used for subsets of horocycle orbits.
The projection π : T 1 S → S is the canonical projection that assigns to each vector in T 1 S its base point. • ϕ α : U α → D × T is a homeomorphism, where D is a disk in R 2 and T is a topological space, and , where λ t αβ is smooth and depends continously on t in the C ∞ topology.
We will always work in the smooth setting unless otherwise stated, although C 3 regularity would be enough for all our purposes.
Each U α is called a foliated chart, a set of the form ϕ −1 α ({x} × T ) being its transversal. The sets of the form ϕ −1 α (D × {t}), called plaques, glue together to form maximal connected surfaces called leaves.
A lamination is said to be minimal if all its leaves are dense.
The tangent bundle of the lamination F is the R 2 -bundle over M which can be trivialized on each foliated chart . It is itself a (noncompact) lamination, whose leaves are the tangent bundles of the leaves of F.
Laminations by hyperbolic surfaces. In each foliated chart we can endow each plaque with a Riemannian metric, in a continuous way. Glueing these local metrics with partitions of unity gives a Riemannian metric on each leaf, which varies continuously in the C ∞ topology as we move from leaf to leaf.
When (M, F) is endowed with such a metric, we define the unit tangent bundle T 1 F of F as the subset of the tangent bundle containing vectors of unit length. It is a circle bundle over M , and it is a lamination whose leaves are the unit tangent bundles of the leaves of F.
Furthermore, we will assume that there is a Riemannian metric for which all leaves are hyperbolic surfaces, that is, they have constant curvature −1. In fact, any given Riemannian metric endows each leaf with a conformal structure, or equivalently, with a Riemann surface structure. If all leaves are uniformized by the disk, then the uniformization is continuous and leaves become hyperbolic surfaces. See [3] and [19]. The existence of these hyperbolic metrics turns out to be a purely topological condition. It is equivalent to every leaf having positive volume entropy. This is independent of the metric because, M being compact, the restriction to leaves of all Riemannian metrics are quasi-isometric.
Once each leaf has a hyperbolic structure, there is a right continuous P SL(2, R)action on T 1 F whose orbits are the unit tangent bundles to the leaves. We will also denote by g t and h s the geodesic and horocycle flows on T 1 F defined by the action of the one-parameter subgroups D and U . The natural right B-action on T 1 F combines these two flows.

The horocycle flow on a minimal foliation by leaves with good pants decomposition
In next sections, we will prove the following (apparently weaker) version of the main theorem which was stated in the Introduction: Theorem 2. Let (M, F) be a minimal compact lamination by hyperbolic surfaces that have a good pants decomposition. Then the horocycle flow on the unitary tangent bundle X = T 1 F is minimal.
First, let us explain what we mean by a good pants decomposition for a noncompact hyperbolic surface: Definition 1. Consider a complete hyperbolic surface L = Γ\H which can be partitioned into countably many pairs of pants whose boundary components are closed geodesics with uniformly bounded lengths, or equivalently whose fundamental group Γ is purely hyperbolic of the first kind. We will say L has a good pants decomposition. See [8] and [16] for the general notion of pants decomposition. Its fundamental group Γ is called tight by S. Matsumoto in [14].
We will consider foliated spaces by hyperbolic surfaces of this type, that are minimal in the sense that all leaves are dense. The best known example of such an object is probably the Hirsch foliation (described in the introduction). Studying the dynamics of the horocycle flow in its unit tangent bundle will involve the understanding of the dynamics of the horocycle flow on a single leaf (i.e. on a single hyperbolic surface with a good pants decomposition), and also considerations about the interplay between the dynamics of the foliation and that of the horocycle flow on individual leaves.
As before, h t will be the horocycle flow and g t the geodesic flow. Their joint action is a right action of the Borel group B, the group of affine transformations of the real line.
is a discrete subgroup of R, so it is either trivial or cyclic.
By hypothesis, C is nontrivial and hence the B-invariant set Therefore, to prove Theorem 2, we have to see that it verifies the hypotheses of Proposition 1.
The first condition, the minimality of the affine action, follows from a result in [13]. The fact that all the leaves have fundamental groups which are neither trivial nor cyclic allows us to use an argument byÉtienne Ghys found in [13,Theorem 2] that proves that the B-action is minimal.
As for the second condition, the transitivity of the horocycle flow, since F is minimal, it is enough to prove that the horocycle flow is transitive when restricted to some leaf L. This is the approach that we will take in the present paper, since the horocycle flow on a non-compact surface is interesting in its own right. Furthermore, it is in studying the dynamics of the flow on a single surface that we will get the third condition in Proposition 1: that any non-trivial minimal set should 'return' under the geodesic flow after a positive time t 0 .
Our strategy is based on geometrical considerations that involve the horocycle flow on a single leaf. Specifically, our aim is to prove that if a non-compact hyperbolic surface S has a good pants decomposition, then for every u ∈ T 1 S, -either h R (u) = T 1 S -or there is a real number t 0 > 0 such that g t0 (u) ∈ h R (u), This dichotomy will be proved in the next two sections. Section 4 is devoted to a Key Lemma where the 'return time' t 0 is shown to exist. Dense orbits appear in Section 5. This allows us to prove Theorem 1 in Section 6.

The horocycle flow on non-compact surfaces with a good pants decomposition
A hyperbolic surface S admits a good pants decomposition if and only if there exist a sequence (S n ) n≥1 of compact hyperbolic surfaces with boundary and two real numbers b ≥ a > 0 such that -S n ⊂ S n+1 for all n and ∪ n≥1 S n = S. -∂S n is composed of finitely many closed geodesics the lengths of which belong to [a, b]. These are, of course, hyperbolic surfaces of infinite area. Furthermore, their fundamental group is a Fuchsian group of the first kind having only hyperbolic elements. Remark 1. Any geodesic ray in S either stays in a compact region or intersects infinitely many closed geodesics whose length is bounded between a and b. When the boundaries of pants components are only bounded from above, a hyperbolic surface having this property has been called weakly tame by Sarig in [16]. Notice that the leaves of a compact lamination are weakly tame if and only if its fundamental groups are tight.
Key Lemma. Let S be a hyperbolic surface and u ∈ T 1 S. If there is a sequence (α n ) n≥1 of closed geodesics in S of lengths (α n ) ∈ [a, b], with b > a > 0, and a sequence of times (t n ) n≥1 such that t n −→ ∞ and π(g tn (u)) ∈ α n , then there exists a positive time t 0 such that g t0 (u) ∈ h R (u).
Proof. The universal cover of S is the hyperbolic plane H, and its unit tangent bundle is T 1 H. If Γ = π 1 (S), we write S = Γ\H. We lift u toũ ∈ T 1 H, that defines a geodesic ray r. Since each geodesic α n is closed on S, it lifts to a geodesicα n on H which is the axis of a hyperbolic element γ n ∈ Γ. Let γ + n and γ − n be the attracting and repelling fixed points of γ n on the boundary at infinity of H. (See Figure 3.) Taking the upper-half-plane model of the hyperbolic plane, we can assume that the geodesic ray directed byũ is the vertical half-line r = {ie t : t ≥ 0}, and that  the axesα n are half-circles orthogonal to the real line. By hypothesis, they intersect r. Furthermore, we will assume without loss of generality that when we orientα n going from γ − n to γ + n , the angle between r andα n is smaller or equal than π/2 for all n. Let r(t n ) be the point ie tn where r intersectsα n .
We will also denote by g and h, respectively, the geodesic and horocycle flows in T 1 H.
Proving that there exists a t 0 > 0 such that g t0 (u) ∈ h R (u) amounts to finding a sequence (γ n ) n in Γ and a sequence of times s n such that γ n h sn (ũ) = g t0 (ũ).
Let H be the horizontal line passing through i, that is, the projection to H of the horocycle orbit of the pointũ. The above condition simply states that γ n (H) approaches the horizontal line passing through ie t0 . (See Figure 4.) Therefore, it suffices to prove that there exist a sequence (γ n ) n in Γ and a constant t 0 > 0 such that (i) γ n (∞) → ∞ and (ii) B γ n (∞) (i, γ n (i)) −→ t 0 , where B is the Busemann function given by B ξ (x, y) = lim z→ξ [d(y, z) − d(x, z)].
The rest of the proof consists in finding such a sequence.
Remark 2. If B γn(∞) (i, γ n (i)) → −∞, then we have that This means that ∞ is a horocyclic point for Γ, which happens if and only if h R (u) is dense in the non-wandering set of h. In particular, h R (u) ⊃ g R (u). See for example [5].
Therefore, we can assume that there exists a constant A ∈ R such that B γn(∞) (i, γ n (i)) ≥ A.
We will prove (i) and (ii) for γ n a convenient subsequence of iterates of γ n , dividing the proof in several steps.
Step 1. We will show that γ n (∞) −→ ∞. Since r(t n ) → ∞ and it belongs to the axisα n of γ n , one of the endpoints γ + n , γ − n ofα n goes to infinity. If γ + n → ξ = ∞, then γ − n → ∞ and the angle at the point r(t n ) between the ray r and the oriented geodesicα n would approach π as n grew. This is inconsistent with our assumption that this angle is bounded above by π 2 . Therefore γ + n → ∞. Step (3) Looking at the dynamics of γ n ∈ P SL(2, R) acting on H ∪ ∂H, we see that γ n ∞ ∈ [γ + n , ∞), therefore γ n (∞) → ∞ as stated.
Step 2. We will show that there exists B ∈ R such that , r(t n )) ≤ b, since both γ −1 n r(t n ) and r(t n ) belong to the closed geodesic α n the length of which is bounded by b, and • B ∞ (γ −1 n i, γ −1 n r(t n )) ≤ d(γ −1 n i, γ −1 n r(t n )) = d(i, r(t n )) = t n . Therefore B γn∞ (i, γ n i) ≤ −t n + b + t n = b.
Step 3. We would like to show that there exists C > 0 such that In fact we will see that there is a positive k such that for every n B γ k n ∞ (i, γ k n i) ≥ C. As before, we consider the decomposition , r(t n )) + B ∞ (r(t n ), i), and we will compute each of these three terms.
For the first term we consider the geodesic ray c n going from r(t n ) to γ n ∞. We have that Take the geodesic path from γ + n to r(t n ) and the one from r(t n ) to i. They form an angle greater or equal to π/2. (See Figure 5.) Then the geodesic path from c n (t) to r(t n ) and the one from r(t n ) to i form an angle bounded away from zero by some constant θ 0 independent of n and t. (See Figure 5.) Therefore there is a constant c(θ 0 ) > 0 such that d(i, c n (t)) ≥ d(i, r(t n )) + d(r(t n ), c n (t)) − c(θ 0 ) = t n + t − c(θ 0 ) ≥ t n − c(θ 0 ), and this proves that B γn∞ (i, r(t n )) ≥ t n − c(θ 0 ). Now we will deal with the second term.
The angle formed by the geodesic path from γ −1 n r(t n ) to r(t n ) and the one from r(t n ) to r(t) is greater or equal to π/2. (See Figure 5.) Therefore, there is a constant c(π/2) > 0 such that , r(t n )) + (t − t n ) − c(π/2). We also know that d(γ −1 n r(t n ), r(t n )) = length(α n ) ≥ a, so we have that d(γ −1 n r(t n ), r(t)) − (t − t n ) ≥ a − c(π/2). As before, the third term is equal to t n . Putting everything together, we have that . To obtain the desired conclusion, it would be enough to verify that a − c(θ 0 ) − c(π/2) > 0. This might not hold, but if we replace γ n with γ k n for a sufficiently large k, we will get B γ k n ∞ (i, γ k n i) ≥ ka − c(θ 0 ) − c(π/2) > 0. An appropriate subsequence of the γ k n satisfies conditions (i) and (ii).

Horocycle flows without minimal sets
In this section, we start by proving the announced dichotomy for each orbit closure of the horocycle flow of any hyperbolic surface with good pants decomposition. As a corollary, we deduce the existence of a 'return time' for any proper h R -minimal set C = ∅ in T 1 S. Note that an h R -minimal set C is an h R -invariant closed subset of T 1 S which is minimal by inclusion.
On the other hand, recall that the horocycle flow is dual to the linear action of the fundamental group Γ, which is now of the first kind, and hence the dynamics of the horocycle flow is represented by this action. In fact, the dichotomy is related to the following classical notion: Definition 2. Let Γ be a non-elementary Fuchsian group of the first kind. A point ξ ∈ ∂H is said to be horocyclic if any horodisc centered at ξ contains points of the orbit Γz for any z ∈ H.
This kind of limit points admits the following well-known characterization (see [5]): Proposition 2. For each v ∈ T 1 H, the following conditions are equivalent: (i) the point ξ = v(+∞) is horocyclic, (ii) the horocycle orbit of the projected point u ∈ T 1 S is dense, (iii) 0 belongs to Γυ where υ is the element of E corresponding to the horocycle passing through v.
Proposition 3. Let S be a non-compact hyperbolic surface having a good pants decomposition. Then, for any tangent vector u ∈ T 1 S, either h R (u) = T 1 S or there is a real number t 0 > 0 such that g t0 (u) ∈ h R (u). In particular, for any proper h R -minimal set C = ∅ in T 1 S, there is t 0 > 0 such that g t0 (C) = C.
Proof. Let u ∈ T 1 S, andũ be a lift of u to T 1 H. If the endpointũ(+∞) of the geodesic directed byũ is a horocyclic point, then, according to Proposition 2, the horocycle orbit h R (u) is dense in T 1 S. Remark that points u which are periodic for the geodesic flow are of this type: u(+∞) is horocyclic. Now, if u directs a geodesic ray that stays in a compact set, consider ω(u), the ω-limit set of u for the geodesic flow. Taking a flow box for the restriction of the flow to ω(u) and the corresponding transversal Σ, let p be the Poincaré first return map to Σ. It is a local Anosov diffeomorphism, and therefore its periodic points are dense. This means that ω(u) contains a closed geodesic, soũ(+∞) must be a horocyclic point.
If u directs a geodesic ray that leaves all compact sets, Key Lemma tells us that there is a positive time t 0 such that g t0 (u) ∈ h R (u), which completes the proof.
Using this proposition, we retrieve a result stated by S. Matsumoto in [14] about the non-existence of minimal sets for the horocycle flow when the hyperbolic surface S admits a good pants decomposition.
Theorem 3 (Matsumoto). For any non-compact hyperbolic surface S having a good pants decomposition, the horocycle flow has no minimal sets.
Proof. Suppose on the contrary that h s admits a proper minimal set C = ∅ in T 1 S. From Proposition 2, this minimal set C contains a horocycle centered at a non-horocyclic limit point ξ of Γ. Now, according to the Proposition 3, for any point u in this horocycle, there is t 0 > 0 such that g t0 (u) ∈ h R (u) = C. Therefore g t0 (C) = C by minimality. Finally, denoting by υ the point of E defined by the horocycle centered at ξ, we deduce by duality that e t0/2 (Γυ) = Γυ and therefore 0 ∈ Γυ, which contradits the assumption that ξ is non-horocyclic.

Foliations with topologically non-trivial leaves
We will begin this section by completing the proof of Theorem 2, and later we will prove Theorem 1.
Proof of Theorem 2. Let M = ∅ be a minimal set for the horocycle flow in T 1 F. To prove that M = T 1 F, we start by observing that both situations described in Proposition 3 arise for any leaf S. The flow h s has a dense orbit in T 1 S, and therefore in T 1 F since F is minimal. On the other hand, for each point u ∈ M, we have a h Rinvariant closed set C induced by M on the unitary tangent bundle T 1 S containing u. If the orbit of u is dense in T 1 S, then C = T 1 S and M = T 1 F. Ohterwise, there is a real number t 0 > 0 such that g t0 (C) = C and hence g t0 (M) = M. Now, we conclude by applying Proposition 1.
In fact, a corollary of the proof is the following remark: Our final purpose is to prove Theorem 1. In fact, we will show that it is equivalent to Theorem 2. This is the content of the following statement: Theorem 4. Let (M, F) be a minimal compact lamination by hyperbolic surfaces. If F admits a leaf whose holonomy covering is not homeomorphic to the plane, then all the leaves admit a good pants decomposition.
Proof. We will divide the proof in three steps, which are the following.
(1) if there exists a leaf L whose holonomy cover is not a plane, then there is a leaf with trivial holonomy that is not a plane.
(2) if there exists a leaf L that is neither a plane or a cylinder and has trivial holonomy, then all leaves have a good pants decomposition. (3) there cannot be a leaf which is a cylinder and has trivial holonomy.
Step 1. We will in fact prove that if there is a leaf L whose holonomy covering is not a plane, then no leaf is a plane. Or equivalently, that if there is a leaf L which is a hyperbolic plane, then the holonomy cover of any leaf L is simply connected.
We will use the following result by P.Lessa, which can be found in [12, Theorem 3.3]: Semicontinuity Theorem. Let (M, F) be a compact lamination and let x n be a sequence converging to a point x ∈ M . If the sequence L xn of pointed leaves passing through the x n converges in the Gromov-Hausdorff topology to a pointed Riemannian manifold, then the limit is a covering of the leaf L x passing through x which is a quotient of the holonomy covering of L x .
As always, F is a minimal lamination by hyperbolic surfaces. Let L be, as above, a leaf which is a hyperbolic plane, and let L be any other leaf. Minimality of the lamination implies that there is a sequence x n of points in L that converges to x ∈ L. Since the hyperbolic plane is homogeneous, as pointed Riemannian manifolds all the (L , x n ) represent the same point in Gromov-Hausdorff space-namely, the pointed hyperbolic plane. The Semicontinuity Theorem implies that the hyperbolic plane is covered by the holonomy cover of L, which must therefore be simply connected.
Step 2. Let L be a leaf without holonomy that is neither a plane nor a cylinder. It contains a pair of pants -that is, an isometric embedding ϕ : P → L, where P is a compact hyperbolic surface with no genus and three boundary components. We will also assume that the boundary components of P are closed geodesics. In fact, we can (and will) take the domain of ϕ to be a slightly larger pair of pants V containing P in its interior.
Since L has trivial holonomy and ϕ(V ) is compact, ϕ extends to a foliated embedding Φ : V × T → M , where T is a transversal to F. Pulling back the leafwise hyperbolic metric from M to V × T , we get a metric on each pair of pants V × {t} depending continously on t ∈ T in the C ∞ topology. Taking a smaller T if necessary, for each t ∈ T there is a pair of pants P t ⊂ V × {t} with geodesic boundary components. Then is a subset of M which is homeomorphic to the product P × T via a homeomorphism which respects the foliated structures.
The continuity of the metric implies that there are uniform bounds b ≥ a > 0 for the lenghts of the boundary components of the Φ(P t ). In fact, since M is compact and the metric on (M, F) is continuous, the injectivity radius of the leaves is bounded away from zero, and we will take a to be a lower bound to the injectivity radius.
Let L be any leaf of F. Since the foliation is minimal, it must pass through W , and in fact W ∩ L is a countable union of pairs of pants that is quasi-isometric to L . This shows, in the first place, that L is neither a plane nor a cylinder. Therefore, it has a decomposition in pairs of pants. The boundary components of the pieces in this decomposition are closed geodesics with lenghts greater than a. We only have to show that their lenghts are bounded from above. But if this were not true, that is, if L had arbitrarily large simple closed geodesics, it would not be quasi-isometric to L ∩ W . This completes the proof of the step 2.
Step 3. Assume on the contrary there exists a leaf L which is a cylinder and has trivial holonomy. It contains a non-trivial separating cycle -that is, an embedding c : S 1 → L which represents a non-trivial homology class in H 1 (L; Z).
Since L has trivial holonomy, c extends to a foliated embedding C : S 1 × T → M , where C| {1}×T embeds a transversal T around the point t 0 = c(1) ∈ L into M . For each t ∈ T , the restriction of C to S 1 × {t} defines a map c t : S 1 → L t which represents a homology class in the homology group H 1 (L t ; Z) of the leaf L t passing through t. In particular, L contains infinite countably many separating cycles C t = c t (S 1 ).
Assume first there are sequences of non-trivial separating cycles C − n = c t − n (S 1 ) and C + n = c t − n (S 1 ) approaching each of two ends of L. For a large enough n, the cycles C − n and C + n bound a compact domain D n in L whose area goes to infinity as n → +∞, but the length of ∂D n = C − n ∪ C + n remains uniformly bounded. As a union of these compact domains D n , the leaf L becomes Følner or closed at infinity, which contradicts the hyperbolicity of F.
Otherwise, except for a finite number of cycles, all the other cycles C t in L are homologically trivial. Thus, we can consider a sequence {t n } converging to t 0 such that all the corresponding cycles C n bound a compact domain D n in L. As before, using an argument by D. Sullivan (see [18,Theorem II.15]), we have that the length of cycles C n = ∂D n is uniformly bounded, but the area of D n goes to infinity as n → +∞. Even if the Følner sequence does not exhaust the leaf, L becomes Følner or closed at infinity.