QUASIGEODESIC FLOWS IN HYPERBOLIC THREE-MANIFOLDS

SÉRGIO FENLEY AND LEE MOSHER

Abstract. Any closed, oriented, hyperbolic 3-manifold with nontrivial second homology has many quasigeodesic flows, where quasigeodesic means that flow lines are uniformly efficient in measuring distance in relative homotopy classes. The flows are pseudo-Anosov flows which are almost transverse to finite depth foliations in the manifold. The main tool is the use of a sutured manifold hierarchy which has good geometric properties.

Introduction

In this article we prove that any closed, oriented, hyperbolic 3-manifold with nontrivial second homology has many quasigeodesic flows, where quasigeodesic means that flow lines are uniformly efficient in measuring distance in relative homotopy classes. The flows are pseudo-Anosov flows which are almost transverse to finite depth foliations in the manifold. The main tool is the use of a sutured manifold hierarchy which has good geometric properties.

The best metric property a flow can have is that all its flow lines are minimal geodesics in their relative homotopy classes, which amounts to being minimal geodesics when lifted to the universal cover. Suspensions of Anosov diffeomorphisms of the torus and geodesic flows on the unit tangent bundle of surfaces of constant negative curvature have this property. Even for these examples one has to choose an appropriate metric to get the minimal property. If the metric is changed the flow lines are only quasigeodesics: when lifted to the universal cover, length along flow lines is a bounded multiplicative distortion of length in the manifold plus an additive constant. The concept of quasigeodesic has the advantage of being independent of the metric in the manifold. We say that a flow is quasigeodesic if all flow lines are quasigeodesics.

Our main interest is in hyperbolic manifolds. In these manifolds, a quasigeodesic in the universal cover is a bounded distance from some minimal geodesic [Th1]. This, among other reasons, makes quasigeodesics extremely useful in studying hyperbolic manifolds [Th1, Th2, Mor, Ca].

A natural question to ask is: how common are quasigeodesic flows? Notice that Zeghib [Ze] proved that there cannot exist a continuous foliation by geodesics in a

Research at MSRI is supported in part by NSF grant DMS-9022140. Both authors were supported by additional NSF grants.

2 SÉRGIO FENLEY AND LEE MOSHER

hyperbolic 3-manifold. On the other hand, in their seminal work [Ca-Th], Cannon and Thurston showed that if a hyperbolic 3-manifold M fibers over the circle, then M has a quasigeodesic flow which is transverse to the fibers. Afterwards Zeghib [Ze] gave a quick and elementary proof that for any compact manifold M which fibers over a circle, any flow transverse to the fibration is quasigeodesic. Mosher [Mo3] produced examples of quasigeodesic flows transverse to a class of depth one foliations in hyperbolic 3-manifolds.

The flow constructed in [Ca-Th] is the suspension of a pseudo-Anosov homeomorphism of the fiber. Hence it is a pseudo-Anosov flow, that is, it has stable and unstable foliations in the same way as Anosov flows do, except that one allows p-prong singularities along finitely many closed orbits. The quasigeodesic and pseudo-Anosov properties are used in an essential way in Cannon-Thurston’s proof that lifts of fibers extend continuously to the sphere at infinity, providing examples of sphere filling curves. The flows constructed in [Mo3] are also pseudo-Anosov. Mosher showed that quasigeodesic pseudo-Anosov flows on hyperbolic manifolds can be used to compute the Thurston norm [Mo1, Mo2].

The quasigeodesic property for Anosov flows in hyperbolic 3-manifolds has also been extensively studied by Fenley who showed that there are many examples which are not quasigeodesic [Fe1]. In addition the quasigeodesic property for Anosov flows is related to the topology of the stable and unstable foliations in the universal cover [Fe2] and implies that limit sets of leaves of these foliations are Sierpinski curves [Fe3].

The main goal of this paper is to show that quasigeodesic flows are quite common. If M^{3} is closed, oriented, irreducible with H_{2}(M) ≠ 0 and if ζ ≠ 0 in H_{2}(M), then Gabai [Ga1] constructed a taut, finite depth foliation F whose set of compact leaves represents ζ. Given such F in a hyperbolic 3-manifold, Mosher [Mo4] constructed pseudo-Anosov flows which are almost transverse to F. Almost transverse means that it will be transverse to F after an appropriate blow up of a finite collection of closed orbits (see detailed definition in section 4).

Main theorem Let M be a closed, oriented, hyperbolic 3-manifold with non zero second betti number and let ζ a nonzero homology class in H_{2}(M). Let F be a taut, finite depth foliation whose compact leaves represent ζ, and Φ a pseudo-Anosov flow which is almost transverse to F. Then Φ is a quasigeodesic flow.

If the compact leaves of F are fibers, the main theorem follows from the above mentioned result of Zeghib by showing that every flow line of Φ hits the compact leaves. If some compact leaf is not a fiber the theorem easily follows from a more general result:

Theorem A Let M be a closed, oriented, hyperbolic 3-manifold with a taut, finite depth foliation F, so that some compact leaf is not a fiber of M over the circle. Let Φ be a flow transverse to F and let Φ̃ be the lifted flow to the universal cover. Then Φ is quasigeodesic if and only if Φ̃ has Hausdorff orbit space.

The orbit space of a flow is the quotient space obtained by collapsing each flow line to a point. We remark that the only if part of theorem A is straightforward. One might also ask whether the condition in theorem A is non void, that is, if there are flows transverse to Reebless finite depth foliations for which the orbit space of Φ̃ is not Hausdorff. Indeed this is possible. Examples are easy to construct containing a flow invariant annulus Z in M, so that flow lines induce in Z a 2-dimensional Reeb foliation. In that situation clearly the orbit space of Φ̃ is not Hausdorff.

When F has a nonfiber compact leaf the main theorem follows from theorem A via two remarks. (1) The important fact is that for the original pseudo-Anosov flow Φ, the covering flow Φ̃ has Hausdorff orbit space. This is where the pseudo-Anosov dynamics plays an essential role. (2) This implies that the blown up flow also has Hausdorff orbit space, and since it is transverse to F it is quasigeodesic by theorem A. As a consequence the original flow is also quasigeodesic.

Here are the key ideas in the proof of theorem A. Some (and hence any) compact leaf of F represents a quasi-Fuchsian subgroup. Therefore this leaf lifts to a quasiisometrically embedded surface in M̃, which has excellent geometric properties. That means that if a flow line in M̃ keeps intersecting lifts of compact leaves, these lifts trap the flow line which then converges to a single point in S_{∞}^{2}.

Next we proceed to extend this argument to all orbits. For that, we use sutured manifold hierarchies and branched surfaces associated to the foliation F. By general principles the sutured manifolds in the hierarchy have good geometric properties, that is, they are quasi-isometrically embedded when lifted to the universal cover and the cutting surfaces in the hierarchy are also quasi-isometrically embedded. The cutting surfaces play the role of compact surfaces in the appropriate sutured manifold in the hierarchy. Using an induction argument with the sutured manifold hierarchy, we can show that for any point in M̃, its flow line in forward time converges to a unique point in the sphere at infinity S_{∞}^{2}, and likewise for the negative direction. This is first shown in the compactified universal cover of the appropriate sutured manifold in the hierarchy and then derived in H^{3} ∪ S_{∞}^{2} by way of the good geometric properties of the sutured manifolds. The existence of a unique limit point of flow lines is a much weaker property than being quasigeodesic: for example horocycles have this property but they are not quasigeodesic.

We also show that the limit point map of flow lines is continuous and that forward and backward limit points in each orbit are distinct. Quasigeodesics satisfy all of these properties. These results only keep track of the asymptotic behavior of a flow line, but a priori do not determine the rough location of the flow line — which must be the case for quasigeodesics. For flows, however, these properties are indeed sufficient to ensure quasigeodesic behavior as proved by:

Theorem B Let Φ be a flow in M^{3} closed hyperbolic. Suppose that:

(a) Each half orbit of Φ̃ has a unique limit point in S_{∞}^{2},

(b) For a given orbit, the forward and backward limit points are distinct,

4 SÉRGIO FENLEY AND LEE MOSHER

(c) The forward and backward limit point maps are continuous. Then Φ is quasigeodesic.

The paper is organized as follows. In section 1 we prove a generalization of theorem B, which applies to closed invariant sets of Φ. This is used in the inductive step of the proof of theorem A. In section 2 we study sutured manifold hierarchies adapted to finite depth foliations and prove the needed geometric properties of the sutured manifolds. In section 3 we prove that conditions (a,b,c) of theorem B hold for flows satisfying the hypothesis of theorem A, and as a consequence derive theorem B. In the final section we study pseudo-Anosov flows and prove the main theorem.

We thank Dick Canary who pointed out to us the needed quasi-isometric properties of finitely generated Kleinian groups and also for many useful suggestions.

1. From continuous extension to quasigeodesic behavior

Here are the basic definitions we need concerning quasi-isometries and quasigeodesics. A good source for foundational material on quasi-isometries is [BKS], especially chapter 10 by Ghys and De la Harpe on “Infinite groups as geometric objects (after Gromov)” and chapter 11 by Cannon on “The theory of negatively curved groups and spaces”.

Definition 1.1. (quasi-isometries and quasigeodesics) Given metric spaces (X, d) and (Y, d′), a map f: X → Y is a quasi-isometric embedding if there is k ≥ 1 so that for any z, w ∈ X, we have

\frac{1}{k} d′(f(z), f(w)) − 1 ≤ d(z, w) ≤ k d′(f(z), f(w)) + k.

Once a metric is fixed we say that f is a k-quasi-isometry.

The spaces X, Y are quasi-isometric if there is a quasi-isometry f: X → Y and a constant k′ such that each point of Y is within distance k′ of the image of f.

Given a metric space X, a quasigeodesic is a map f: R → X which is a quasiisometry. If f is a k-quasi-isometry we say that f is a k-quasigeodesic.

A flow on a manifold X is a continuous action of R on X, i.e. a continuous map ψ: X × R → X written (x, t) → ψ_{t}(x), such that

ψ_{s+t}(x) = ψ_{s}(ψ_{t}(x)) for all x ∈ X, s, t ∈ R.

A semiflow on X is defined similarly, except that the domain of ψ is a closed subset D ⊂ X × R such that for each x ∈ X, (x × R) ∩ D = x × J for some closed, connected set J ⊂ R containing 0, and the above equation holds whenever the two sides are defined. All flows and semiflows in this article are generated by nonzero, continuous vector fields, i.e. flow lines are smooth immersions and the tangent vector field is continuous on X. When ψ is understood, we often write x · t = ψ_{t}(x), and if J ⊂ R is an interval we write x · J = ψ_{J}(x). If y = x · t let [x, y] = x · [0, t] and (x, y) = [x, y] − {x, y}; also let τ(x, y) = |t|.

Definition 1.2. (uniformly quasigeodesic flows) Let Φ be a flow on a compact manifold M. Let Φ̃ be the lifted flow on the universal cover M̃. We say that Φ is uniformly quasigeodesic if there exists a constant k ≥ 1 such that for each x ∈ M̃, the map t → x · t is a k-quasigeodesic.

Since M is compact this property is independent of the parameterization of Φ. Henceforth flowlines are always parameterized by arc length. Given this, a flow line in M̃ is k-quasigeodesic if for any x, y in the flow line, τ(x, y) ≤ k d(x, y) + k. We sometimes say that flow lines of Φ are quasigeodesic.

We first show that there is a weak form of quasigeodesic behavior which follows from purely topological properties of the orbit space of Φ̃. Given ϵ, T > 0 an ϵ, T cycle of Φ is a closed loop in M obtained from an orbit segment of length > T with endpoints less than ϵ apart, closed up by an arc of length < ϵ. The following lemma does not assume that the manifold is hyperbolic. This lemma is not logically necessary for proving the main theorem and theorems A, B; but it describes additional hypotheses (which are quite common) under which the proofs of these theorems can be simplified.

Lemma 1.3. (weak quasigeodesics) Let Y be a closed invariant set of a flow Φ in a closed manifold M. Let Ỹ be the lift of Y to M̃. Suppose that

(a) Φ̃|Ỹ has Hausdorff orbit space,

(b) There are ϵ, T > 0 so that any ϵ, T cycle of Φ|_{Y} is not null homotopic in M. Then for any b > 0 there is c_{b} > 0 (depending only on b) so that if x, y are in an orbit γ of Φ̃ and τ(x, y) > c then d(x, y) > b.

Proof. Otherwise there is b > 0 and x_{i}, y_{i} ∈ Ỹ, with x_{i}, y_{i} in the same orbit γ_{i} of Φ̃ and τ(x_{i}, y_{i}) > i but d(x_{i}, y_{i}) < b. Since M is compact, then up to covering translations and taking subsequences we may assume that x_{i} → x and y_{i} → y. Notice that x, y ∈ Ỹ since Ỹ is closed.

By hypothesis (a) it follows that x and y are in the same orbit of Φ̃ so y = x · t for some t ∈ R. By the local product structure of the flow along compact orbit segments, there are t_{i} → t so that z_{i} = x_{i} · t_{i} → y. If i is big enough then τ(z_{i}, y_{i}) > T and d(z_{i}, y_{i}) < ϵ, thereby producing ϵ, T -cycles of Φ̃. These project to null homotopic ϵ, T -cycles of Φ, contradiction.

Remarks: (1) Notice in particular that condition (b) implies that orbits of Φ̃ are never periodic and are properly embedded in M̃. When (b) holds we say that Φ|_{Y} satisfies the ϵ, T -cycles condition.

(2) Quasigeodesic behavior is the additional property that c_{b} is bounded by an affine function of b. In general conditions (a),(b) are not sufficient to ensure quasigeodesic behavior. For instance Anosov flows always satisfy these conditions [Fe1],

6 SÉRGIO FENLEY AND LEE MOSHER

but there are many examples of Anosov flows in hyperbolic manifolds which are not quasigeodesic [Fe1].

(3) When Y = M conditions (a),(b) together are equivalent to the orbit space O of Φ̃ being homeomorphic to either the plane R^{2} or the sphere S^{2}. (a) implies that O is a 2-dimensional manifold, and it is Hausdorff. Since O is simply connected and has no boundary it is either S^{2} or R^{2}. This means that the flow Φ̃ is topologically a product flow in M̃. Lemma 1.3 means that topological product flows always satisfy a weak quasigeodesic property. This is reminiscent of the same situation for codimension one foliations which was studied in [Fe4]. Finally we remark that our main interest is in closed hyperbolic manifolds, where π_{2}(M) is trivial. In that case conditions (a),(b) of lemma 1.3 are equivalent to Φ̃ having orbit space homeomorphic to R^{2}.

For flows in hyperbolic 3-manifolds, we now develop a method to upgrade information about asymptotic behavior of flow lines of Φ̃ into metric efficiency of flow lines. This will be the key tool to prove uniform quasigeodesic behavior for a large class of flows in hyperbolic 3-manifolds.

Theorem 1.4. Let Y be a closed invariant set of a non singular flow Φ in M^{3} closed hyperbolic. Suppose that

(a) half orbits converge: for any x ∈ Ỹ each of the two rays of x · R accumulate in a single point of S_{∞}^{2}, that is, the following limits exist:

lim_{t→+∞} x · t = η_{+}(x) ∈ S_{∞}^{2} and lim_{t→−∞} x · t = η_{−}(x) ∈ S_{∞}^{2}.

(b) For each x ∈ Ỹ, η_{+}(x) ≠ η_{−}(x).

(c) The maps η_{+}, η_{−} : Ỹ → S_{∞}^{2} are continuous. Then the orbits in Φ̃|Ỹ are uniformly quasigeodesics: there is k > 0 so that for any orbit γ of Φ̃ in Ỹ and for any x, y ∈ γ, τ(x, y) < kd(x, y) + k.

Notice that conditions (a), (b) and (c) are necessary to get quasigeodesic behavior. Conditions (a) and (b) follow directly from the fact that single flow lines are quasigeodesics [Th1, Gr, Gh-Ha]. Condition (c) is not true for an arbitrary collection of quasigeodesics, but holds for uniformly quasigeodesic closed invariant sets of flows. Since we will use this last fact in the proof of theorem A, we provide a proof in section 3.

Proof. Since η_{+}(x) ≠ η_{−}(x) for any x ∈ Ỹ, let g_{x} be the unique geodesic in H^{3} with endpoints η_{+}(x), η_{−}(x) and let ∂g_{x} ⊂ S_{∞}^{2} be the ideal points of g_{x}. Given a > 0 let U_{a}(g_{x}) ⊂ H^{3} be the neighborhood of radius a around g_{x}. We first show:

(∗) there exists a > 0 so that for any x ∈ Ỹ we have x · R ⊂ U_{a}(g_{x}). Otherwise, let x_{i} ∈ Ỹ with d(x_{i}, g_{x_{i}}) → +∞. Up to covering translations and taking a subsequence assume that x_{i} → x. Hence x ∈ Ỹ and by (b), (c) we have

lim η_{+}(x_{i}) = η_{+}(x) ≠ η_{−}(x) = lim η_{−}(x_{i})

But d(x, g_{x_{i}}) → +∞, so up to taking a subsequence we may assume that there exists p ∈ S_{∞}^{2} such that lim(g_{x_{i}} ∪ ∂g_{x_{i}}) = p in the Hausdorff topology on closed subsets of H^{3} ∪ S_{∞}^{2} ≈ B^{3}. Therefore p = lim ∂g_{x_{i}} = lim{η_{+}(x_{i}), η_{−}(x_{i})}, contradicting that lim η_{+}(x_{i}) ≠ lim η_{−}(x_{i}).

We now assume that Φ̃|Ỹ is not uniformly quasigeodesic and derive a contradiction. There are two steps in the argument:

Step(1) Using (∗) we show that there are x_{i}, y_{i} in the same orbit of Ỹ with d(x_{i}, y_{i}) bounded but τ(x_{i}, y_{i}) → +∞.

If in addition one knows that Φ|_{Y} satisfies the ϵ, T -cycles condition and Φ̃|Ỹ has Hausdorff orbit space, then the conclusion of step (1) is disallowed by lemma 1.3, finishing the proof. However, these additional assumptions are not necessary because:

Step (2) The conclusion of step (1) is disallowed by conditions (a), (b) and (c). We need the following definitions: given w ∈ H^{3} and an oriented geodesic g ⊂ H^{3},

let P (w, g) be the hyperplane of H^{3} containing w and perpendicular to g. Let F (w, g) be the component of H^{3} − P (w, x) containing the positive endpoint of g in its closure and let B(w, x) be the other component. For w ∈ g and b > 0 let w + b be the point of g with d(w, w + b) = b and w + b in the ray from w to the positive endpoint of g.

For any x ∈ Ỹ we define ρ_{x}: H^{3} → g_{x} to be the closest point projection onto g_{x}. Then for any x, y in the same orbit of Φ̃ in Ỹ,

d(ρ_{x}(x), ρ_{x}(y)) ≤ d(x, y) ≤ d(ρ_{x}(x), ρ_{x}(y)) + 2a  (1)

because x, y ∈ U_{a}(g_{x}).

To prove step (1), because we have assumed that orbits are not uniformly quasigeodesic, for any C > 0 there is an orbit segment γ = [x, y] in Ỹ which satisfies Length(γ)/d(x, y) > 2C and Length(γ) > C. Assume first that d(x, y) ≥ 1 + 2a. Hence d(ρ_{x}(x), ρ_{x}(y)) ≥ 1 by (1). Also d(x, y) ≥ d(ρ_{x}(x), ρ_{x}(y)), so

\frac{Length(γ)}{d(ρ_{x}(x), ρ_{x}(y))} ≥ \frac{Length(γ)}{d(x, y)} > 2C ≥ C + \frac{C}{d(ρ_{x}(x), ρ_{x}(y))}

and therefore

\frac{Length(γ)}{C} > d(ρ_{x}(x), ρ_{x}(y)) + 1 > ⌈d(ρ_{x}(x), ρ_{x}(y))⌉

where ⌈x⌉ denotes the “ceiling” function, that is, the least integer ≥ x. Set n_{0} = ⌈d(ρ_{x}(x), ρ_{x}(y))⌉, so Length(γ) > n_{0}C. Also, (n_{0} − 1) < d(ρ_{x}(x), ρ_{x}(y)) ≤ n_{0}, so we can find consecutive points ρ_{x}(x) = z_{0}, z_{1}, …, z_{n_{0}} = ρ_{x}(y) on g_{x} such that d(z_{n−1}, z_{n}) = 1 if 1 ≤ n < n_{0} and 0 < d(z_{n_{0}−1}, z_{n_{0}}) ≤ 1. Let x = x_{0} and for 1 ≤ n ≤ n_{0} let x_{n} be the last point on γ = [x, y] such that ρ_{x}(x_{n}) = z_{n}, so the points x_{0}, x_{1}, …, x_{n_{0}} are consecutive on γ and they partition γ into subsegments γ = γ_{1} ∗ ⋯ ∗ γ_{n_{0}} with γ_{n} = [x_{n−1}, x_{n}]. Since

∑_{n=1}^{n_{0}} Length(γ_{n}) = Length(γ) > n_{0}C

then for some n we have Length(γ_{n}) > C.

This shows that regardless of whether d(x, y) ≥ 1 + 2a or not, we produce a pair x′, y′ so that d(x′, y′) < 1 + 2a but τ(x′, y′) > C. This finishes the proof of step 1.

We now prove step 2. Let then x_{i}, y_{i} with d(x_{i}, y_{i}) < 1 + 2a but τ(x_{i}, y_{i}) → +∞. Without loss of generality assume that y_{i} = x_{i} · t_{i} with t_{i} > 0. \underline{Case 1} — The sequence of intervals [x_{i}, y_{i}] has a subsequence with bounded diameter, which we may assume is the original sequence.

Let v_{i} be the middle point in [x_{i}, y_{i}] (with respect to the parametrization) and up to subsequence and covering translations assume that v_{i} → v ∈ Ỹ. For any t ∈ R, v_{i} · t → v · t and for i big enough v_{i} · t ∈ [x_{i}, y_{i}]. Since d(v_{i} · t_{i}, v_{i}) is bounded, this shows that d(v · t, v) is also bounded. It follows that v · R is contained in a bounded set in H^{3}, hence it accumulates in H^{3} contradicting condition (a) of the theorem.

\underline{Case 2} — There exist x_{i}, y_{i} ∈ Ỹ such that d(x_{i}, y_{i}) < 1 + 2a and diameter([x_{i}, y_{i}]) → +∞.

Let g_{i} = g_{x_{i}} and let ρ_{i}: H^{3} → g_{i} be the closest point projection. Choose w_{i} ∈ [x_{i}, y_{i}] so that

d(ρ_{i}(x_{i}), ρ_{i}(w_{i})) = max_{w∈(x_{i},y_{i})} d(ρ_{i}(x_{i}), ρ_{i}(w)).

Since diam[x_{i}, y_{i}] → +∞ and d(x, y) < 1 + 2a, then d(w_{i}, x_{i}) >> d(x_{i}, y_{i}). Assume first that w_{i} ∈ F (x_{i}, g_{i}), hence w_{i} ∈ F (y_{i}, g_{i}). If x_{i} ∈ B(y_{i}, g_{i}), then as P (y_{i}, g_{i}) separates x_{i} from w_{i}, we can find z_{i}, v_{i} ∈ [x_{i}, y_{i}] so that:

w_{i} ∈ (z_{i}, v_{i}), ρ_{i}(z_{i}) = ρ_{i}(v_{i}) = ρ_{i}(y_{i}) and ρ_{i}(w) ≠ ρ_{i}(y_{i}), ∀w ∈ (z_{i}, v_{i}).

Similarly if x_{i} ∈ F (y_{i}, g_{i}) ∪ P (y_{i}, g_{i}), we can find z_{i}, v_{i} satisfying the conditions above except that ρ_{i}(z_{i}) = ρ_{i}(v_{i}) = ρ_{i}(x_{i}). We obtain a similar statement if w_{i} ∈ B(x_{i}, g_{i}).

These arguments show that there are x_{i}, y_{i} so that ρ_{i}(x_{i}) = ρ_{i}(y_{i}), ρ_{i}(w) ≠ ρ_{i}(x_{i}) for any w ∈ (x_{i}, y_{i}) and diam[x_{i}, y_{i}] → +∞. If (x_{i}, y_{i}) ⊂ F (x_{i}, g_{i}), let w_{i} be defined as above. Since η_{+}(x_{i}) is in the closure of F (ρ_{i}(x_{i}) + 1, g_{i}) in H^{3} ∪ S_{∞}^{2} (where ρ_{i}(x_{i}) + 1 is computed in g_{i}) and y_{i} = w_{i} · r′_{i} with r′_{i} > 0, there must be a point u_{i} = w_{i} · s_{i}, with s_{i} > 0 and ρ_{i}(u_{i}) = ρ_{i}(w_{i}). In addition we may assume that (w_{i}, u_{i}) ⊂ B(w_{i}, g_{i}) and diam[w_{i}, u_{i}] → +∞.

Since d(w_{i}, x_{i}) → +∞ these arguments show that in any case there are x_{i}, y_{i} ∈ Ỹ, with ρ_{i}(x_{i}) = ρ_{i}(y_{i}), (x_{i}, y_{i}) ⊂ B(x_{i}, g_{i}) and diam[x_{i}, y_{i}] → +∞, see figure 1. Up to covering translations and taking a subsequence we may assume that x_{i} → x ∈ Ỹ.

Figure 1: Too much backtracking eventually traps orbits in the negative direction.

Since g_{i} = g_{x_{i}} → g_{x} and ρ_{i}(x_{i}) = ρ_{x_{i}}(x_{i}) → ρ_{x}(x), then

\overline{P (ρ_{x_{i}}(x_{i}) + 2, g_{i})} → \overline{P (ρ_{x}(x) + 2, g_{x})}

in the topology of closed sets of H^{3} ∪ S_{∞}^{2}. Then

η_{+}(x) ∈ \overline{F (ρ_{x_{i}}(x_{i}) + 1, g_{i})}

for i big enough. Hence there is t_{0} > 0 so that for i big enough x · [t_{0}, +∞) ⊂ F (ρ_{x_{i}}(x_{i}), g_{i}). It therefore follows that for i big enough there is r_{i} with r_{i} < t_{0} + 1 and x_{i} · r_{i} ∈ F (ρ_{x_{i}}(x_{i}), g_{i}). This contradicts the facts (x_{i}, y_{i}) ⊂ B(ρ_{x_{i}}(x_{i}), g_{i}) with y_{i} = x_{i} · t_{i} and t_{i} → +∞. This finally finishes the proof of theorem 1.4.

Remarks:

(1) The above proof works verbatim for M^{n} closed hyperbolic.

(2) Without much more work, one can also prove this result under more general assumptions:

(i) M is any manifold and π_{1}(M) is word hyperbolic in the sense of Gromov [Gr]. In that case instead of S_{∞}^{2} one uses the appropriate boundary at infinity ∂_{∞}M̃ — notice that if M is an oriented, irreducible 3-manifold then ∂_{∞}M̃ is still a 2-dimensional sphere [Be-Me].

(ii) Y is a compact space on which a flow Φ is defined, and instead of Y being a subset of M we have instead a continuous function f: Y → M. By the pullback construction we have f: Ỹ → M̃ and a flow Φ̃ on Ỹ, and theorem 1.4 applies.

2. Hierarchies

Finite depth foliations are closely related to sutured manifold hierarchies [Ga1]. In [Ga3] an “internal” version of a sutured manifold hierarchy was defined, in terms of branched surfaces. We review this subject here, providing some proofs of known but unpublished information, and we add some geometric information. For detailed definitions concerning foliations and laminations on 3-manifolds see [Ga1] and [Ga-Oe].

A 2-dimensional foliation of a 3-manifold M is a decomposition of M into 2dimensional manifolds called leaves which fit together in a local product structure. A foliation F is taut if it is transversely oriented, no leaf is a sphere, and each leaf of F intersects some closed curve in M which is transverse to F. The leaves of a taut foliation are π_{1}-injective [No].

A lamination of M is a foliation of a closed subset of M, covered by charts of the form D^{2} × [0, 1] so that each component of a leaf intersected with the chart has the form D^{2} × t for some t ∈ [0, 1]. A lamination Λ is essential if it has no sphere leaves, no Reeb components, and if M_{Λ} is the metric completion of M − Λ, then M_{Λ} is irreducible, boundary incompressible, and end incompressible; the latter condition means intuitively that ∂M_{Λ} has no “infinite folds”. The leaves of an essential lamination Λ are π_{1}-injective in M, and the components of M − Λ are π_{1}-injective.

A taut foliation is obviously an essential lamination. It is an exercise in the results of [Ga-Oe] to show that every sublamination of an essential lamination is essential.

Given a foliation F in a closed manifold M we say that a leaf L of F is proper if \overline{L} − L is a closed subset of M and F is proper if all leaves are proper. In that case Zorn’s lemma implies that F has compact leaves, which are then the depth 0 leaves. The depth is an ordinal ω defined by induction: a leaf L is at depth ω if \overline{L} − L is contained in the union of leaves of depth < ω, but not contained in the union of leaves < ω_{1} for any ω_{1} < ω. A proper foliation has finite depth n if n is the maximum of the depth of its leaves.

A branched surface in a closed 3-manifold M is a smooth 2-complex B ⊂ M such that for each x ∈ B, there exists a smoothly embedded open disc in B containing x, and all such discs are tangent at x, therefore determining a well-defined tangent plane T_{x}B. The set of points where B is not locally a surface is called the branch locus. A sector of B is a complementary component of the branch locus. In this paper, all branched surfaces will be transversely oriented. A flow (or semiflow) is transverse to B if, for each x ∈ B, the tangent vector to the flow points to the same side of T_{x}B as the transverse orientation.

Given a branched surface B ⊂ M, a semiflow neighborhood U(B) is a piecewise smooth regular neighborhood equipped with a semiflow φ satisfying the following conditions. Each orbit of φ is compact, transverse to B, and pierces B in at least one point. There is a decomposition of ∂U(B) into subsurfaces with disjoint interior

∂U(B) = ∂_{v}U(B) ∪ ∂_{h}^{−}U(B) ∪ ∂_{h}^{+}U(B)

Figure 2: Boundary tangencies of semiflows

such that orbits of φ point inward along ∂_{h}^{−}U(B), outward along ∂_{h}^{+}U(B), and are internally tangent along ∂_{v}U(B) as shown in figure 2. Each component of ∂_{v}U(B) is an annulus and the orbits of φ define a fibration of the annulus over the circle with interval fiber. If one forgets the orientation and parameterization of φ, one obtains an I-bundle neighborhood of B as in [Ga-Oe].

We shall regard the exterior of U(B) as a sutured manifold, as follows.

A sutured manifold is an oriented 3-manifold P whose boundary is decomposed into surfaces with disjoint interior as ∂P = R_{−}P ∪ R_{+}P ∪ AP, where R_{−}P and R_{+}P are disjoint, AP is a collection of annuli called the sutures, R_{−}P ∩ AP = ∂(R_{−}P) ⊂ ∂(AP), R_{+}P ∩ AP = ∂(R_{+}P) ⊂ ∂(AP), and each component of AP has one boundary component in R_{−}P and the other in R_{+}P. Let R_{±}P = R_{−}P ∪R_{+}P. A sutured manifold P is a product if there is a homeomorphism (P, AP) ≈ (S×I, ∂S×I) for some compact surface S. Given a semiflow on a sutured manifold P, we say that P is an isolating block for the semiflow if orbits point outward along R_{+}P, inward along R_{−}P, and are externally tangent along γP as shown in figure 2. A foliation of a sutured manifold P is required to be tangent to R_{±}P and transverse to AP. Near AP, a transverse foliation and semiflow appear in cross section as in figure 3.

If B ⊂ M is a branched surface then P (B) = cl(M − U(B)) has the structure of a sutured manifold where R_{+}P (B) = ∂_{h}^{−}U(B), R_{−}P (B) = ∂_{h}^{+}U(B), and AP (B) = ∂_{v}U(B). Given a flow φ on M transverse to B, there exists a regular neighborhood U(B) so that the restriction of φ makes U(B) into a semiflow neighborhood, and so that P (B) is an isolating block for φ. We say that U(B) is adapted to φ.

A hierarchy on M, in the sense of Gabai, is a sequence of branched surfaces B_{0} ⊂ B_{1} ⊂ ⋯ ⊂ B_{N} such that B_{0} is a surface, B_{n} − B_{n−1} is a union of sectors of B_{n} for each 1 ≤ n ≤ N, and P (B_{N}) is a product. Let S(B_{n}) = B_{n} ∩ P (B_{n−1}); this is a transversely oriented surface properly embedded in P (B_{n−1}), and it is obtained from

Figure 3: A foliation and semiflow on a sutured manifold.

the sectors B_{n} − B_{n−1} by removing an open collar of each boundary component. We may assume that each component c of ∂S(B_{n}) satisfies the property that either c is a core curve of some suture of P (B_{n−1}) or c is transverse to the sutures, meaning that each component of c ∩ AP (B_{n−1}) is an arc connecting opposite boundaries of some component of AP (B_{n−1}). This property is immediate if each component of the boundary of B_{n} − B_{n−1} either is in general position with respect to the branch locus of B_{n−1}, or is contained in the branch locus. Otherwise, this property can be achieved by first choosing U(B_{n−1}) so that the annuli of ∂_{v}U(B_{n−1}) are extremely thin, and then doing a small isotopy of S(B_{n}). Note that P (B_{n}) is obtained by doing a sutured manifold decomposition of P (B_{n−1}) along S(B_{n}), in the sense of [Ga1].

Let F be a transversely oriented foliation in M, with a fixed transversal flow φ. Given a saturated open set W, let Ŵ be its metric completion. The inclusion ι: W ↪ M extends to an immersion ι̂: Ŵ → M carrying each component of ∂Ŵ diffeomorphically onto a leaf of F, but ι̂ may identify some of these boundary components pairwise. In addition the structure of the foliation in Ŵ is as follows [Di, Ca-Co]: Ŵ = Q ∪ (Σ × I), where Q is a compact (possibly empty) sutured manifold, each component of Σ is a noncompact surface with compact connected boundary, Q is glued to Σ × I by identifying AQ with ∂Σ × I, and the restrictions of F and φ to Q and to Σ × I are well-behaved as follows. The restrictions to Q give a sutured manifold foliation and semiflow; and the restriction of φ to Σ × I is a product flow, that is, orbits have the form x × I. Note that the restriction of F to Σ × I need not have leaves of the form Σ×t. If a connected open saturated set W has Q = ∅, then Ŵ (or W) is called a foliated product. An important fact is that for any saturated open set W, at most finitely many components of W are not foliated products [Di, Ca-Co].

The results above are also true for foliations of sutured manifolds. To see this, first double along the sutures and then double along the remaining boundary and apply the result to the final closed manifold.

Proposition 2.1. (hierarchy exists) If F is a taut finite depth foliation on M and φ is a flow transverse to F, then there exists a hierarchy B_{0} ⊂ ⋯ ⊂ B_{N} transverse to φ with neighborhoods U(B_{0}) ⊂ ⋯ ⊂ U(B_{N}) adapted to φ, such that the sutured manifolds P (B_{n}) and the surfaces S(B_{n}) are π_{1}-injective in M, and each component of B_{0} is isotopic to a compact leaf of F.

The construction of finite depth foliations in theorem 5.1 of [Ga1] proceeds by first constructing a hierarchy and then using it to produce the foliation. The point of this proposition is that all finite depth foliations arise by this construction; and we obtain additional information about flows. Since we cannot find a published proof we provide one, though the ideas are well-known.

Proof. Let F_{n} be the union of leaves of F of depth at most n. Since F is proper, F_{n} is closed, hence it is a sublamination of F, therefore an essential lamination in M. Let M_{n} = M − F_{n}, and M̂_{n} its metric completion. Then at most finitely many components of M_{n} are not foliated products. Now we may alter F without altering φ, collapsing foliated product components of M_{n}; do this inductively for each n. After the alterations, only finitely many leaves are isolated, where a leaf L at depth n is said to be isolated if L is isolated in F_{n}. In addition if L at depth n is not isolated, then the component of F_{n} − F_{n−1} containing L fibers over the circle with fiber L. Next fatten up each of the finitely many isolated leaves into a fibration over a closed interval, again altering F without altering φ. Each connected component of F_{n}−F_{n−1} is now either a fibration over a closed interval, or a fibration over the circle; the latter are called “circular” components of F_{n} − F_{n−1}. Under these condition M̂_{n} is a 1-1 immersed submanifold of M whose boundary components are leaves of F_{n}.

Now we construct by induction a hierarchy B_{0} ⊂ ⋯ ⊂ B_{N} and neighborhoods U(B_{0}) ⊂ ⋯ ⊂ U(B_{N}) adapted to φ so that if 0 ≤ n ≤ N then:

1. U(B_{n}) contains every noncircular component of F_{n} − F_{n−1}, as well as an interval’s worth of leaves in every circular component.

2. F restricts to a foliation of the sutured manifold P (B_{n}) = \overline{(M − U(B_{n}))}. It may be a product foliation in some components of P (B_{n}).

3. The embedding P (B_{n}) ↪ P (B_{n−1}) is π_{1}-injective.

4. S(B_{n}) = B_{n} ∩ P (B_{n−1}) is π_{1}-injective in P (B_{n−1}).

If F is not a fibration over S^{1}, let U(B_{0}) to be the union of the compact leaves, otherwise let U(B_{0}) be a closed interval of leaves. Properties (1–2) are evident. We interpret (3) and (4) by setting P (B_{−1}) = M, and these properties follow because the compact leaves of F are incompressible surfaces in M.

To continue the induction, given a component Z of P (B_{n}) on which the restriction of F and φ does not already give a product structure, we describe the intersections with Z of P (B_{n+1}), U(B_{n+1}), and B_{n+1} itself. Let V (∂_{±}Z) be a neighborhood so that the restriction of ϕ has compact orbits, defining a product structure with projection map q: V (∂_{±}Z) → ∂_{±}Z.

Let F′ be the restriction of F_{n+1} to Z. This lamination has depth ≤ 1. Its compact leaves consist of ∂_{±}Z plus foliated products F × I with ∂F × I ⊂ AZ; all but finitely many of these foliated products have leaves parallel to a component of

Figure 4: Isotopic junctures

∂_{±}Z. The noncompact leaves of F′ fall into a finite collection of circular components and foliated products, and each end E of a noncompact leaf L spirals into some component F of ∂_{±}Z in the following manner: there is a nonseparating, properly embedded, connected 1-manifold c ⊂ F called a juncture for E, there is a Z-covering space F̃ → F to which c lifts homeomorphically, there is a subset L_{E} ⊂ L representing E and contained in V (F) (the component of V (∂_{±}Z) containing F), and there is an embedding L_{E} → F̃, such that the maps L_{E} → F̃ → F and L_{E} ↪ V (F) → F are the same, and the image of ∂L_{E} is the curve c. If E, E′ are ends of noncompact leaves spiralling into the same component F of ∂_{±}Z, and if E, E′ are not in the same end of a circular component of F′, we may assume that the junctures for E, E′ are disjoint but isotopic curves in F.

Let F′′ be the union of the finitely many foliated products in F′ which are not contained in V (∂_{±}Z), plus a closed interval of leaves in each circular component of F′, plus ∂_{±}Z. Since Z −F′′ is an open subset of Z saturated by F | Z, letting \overline{Z − F′′} be the metric completion of Z − F′′, there is a decomposition \overline{Z − F′′} = Q ∪ (Σ × I) where Q is a compact sutured manifold and Σ × I is a foliated product so that each noncompact component of Σ has compact, connected boundary. We can choose Q big enough so that Σ × I is contained in V (∂_{±}Z), and so that the boundary of each component of Σ projects to some juncture, thus we can associate a juncture to each component of Σ. Moreover, if Σ_{1}, …, Σ_{n} are components of Σ with isotopic junctures c_{1}, …, c_{n} lying in a component F of ∂_{±}Z, then we can choose Q so that the picture in figure 4 holds: the curves c_{1}, …, c_{n} are ordered so that c_{i} ∪c_{i+1} bounds an annulus disjoint from the other curves; and the “first hitting” map Σ_{i+1} → Σ_{i} is continuous for i = 1, …, n − 1, where this map is defined by starting from a point in Σ_{i+1}, going along an orbit of ϕ towards F, and stopping at the first point of Σ_{i}.

Now define P (B_{n+1})∩Z = Q. Notice that F′′ is an essential lamination of Z, hence Z − F′′ is π_{1}-injective in Z. Since each noncompact component of Σ has compact, connected boundary, it follows that Q is π_{1}-injective in Z − F′′, and property (3) for P (B_{n+1}) follows. Property (2) is easily checked, as is property (1) for U(B_{n+1}) = \overline{(M − P (B_{n+1}))}.

To define the sectors of B_{n+1} intersecting Z, note that the foliation and flow define a product structure F′′−∂_{±}Z = G×[0, 1] for some surface G, not necessarily connected. Each end of G×0 or G×1 is eventually contained in Σ×I. Let L be the set of x ∈ G such that x × [0, 1] is disjoint from Σ × I. Then L is a subsurface of G, and we may regard L as being embedded in F′′ transverse to the flow. Homotop ∂L to a map into B_{n} as described below, and extend the homotopy over L. The image of L after the homotopy is the union of sectors of Bn + 1 − B_{n} intersecting Z. To describe the homotopy of ∂L, each component β of ∂L − AZ lies in V (∂_{±}Z), and β projects via ϕ to a juncture; homotop ∂L along orbits of ϕ until β lies on that juncture. Extend the homotopy of ∂L − AZ over all of ∂L, so that each arc component of ∂L ∩ AZ maps to an arc in B_{n} crossing the branch locus in a single transverse intersection point, and each circle component of ∂L ∩ AZ maps to a circle in the branch locus of B_{n}. Because of the properties of junctures given in figure 4, this homotopy may be carried out so that L − ∂L is embedded in M − B_{n}, and hence the image of L after the homotopy consists of sectors of B_{n+1} −B_{n} whose boundary lies in B_{n}, as required for a hierarchy.

Property (4) for S(B_{n+1}) follows because each component of S(B_{n+1}) corresponds to a component of L as above, but components of L are π_{1}-injective in in P (B_{n}).

The only nonobvious points remaining in the proof of the lemma are the statements on π_{1}-injectivity. Using (3) it follows by induction that P (B_{n}) is π_{1}-injective in M. Using (4), and the fact that P (B_{n−1}) is π_{1}-injective in M, it follows that S(B_{n}) is π_{1}-injective in M.

We also need some geometric information about hierarchies. Consider a closed, oriented 3-manifold M and a nonseparating incompressible surface F. We say F is a fiber if F is a leaf of a fibration of M over S^{1}. If M is hyperbolic we say that F is quasi-fuchsian if the representation π_{1}F → π_{1}M → Isom(H^{3}) is quasi-fuchsian. A deep fact due to Marden [Ma], Thurston [Th1] and Bonahon [Bo], is that when M is hyperbolic then F is either a fiber or quasi-fuchsian (if F is separating there is another option, namely that F is a “virtual fiber”).

Recall from [Gr, Gh-Ha] that a locally compact path metric space X is negatively curved in the large if it satifies the “thin triangles” condition: there exists δ > 0 such that for every geodesic triangle with sides A_{1}, A_{2}, A_{3} we have A_{1} ⊂ U_{δ}(A_{2} ∪ A_{3}). In this case there is a canonically defined ideal boundary at infinity denoted ∂_{∞}X, and there is a canonical compactification X ∪ ∂_{∞}X. If X is negatively curved in the large, so is any space quasi-isometric to X. If X, Y are negatively curved in the large and f: X → Y is a quasi-isometry then f extends canonically to a continuous map f̂: X ∪∂_{∞}X → Y ∪∂_{∞}Y that restricts to an embedding ∂_{∞}X ↪ ∂_{∞}Y. For example, hyperbolic space H^{n} is negatively curved in the large, and its ideal boundary in the sense of Gromov is the same as the sphere at infinity S_{∞}^{n−1}. Also, any closed convex subset X ⊂ H^{n} is negatively curved in the large, and ∂_{∞}X coincides with the set of limit points of X in S_{∞}^{n−1}.

16 SÉRGIO FENLEY AND LEE MOSHER

Given a group G with a finite generating set A, the Cayley graph C is the graph with a vertex for each g ∈ G, and an edge from g to ga for each a ∈ A. By taking each edge to be a path of length 1 we make the Cayley graph a path metric space. For any two finite generating sets, the Cayley graphs are quasi-isometric. The group G is word hyperbolic if some (and hence any) Cayley graph of G is negatively curved in the large. Then we let ∂_{∞}G = ∂_{∞}C. Consider a compact Riemannian manifold M with G = π_{1}(M, x), choose a lift x_{0} of x in the universal cover M̃ and choose loops in M representing a finite generating set for π_{1}(M). Lifting to M̃ we have a map of the Cayley graph of G to M̃, which gives a quasi-isometry between G and M̃ [Mi]. Therefore G = π_{1}M is word hyperbolic if and only if M̃ is negatively curved in the large, in which case there is a canonical identification ∂_{∞}G ≈ ∂_{∞}M̃.

Proposition 2.2. Let M be a closed, orientable hyperbolic 3-manifold, B_{0} ⊂ ⋯ ⊂ B_{N} a hierarchy in M such that P (B_{n}) and S(B_{n}) = B_{n} ∩ P (B_{n−1}) are π_{1}-injective for all n. Suppose that some component F of B_{0} is a quasi-fuchsian surface in M. Then:

(1) Each group π_{1}(P (B_{n})), π_{1}(S(B_{n})) is word hyperbolic.

(2) Each embedding S̃(B_{n}) ↪ P̃(B_{n−1}) is a quasi-isometry which takes ∂S̃(B_{n}) into ∂P̃(B_{n}). Therefore it extends to an embedding S̃(B_{n}) ∪ ∂_{∞}S̃(B_{n}) ↪ P̃(B_{n−1}) ∪ ∂_{∞}P̃(B_{n−1}).

(3) Each inclusion P̃(B_{n}) ↪ P̃(B_{n−1}) is a quasi-isometry in the path metrics induced from M̃. Therefore it extends to an embedding P̃(B_{n}) ∪ ∂_{∞}P̃(B_{n}) ↪ P̃(B_{n−1}) ∪ ∂_{∞}P̃(B_{n−1}).

Remark: The group π_{1}(S(B_{n})) may be cyclic, when S(B_{n}) is an annulus; or trivial, when S(B_{n}) is a disc.

Proof. Let ρ: π_{1}(M) → P SL_{2}(C) be the holonomy representation of the hyperbolic structure. Consider the subgroup G = ρ(π_{1}(M − F)) with limit set ΛG ⊂ S_{∞}^{2}, let H(ΛG) ⊂ H^{3} ∪ S_{∞}^{2} be its convex hull, let H_{ϵ}(ΛG) ⊂ H^{3} ∪ S_{∞}^{2} be the union of H(ΛG) with its ϵ-neighborhood in H^{3}, and let X_{G} = H_{ϵ}(ΛG)∩H^{3}. Since F is quasi-Fuchsian, the group G is convex cocompact, that is G acts cocompactly on X_{G}; equivalently, G is geometrically finite [Ma, Th1]. It follows that any finitely generated subgroup H < G is convex cocompact [Th1, Cn]. Since X_{H} is negatively curved in the large, and the action of H on X_{H} is properly discontinuous and cocompact, it follows that H is word hyperbolic, proving (1).

Suppose P ⊂ M − F is a compact, π_{1}-injective submanifold and H = π_{1}(P). We show that the natural inclusion j: P̃ → H^{3} is a quasi-isometric embedding, where P̃ has the path metric induced from M̃.

The natural inclusion ι: X_{H} ↪ H^{3} is a quasi-isometric embedding. Let Γ_{H} be the Cayley graph of H, and consider the two quasi-isometries α: Γ_{H} → X_{H}, β: Γ_{H} → P̃ given by [Mi]. The maps ι ∘ α, j ∘ β: Γ_{H} → H^{3} differ by a bounded amount, therefore j ∘ β is a quasi-isometric embedding. There is a quasi-isometry β̄: P̃ → Γ_{H} which is an inverse of β in the quasi-isometric category, so in particular β ∘ β̄: P̃ → P̃ differs from the identity by a bounded amount. Thus we have a quasi-isometric embedding j ∘ β ∘ β̄: P̃ → H^{3} which differs from j by a bounded amount, so j is a quasiisometric embedding. Hence each P̃(B_{n}) ↪ H^{3} extends to a continuous embedding P̃(B_{n}) ∪ ∂_{∞}P̃(B_{n}) ↪ H^{3} ∪ S_{∞}^{2}.

To prove (3), the quasi-isometric embedding P̃(B_{n}) ↪ H^{3} factors through the natural embedding P̃(B_{n}) ↪ P̃(B_{n−1}), so P̃(B_{n}) ↪ P̃(B_{n−1}) is a quasi-isometric embedding. The proof of (2) is similar.

Remark: Although not logically necessary for our results, it is helpful to keep in mind the following additional facts:

(4) Each compactified universal cover P̃(B_{n}) ∪ ∂_{∞}P̃(B_{n}) is a 3-ball.

(5) Each S̃(B_{n}) ∪ ∂_{∞}S̃(B_{n}) is a 2-disc properly embedded in the above 3-ball. To prove (4), let P = P (B_{n}), let H = π_{1}(P), and consider the action of H on

H^{3}. In the manifold H^{3}/H, both of the compact manifolds X_{H}/H and P embed as deformation retracts. By [Mc-MS] it follows that there is a homeomorphism X_{H}/H ≈ P in the correct homotopy class. This homeomorphism lifts to an Hequivariant quasi-isometric homeomorphism X_{H} ≈ P̃, which extends to a homeomorphism H_{ϵ}(ΛH) ≈ P̃ ∪ ∂_{∞}P̃, and H_{ϵ}(ΛH) is obviously a 3-ball.

The proof of (5) when S(B_{n}) is a disc or annulus is easy. Otherwise there is a hyperbolic metric with geodesic boundary on S(B_{n}), and S̃(B_{n}) ∪ ∂_{∞}S(B_{n}) is therefore a 2-disc, which by (2) is properly embedded in the 3-ball P̃ ∪ ∂_{∞}P̃.

3. Inductive proof of theorem A

To set up the induction, apply proposition 2.1 to obtain a hierarchy B_{0} ⊂ ⋯ ⊂ B_{N}. We use the following notation. If 0 ≤ n ≤ N let P_{n} = P (B_{N−n}), and let P_{N+1} = M, so P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{N} ⊂ P_{N+1}; the indexing is reversed to facilitate the induction proof. If 1 ≤ n ≤ N + 1 let S_{n} = S(B_{N−n+1}) = B_{N−n+1} ∩ P_{n}, so S_{n} is properly embedded in P_{n}, and P_{n−1} is obtained from P_{n} by removing a regular neighborhood of S_{n}. Since some (and hence all) compact leaves of F are quasi-fuchsian, the same is true for components of B_{0}, hence proposition 2.2 applies.

Given a flow line x · R of Φ, define the codepth to be the minimal integer n such that x · R ⊂ P_{n}, and define Ω_{n} to be the union of all flow lines of codepth at most n. Note that Ω_{n} is the set of all flow lines contained in P_{n}, and Ω_{n} is closed. As special cases, Ω_{0} = ∅ since P_{0} is a product, and Ω_{N+1} = M.

The program for the proof of theorem A is to assume that orbits in Ω_{n−1} are uniformly quasigeodesic and then show that orbits in Ω_{n} are also uniformly quasigeodesic (with a bigger quasigeodesic constant). Since Ω_{0} = ∅ and Ω_{N+1} = M, induction will

18 SÉRGIO FENLEY AND LEE MOSHER

show that all orbits are uniformly quasigeodesic. This will complete the proof of theorem A.

For notational convenience, throughout this section we write P = P_{n}, Ω = Ω_{n}, S = S_{n}, P′ = P_{n−1}, and Ω′ = Ω_{n−1}. Let π: P̃ → P be the universal covering, and let Ω̃ = π^{−1}(Ω).

Fix a connected lift P̃′ ↪ P̃, and let Ω̃′ = π^{−1}(Ω′) ∩ P̃′. The induction hypothesis says that orbits in Ω̃′ are uniform quasigeodesics in P̃′. Since P̃′ ↪ P̃ is a quasiisometry, then orbits in Ω̃′ are uniformly quasigeodesic in P̃; using the action of π_{1}P by isometries on P̃, the same is true for orbits in π^{−1}(Ω′). Recall that the hypothesis in theorem A is that Φ̃ has Hausdorff orbit space. This will be used in verifying conditions (b) and (c) of theorem 1.4.

We will need the following well known simple result [Co]:

Lemma 3.1. Let W be a compact metric space with a nonsingular semiflow φ parameterized by arc length. Let Ω be the set of points x for which φ_{t}(x) is defined for all t ∈ R. Then given any δ > 0, there is a > 0 so that any orbit γ of φ is in the δ-neighborhood of Ω except perhaps for an initial segment of length < a and another final segment of length < a.

We will also use the following localization property of quasigeodesics.

Proposition 3.2. [Gr, Gh-Ha] Let M be a compact manifold with negatively curved π_{1}(M). Then for any K > 0 there is L(K) > 0 (usually L(K) >> K) satisfying: if γ is an embedded curve so that any subarc of length ≤ L(K) is a K quasigeodesic then γ is a 2K-quasigeodesic.

The following essential fact which is a direct consequence of proposition 2.2 will be used explicitly or implicitly throughout this section: if γ is a curve contained in P̃′, then γ is a quasigeodesic in P̃′ if and only if γ is a quasigeodesic in P̃ and also if and only if γ is a quasigeodesic in H^{3}. The quasigeodesic constants may differ. We caution the reader that some arguments are done in H^{3} while others are done in P̃. The context makes it clear.

For flow segments disjoint from S, the following lemma establishes the quasigeodesic property directly.

Lemma 3.3. There is K > 0 so that all flow segments, half orbits or full orbits of Φ̃ contained in P̃′ are K-quasigeodesics of P̃; translating by the action of π_{1}(P), the same is true in any lift of P′.

Proof. The full orbits staying in P̃′ are precisely the orbits in Ω̃′, which are kquasigeodesic for some uniform k. Fix δ_{0} > 0, with 2kδ_{0} < 1. Let L = L(k + 1) be given by proposition 3.2. Choose δ_{1} > 0 so that if x, y ∈ P̃ and d(x, y) < δ_{1} then d(x · t, y · t) < δ_{0} for any |t| < L. Choose a > 0 so that any orbit in P′ is in the δ_{1}-neighborhood of Ω′, except perhaps for initial and final segments of length < a.

Let now x, y ∈ P̃′ with y = x · t. Choose a subsegment γ = [z, w] ⊂ [x, y], with z = x·t_{1}, w = y·t_{2}, 0 ≤ t_{1} < a and −a < t_{2} ≤ 0, so that γ ⊂ U_{δ_{1}}(Ω̃′). Let α = [z_{0}, w_{0}] be a subarc of γ with 0 < t_{3} = τ(z_{0}, w_{0}) ≤ L. By the choice of δ_{1} it follows that there is z_{1} ∈ Ω̃′, with d(z_{0} · t, z_{1} · t) < δ_{0}, for any |t| < L. Since z_{1} · R is a k-quasigeodesic then

τ(z_{0}, w_{0}) = t_{3} = τ(z_{1}, z_{1} · t_{3}) ≤ k d(z_{1}, z_{1} · t_{3}) + k
≤ kd(z_{0}, w_{0}) + 2kδ_{0} + k < kd(z_{0}, w_{0}) + (k + 1).

Clearly this also works for any subsegment of α, hence α is a (k + 1)-quasigeodesic. By the previous proposition, one concludes that γ is a (2k + 2)-quasigeodesic. This implies that

τ(x, y) < 2a + τ(z, w) ≤ (2k + 2)d(z, w) + (2k + 2 + 2a)
≤ (2k + 2)(d(x, y) + 2a) + (2k + 2 + 2a)
= (2k + 2)d(x, y) + (2k + 2 + 2a + (2k + 2)2a)

Hence any piece of orbit of Φ̃ contained in P̃′ is a (4ka+2k +6a+2)-quasigeodesic of P̃′. Since P̃′ is quasi-isometrically embedded in P̃′, there is K > 0 so that any piece of orbit of Φ̃ contained in P̃ is a K-quasigeodesic of P̃.

Now we prepare the ground for applying theorem 1.4 to show that full orbits in Ω̃ are uniformly quasigeodesic. We must study how orbits in Ω̃ − Ω̃′ cross lifts of S in P̃, so we embark on a study of these lifts.

Let E be the collection of lifts of S to P̃. Any element E ∈ E is transversely oriented and separates P̃. The front of E is the component Fr(E) of P̃ − E on the positive side of E, and the closure of Fr(E) in P̃ ∪ ∂_{∞}P̃, is denoted σ^{f}(E). The back Bc(E) and its closure σ^{b}(E) are similarly defined. Define a strict partial order on E where E < E′ if Bc(E) ∩ Fr(E′) = ∅ and E ≠ E′, hence in particular E′ ⊂ Fr(E).

Notice that since S is compact, a bounded subset of P̃ intersects only finitely many E ∈ E. The following lemmas strengthen this fact, by showing that sequences in E are limited in how they may accumulate in the ball P̃ ∪ ∂_{∞}P̃; these lemmas will be useful in analyzing flow lines that cross S many times. If ρ : π_{1}(M) → P SL_{2}(C) is the holonomy representation, then ρ(π_{1}(P)) is a Kleinian group which is convex cocompact.

Lemma 3.4. There is J_{0} > 0 such that if E_{1}, …, E_{J_{0}} are distinct elements of E, then ∂_{∞}(E_{1})∩⋯∩∂_{∞}(E_{J_{0}}) = ∅. If moreover E_{1} < ⋯ < E_{J_{0}} then σ^{b}E_{1} ∩σ^{f}E_{J_{0}} = ∅.

Proof. First we need the fact that if H_{1}, …, H_{n} are geometrically finite Kleinian groups that generate a discrete group, then ΛH_{1} ∩ ⋯ ∩ ΛH_{n} = Λ(H_{1} ∩ ⋯ ∩ H_{n}). When n = 2 this is proved in [Su], and the statement for finitely many subgroups follows by induction.

20 SÉRGIO FENLEY AND LEE MOSHER

To prove the first statement, suppose that ∂_{∞}(E_{1})∩⋯∩∂_{∞}(E_{J}) ≠ ∅. Let H_{i} ⊂ π_{1}P be the stabilizer subgroup of E_{i}, so Λ(H_{i}) = ∂_{∞}(E_{i}), and by the above argument using Susskind’s theorem it follows that H_{1} ∩ ⋯ ∩ H_{J} ≠ ∅. Let f be in the intersection, and let Axis_{f} be an axis for f in H^{3}. By conjugation, we may assume that Axis_{f} intersects a fixed fundamental domain D for π_{1}M.

Since E is quasi-isometrically embedded in H^{3}, it is R-quasiconvex, that is, for any x, y ∈ E ∪ ∂_{∞}E, the geodesic in E connecting them is at most R distant from the hyperbolic geodesic connecting them [Gr, Gh-Ha]. The R is independent of the lift E of S. Thus each E_{i} intersects U_{R}(D), the open R-neighborhood of D. This shows that J ≤ J_{0} where J_{0} is an upper bound for the number of distinct E ∈ E intersecting the bounded set U_{R}(D).

To prove the second statement, suppose that σ^{b}E_{1} ∩ σ^{f}E_{J_{0}} ≠ ∅. It follows that ∂_{∞}E_{1} ∩ ∂_{∞}E_{J_{0}} ≠ ∅; let ξ be a point in this intersection. If for some 1 ≤ i ≤ n, ξ ∉ ∂_{∞}E_{i}, let V be a neighborhood of ξ in P̃ ∩ ∂_{∞}P̃ with E_{i} ∩ (V ∩ P̃) = ∅. However since ξ ∈ ∂_{∞}E_{0} ∩ ∂_{∞}E_{J_{0}}, there are x ∈ E_{0} ∩ V and y ∈ E_{J_{0}} ∩ V. Hence x can be connected to y in V ∩ P̃, contradicting the fact that E_{i} separates E_{0} from E_{J_{0}}. But then we have proved ∂_{∞}E_{1} ∩ ⋯ ∩ ∂_{∞}E_{J_{0}} ≠ ∅, contradicting the first statement of the lemma.

The J_{0} given by the previous lemma is fixed from now on. The next lemma gives an even stronger accumulation property.

Lemma 3.5. Given an infinite sequence E_{1}, E_{2}, … ∈ E such that E_{i} ≠ E_{j} if i ≠ j, suppose there exists E_{0} ∈ E such that E_{0} < E_{i} for all i ≥ 1. Then there is a subsequence E_{i_{n}} such that σ^{f}(E_{i_{n}}) converges to a single point ξ ∈ ∂_{∞}P̃, in the Hausdorff topology on closed subsets of P̃ ∪ ∂_{∞}P̃.

Proof. Choose covering translations f_{i} ∈ π_{1}P such that f_{i}(E_{0}) = E_{i}. Then f_{i} ≠ f_{j} when i ≠ j, so by the convergence group property for Kleinian groups applied to ρ(π_{1}(M)) (see pg. 22 of Maskit’s book [Ma]), we may pass to a subsequence so that there is a pair of points ξ_{−}, ξ_{+} ∈ ∂_{∞}P called a source and sink for the sequence of functions f_{i}, meaning that f_{i} | (P̃ ∪ ∂_{∞}P̃) − ξ_{−} converges uniformly on compact sets to the constant map with value ξ_{+}, and f_{i}^{−1} | (P̃ ∪ ∂_{∞}P̃) − ξ_{+} converges similarly to ξ_{−}. Since f_{i}^{−1}(σ^{b}(E_{i})) = σ^{b}(E_{0}) for all i, it follows that f_{i}^{−1}(σ^{b}(E_{0})) ⊂ σ^{b}(E_{0}), and hence ξ_{−} ∈ σ^{b}(E_{0}).

If ξ_{−} ∈ σ^{b}E_{0} − ∂_{∞}E_{0} then we are done, because then ξ_{−} ∉ σ^{f}(E_{0}), so the sequence of maps f_{i} | σ^{f}(E_{0}) converges uniformly to the constant map with value ξ_{+}, and hence the images σ^{f}(E_{i}) of these maps converge in the Hausdorff topology to ξ_{+}.

Suppose on the other hand that ξ_{−} ∈ ∂_{∞}E_{0}. By the previous lemma, for all but finitely many E_{i} we have ξ_{−} ∉ ∂_{∞}E_{i}; in particular this is true for some i = i_{0}. We also have ξ_{−} ∉ σ^{f}(E_{i_{0}}), because ξ_{−} ∈ σ^{b}(E_{0}). Note that ξ_{−}, ξ_{+} is a source, sink pair for the sequence f_{i} ∘ f_{i_{0}}^{−1}, so the maps f_{i} ∘ f_{i_{0}}^{−1} | σ^{f}(E_{i_{0}}) converge uniformly to the constant map with value ξ_{+}, and hence their images σ^{f}(E_{i}) converge in the Hausdorff topology to ξ_{+}.

Now we show that all flow lines in Ω̃ extend continuously to ∂_{∞}P̃, verifying condition (a) of theorem 1.4.

Proposition 3.6. (extension of orbits) If x ∈ Ω̃, then the following limits exist:

lim_{t→+∞} x · t = η_{+}(x) ∈ ∂_{∞}P̃ and lim_{t→−∞} x · t = η_{−}(x) ∈ ∂_{∞}P̃

Proof. We only consider forward limits. There are two cases: \underline{Case 1} — x · [0, +∞) eventually stops intersecting π^{−1}(S) (this includes x ∈ Ω̃′).

Let y be the last intersection of x · [0, +∞) with π^{−1}(S) if any exists, otherwise let y = x. Then π(y) · [0, +∞) is contained in P′ hence y · [0, +∞) is a K-quasigeodesic in P̃ by lemma 3.3. Therefore it has a unique limit point. \underline{Case 2} — x · [0, +∞) keeps intersecting π^{−1}(S).

Let E_{0} < E_{1} < ⋯ ∈ E be the elements that x·[0, +∞) intersects. Thus, all accumulation points of x·[0, +∞) are contained in σ^{f}(E_{1})∩σ^{f}(E_{2})∩· · ·. Since this is a nested intersection, it is the same as the Hausdorff limit of the sets σ^{f}(E_{1}), σ^{f}(E_{2}), ⋯, which by lemma 3.5 is a single point.

Next we verify condition (b) of theorem 1.4. We use the following facts: if α and β are two quasigeodesic rays in H^{3} with the same ideal point, then there is R > 0 (depending on α and β) so that α is in the R neighborhood of β and vice versa. For a pair of bi-infinite quasigeodesics with the same ideal points there is also a bound, which depends only on the quasigeodesic constant. We also need the following very useful result:

Lemma 3.7. Suppose that γ_{i} = x_{i} · [0, t_{i}) and γ = x · [0, +∞) are k_{1}-quasigeodesic flow segments or rays, where x_{i} → x in P̃ and t_{i} → +∞ in [0, +∞]; we allow t_{i} = +∞. Then x_{i} · t_{i} → η_{+}(x) in P̃ ∪ ∂_{∞}P̃, where x_{i} · t_{i} means η_{+}(x_{i}) if t_{i} = +∞.

Proof. Otherwise, passing to a subsequence we have x_{i} · t_{i} → w ∈ P̃ ∪ ∂_{∞}P̃ with w ≠ η_{+}(x). Choose R > 0 so that any k_{1}-quasigeodesic is at most R distant from a corresponding minimal geodesic.

Choose disjoint small neighborhoods W_{0} of η_{+}(x) in P̃ ∪∂_{∞}P̃, W_{1} of w in P̃ ∪∂_{∞}P̃ and W_{2} of x in P̃ so that for any minimal geodesic segment or ray β starting in W_{2} and with endpoint in W_{1} then

U_{R}(β) ∩ (W_{0} ∩ P̃) = ∅.

For i big enough x_{i} ∈ W_{2} and x_{i} · t_{i} ∈ W_{1}, so since γ_{i} is at most R distant from the corresponding minimal geodesic, then γ_{i} ∩ (W_{0} ∩ P̃) = ∅. But since x_{i} → x then by continuity of the flow γ_{i} ∩ (W_{0} ∩ P̃) ≠ ∅ for i big enough, contradiction.

Figure 5: Identification of ideal points in an orbit produces two quasigeodesic orbits with same pair of ideal points.

Proposition 3.8. (distinct limit points) For any x ∈ Ω̃, η_{+}(x) ≠ η_{−}(x).

Proof. There are two cases: \underline{Case 1} — x · R intersects π^{−1}(S) only finitely many times.

Then there are t_{0} < t_{1} ∈ R, with γ_{0} = x · (−∞, t_{0}] and γ_{1} = x · [t_{1}, +∞), so that γ_{0} ∩ π^{−1}(S) = γ_{1} ∩ π^{−1}(S) = ∅. By lemma 3.2, γ_{0} and γ_{1} are K-quasigeodesics in H^{3}. Assume they have the same ideal point p. Then there are sequences v_{i} ∈ γ_{0}, u_{i} ∈ γ_{1}, such that u_{i} → p, v_{i} → p, and d(u_{i}, v_{i}) is bounded. But τ(u_{i}, v_{i}) → ∞.

Let y = x · t_{0} and z = x · t_{1}. Up to taking a subsequence assume that π(u_{i}) converges in M. Then there are covering translations h_{i} so that h_{i}(u_{i}) → u. Since u_{i} ∈ P̃′, then π(u) ∈ P′. We can also assume that h_{i}(v_{i}) → v and hence π(v) ∈ P′.

Since [z, u_{i}] is contained in P̃′, it is a K-quasigeodesic by lemma 3.3. Since τ(z, u_{i}) → +∞, it then follows that d(z, u_{i}) → +∞, so d(h_{i}(z), h_{i}(u_{i})) → +∞. Hence up to taking a further subsequence we may assume that h_{i}(z) converges to some point in S_{∞}^{2}. Since τ(h_{i}(z), h_{i}(u_{i})) → +∞ and h_{i}(u_{i}) → u, lemma 3.7 implies that h_{i}(z) → η_{−}(u). In addition since h_{i}(z) · [0, +∞) contains h_{i}(u_{i}) → u ∈ M̃ and h_{i}(u_{i}) = h_{i}(z) · r_{i} with r_{i} → +∞, it follows that every point in u · R is obtained as a limit of points q_{i} ∈ h_{i}(z) · [0, +∞). Since π^{−1}(P′) is a closed set in M̃, this shows that π(u · R) is contained in P′ and therefore u · R is a K-quasigeodesic. In the same way h_{i}(y) → η_{+}(v) and v · R is obtained as a limit of h_{i}(y) · (−∞, 0], hence v · R is also a K-quasigeodesic, see fig. 5.

As d(h_{i}(z), h_{i}(y)) is a constant, it follows that η_{+}(v) = η_{−}(u). Since both v · R and u · R are K-quasigeodesics, this equality implies that u and v are not in the same flow line of Φ̃. But for each i, h_{i}(v_{i}) and h_{i}(u_{i}) are in the same orbit of Φ̃, h_{i}(v_{i}) → v, and h_{i}(u_{i}) → u. This contradicts the Hausdorff orbit space condition. We conclude that γ_{0} and γ_{1} do not have the same ideal point.

\underline{Case 2} — x · R intersects π^{−1}(S) infinitely many times.

Then there is a sequence E_{1} < ⋯ < E_{J_{0}} ∈ E such that x·R intersects each, and by lemma 3.4 we have σ^{b}(E_{1}) ∩ σ^{f}(E_{J_{0}}) = ∅. But η_{−}(x) ∈ σ^{b}(E_{1}) and η_{+}(x) ∈ σ^{f}(E_{J_{0}}) so η_{−}(x) ≠ η_{+}(x).

Finally we prove property (c) of 1.4.

Proposition 3.9. (continuity of extension) The map η_{+} : Ω̃ → S_{∞}^{2} is continuous and similarly for η_{−}.

Proof. We only consider η_{+}. \underline{Case 1} — x · [0, +∞) intersects π^{−1}(S) infinitely often.

Let E_{1} < E_{2} < ⋯ be the elements of E that x · [0, +∞) intersects. From lemma 3.5, ⋂_{i∈N} σ^{f}(E_{i}) is a single point p in ∂_{∞}P̃. Since σ^{f}(E_{i}) ∩ ∂_{∞}P̃ is a decreasing sequence of compact sets in ∂_{∞}P̃, then for each δ > 0, there is i_{0} so that ∀i > i_{0}, the diameter of σ^{f}(E_{i})∩∂_{∞}P̃ < δ. If z is sufficiently near x then z ·[0, +∞) will intersect E_{i_{0}} therefore η_{+}(z) ∈ σ^{f}(E_{i}) ∩ ∂_{∞}P̃. Hence the distance from η_{+}(z) to η_{+}(x) < δ. This implies that η_{+} is continuous at x. \underline{Case 2} — x · [0, +∞) has finite intersection with π^{−1}(S).

There is s_{1} ≥ 0 with x · [s_{1}, +∞) ∩ π^{−1}(S) = ∅. Let x′ = x · s_{1}. Since

x_{i} → x ⟺ x_{i} · s_{1} → x′,

we may assume that x · [0, +∞) ∩ π^{−1}(S) = ∅. Consider a sequence x_{i} → x, and let

m_{i} = |x_{i} · [0, +∞) ∩ π^{−1}(S)|

where |V | denotes cardinality of V. Passing to a subsequence, either m_{i} takes a constant value ≤ J_{0} for all i, or m_{i} > J_{0} for all i. \underline{Case 2.1} — m_{i} = 0 for all i.

We may assume all γ_{i} = x_{i}·[0, +∞) and γ_{0} = x·[0, +∞) are in the δ_{1}-neighborhood of Ω. By lemma 3.3, γ_{0} and each γ_{i} are K-quasigeodesic. The proof in this case is finished by lemma 3.7.

Before addressing the case m_{i} > 0 we need the following lemma.

Lemma 3.10. Suppose m_{i} ≥ j − 1 for all i. Let v_{i} be the j-th intersection of x_{i} · [0, +∞) with π^{−1}(S), if it exists, otherwise let v_{i} = η_{+}(x_{i}). Then v_{i} converges to η_{+}(x) in P̃ ∪ ∂_{∞}P̃.

Proof. First suppose j = 1, and let v_{i} = x_{i} · t_{i} with t_{i} ∈ [0, +∞]. By lemma 3.3 the segments or rays [x_{i}, v_{i}) are all K-quasigeodesics. In addition t_{i} → +∞, since x · [0, +∞) ∩ π^{−1}(S) = ∅ and x_{i} → x. By lemma 3.7 we conclude that z_{i} → η_{+}(x).

Figure 6: Successive intersections with E.

Suppose now that m_{i} ≥ j − 1 for all i, and let w_{i} be the (j − 1)-th intersection. We assume by induction that w_{i} → η_{+}(x). Let ζ_{i} = [w_{i}, v_{i}], so each ζ_{i} is a Kquasigeodesic, see fig. 6. If ζ_{i} is not escaping compact sets in P̃, then passing to a subsequence ζ_{i} accumulates on some u ∈ P̃, leading to a contradiction as follows: if u ∈ x · R, then since each ζ_{i} − {w_{i}} is separated from x by at least one element of E, this would imply that x·R_{+} has to intersect E contradiction. In the other case, x and u are not on the same flow line, contradicting the Hausdorff orbit space condition.

The paths ζ_{i} therefore escape to infinity in P̃. These paths are K-quasigeodesics, so their diameters are shrinking to zero in a topological metric on P̃ ∪ ∂_{∞}P̃. Since w_{i} → η_{+}(x) it then follows that v_{i} → η_{+}(x). Induction now completes the proof of the lemma.

\underline{Case 2.2} — m_{i} = m is a constant with 0 < m ≤ J_{0}.

Applying lemma 3.10 with j = m − 1 it follows that η_{+}(x_{i}) → η_{+}(x).

\underline{Case 2.3} — m_{i} > J_{0} for every i.

Let y_{i} be the corresponding (J_{0} + 1)-th intersection. Then y_{i} → η_{+}(x). Let C_{i} ∈ E be the lift of S containing y_{i}.

If there is an infinite subsequence i_{l} with C_{i_{l}} = C for all l, then because z_{i_{l}} → η_{+}(x), it follows that η_{+}(x) ∈ ∂_{∞}C. But for any i, the flow line through x_{i} intersects a sequence E_{1} < ⋯ < E_{J_{0}} ∈ E before intersecting C_{i}. Since x · [0, +∞) ⊂ σ^{b}E_{1} then η_{+}(x) ∈ σ^{b}E_{1}, but also η_{+}(x) ∈ ∂_{∞}C ⊂ σ^{f}E_{J_{0}}, and therefore σ^{b}E_{1} ∩ σ^{f}E_{J_{0}} ≠ ∅ contradicting lemma 3.4. We may therefore assume up to taking a further subsequence that all C_{i} are distinct from each other.

Note that there is E_{0} ∈ E such that E_{0} < C_{i} for all i; take any E_{0} so that x ∈ Fr(E_{0}). Applying lemma 3.5, there is a subsequence i_{l} so that that σ^{f}C_{i_{l}} approaches a single point of ∂_{∞}P in the Hausdorff topology. This point must be η_{+}(x), since z_{i_{l}} ∈ σ^{f}(C_{i_{l}}) and z_{i_{l}} → η_{+}(x). Therefore the sequence η_{+}(x_{i_{l}}) ∈ σ^{f}(C_{i_{l}}) approaches η_{+}(x).

The arguments of cases 2.1, 2.2 and 2.3, show that given any sequence y_{i} → x, there is always a subsequence y_{i_{l}} for which η_{+}(y_{i_{l}}) → η_{+}(x). This implies that the original sequence η_{+}(y_{i}) → η_{+}(x). This finally finishes the proof of proposition 3.9.

Propositions 3.6, 3.8 and 3.9 show that orbits in Ω_{n+1} satisfy conditions (a,b,c) of theorem 1.4 as orbits in P̃ ∪ ∂_{∞}P̃. Since P̃ ∪ ∂_{∞}P̃ embeds continuously in H^{3} ∪ S_{∞}^{2}, it follows that they also satisfy these conditions as seen in H^{3} ∪ S_{∞}^{2}. Theorem 1.4 then implies that orbits of Φ̃ in P̃ are uniformly quasigeodesic in H^{3} (hence also in P̃). Induction now finishes the proof of theorem A.

Remark: In this article we use the inductive step of this section to study flows transverse to Reebless finite depth foliations. By Novikov’s theorem [No], Φ satisfies the ϵ, T -cycles condition, hence lemma 1.3 can be applied, simplifying the proof of case 1 in proposition 3.8. However, the inductive proof we give in this section has more general hypothesis, namely: (1) Φ̃ has Hausdorff orbit space, (2) Φ is well adapted to a partial sutured manifold hierarchy, that is, the smallest sutured manifold in the hierarchy, call it P may not be a product sutured manifold; and (3) orbits entirely contained in P are uniform quasigeodesics. There may be flows satisfying these properties, which are not transverse to finite depth foliations.

4. Quasigeodesic pseudo-Anosov flows

Pseudo-Anosov flows are a generalization of suspension flows of pseudo-Anosov surface homeomorphisms. These flows behave much like Anosov flows, but they have finitely many singular orbits with a prescribed behavior. In order to define pseudo-Anosov flows, first we recall singularities of pseudo-Anosov surface homeomorphisms.

Given n ≥ 2, the quadratic differential z^{n−2}dz^{2} on the complex plane C (see [St] for quadratic differentials) has a horizontal singular foliation f^{u} with transverse measure μ^{u}, and a vertical singular foliation f^{s} with transverse measure μ^{s}. These foliations have n-pronged singularities at the origin, and are regular and transverse to each other at every other point of C. Given λ > 1, there is a map ψ: C → C which takes f^{u} and f^{s} to themselves, preserving the singular leaves, stretching the leaves of f^{u} and compressing the leaves of f^{s} by the factor λ. Let R_{θ} be the homeomorphism z → e^{2πθ}z of C. If 0 ≤ k < n the map R_{k/n} ∘ ψ has a unique fixed point at the origin, and this defines the local model for a pseudohyperbolic fixed point, with n-prongs and with rotation k. Let d_{E} be the singular Euclidean metric on C associated to the

26 SÉRGIO FENLEY AND LEE MOSHER

quadratic differential z^{n−2}dz^{2}, given by

d_{E}^{2} = μ_{u}^{2} + μ_{s}^{2}

Note that

(R_{k/n} ∘ ψ)^{∗}d_{E}^{2} = λ^{−2}μ_{u}^{2} + λ^{2}μ_{s}^{2}

Now consider the mapping torus N = C × R/(z, r + 1) ∼ (R_{k/n} ∘ ψ(z), r), with suspension flow Ψ arising from the flow in the R direction on C × R. The suspension of the origin defines a periodic orbit γ in N, and we say that (N, γ) is the local model for a pseudohyperbolic periodic orbit, with n prongs and with rotation k. The suspension of the foliations f^{u}, f^{s} define 2-dimensional foliations on N, singular along γ, called the local weak unstable and stable foliations. Note that there is a singular Riemannian metric ds on C × R that is preserved by the gluing homeomorphism (z, r + 1) ∼ (R_{k/n} ∘ ψ(z), r), given by the formula

ds^{2} = λ^{−2t}μ_{u}^{2} + λ^{2t}μ_{s}^{2} + dt^{s}

The metric ds descends to a metric on N denoted ds_{N}.

Let Φ be a flow on a closed, oriented 3-manifold M. We say that Φ is a pseudo-Anosov flow if the following are satisfied:

• For each x ∈ M, the flow line t → Φ(x, t) is C^{1}, and the tangent vector bundle D_{t}Φ is C^{0}.

• There is a finite number of periodic orbits {γ_{i}}, called singular orbits, such that the flow is smooth off of the singular orbits.

• Each singular orbit γ_{i} is locally modelled on a pseudo-hyperbolic periodic orbit. More precisely, there exist n, k with n ≥ 3 and 0 ≤ k < n, such that if (N, γ) is the local model for an pseudo-hyperbolic periodic orbit with n prongs and with rotation k, then there are neighborhoods U of γ in N and U_{i} of γ_{i} in M, and a diffeomorphism f: U → U_{i}, such that f takes orbits of the semiflow R_{k/n} ∘ψ | U to orbits of Φ | U_{i}.

• There exists a path metric d_{M} on M, such that d_{M} is a smooth Riemannian metric off of the singular orbits, and for a neighborhood U_{i} of a singular orbit γ_{i} as above, the derivative of the map f: (U − γ) → (U_{i} − γ_{i}) has bounded norm, where the norm is measured using the metrics ds_{N} on U and d_{M} on U_{i}.

• On M − ⋃ γ_{i}, there is a continuous splitting of the tangent bundle into three 1-dimensional line bundles E^{u} ⊕ E^{s} ⊕ T Φ, each invariant under Φ, such that T Φ is tangent to flow lines, and for some constants ν > 1, θ > 1 we have

1. If v ∈ E^{u} then |DΦ_{t}(v)| ≤ θν^{t}|v| for t < 0

2. If v ∈ E^{s} then |DΦ_{t}(v)| ≤ θν^{−t}|v| for t > 0 where norms of tangent vectors are measured using the metric d_{M}.

With the definition formulated in this manner, the entire theory of Anosov flows can be mimicked for pseudo-Anosov flows. In particular, a pseudo-Anosov flow Φ has a 2-dimensional weak unstable foliation F^{u} tangent to E^{u} ⊕ T Φ away from the singular orbits, and a 2-dimensional weak stable foliation F^{s} tangent to E^{s} ⊕ T Φ.

These foliations are singular along the singular orbits of Φ, and regular everywhere else. In the neighborhood U_{i} of an n-pronged singular orbit γ_{i}, the images of F^{s} and F^{u} in the model manifold N are identical with the local weak stable and unstable foliations.

The restriction of F^{s} to M− (singular orbits) defines a true foliation; a complete leaf of this foliation is called a nonsingular leaf of F^{s}; an incomplete leaf may be completed by adding a singular orbit γ of Φ, and the result is called a singular leaf of F^{s} abutting γ. Singular and nonsingular leaves of F^{u} are similarly defined. The bare term “leaf” means either a nonsingular or a singular leaf. For some small neighborhood W of any point x lying on a singular orbit γ, the singular leaves of F^{s} divide W into n sectors, each sector parameterized by D^{2} × I, so that the restriction of F^{s} to the sector agrees with the foliation by level discs D^{2} × t; and similarly for F^{u}. All the terms defined here apply as well to the lifted singular foliations F̃^{s}, F̃^{u} in M̃.

Proposition 4.1. If Φ is a pseudo-Anosov flow in M^{3} then the orbit space O of Φ̃ is homeomorphic to R^{2}.

Proof. Let F^{s}, F^{u} be the singular stable and unstable foliations of Φ. A short embedded path α is a quasi-transversal of F^{s} if one of the following happens: either α is transverse to F^{s}; or there is a singular orbit γ and x ∈ int(α) ∩ γ such that the closure of each half of α − x either is transverse to F^{s} or lies on a singular leaf of F^{s}, and the two halves do not both lie in the closure of any sector of F^{s} at x. Quasitransversals of F^{u} are similarly defined. We need the following facts about Φ, F^{s} and F^{u}.

1. Each periodic orbit of Φ is homotopically nontrivial.

2. A quasitransversal ρ of F^{u} is not path homotopic into a leaf of F^{u}; and similary for F^{s}.

3. The inclusion map of every leaf of F̃^{s} or F̃^{u} into M̃ is proper.

These facts are proved using the theory of essential laminations [Ga-Oe]; we give the argument for F^{s}. Split F^{s} along singular leaves to produce a lamination L^{s}. There is one complementary component T_{γ} of L^{s} for each singular orbit γ of Φ, and if T̂_{γ} is the metric completion of T_{γ} then T̂_{γ} is homeomorphic to a solid torus with an (n, k) torus knot removed from the boundary, where γ has n prongs and rotation k. Since n ≥ 3 for all γ it follows that L^{s} is an essential lamination. There is a map f: M → M homotopic to the identity taking L^{s} onto F^{s}, taking each leaf of L^{s} not on the boundary of some T_{γ} homeomorphically onto a non-singular leaf of F^{s}, and taking each leaf on the boundary of each T_{γ} onto a union of singular leaves abutting γ. Given a leaf L ⊂ ∂T_{γ} if gcd(n, k) ≠ 1 then L is mapped onto a union of two singular leaves and the map is 1-1 off of γ; and if gcd(n, k) = 1 then L maps onto one singular leaf and the map is 2-1 off of γ; in either case, f^{−1}(γ) ∩ L is a core curve of the annulus L.

28 SÉRGIO FENLEY AND LEE MOSHER

Properties (1–3) follow from results of [Ga-Oe]. Property (1) follows because each periodic orbit of Φ is homotopically nontrivial in a leaf of F^{s}, and because leaves of L^{s} are π_{1}-injective. Property (2) follows because if ρ′ is a path transverse to L^{s}, and if the closure of each component of ρ′ − L^{s} is not path homotopic into a leaf of L^{s}, then ρ′ is not path homotopic into a leaf of L^{s}. Property (3) follows because leaves of L̃^{s} and L̃^{u} include properly in M̃.

In order to prove the proposition, the key facts to prove are that O is a two dimensional manifold and that it is Hausdorff.

Suppose there are ϵ, T -cycles of Φ for ϵ arbitrarily small and T arbitrarily big. We can then choose x_{i} → x ∈ M̃ and t_{i} → +∞ so that y_{i} = x_{i}·t_{i} → x. Let α_{i} = x_{i}·[0, t_{i}]. If x_{i}, y_{i} are in the same local orbit of Φ̃ near x, then α_{i} can be easily completed to a closed orbit, contradicting property (1) above. If the endpoints of α_{i} are not in the same local orbit and ϵ is sufficiently small, then the endpoints of α_{i} may be joined by a path β which is a quasi-transversal to either F̃^{s} or F̃^{u}; but β is path homotopic to α_{i}, contradicting property (2) above.

This implies that O is locally 2-dimensional. We next prove that it is Hausdorff. The proof given for Anosov flows in [Fe1] goes through almost verbatim, with slight changes to take pseudo-Anosov behavior into account, or we may proceed as follows.

Let x_{i} → x, y_{i} = x_{i} · t_{i} → y in M̃. We first want to show that t_{i} is bounded. Assume then up to taking a subsequence that t_{i} → +∞. If x lies on a singular orbit γ of Φ, pass to a subsequence so that x_{i} lies in a single sector of F̃^{s} near x. Now vary x_{i} to nearby z_{i}, so that z_{i} → x, z_{i} ∈ W̃^{s}(x_{i}) ∩ W̃^{u}(x). Since flow lines in the stable foliation converge together exponentially it follows that there are s_{i} → +∞ with w_{i} = z_{i} · s_{i} → y. The sequence z_{i} forms a bounded subset of a leaf L of W̃^{u}(x), and hence the sequence z_{i} · s_{i} leaves every compact subset of L, because flow lines are properly embedded in the leaf L — the flow induces a product structure in L. But by (3) above the inclusion map L ↪ M̃ is proper, contradicting that z_{i} · s_{i} converges in M̃. This contradiction shows that the t_{i} are bounded.

As the t_{i} are bounded, we may assume up to subsequence that t_{i} → t_{0}. Then y_{i} → x·t_{0} = y. This shows that O is Hausdorff and it follows that it is homeomorphic to R^{2}.

Given any closed, irreducible, orientable 3-manifold M with H_{2}(M) ≠ 0, Gabai [Ga1] constructed many finite depth foliations in M: for any ζ ≠ 0 in H_{2}(M) there is a taut, finite depth foliation F in M so that the compact leaves of F represent ζ.

If in addition M is atoroidal, then Mosher [Mo4] constructed pseudo-Anosov flows in M which are almost tranverse to F. These flows become transverse to F after blow up of finitely many singular orbits γ_{i}, 1 ≤ i ≤ i_{0} of Φ. The blown up flow is denoted by Φ^{b}. The blow up transforms the orbit γ_{i} into Z_{i} = (T_{i} × [0, 1])/f, where T_{i} is a finite simplicial tree, and f is a homeomorphism of T_{i} × {1} to T_{i} × {0} which sends edges to edges. Each edge E of T_{i} eventually returns to itself, producing an annulus A in Z_{i}. The flow in A is as follows: the boundary consists of two orbits coherently oriented, and the interior orbits spiral from one boundary circle to the other without forming a two dimensional Reeb component. Clearly there is a global section to Φ^{b} restricted to Z_{i}. The flow Φ^{b} is transverse to F.

In addition Φ^{b} is semiconjugate to Φ: there is a continuous map q: M → M, so that

• q is homotopic to the identity,

• q takes orbits of Φ^{b} to orbits of Φ preserving orientation,

• q is C^{1} along orbits of Φ^{b},

• q is one to one except in ∪_{1≤i≤i_{0}}Z_{i},

• q(Z_{i}) = γ_{i}.

Proposition 4.2. The orbit space O^{b} of Φ̃^{b} is homeomorphic to R^{2}.

Proof. Since Φ^{b} is transverse to F, then if ϵ > 0 is very small and T > 0 very large, it follows that an ϵ, T cycle can be perturbed to be transverse to F, hence it is not null homotopic by Novikov’s theorem [No]. This shows that O^{b} is locally two dimensional. We now show that O^{b} is Hausdorff.

Lift q to a map q̃: M̃ → M̃, which is boundedly homotopic to the identity map. Let τ(x, y) be the difference in flow parameter between x, y on the same flow line of Φ̃, and let τ^{b}(x, y) be similarly defined for Φ̃^{b}.

\underline{Claim} — There are m_{0}, m_{1} > 0 so that if x, y are in the same flow line of Φ̃^{b}, then m_{0}τ^{b}(x, y) < τ(q̃(x), q̃(y)) < m_{1}τ^{b}(x, y).

Given z ∈ M, let g(z) be the derivative at z of q restricted to the flow line through z. Then since g(z) is continuous and positive and M compact, there are m_{0} and m_{1} > 0 which are the maxima and minima of g in M. The claim follows.

Let now x_{i} ∈ M̃, y_{i} = Φ̃_{t_{i}}^{b}(x_{i}) so that y_{i} → y and x_{i} → x. Then q̃(x_{i}) → q̃(x) and q̃(y_{i}) → q̃(y). Since Φ is pseudo-Anosov, the previous proposition shows that the orbit space O of Φ̃ is Hausdorff, hence q̃(y) = Φ̃_{s}(q̃(x)). Since O is homeomorphic to R^{2}, it follows that q̃(y_{i}) = Φ̃_{s_{i}}(q̃(x_{i})) where s_{i} → s.

Partitioning s_{i} into a uniformly bounded number of subsegments of length ≤ 1, and using the claim above, it follows that y_{i} = Φ̃_{r_{i}}^{b}(x_{i}) where r_{i} is uniformly bounded. Up

to taking a subsequence, assume that r_{i} converges to some r, and hence y_{i} → Φ̃_{r}^{b}(x). Consequently y = Φ̃_{r}^{b}(x). This shows that O^{b} is Hausdorff. It follows that O^{b} is homeomorphic to R^{2}.

Now we prove the main theorem:

Main theorem Let M^{3} be closed, orientable, irreducible with non zero second betti number. Let ζ ≠ 0 ∈ H_{2}(M) and F be a taut finite depth foliation with compact leaves representing ζ. Let Φ be a pseudo-Anosov flow which is almost transverse to F. Then Φ is quasigeodesic.

Proof. By the previous proposition the flow Φ̃^{b} on M̃ has Hausdorff orbit space.

Suppose first that F has a compact leaf which is not a fiber. Then Φ^{b} is quasigeodesic by theorem A. The semiconjugacy q̃ from Φ̃^{b} to Φ̃ moves points a uniformly bounded distance in M̃. Since there is also a bound on how much the map q̃ expands the lengths of flow lines (given by the claim in the previous proposition), it follows that flow lines of Φ̃ are uniform quasigeodesics.

Now suppose that each compact leaf of F is a fiber. If F is a fibration over the circle, then Φ^{b} is quasigeodesic by [Ze], and the arguments of the previous paragraph show that Φ is quasigeodesic.

Suppose F is not a fibration over the circle. We claim that F can be replaced by another finite depth foliation of smaller depth that is still transverse to Φ^{b}; continuing by induction, eventually Φ^{b} is transverse to a fibration over the circle, and we are done.

We sketch a proof of the claim. Let F have depth n. Note that each component of F − F^{0} is homeomorphic to the product of a depth 0 leaf crossed with an interval; similarly each component of F − F^{k} is homeomorphic to the product of a depth k leaf crossed with an interval. Note also that the restriction of F to a component of F − F^{n−1} is a fibration over the circle with fiber a depth n leaf L. It follows that if f: L → L is the first return map of Φ^{b}, then there is a translation map g: L → L, i.e. a map which generates a properly discontinuous, free action of Z, such that f is isotopic to g by a compactly supported isotopy. Any compact subset of L invariant under f is contained in the support of the isotopy from f to g, and hence there is a maximal compact invariant set C of f. Using the fact that Φ^{b} is a blown up pseudo-Anosov flow, the set C is nonempty if and only if f has periodic points. Moreover, any periodic points are non-removable in the proper isotopy class of f; the proof given for pseudo-Anosov surface homeomorphisms in [Bi-Ki] works just as well in the present context. Since g has no periodic points, it follows that C = ∅, and therefore f is itself a translation map. The component of F −F^{n−1} containing L may therefore be refoliated by leaves of depth n − 1, staying transverse to Φ^{b}. Doing this for each component of F − F^{n−1} proves the claim.

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