

\section*{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 \cite{Th1}. This, among other reasons, makes
quasigeodesics extremely useful in studying hyperbolic manifolds
\cite{Th1,Th2, Mor,Ca}.

A natural question to ask is: how common are quasigeodesic flows? Notice
that Zeghib \cite{Ze} proved that there cannot exist a continuous
foliation by geodesics in a hyperbolic $3$-manifold. On the other hand,
in their seminal work \cite{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
\cite{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 \cite{Mo3} produced examples of quasigeodesic 
flows transverse to a class of depth one foliations in hyperbolic
$3$-manifolds.

The flow constructed in \cite{Ca-Th} is the suspension of a
pseudo-Anosov homeomorphism of the fiber. Hence it is a {\em
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 \cite{Mo3} are also pseudo-Anosov. Mosher showed
that quasigeodesic pseudo-Anosov flows on hyperbolic manifolds can be
used to compute the Thurston norm \cite{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 \cite{Fe1}. In addition the
quasigeodesic property for Anosov flows is related to the topology of
the stable and unstable foliations in the universal cover \cite{Fe2}
and implies that limit sets of leaves of these foliations are Sierpinski
curves \cite{Fe3}. 

The main goal of this paper is to show that quasigeodesic flows are
quite common. If $M^3$ is closed, oriented, 
irreducible with $H_2(M) \not = 0$ and
if $\zeta \not = 0$ in $H_2(M)$, then Gabai \cite{Ga1} constructed a
taut, finite depth foliation $\fol$ whose set of compact leaves
represents $\zeta$. Given such $\fol$ in a hyperbolic $3$-manifold,
Mosher \cite{Mo4} constructed pseudo-Anosov flows which are almost
transverse to $\fol$. Almost transverse means that it will be transverse
to $\fol$ after an appropriate blow up of a finite collection of closed
orbits (see detailed definition in section $4$).


\medskip
\noindent {\bf Main theorem} \ {\em Let $M$ be a closed, oriented,
hyperbolic
$3$-manifold with non zero  second betti number
and let $\zeta$ a nonzero homology
class in $H_2(M)$. 
Let $\fol$ be a taut, finite depth foliation whose compact
leaves represent $\zeta$, and $\Phi$ a pseudo-Anosov flow which is
almost transverse to $\fol$. Then $\Phi$ is a quasigeodesic flow.}
\medskip

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

\medskip
\noindent {\bf Theorem A} \
{\em Let $M$ be a closed, oriented,  hyperbolic $3$-manifold with
a taut, finite depth foliation $\fol$, so that some 
compact leaf is not a fiber of $M$ over the circle. Let $\Phi$
be a flow transverse to $\fol$ and let $\wwp$ be the
lifted flow to the universal cover.
Then $\Phi$ is quasigeodesic if and only if
$\wwp$ has Hausdorff orbit space.
}
\medskip


The orbit space of a flow is the quotient space obtained by collapsing
each flow line to a point. 
%It is easy to prove that if $\Phi$ is transverse to a Reebless
%foliation then $\wwp$ having Hausdorff orbit space is 
%equivalent to having orbit space homeomorphic to the plane $\rrrr^2$.
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 $\wwp$ 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 $\wwp$ is
not Hausdorff.

When $\fol$ 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 $\Phi$, the covering flow
$\wwp$ 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
$\fol$ 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
$\fol$ 
represents a quasi-Fuchsian subgroup. Therefore this leaf lifts to a
quasi-isometrically embedded surface in $\mi$, which has excellent
geometric properties. 
That means that if a 
flow line in $\mi$ keeps intersecting lifts
of compact leaves, these lifts trap the flow line
which then converges to a single
point in $\si$.


Next we proceed to extend this argument to all orbits.
For that, we use sutured manifold hierarchies
and branched surfaces associated to the foliation $\fol$. 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 $\mi$, its
flow line in forward time converges to a unique point in the sphere at
infinity $\si$, 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 $\hhh \cup \si$ 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: 


\medskip
\noindent {\bf Theorem B} \
{\em 
Let $\Phi$ be a flow in $M^3$ closed hyperbolic.
Suppose that:

(a) Each half orbit of $\wwp$ has a unique limit point in $\si$,

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

(c) The forward and backward limit point maps are continuous.

\noindent
Then $\Phi$ is quasigeodesic.
}
\medskip

The paper is organized as follows.
In section $1$ we prove a generalization of theorem $B$, 
which applies to closed
invariant sets of $\Phi$. 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.

