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\begin{document}
\title{Geometry and topology of $\mathbb{R}$-covered foliations}

\author{Danny Calegari}
\address{Department of Mathematics, UC Berkeley, Berkeley, CA 94720}
\email{dannyc@math.berkeley.edu}

\commby{Walter Neumann}
\date{May 7, 1999}


\subjclass[2000]{Primary 57M50}
\keywords{Foliations, laminations, $3$-manifolds, geometrization,
$\mathbb{R}$-covered, product-covered, group actions on $\mathbb{R}
$ and $S^1$}

\begin{abstract}
An $\mathbb{R}$-covered foliation is a special type of taut foliation on a
$3$-manifold: one for which holonomy is defined for all transversals
and all time. The universal cover of a manifold $M$ with such a
foliation can be partially compactified by a cylinder at infinity,
somewhat analogous to the sphere at infinity of a hyperbolic manifold.
The action of $\pi_1(M)$ on this cylinder decomposes into a product
by elements of $\text{Homeo}(S^1)\times\text{Homeo}(\mathbb{R})$.
 The action on the 
$S^1$ factor of this cylinder is rigid under deformations of the
foliation through $\mathbb{R}$-covered foliations. Such a foliation admits
a pair of transverse genuine laminations whose complementary
regions are solid tori with finitely many boundary leaves, which
can be blown down to give a transverse regulating pseudo-Anosov
flow. These results all fit in an essential way into Thurston's
program to geometrize manifolds admitting taut foliations.
\end{abstract}
\maketitle

\section{Holonomy}

\subsection{Notation} 

Our notational convention in this paper will be that if $\Sigma$ is some
structure or object on or in a manifold $M$, then $\til{\Sigma}$
denotes the pullback of $\Sigma$ to the universal cover of $M$, which
is consequently denoted by $\til{M}$.

\subsection{Collating holonomy}

\begin{defn}
A {\em taut foliation} $\F$ of a $3$-manifold $M$ is a codimension one
foliation such that there is a circle in $M$ transverse to $\F$ which
intersects every leaf.
\end{defn}

For a taut foliation, every leaf is incompressible, and every transverse
loop is homotopically essential by \cite{sN65}. Furthermore, there is a
metric on $M$ such that every leaf of $\F$ is a minimal surface by \cite{dS79}.

In this paper we will discuss a particular special class of taut foliations,
known as $\R$-covered foliations. 

\begin{defn}
An $\R$-covered foliation $\F$ is a taut, codimension one foliation of a
$3$-manifold $M$ such that the pulled-back foliation $\til{\F}$ of
$\til{M}$ is the product foliation of $\R^3$ by
horizontal $\R^2$'s.
\end{defn}

The definition of $\R$-covered given above is global in nature. 
It would be nice to characterize $\R$-covered foliations by some 
local property, but this seems to be very hard to do. It is easy to
come up with examples of taut foliations which are not $\R$-covered.
For example, if $\F$ contains a compact leaf which is not the fiber
of a fibration over $S^1$, $\F$ is not $\R$-covered.

$\R$-covered foliations have an intuitive description in terms of holonomy.
Suppose we have a square $S$ transverse to $\F$ whose top and bottom sides
are tangent to $\F$, and whose left and right sides are therefore transverse
to $\F$. Then the intersection of $\F$ with the square $S$ induces a
codimension $1$ foliation of $S$, giving it the form of a product 
$I \times I$, where each vertical interval $I \times \text{point}$ is
transverse to $\F$, and each horizontal interval $\text{point} \times I$ is
tangent to a leaf of $\F$. This product structure allows one to define a
homeomorphism from the left side of $S$ to the right side of $S$, by
\begin{equation*}H_{[S]}:(t,0) \to (t,1)\end{equation*} 
called the {\em holonomy transport}. One observes that
this homeomorphism only depends on the homotopy class of $S$ through
transverse squares with left and right sides fixed. 

For a general pair of transversals $\tau,\tau'$ and a homotopy class of
path $\gamma$ from a point in $\tau$ to a point in $\tau'$, one cannot
always find a square $S$ as above containing $\tau$ and $\tau'$ in its
left and right sides, and $\gamma$ as a horizontal leaf, {\em unless the
foliation is $\R$-covered}.
This has as a consequence that there is a well-defined
{\em holonomy representation} of $\pi_1(M)$ in $\myhom(L)$, where $L$
denotes the leaf space of $\til{\F}$. Of course, $L = \R$.

\begin{thm}[folklore]
The holonomy groupoids over all leaves of an $\R$-covered foliation
can be collated into a representation
\begin{equation*}\rho_H:\pi_1(M) \to \myhom(\R)\end{equation*}
called the {\em holonomy representation}.
\end{thm}

Philosophically, it might be reasonable to extend the definition to
include foliations
covered by the product foliation of $S^2\times\R$ by horizontal $S^2$'s since
these too have a holonomy representation, but conventionally one does 
not do so.

We make the simplifying assumption in what follows that our $3$-manifold
$M$ is orientable, and our foliation $\F$ is co-orientable. It follows that
the holonomy representation acts by orientation-preserving homeomorphisms of
the leaf space $\R$. This is not a serious restriction, since it can be
achieved by passing to a finite cover.

\section{Confined leaves and slitherings}

In \cite{wT97}, Thurston makes the following definition:

\begin{defn}
A manifold $M$ {\em slithers over $S^1$} if there is a fibration
$\phi:\til{M} \to S^1$ such that $\pi_1(M)$ acts on this fibration by
bundle maps. A slithering induces a foliation of $\til{M}$ by the
connected components of preimages of points in $S^1$ under the slithering
map, and this descends to an $\R$-covered foliation of $M$.
\end{defn}

By compactness of $M$ and $S^1$, it is clear that the leaves of
$\til{\F}$ stay within bounded neighborhoods of each other for a foliation
$\F$ obtained from a slithering. That prompts
us to define a confined leaf.

\begin{defn}
A leaf $\lambda$ of the pulled-back foliation $\til{\F}$ of $\til{M}$ is
{\em confined} if there is a neighborhood $U$ of $\lambda$ in the leaf
space $L$ such that for any $\mu$ in $U$ there is a $\delta$ with the
property that
\begin{equation*}\mu \subset N_\delta(\lambda),\end{equation*}
where $N_\delta(*)$ denotes the $\delta$-neighborhood of a set.
\end{defn}

In \cite{dC99} we show that this condition is {\em symmetric}. That is,
we show

\begin{thm}
For $\lambda,\mu$ leaves of $\til{\F}$, where $\F$ is $\R$-covered, there
exists $\delta$ such that $\mu \subset N_\delta(\lambda)$ iff there exists
$\delta'$ such that $\lambda \subset N_{\delta'}(\mu)$.
\end{thm}

It follows that the confined leaves of $\F$ (i.e. those whose covers in
$\til{\F}$ are confined) are an open subset of $M$, and the
unconfined leaves are a minimal set. Having established this technical
theorem, one can show that either every leaf is confined, or else the
confined leaves of $\til{\F}$ fall into a collection of pockets of
uniformly bounded transverse width, and they can be ``blown down'' (i.e.
collapsed) in a standard way. In fact, we have the dichotomy:

\begin{cor}
Let $\F$ be an $\R$-covered foliation. Then after possibly blowing down some
confined pockets of leaves, we may assume that every leaf is dense. If every
leaf is dense, then either $\til{\F}$ contains no confined leaves 
or $\F$ arises from a slithering of $M$ over $S^1$.
\end{cor}

This dichotomy can be translated into information about the holonomy
representation as follows:

\begin{cor}
Let $\F$ be an $\R$-covered foliation. Then after possibly blowing down some
confined pockets of leaves, the holonomy representation is one of the
following two kinds:
\begin{enumerate}
\item{$\rho_H(\pi_1(M))$ is conjugate to a subgroup of 
$\til{\myhom(S^1)}$, the universal central extension of $\myhom(S^1)$.}
\item{For any two compact intervals $I,J \subset L$ there is an 
$\alpha \in \pi_1(M)$ such that 
\begin{equation*}\rho_H(\alpha)(I) \subset J.\end{equation*}}
\end{enumerate}
\end{cor}

In \cite{wT97} Thurston conjectured that for an atoroidal
$M$, every $\R$-covered foliation should arise from a slithering. This
would make our corollary quite uninteresting. However, this conjecture
is false, and in \cite{dC99a} we construct many examples of $\R$-covered
foliations of hyperbolic $3$-manifolds which do not arise from 
slitherings.

\section{Topology at infinity}

For $M$ atoroidal and not covered by $S^2 \times S^1$, any taut foliation
may be assumed to have leaves of exponential growth. That is, the leaves
look coarsely like surfaces of negative curvature. The classical uniformization
theorem has been generalized by Candel to this context, and in fact one
knows from \cite{aC93}:

\begin{thm}[Candel]
Let $\F$ be a taut foliation of an atoroidal, irreducible $3$-manifold $M$. 
Then there is a metric on $M$ such that the induced metric on leaves 
has constant curvature $-1$. 
\end{thm}

We say that such a foliation has 
{\em hyperbolic leaves}. In the sequel, we assume
that our $\R$-covered foliation has hyperbolic leaves. In the universal
cover, leaves of $\til{\F}$ are isometric to copies of the hyperbolic
plane $\myH^2$, and therefore they each have a circle at infinity. For a
leaf $\lambda$ of $\til{\F}$, let $S^1_\infty(\lambda)$ denote its circle
at infinity.

In \cite{lG83} it is shown that under very general circumstances, a foliation
admits a {\em harmonic measure}. That is, a measure $m$ on the foliated
manifold, such that for any measurable, leafwise $C^\infty$ function $f$,
we have \begin{equation*}\int_M \Delta_\F f \thickspace dm = 0,\end{equation*} 
where $\Delta_\F$ denotes the leafwise
Laplacian. The leafwise Laplacian is a generalization of the standard
Laplacian for a Riemannian manifold. Informally, if we imagine that each
leaf of our foliation is painted on either side with a perfect coat of
insulating paint so that heat can only flow {\em within} a leaf and not
transverse to it, then the leafwise Laplacian governs the constrained
heat flow for this system, exactly as the usual Laplacian governs the
heat flow for a Riemannian manifold. In a local coordinate patch, where
$T\F$ is spanned by an orthonormal basis
$\frac {\partial} {\partial x_i}$ for $1 \le i \le k$
and $\frac {\partial} {\partial x_i}$ for $k < i \le n$ are transverse,
the leafwise Laplacian looks like
\begin{equation*}\Delta_\F = \sum_{i=1}^k \frac {\partial^2} {\partial x_i^2} + \text{higher
order terms}.\end{equation*}


For foliations with every leaf dense, the measure $m$ disintegrates
into the product of the leafwise Lebesgue measure with a measure on
normal transversals which is infinitesimally harmonic, thought of as a 
function on the leaves.
Using this structure, Thurston shows in
\cite{wT98} that exactly one of the following possibilities occurs:

\begin{itemize}
\item{The transverse measure can be decomposed nontrivially
as a sum of transverse
measures $m_h + m_i$ where $m_h$ is harmonic and
$m_i$ is {\em invariant} --- that is, its value on a transversal is
invariant under holonomy transport of that transversal.}
\item{For each pair of leaves $\lambda,\mu$ which are distance
$\epsilon$ apart at $p \in \lambda$, a random walk on $\lambda$ will
shrink the distance between $\lambda$ and $\mu$ to $0$ with probability
$1-\delta$, where $\delta \to 0$ as $\epsilon \to 0$.}
\end{itemize}

A random walk on $\myH^2$ determines a unique point at 
infinity with probability $1$, so this shows that a subset of 
$S^1_\infty(\lambda)$ of visual measure close to $2\pi$ (for small $\epsilon$)
can be compared with a similar subset of $S^1_\infty(\mu)$.

These directions at infinity where leaves converge allow one to partially
define a connection on the union of circles $\coprod S^1_\infty(\lambda)$.
That is, one has a collection of transversals to the $S^1$ direction which
allow us to identify subsets of $S^1_\infty(\mu)$ with $S^1_\infty(\lambda)$. 
These transversals are dense in any circle, and can be used to topologize the 
union of circles. 

As a basis for this topology, take sets $U$ defined as follows: for a
point $p \in S^1_\infty(\lambda)$, there are nearby transversals $\tau_l,
\tau_r$ which run between $S^1_\infty(\lambda^+)$ and $S^1_\infty(\lambda^-)$
where $\lambda$ is between $\lambda^+$ and $\lambda^-$ in $L$. Then
the points of $U$ are the points in the circles $S^1_\infty(\mu)$ for
$\mu$ between $\lambda^+$ and $\lambda^-$ which are squeezed between the
transverse points $\tau_l \cap \mu$ and $\tau_r \cap \mu$. These open sets
generate a topology for which $C_\infty = \coprod S^1_\infty(\lambda)$
is homeomorphic to a cylinder. The transversals sit in this cylinder as
a collection of vertical lines of various lengths, whose closure is the
whole of $C_\infty$.

There is an alternate description of the topology on $C_\infty$. For
each small transversal $\tau$ to $\til{\F}$, the unit tangent bundle to
$\til{\F}$ restricts to a circle bundle over $\tau$, whose total space is
a cylinder. Each point in this cylinder determines a unique point in
$C_\infty$, by following the corresponding leafwise 
geodesic ray emanating from $\tau$ until it lands at a point in the circle
at infinity of that leaf. This map is a homeomorphism for the topology on
$C_\infty$ defined above.

As mentioned above, the collection of transversals sitting in this 
cylinder $C_\infty$ is
like a partially defined connection, which one would like to extend to
a complete connection and thereby ($\pi_1(M)$-equivariantly)
trivialize $C_\infty$ as a product. One cannot integrate along this
partial connection for all time --- one falls off the end of an integral curve.
Nevertheless, one can define, canonically, a collection of trajectories which
are integral curves of the partial connection where it is defined. These are
simply the {\em leftmost trajectories} --- those transversals to the circles
in $C_\infty$ which rotate clockwise as much as possible without ever
crossing any of the transversals defined by the harmonic measure. Starting
from an initial point, the leftmost trajectory is defined for all time, but
it is possible that two (initially distinct) leftmost trajectories may run
into each other in finite time. However, using the property of
$\rho_H(\pi_1(M))$ that it can blow up any interval in $L$ as large as
we like, one can find leftmost trajectories which are distinct for as
long as required. Extracting a limit, one of the two
possibilities must occur:

\begin{enumerate}
\item{Leftmost trajectories never run into each other.}
\item{There is an invariant {\em spine} --- a bi-infinite properly embedded
arc in $C_\infty$ transverse to the $S^1$'s which is invariant under the
action of $\pi_1(M)$.}
\end{enumerate}

In \cite{dC99} we show

\begin{thm}
If $\F$ is an $\R$-covered foliation with hyperbolic leaves and $C_\infty$
contains an invariant spine, then (after blowing down) 
either $\F$ comes from a slithering, or
$M$ is solv and $\F$ is the suspension of
the stable or unstable foliation of an Anosov automorphism of a torus.
\end{thm}
If $\F$ is the suspension of the stable or unstable
foliation of an Anosov automorphism of a torus, then in fact
$C_\infty$ is a product, and leftmost trajectories are distinct for all
time.

Thurston shows in \cite{wT97} that for foliations coming from a slithering,
the universal cylinder is a product. In fact this is easy to see, since
when every pair of leaves $\mu,\lambda$ in $\til{\F}$ 
are a uniformly bounded distance
apart, a geodesic on $\mu$ stays a bounded distance away from a unique
geodesic on $\lambda$, allowing us to canonically identify all the circles
at infinity. In short, we have

\begin{thm}
Let $\F$ be an $\R$-covered foliation with hyperbolic leaves. Then the
circles $S^1_\infty$ can be collated in a cylinder 
$C_\infty = S^1_{\myu} \times \R$ such that $\pi_1(M)$ acts by elements of
$\myhom(S^1) \times \myhom(\R)$.
\end{thm}

If $M$ is not solv, and $\F$ with dense hyperbolic leaves does not 
come from a slithering, we say
that $\F$ is {\em ruffled}. This name reflects the fact that for such
a foliation, each pair of leaves in $\til{\F}$ have a dense set of
points at infinity where they are asymptotic to each other, and a
dense set of points at infinity where they diverge from each other.

The proof that $C_\infty$ for a ruffled foliation is a product is very
robust, and may be greatly generalized. For many invariant structures 
at infinity one can blow up the vertical direction by the action of
$\pi_1(M)$ to show that the structure is either vertical, or
else there is an invariant spine and $M$ is solv. An example shows how
genuine laminations in $M$ interact with a ruffled foliation.

We begin with the definition of a genuine lamination (see \cite{dGuO87}).
\begin{defn}
A {\em lamination} in a $3$-manifold is a foliation of a closed subset of $M$
by $2$ dimensional leaves. The complement of this closed subset falls into
connected components, called {\em complementary regions}. 
A lamination is {\em essential} if it contains
no spherical leaf or torus leaf bounding a solid torus, and furthermore if
$C$ is the closure (with respect to the path metric in $M$)
of a complementary region, then $C$ is irreducible and
$\partial C$ is both incompressible and {\em end incompressible} in $C$.
Here an end compressing disk is a properly embedded $(D^2 - (\text{closed
arc in } \partial D^2))$ in $C$ which is not properly isotopic rel $\partial$
in $C$ to an embedding into a leaf. Finally, an essential lamination is
{\em genuine} if it has some complementary region which is not an $I$-bundle.
\end{defn}

Each complementary region falls into two pieces: the {\em guts}, which
carry the essential topology of the complementary region, and the {\em
interstitial regions}, which are just $I$ bundles over noncompact surfaces,
which get thinner and thinner as they go away from the guts. The interstitial
regions meet the guts along annuli. Ideal polygons can be properly embedded
in complementary regions, where the cusp neighborhoods of the ideal points
run up the interstitial regions as $I \times \R^+$. An end compressing disk
is just a properly embedded ideal monogon which is not isotopic 
rel $\partial$ into a leaf.

In \cite{dGuO87} many important properties of genuine laminations are
established, including the fact that their leaves are incompressible.

\begin{defn}
A lamination of a circle $S^1$ is a closed subset of the space of
unordered pairs of distinct points in $S^1$ such that no two pairs
link each other.
\end{defn}
Thinking of $S^1$ as the circle at infinity of $\myH^2$, a lamination of
$S^1$ gives rise to a lamination of $\myH^2$, by joining each pair of points
in $S^1$ by the unique geodesic in $\myH^2$ connecting them. A lamination
$\Lambda_{\myu}$ of $S^1_{\myu}$ invariant under the action of $\pi_1(M)$ 
determines a lamination
in each leaf of $\til{\F}$, and the union of these laminations sweep out
a lamination $\til{\Lambda}$
of $\til{M}$ which, by equivariance of the construction,
covers a lamination $\Lambda$ 
in $M$. By examining $\til{\Lambda}$, one sees that $\Lambda$ is genuine.

\begin{thm}
Let $\Lambda$ be a genuine lamination transverse to a ruffled foliation,
so that the intersection of any leaf of $\til{\Lambda}$ with any leaf of
$\til{\F}$ is a quasigeodesic in that leaf of $\til{\F}$. Then $\Lambda$ arises
from a $\pi_1(M)$-equivariant lamination $\Lambda_{\myu}$ on $S^1_{\myu}$. 
\end{thm}

If the intersection is not quasigeodesic in each leaf of $\til{\F}$, we
can nevertheless straighten out the intersections leafwise. If $\lambda$
is a leaf of $\til{\Lambda}$ and $\mu$ is a leaf of $\til{\F}$ which intersect
in a properly embedded arc $\alpha \subset \mu$, then the two ends of
$\alpha$ determine two subsets $\alpha^\pm$
of $S^1_\infty(\mu)$. We replace $\alpha$
by $\bar{\alpha}$ which runs between the most anticlockwise points in
$\alpha^\pm$. This straightens out $\Lambda$ to a new lamination 
$\bar{\Lambda}$, but in performing the straightening it is possible
that we collapse some of $\Lambda$, and we do not necessarily know that
our new lamination is isotopic to the old one.

\section{Regulating vector fields}

\begin{defn}
A vector field $X$ transverse to an $\R$-covered foliation $\F$ is
{\em regulating} if in the universal cover, every integral curve of
$\til{X}$ intersects every leaf of $\til{\F}$ exactly once.
\end{defn}

By analyzing the action of $\pi_1(M)$ on $S^1_{\myu}$ defined in the previous
section, following Thurston one can construct a pair of 
laminations $\Lambda^\pm_{\myu}$ on $S^1_{\myu}$
invariant under the action of $\pi_1(M)$.
We show in \cite{dC99} that any such invariant lamination has 
complementary regions which are {\em finite-sided} 
ideal polygons.
As before, an invariant lamination on $S^1_{\myu}$ determines a lamination in
each leaf of $\til{\F}$ by the construction outlined at the end of the 
previous section.

If $M$ is atoroidal, the pair of 
laminations {\em bind} each leaf of $\til{\F}$; that is,
they cut up the leaves into finite-sided geodesic polygons of bounded
diameter. The combinatorics of this decomposition is constant as we
vary from leaf to leaf, and it therefore allows us to equivariantly define
identifications between leaves of $\til{\F}$. This identification can be
easily made on the intersections of $\Lambda^+$ and $\Lambda^-$, and
then inductively extended to the complementary intervals and polygons. 
This lets us canonically identify each leaf $\lambda$ of
$\til{\F}$ with a copy of $\myH^2$ in a $\pi_1(M)$ equivariant way. The
fibers of this identification give a $1$-dimensional foliation transverse
to $\til{\F}$, and the tangent vectors to this foliation descend to a
regulating vector field on $M$. One can arrange for this vector
field to have some closed orbits, corresponding to the centers of the polygonal
regions in each leaf. If $M$ is toroidal but $\F$ has hyperbolic leaves,
the tori can be shown to be regulating, and therefore we can inductively
decompose $M$ along regulating tori to show the following:

\begin{thm}
An $\R$-covered foliation with hyperbolic leaves admits a regulating 
transverse vector field which can be chosen to have some closed orbits.
\end{thm}

\begin{cor}
If $M$ admits an $\R$-covered foliation with hyperbolic leaves, then $\til{M}$
has the structure of a product $\R^2 \times \R$ such that $\pi_1(M)$ acts
by elements of $\myhom(\R^2) \times \myhom(\R)$. Moreover, there are elements
$\alpha \in \pi_1(M)$ such that $\rho_H(\alpha)$ has no fixed points, and
is conjugate to a translation.
\end{cor}

Following \cite{wT98b} and \cite{lM00}, one can collapse $\til{M}$ to the
intersection $\til{\Lambda}^+ \cap \til{\Lambda}^-$ to give a 
slightly different
transverse regulating flow. The flow that one constructs is {\em topologically
pseudo-Anosov}, as defined by Mosher in \cite{lM00}. The issue of whether
or not the flow is literally pseudo-Anosov comes down to a smoothability
issue.

Combined with the result at the end of the previous section, we can see that
the universal circle is stable under perturbations through $\R$-covered
foliations:

\begin{cor}
Let $\F_t$ for $-1
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\bigskip
\noindent{As remarked earlier, all the possibilities suggested by this 
flowchart actually occur, by the construction in \cite{dC99a}.}



\begin{thebibliography}{99}
\bibitem{dC99}
	D. Calegari,
	\emph{The geometry of $\R$-covered foliations I},
	math.GT/9903173.
\bibitem{dC99a}
	D. Calegari,
	\emph{$\R$-covered foliations of hyperbolic $3$-manifolds},
	Geometry and Topology {\bf 3} (1999), pp. 137--153.
\MR{2000c:57038}
\bibitem{dC00}
	D. Calegari,
	\emph{Foliations with one-sided branching},
	preprint.
\bibitem{aC93}
	A. Candel,
	\emph{Uniformization of surface laminations},
	Ann. Sci. Ec. Norm. Sup. (4) {\bf 26} (1993), pp. 489--516.
\MR{94f:57025}
\bibitem{dGwK93}
	D. Gabai and W. Kazez,
	\emph{Homotopy, isotopy and genuine laminations of $3$-manifolds},
	in {\em Geometric Topology} (ed. W. Kazez) proceedings of
	the 1993 Georgia International Topology Conference,
	Vol. 1, pp. 123--138. \MR{98k:57026}
\bibitem{dGuO87}
	D. Gabai and U. Oertel,
	\emph{Essential laminations in $3$-manifolds},
	Ann. Math. (2) {\bf 130} (1989), pp. 41--73. \MR{90h:57012}
\bibitem{lG83}
	L. Garnett,
	\emph{Foliations, the ergodic theorem and Brownian motion},
	J. Func. Anal. {\bf 51} (1983), pp. 285--311. \MR{84j:58099}
\bibitem{lM00}
	L. Mosher,
	\emph{Laminations and flows transverse to finite depth foliations, Part I:
        Branched surfaces and dynamics}, preprint.
\bibitem{sN65}
	S. Novikov,
	\emph{Topology of foliations},
	Trans. Mosc. Math. Soc. (1965), pp. 268--304. \MR{34:824}
\bibitem{dS79}
	D. Sullivan,
	\emph{A homological characterization of foliations consisting 
of minimal surfaces}, 
	Comm. Math. Helv. {\bf 54} (1979), pp. 218--223. \MR{80m:57022}
\bibitem{wT97}
	W. Thurston,
	\emph{$3$-manifolds, foliations and circles I},
	math.GT/9712268.
\bibitem{wT98}
	W. Thurston,
	\emph{$3$-manifolds, foliations and circles II},
	preprint.
\bibitem{wT98b}
	W. Thurston,
	\emph{Hyperbolic structures on $3$-manifolds II: Surface groups and
$3$-manifolds which fiber over the circle}, math.GT/9801045.
\end{thebibliography}


\end{document}