\newcommand{\R}{{\mathcal R}}
\newcommand{\Fol}{\fol}
\newcommand{\Annuli}{{\mathcal A}}
\newcommand{\bdyv}{\bdy_{v}}
\newcommand{\bdyin}{\bdy_{h}^-}
\newcommand{\bdyout}{\bdy_{h}^+}
\newcommand{\Hull}{{\mathcal H}}
\newcommand{\compact}[1]{\cover #1 \union \bdy_\infinity \cover #1}
\newcommand{\inject}{\hookrightarrow}
\newcommand{\Ker}{\func{Ker}}
\newcommand{\subgroup}{<}
\newcommand{\cover}{\widetilde}
\newcommand{\Rplus}{[0,+\infinity)}
\newcommand{\RPlus}{\Rplus}


\section{Hierarchies}

Finite depth foliations are closely related to sutured manifold
hierarchies \cite{Ga1}. In \cite{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review this subject here, providing some proofs of known but unpublished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
\cite{Ga1} and \cite{Ga-Oe}.

A $2$-dimensional {\em foliation} of a 3-manifold $M$ is a decomposition
of $M$ into $2$-dimensional manifolds called {\em leaves} which fit
together in a local product structure. A foliation $\fol$ is {\em taut}
if it is  transversely oriented, no leaf is a sphere, and each leaf of
$\fol$  intersects some closed curve in $M$ which is transverse to
$\fol$. The leaves of a taut foliation are $\pi_1$-injective \cite{No}.

A {\em lamination} of $M$ is a foliation of a closed subset of $M$,
covered by charts of the form $D^2 \cross [0,1]$ so that each component of a
leaf intersected with the chart has the form $D^2 \cross t$ for some
$t \in [0,1]$. A lamination $\Lambda$ is {\em essential} if it has no sphere
leaves, no Reeb components, and if $M_\Lambda$ is the metric completion
of $M - \Lambda$, then $M_\Lambda$ is irreducible, boundary
incompressible, and end incompressible; the latter condition means
intuitively that $\bdy M_\Lambda$ has no ``infinite folds''. The leaves
of an essential lamination $\Lambda$ are $\pi_1$-injective in $M$, and
the components of  $M - \Lambda$ are $\pi_1$-injective.

A taut foliation is obviously an essential lamination. It is an exercise
in the results of \cite{Ga-Oe} to show that every sublamination of an
essential lamination is essential.

Given a foliation $\fol$ in a closed manifold $M$  we say that a leaf $L$
of $\fol$ is {\em proper} if $\overline L - L$ is a closed subset  of $M$
and $\fol$ is {\em proper} if  all leaves are proper. In that case Zorn's
lemma implies that $\fol$ has compact leaves, which are then the depth  0
leaves. The {\em depth} is an ordinal $\omega$ defined by induction: a
leaf $L$ is at {\em depth} $\omega$ if $\overline L - L$ is contained in
the union of leaves of depth $< \omega$, but not contained in the union
of leaves $< \omega_1$ for any $\omega_1 < \omega$. A proper foliation
has {\em finite depth} $n$  if $n$ is the maximum of the depth of its
leaves.


A {\em branched surface} in a closed 3-manifold $M$ is a smooth
2-complex $B \subset M$ such that for each $x \in B$, 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 {\em branch locus}. A {\em sector} of $B$ is a complementary
component of the branch locus. In this paper, {\em all branched surfaces
will be transversely oriented}. A flow (or semiflow) is transverse to
$B$ if, for each $x \in B$, the tangent vector to the flow points
to the same side of $T_x B$ as the transverse orientation.

Given a branched surface $B \subset M$, a {\em semiflow
neighborhood} $U(B)$ is a piecewise smooth regular neighborhood
equipped with a semiflow $\varphi$ satisfying the following conditions.
Each orbit of $\varphi$ is compact, transverse to $B$, and pierces $B$
in at least one point. There is a decomposition of $\bdy U(B)$ into
subsurfaces with disjoint interior
$$\bdy U(B) = \bdyv U(B) \union \bdyin U(B) \union \bdyout U(B)
$$
such that orbits of $\varphi$ point inward along $\bdyin
U(B)$, outward along $\bdyout U(B)$, and are {\em internally
tangent} along
$\bdyv U(B)$ as shown in figure \ref{BoundaryTan}. Each component of
$\bdyv U(B)$ is an annulus and the orbits of $\varphi$ define a fibration
of the annulus over the circle with interval fiber. If one forgets the
orientation and parameterization of $\varphi$, one obtains an $I$-bundle
neighborhood of $B$ as in \cite{Ga-Oe}.

\begin{figure}
\centeredepsfbox{Tangencies.eps}
\caption{Boundary tangencies of semiflows}
\label{BoundaryTan}
\end{figure}

We shall regard the {\em exterior} of $U(B)$ as a sutured manifold, as
follows.

A {\em sutured manifold} is an oriented 3-manifold $P$ whose boundary
is decomposed into surfaces with disjoint interior as $\bdy P = \R_- P
\union \R_+ P \union \Annuli P$, where $\R_- P$ and $\R_+ P$ are
disjoint, $\Annuli P$ is a collection of annuli called the
{\em sutures}, $\R_- P \intersect \Annuli P = \bdy(\R_- P) \subset \bdy
(\Annuli P)$, $\R_+ P \intersect \Annuli P = \bdy(\R_+ P) \subset \bdy
(\Annuli P)$, and each component of $\Annuli P$ has one boundary component
in $\R_- P$ and the other in $\R_+ P$. Let $\R_\pm P = \R_- P \union \R_+
P$. A sutured manifold $P$ is a {\em product} if there is a homeomorphism
$(P,\Annuli P) \homeo (S \cross I, \bdy S \cross I)$ for some compact
surface $S$. Given a semiflow on a sutured manifold $P$, we say that $P$
is an {\em isolating block} for the semiflow if orbits point outward
along $\R_+ P$, inward along $\R_- P$, and are externally tangent along
$\gamma P$ as shown in figure \ref{BoundaryTan}. A {\em foliation} of a
sutured manifold $P$ is required to be tangent to $\R_\pm P$ and
transverse to $\Annuli P$.  Near $\Annuli P$, a transverse foliation and
semiflow appear in cross section as in figure \ref{SuturedFoliation}.

\begin{figure}
\centeredepsfbox{FoliationAndFlow.eps}
\caption{A foliation and semiflow on a sutured manifold.}
\label{SuturedFoliation}
\end{figure}

If $B \subset M$ is a branched surface then $P(B) = \closure(M - U(B))$
has the structure of a sutured manifold where $R_+ P(B) = \bdyin U(B)$,
$R_- P(B) = \bdyout U(B)$, and $\Annuli P(B) = \bdyv U(B)$. Given a flow
$\varphi$ on $M$ transverse to $B$, there exists a regular neighborhood
$U(B)$ so that the restriction of $\varphi$ makes $U(B)$ into a semiflow
neighborhood, and so that $P(B)$ is an isolating block for $\varphi$. We
say that $U(B)$ is {\em adapted} to $\varphi$.

A {\em hierarchy} on $M$, in the sense of Gabai, is a sequence of
branched surfaces $B_0 \subset B_1 \subset \cdots \subset B_{N}$ such
that $B_0$ is a surface, $B_n - B_{n-1}$ is a union of sectors of
$B_n$ for each $1 \le n \le N$, and $P(B_{N})$ is a product. Let
$S(B_n) = B_n \intersect P(B_{n-1})$; this is a transversely oriented
surface properly embedded in $P(B_{n-1})$, and it is obtained from the
sectors $B_n - B_{n-1}$ by removing an open collar of each boundary
component. We may assume that each component $c$ of $\bdy S(B_n)$
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 $c \intersect \Annuli P(B_{n-1})$ is an arc connecting
opposite boundaries of some component of $\Annuli P(B_{n-1})$. 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
$\bdyv U(B_{n-1})$ 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 \cite{Ga1}.


Let $\fol$ be a transversely oriented foliation in $M$, with a fixed
transversal flow $\varphi$. Given a saturated open set $W$, let $\widehat
W$ be its metric completion. The inclusion $\iota \from W \inject M$
extends to an immersion $\hat \iota \from \widehat W \rightarrow M$
carrying each component of $\partial \widehat W$ diffeomorphically onto a
leaf of $\fol$, but $\hat\iota$ may identify some of these boundary
components pairwise. In addition the structure of the foliation in
$\widehat W$ is as follows \cite{Di,Ca-Co}: $\widehat W = Q \union 
(\Sigma \times I)$, where $Q$ is a compact (possibly empty) sutured
manifold, each component of $\Sigma$ is a noncompact surface with compact
connected boundary, $Q$ is glued to $\Sigma \cross I$ by identifying
$\Annuli Q$ with $\bdy \Sigma \cross I$, and the restrictions of $\fol$
and $\varphi$ to $Q$ and to $\Sigma \cross I$ are well-behaved as
follows. The restrictions to $Q$ give a sutured manifold foliation and
semiflow; and the restriction of $\varphi$ to $\Sigma \cross I$ is a
product flow, that is, orbits have the form $x \cross I$. Note that
the restriction of $\fol$ to $\Sigma \cross I$ {\em need not} have
leaves of the form $\Sigma \cross t$. If a connected
open saturated set $W$ has $Q = \emptyset$, then $\widehat W$ (or $W$)
is called a {\em foliated  product}. An important fact is that for any
saturated open set $W$, at most finitely many components of $W$ are not
foliated products \cite{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.






\begin{proposition}{(hierarchy exists)}{} 
If $\Fol$ is a taut finite depth foliation on $M$ and $\varphi$ is a flow
transverse to $\Fol$, then there exists a hierarchy $B_0 \subset \cdots
\subset B_N$ transverse to $\varphi$ with neighborhoods $U(B_0) \subset
\cdots \subset U(B_N)$ adapted to $\varphi$, such that the sutured
manifolds $P(B_n)$ and the surfaces $S(B_n)$ are $\pi_1$-injective in
$M$, and each component of $B_0$ is isotopic to a compact leaf of $\Fol$.
\label{hierarchy}
\end{proposition}

The construction of finite depth foliations in theorem 5.1 of
\cite{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.

\begin{proof}{} Let $\Fol_n$ be the union of leaves of $\Fol$ of depth at
most $n$. Since $\fol$ is proper, $\fol_n$ is closed, hence it is a
sublamination of $\fol$, therefore an essential lamination in $M$. Let
$M_n = M - \fol_n$, and $\widehat M_n$ its metric completion. Then at
most finitely many components of $M_n$ are not foliated products. Now we
may alter $\fol$ without altering $\varphi$, 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 $\fol_n$. In
addition if $L$ at depth $n$ is not isolated, then the component of
$\fol_n - \fol_{n-1}$  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 $\fol$ without altering
$\varphi$. Each connected component of $\Fol_n - \Fol_{n-1}$ is now
either a fibration over a closed interval, or a fibration over the
circle; the latter are called ``circular'' components of $\Fol_n -
\Fol_{n-1}$.  Under these condition $\widehat M_n$ is a 1-1 immersed
submanifold of $M$ whose boundary components are leaves of $\Fol_n$. 

Now we construct by induction a hierarchy $B_0 \subset \cdots \subset
B_N$ and neighborhoods $U(B_0) \subset \cdots \subset U(B_N)$ adapted to
$\varphi$ so that if $0 \le n \le N$ then:

\begin{enumerate}
\item $U(B_n)$ contains every noncircular component of $\Fol_n -
\fol_{n-1}$, as well as an interval's worth of leaves in every circular
component.

\item $\Fol$ restricts to a foliation of the sutured manifold 
$P(B_n) = \overline{\bigl(M - U(B_n)\bigr)}.
$
It may be a product foliation
in some components of  $P(B_n)$.

\item The embedding $P(B_n) \inject P(B_{n-1})$ is $\pi_1$-injective.

\item $S(B_n) = B_n \intersect P(B_{n-1})$ is $\pi_1$-injective in
$P(B_{n-1})$.
\end{enumerate}

If $\Fol$ 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 $\Fol$ are incompressible surfaces in $M$. 

To continue the induction, given a component $Z$ of $P(B_n)$ on which
the restriction of $\Fol$ and $\varphi$ 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 $V(\bdy_\pm Z)$
be a neighborhood so that the restriction of $\phi$ has compact
orbits, defining a product structure with projection map $q \from
V(\bdy_\pm Z) \to \bdy_\pm Z$. 

Let $\fol'$ be the restriction of $\fol_{n+1}$ to $Z$. This lamination
has depth $\le 1$. Its compact leaves consist of $\bdy_\pm Z$ plus
foliated products $F \cross I$ with $\bdy F \cross I \subset \Annuli
Z$; all but finitely many of these foliated products have leaves
parallel to a component of $\bdy_\pm Z$. The noncompact leaves of
$\fol'$ 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 $\bdy_\pm Z$ in the following
manner: there is a nonseparating, properly embedded, connected
1-manifold $c \subset F$ called a {\em juncture} for $E$, there is a
$\integers$-covering space
$\tilde F \to F$ to which $c$ lifts homeomorphically, there is a
subset $L_E \subset L$ representing $E$ and contained in $V(F)$ (the
component of $V(\bdy_\pm Z)$ containing $F$), and there is an embedding
$L_E \to\tilde F$, such that the maps $L_E \to \tilde F \to F$ and $L_E
\inject V(F) \to F$ are the same, and the image of $\bdy L_E$ is the
curve $c$. If $E, E'$ are ends of noncompact leaves spiralling into
the same component $F$ of $\bdy_\pm Z$, and if $E,E'$ are not in the
same end of a circular component of $\fol'$, we may assume that the
junctures for $E,E'$ are disjoint but isotopic curves in~$F$.

Let $\fol''$ be the union of the finitely many foliated products in
$\fol'$ which are not contained in $V(\bdy_\pm Z)$, plus a closed
interval of leaves in each circular component of
$\fol'$, plus $\bdy_\pm Z$. Since $Z - \fol''$ is an open subset of
$Z$ saturated by $\fol \restrictedto Z$, letting ${\overline {Z -
\fol''}}$ be the metric completion of $Z - \fol''$, there is a
decomposition ${\overline {Z - \fol''}} = Q \union (\Sigma \cross I)$
where $Q$ is a compact sutured manifold and $\Sigma \cross I$ is a
foliated product so that each noncompact component of $\Sigma$ has
compact, connected boundary. We can choose $Q$ big enough so that
$\Sigma \cross I$ is contained in $V(\bdy_\pm Z)$, and so that the
boundary of each component of $\Sigma$ projects to some juncture, thus
we can associate a juncture to each component of $\Sigma$. Moreover, if
$\Sigma_1,\ldots,\Sigma_n$ are components of $\Sigma$ with isotopic
junctures $c_1,\ldots, c_n$ lying in a component $F$ of $\bdy_\pm
Z$, then we can choose $Q$ so that the picture in figure
\ref{junctures} holds: the curves $c_1,\ldots,c_n$ are ordered so
that $c_i \union c_{i+1}$ bounds an annulus disjoint from the other
curves; and the ``first hitting'' map $\Sigma_{i+1} \to \Sigma_{i}$ is
continuous for $i=1,\ldots,n-1$, where this map is defined by starting
from a point in $\Sigma_{i+1}$, going along an orbit of $\phi$ towards
$F$, and stopping at the first point of $\Sigma_{i}$.


\begin{figure}
\centeredepsfbox{junctures.eps}
\caption{Isotopic junctures}
\label{junctures}
\end{figure}

Now define $P(B_{n+1}) \intersect Z = Q$. Notice that $\fol''$ is an
essential lamination of $Z$, hence $Z - \fol''$ is
$\pi_1$-injective in $Z$. Since each noncompact component of $\Sigma$
has compact, connected boundary, it follows that $Q$ is
$\pi_1$-injective in $Z - \fol''$, 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
$\fol'' - \bdy_\pm Z = G \cross [0,1]$ for some surface $G$, not
necessarily connected. Each end of $G \cross 0$ or $G \cross 1$ is
eventually contained in $\Sigma \cross I$. Let $L$ be the set of $x \in
G$ such that $x \cross [0,1]$ is disjoint from $\Sigma \cross I$.
Then $L$ is a subsurface of $G$, and we may regard $L$ as being
embedded in $\fol''$ transverse to the flow. Homotop $\bdy L$ 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 $B{n+1} -
B_n$ intersecting $Z$. To describe the homotopy of $\bdy L$, each
component $\beta$ of $\bdy L - \Annuli Z$ lies in
$V(\bdy_\pm Z)$, and $\beta$ projects via $\phi$ to a juncture;
homotop $\bdy L$ along orbits of $\phi$ until $\beta$ lies on that
juncture. Extend the homotopy of $\bdy L - \Annuli Z$ over all of
$\bdy L$, so that each arc component of $\bdy L \intersect \Annuli Z$
maps to an arc in $B_n$ crossing the branch locus in a single
transverse intersection point, and each circle component of $\bdy L
\intersect \Annuli Z$ maps to a circle in the branch locus of $B_n$.
Because of the properties of junctures given in figure
\ref{junctures}, this homotopy may be carried out so that $L - \bdy
L$ 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 $\pi_1$-injective in in $P(B_n)$.

The only nonobvious points remaining in the proof of the lemma are the
statements on $\pi_1$-injectivity. Using (3) it follows by induction that
$P(B_n)$ is $\pi_1$-injective in $M$. Using (4), and the fact that
$P(B_{n-1})$ is $\pi_1$-injective in $M$, it follows that $S(B_n)$ is
$\pi_1$-injective in $M$.
\end{proof}



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 {\em fiber} if $F$ is a leaf of a
fibration of $M$ over $S^1$. If $M$ is hyperbolic we say that $F$ is
{\em quasi-fuchsian} if the representation $\pi_1 F \to \pi_1 M \to
\Isom(\hyperbolic^3)$ is quasi-fuchsian. A deep fact due to Marden
\cite{Ma}, Thurston \cite{Th1} and Bonahon \cite{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 \cite{Gr,Gh-Ha} that a locally compact path metric space
$X$ is {\em negatively curved in the large} if it satifies the ``thin
triangles'' condition: there exists $\delta>0$ such that for every
geodesic triangle with sides $A_1,A_2, A_3$  we have $A_1
\subset U_\delta(A_2 \union A_3)$. In this case there is a
canonically defined ideal boundary at infinity denoted
$\bdy_\infinity X$, and there is a canonical compactification $X
\union \bdy_\infinity X$. 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 $f \from X \to Y$ is a quasi-isometry then
$f$ extends canonically to a continuous map $\hat f \from X \cup
\bdy_{\infty} X \rightarrow Y \cup \bdy_{\infty} Y$ that
restricts to an embedding $\bdy_\infinity X \inject \bdy_\infinity Y$.
For example, hyperbolic space $\hyperbolic^n$ is negatively curved in the
large, and its ideal boundary in the sense of Gromov is the same as the
sphere at infinity $S^{n-1}_\infinity$. Also, any closed convex subset $X
\subset \hyperbolic^n$ is negatively curved in the large, and
$\bdy_\infinity X$ coincides with the set of limit points of $X$ in
$S^{n-1}_\infinity$. 

Given a group $G$ with a finite generating set $A$, the {\em Cayley
graph} $C$ is the graph with a vertex for each $g \in G$, and an edge from
$g$ to $ga$ for each $a \in A$. 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
{\em word hyperbolic} if some (and hence any) Cayley graph of $G$ is
negatively curved in the large. 
Then we let $\partial _{\infty} G = \partial _{\infty} C$.
Consider a compact Riemannian
manifold $M$ with $G = \pi_1(M,x)$, choose a lift $x_0$ of $x$ in the
universal cover $\widetilde M$ and choose loops in $M$ representing a
finite generating set for $\pi_1 (M)$. Lifting to $\widetilde M$ we have a
map of the Cayley graph of $G$ to $\widetilde M$, which gives a 
quasi-isometry between $G$ and $\widetilde M$ \cite{Mi}. Therefore $G
= \pi_1 M$ is word hyperbolic if and only if $\widetilde M$ is
negatively curved in the large, in which case there is a canonical
identification
$\bdy_\infinity G \homeo \bdy_\infinity \widetilde M$.

\begin{proposition}{}{} Let $M$ be a closed, orientable hyperbolic
3-manifold, $B_0 \subset \cdots \subset B_N$ a hierarchy in $M$ such
that $P(B_n)$ and $S(B_n) = B_n \intersect P(B_{n-1})$ are
$\pi_1$-injective for all $n$. Suppose that some component $F$ of
$B_0$ is a quasi-fuchsian surface in $M$. Then:

\smallskip
\noindent
(1) Each group $\pi_1(P(B_n))$, $\pi_1(S(B_n))$ is word hyperbolic.

\noindent 
(2) Each embedding $\cover S(B_n) \inject \cover P(B_{n-1})$ is a
quasi-isometry which takes $\bdy \cover S(B_n)$ into
$\bdy \cover P(B_n)$. Therefore it extends to an embedding $\cover
S(B_n) \union \bdy_{\infty} \cover S(B_n) \inject \cover P(B_{n-1})
\union \bdy_\infinity \cover P(B_{n-1})$.

\noindent
(3) Each inclusion $\cover P(B_{n}) \inject \cover P(B_{n-1})$ is a
quasi-isometry in the path metrics induced from $\mi$.
Therefore it extends to an embedding
$\cover P(B_n) \cup \partial _{\infty} \cover P(B_n) \inject
\cover P(B_{n-1}) \cup \partial _{\infty} \cover P(B_{n-1})$.

\label{DiscsAndBalls}
\end{proposition}

\noindent
Remark: The group $\pi_1 (S(B_n))$ may be cyclic, when $S(B_n)$
is an annulus; or trivial, when $S(B_n)$ is a disc.

\begin{proof}{}  
Let $\rho \from \pi_1(M) \rightarrow PSL_2({\bf C})$ be the holonomy
representation of the hyperbolic structure. Consider the subgroup $G =
\rho(\pi_1(M-F))$ with limit set $\Lambda G \subset S^2_\infinity$, let
$\Hull (\Lambda G) \subset \hyperbolic^3 \union S^2_\infinity$ be its convex
hull, let $\Hull_\epsilon (\Lambda G) \subset \hyperbolic^3 \union
S^2_\infinity$ be the union of $\Hull (\Lambda G)$ with its
$\epsilon$-neighborhood in $\hyperbolic^3$, and let $X_G =
\Hull_\epsilon (\Lambda G) \intersect \hyperbolic^3$. Since
$F$ is quasi-Fuchsian, the group $G$ is convex cocompact, that is
$G$ acts cocompactly on $X_G$; equivalently, $G$ is geometrically finite
\cite{Ma,Th1}. It follows that any finitely generated subgroup $H
\subgroup G$ is convex cocompact \cite{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 $P \subset M-F$ is a compact, $\pi_1$-injective submanifold and $H
= \pi_1 (P)$. We show that the natural inclusion $j \from \cover P \to
\hyperbolic^3$ is a quasi-isometric embedding, where $\widetilde P$ 
has the path metric induced from $\mi$.

The natural inclusion
$\iota \from X_H \inject \hyperbolic^3$ is a quasi-isometric embedding.
Let $\Gamma_H$ be the Cayley graph of $H$, and consider the two
quasi-isometries $\alpha \from\Gamma_H \to X_H$, $\beta \from \Gamma_H
\to
\cover P$ given by \cite{Mi}. The maps $\iota\composed \alpha, j
\composed \beta \from \Gamma_H \to\hyperbolic^3$ differ by a bounded
amount, therefore $j \composed \beta$ is a quasi-isometric embedding.
There is a quasi-isometry $\bar\beta \from \cover P \to \Gamma_H$
which is an inverse of $\beta$ in the quasi-isometric category, so in
particular
$\beta \composed \bar\beta \from \cover P \to \cover P$ differs from
the identity by a bounded amount.  Thus we have a quasi-isometric
embedding
$j \composed \beta \composed \bar\beta \from \cover P \to
\hyperbolic^3$ which differs from $j$ by a bounded amount, so $j$ is a
quasi-isometric embedding.  Hence each $\widetilde P(B_n) \inject
\hhh$ extends to a  continuous embedding $\cover P(B_n) \cup \partial
_{\infty} 
\cover P(B_n)  \inject \hhh \cup \si$.

To prove (3), the quasi-isometric embedding $\cover P(B_n) \inject
\hyperbolic^3$ factors through the natural embedding $\cover P(B_n)
\inject \cover P(B_{n-1})$, so $\cover P(B_n) \inject \cover
P(B_{n-1})$ is a quasi-isometric embedding. The proof of (2) is
similar.
\end{proof}

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

\smallskip
\noindent
(4) Each compactified universal cover $\widetilde P(B_n) \cup
\bdy_{\infty} \widetilde P(B_n)$ is a 3-ball.

\noindent
(5) Each $\widetilde S(B_n) \union \bdy_{\infty} \widetilde
S(B_n)$ is a 2-disc properly embedded in the above 3-ball. 
\smallskip

To prove (4), let $P = P(B_n)$, let $H = \pi_1(P)$, and consider
the action of $H$ on $\hyperbolic^3$. In the manifold $\hyperbolic^3 /
H$, both of the compact manifolds $X_H / H$ and $P$ embed as deformation
retracts. By \cite{Mc-MS} it follows that there is a homeomorphism $X_H /
H \homeo P$ in the correct homotopy class. This homeomorphism lifts to an
$H$-equivariant quasi-isometric homeomorphism $X_H \homeo \cover P$, which
extends to a homeomorphism $\Hull_\epsilon (\Lambda H) \homeo \compact P$,
and $\Hull_\epsilon (\Lambda H)$ 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
$\cover S(B_n) \union \bdy_\infinity S(B_n)$ is therefore a 2-disc,
which by (2) is properly embedded in the 3-ball $\compact P$. 
