Summary: Bundles and finite foliations.
D. Cooper \Lambda D. D. Long y A. W. Reid z
By a hyperbolic 3manifold, we shall always mean a complete orientable hyperbolic 3manifold of
finite volume. We recall that if \Gamma is a Kleinian group then it is said to be geometrically finite if
there is a finitesided convex fundamental domain for the action of \Gamma on hyperbolic space. Otherwise,
\Gamma is geometrically infinite. If \Gamma happens to be a surface group, then we say it is quasiFuchsian if
the limit set for the group action is a Jordan curve C and \Gamma preserves the components of S 2
1 n C.
The starting point for this work is the following theorem, which is a combination of theorems due
to Marden , Thurston  and Bonahon .
Theorem 1.1 Suppose that M is a closed orientable hyperbolic 3manifold. If g : S # M is a
ß 1 injective map of a closed surface into M then exactly one of the two alternatives happens:
ffl The geometrically infinite case: there is a finite cover ~
M of M to which g lifts and can be
homotoped to be a homeomorphism onto a fiber of some fibration of ~
M over the circle.
ffl The geometrically finite case: g \Lambda ß 1 (S) is a quasiFuchsian group.
The dichotomy between geometrically finite and geometrically infinite is fundamental and despite
the fact that these two cases exhibit widely different behaviour, it seems to be a very difficult