Summary: Bundles and nite foliations.
D. Cooper D. D. Long y
A. W. Reid z
By a hyperbolic 3-manifold, we shall always mean a complete orientable hyperbolic 3-manifold of
nite volume. We recall that if ; is a Kleinian group then it is said to be geometrically nite if
there is a nite-sided convex fundamental domain for the action of ; on hyperbolic space. Otherwise,
; is geometrically in nite. If ; happens to be a surface group, then we say it is quasi-Fuchsian if
the limit set for the group action is a Jordan curve C and ; preserves the components of S2
1 n C.
The starting point for this work is the following theorem, which is a combination of theorems due
to Marden 10], Thurston 14] and Bonahon 1].
Theorem 1.1 Suppose that M is a closed orientable hyperbolic 3-manifold. If g : S # M is a
1-injective map of a closed surface into M then exactly one of the two alternatives happens:
The geometrically in nite case: there is a nite cover ~M of M to which g lifts and can be
homotoped to be a homeomorphism onto a ber of some bration of ~M over the circle.
The geometrically nite case: g 1(S) is a quasi-Fuchsian group.
The dichotomy between geometrically nite and geometrically in nite is fundamental and despite
the fact that these two cases exhibit widely di erent behaviour, it seems to be a very di cult
problem in general to nd a criterion in terms of the image g(S) which distinguishes them.