I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the circle. I could not find the same result stated for finite volume, possibly cusped, hyperbolic $3$-manifolds. If such a manifold has a virtually special fundamental group then a result of Agol implies the virtual fibering property.
In order to prove that the fundamental group of a closed hyperbolic $3$-manifold is virtually special, one relies on the following:
- Kahn-Markovic's result on the existence of immersed almost geodesic surfaces in closed hyperbolic $3$-manifolds, which allows to build a CAT(0) cube complex on which the fundamental group acts properly discontinuously and cocompactly.
- Agol's result which states that a hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex is virtually special.
The first result is stated only for closed hyperbolic $3$-manifolds. The second one holds for (word)-hyperbolic groups, but the fundamental group of cusped hyperbolic $3$-manifolds is hyperbolic relative to the cusp subgroups.
I would be interested in knowing if anyone has managed to get around these issues.