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:

  1. 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.
  2. 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.


1 Answer 1


It does hold - Wise proved that finite-volume non-compact hyperbolic 3-manifolds are virtually special, hence virtually RFRS and so virtually fibred by one of Agol's results. Details and references are contained in this survey.

  • $\begingroup$ Also, I the non-compact case one does not even need Kahn and Markovich. $\endgroup$
    – Misha
    Feb 13, 2014 at 4:50
  • $\begingroup$ Or Agol's (other) theorem! $\endgroup$
    – HJRW
    Feb 13, 2014 at 5:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.