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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.

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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.

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  • $\begingroup$ Also, I the non-compact case one does not even need Kahn and Markovich. $\endgroup$ – Misha Feb 13 '14 at 4:50
  • $\begingroup$ Or Agol's (other) theorem! $\endgroup$ – HJRW Feb 13 '14 at 5:25

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