 
Summary: CRITERIA FOR VIRTUAL FIBERING
IAN AGOL
Abstract. We prove that an irreducible 3manifold whose fundamental group satisfies
a certain grouptheoretic property is virtually fibered. As a corollary, we show that 3
dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic
form virtually fiber. Moreover, we show that a taut sutured compression body has a finite
sheeted cover with a taut orientable depth one foliation.
1. Introduction
Thurston proposed the question of whether a hyperbolic 3manifold has a finitesheeted
cover fibering over S1
[29]. A fibered manifold must be irreducible, and there exist non
fibered irreducible manifolds which have no cover which fibers over S1
, the simplest exam
ples being Seifert fibered spaces with nonzero Euler class and base orbifold of nonzero
Euler characteristic [26]. Since Thurston posed this question, there have been many ex
amples given proving this for various classes of hyperbolic 3manifolds. Thurston gave an
example of an orbifold which virtually fibers, namely the reflection group in a rightangled
hyperbolic dodecahedron.
Gabai gave an example of a union of two Ibundles over nonorientable surfaces, which
has a 2fold cover which fibers, but in some sense this example is too simple since it fibers
