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CRITERIA FOR VIRTUAL FIBERING Abstract. We prove that an irreducible 3-manifold whose fundamental group satisfies

Abstract. We prove that an irreducible 3-manifold whose fundamental group satisfies
a certain group-theoretic 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 3-manifold has a finite-sheeted
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 non-zero Euler class and base orbifold of non-zero
Euler characteristic [26]. Since Thurston posed this question, there have been many ex-
amples given proving this for various classes of hyperbolic 3-manifolds. Thurston gave an
example of an orbifold which virtually fibers, namely the reflection group in a right-angled
hyperbolic dodecahedron.
Gabai gave an example of a union of two I-bundles over non-orientable surfaces, which
has a 2-fold cover which fibers, but in some sense this example is too simple since it fibers


Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago


Collections: Mathematics