Criteria for virtual fibering

By Ian Agol. 2007

217 citations recorded by the source corpus.

We prove that an irreducible 3‐manifold with fundamental group that satisfies a certain group‐theoretic property called RFRS is virtually fibered. As a corollary, we show that 3‐dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi groups. Moreover, we show that a taut‐sutured compression body has a finite‐sheeted cover with a depth one taut‐oriented foliation.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e6b57d6e218f18a4027545130fd0e4694b16d389

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In Applied Mathematics

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Citations in the corpus

The asymptotic growth of torsion homology for arithmetic groups (2010, background): “In particular, Kl[N ] = Kl[1] if l does not divide N .”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “For X = Hd, it is known that for all d, there are closed hyperbolic d-manifolds with arbitrarily short systoles, [9], [22], [16].”
Rank gradient, cost of groups and the rank versus Heegaard genus problem (2007, background): “Agol [5] has proved that all the Bianchi groups are virtually free-by-cyclic.”
A boundary criterion for cubulation (2009, background): “Agol shows even more: if π1M is virtually special then M is virtually fibered [1] in the sense that.”
Embedability between right-angled Artin groups (2011, background): “It is an outstanding question whether the fundamental group of an arbitrary closed hyperbolic 3-manifold virtually embeds into some right-angled Artin group [16, 1].”
The Thurston norm and twisted Alexander polynomials (2012, methodology): “By picking one class in the cone above each of these faces, and repeatedly applying Agol’s theorem to the (transfer of) each such class, we obtain after finitely many steps a finite regular cover, that we will denote as well by p : M → N , such that p∗φ is quasi–fibered for any φ ∈ H1(N ;R).”
Twisted Alexander polynomials detect fibered 3-manifolds (2008, methodology): “[Gr70], [AHKS07]). But in our 3–dimensional setting we can use a recent result of Agol [ Ag08 ] to prove the following theorem.”
Virtual cubulation of nonpositively curved graph manifolds (2011, background): “This implies virtual fibering by Agol’s criterion.”
Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (2010, background): “Agol proves that right-angled Artin groups are virtually RFRS (see [1]).”
Graph manifolds with boundary are virtually special (2011, unknown): “Hence, by Agol’s virtual fibring criterion [1] such a manifold cannot have virtually special fundamental group.”
RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY (2009, background): “The key to proving the virtual fibering problem is the following beautiful result which weaves together several important ideas from 3-manifold topology [2]:”