MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 6 Added update on the proofs of these conjectures!

ADDED (29 May, 2013)

As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaking work of Agol, Kahn--Markovic and Wise. Here's a brief summary of some of the highlights. (Shameless self-promotion: see this survey article for too many further details, including definitions of some of the terms.)

  • Haglund--Wise define the notion of special (non-positively curved) cube complex. If a closed hyperbolic 3-manifold $M$ is homotopy equivalent to a special cube complex then $M$ satisfies L (largeness, defined below).

  • Agol proves that if $M$ is homotopy equivalent to a special cube complex then $M$ also satisfies VFC (the Virtually Fibred Conjecture, also defined below).

  • Kahn--Markovic prove SSC (the Surface Subgroup Conjecture, also defined below), using mixing properties of the geodesic flow. In fact, they construct enough surfaces to show that $M$ is homotopy equivalent to a cube complex.

  • Wise proves (independently of Kahn--Markovic) that if $M$ contains an embedded, geometrically finite surface then $M$ is special.

  • Agol uses a very deep theorem of Wise (the Malnormal Special Quotient Theorem) to prove a conjecture (also of Wise), which states that word-hyperbolic fundamental groups of non-positively curved cube complexes are special. All the properties below follow.

    • It's quite a story, and many other names have gone unmentioned. There were also very important contributions by Sageev (who's thesis initiated the programme of using cube complexes to attack these problems), Groves--Manning, Bergeron--Wise, Hsu--Wise and another very deep paper of Haglund--Wise. To extend these results to the cusped hyperbolic case you need results of Hruska--Wise and Sageev--Wise. Finally, it turns out that similar results hold for all non-positively curved 3-manifolds, a result established by Liu and Przytycki--Wise.

    The Surface Subgroup Conjecture (SSC). $\pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. (Recently proved by Kahn and Markovic.)
    show/hide this revision's text 5 fixed mathscinet links

    Let $M$ be a finite-volume hyperbolic 3-manifold. (Some of these extend, suitably restated, to larger classes of 3-manifolds. But it follows from Geometrisation that the hyperbolic case is often the most interesting. These are all trivial or trivially false in the elliptic case, for example.)

    The Surface Subgroup Conjecture (SSC). $\pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed surface. (Recently proved by Kahn and Markovic.)

    The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space with an embedded incompressible subsurface.

    Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $\widehat{M}$ with $b_1(\widehat{M})\geq 1$.

    Virtually infinite first Betti number (VIFB). $M$ has finite-sheeted covering spaces $\widehat{M}_k$ with $b_1(\widehat{M}_k)$ arbitrarily large.

    Largeness (L). $\pi_1(M)$ has a finite-index subgroup that surjects a non-abelian free group.

    The Virtually Fibred Conjecture (VFC). $M$ has a finite sheeted cover that is homeomorphic to the mapping torus of a (necessarily pseudo-Anosov) surface automorphism. This is false for graph manifolds. There are fairly easy implications

    $L\Rightarrow VIFB \Rightarrow VPFB \Rightarrow VHC \Rightarrow SSC$.

    Also, a fortiori,

    $VFC\Rightarrow VPFB$.

    Recently, Daniel Wise announced a proof that $VHC\Rightarrow VFC$. His proof also shows that, if $M$ has an embedded geometrically finite subsurface, then we get $L$ and other nice properties.

    This list is similar to the one that Agol links to in the comments. Also, I suppose it's exactly what Daniel Moskovich meant by 'The Virtually Fibred Conjecture, and related problems'. I thought some people might be interested in a little more detail.

    Paul Siegel asks in comments: 'Would it be correct to guess that the "virtually _ conjecture" problems can be translated into a question about the large scale geometry of the fundamental group?'

    Certainly, it's true that most of these can be translated into an assertion about how (some finite-index subgroup of) $\pi_1M$ splits as an amalgamated product, HNN extension or, more generally, as a graph of groups. The equivalence uses the Seifert--van Kampen Theorem in one direction, and something like Proposition 2.3.1 of Culler--Shalen in the other. Rephrased like this, some of the above conjectures turn out as follows.

    The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits.

    Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as an HNN extension.

    Largeness (L). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as a graph of groups with underlying graph of negative Euler characteristic.

    The Virtually Fibred Conjecture (VFC). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits can be written as a semi-direct product

    $\pi_1(\widehat{M}) \cong K\rtimes\mathbb{Z}$

    with $K$ finitely generated. (Here we invoke Stallings' theorem that a 3-manifold whose fundamental group has finitely generated commutator subgroup is fibred.)

    I don't think I know a way to rephrase $VIFB$ in terms of splittings of $\pi_1$.

    Often, when people say 'the large scale geometry of $\pi_1$' they're talking about properties that are invariant under quasi-isometry. I'm really not sure whether these splitting properties (or, more exactly, 'virtually having these splitting properties') are invariant under quasi-isometry. Perhaps something like the work of Mosher--Sageev--Whyte does the trick?
    show/hide this revision's text 4 missed a hat

    Let $M$ be a finite-volume hyperbolic 3-manifold. (Some of these extend, suitably restated, to larger classes of 3-manifolds. But it follows from Geometrisation that the hyperbolic case is often the most interesting. These are all trivial or trivially false in the elliptic case, for example.)

    The Surface Subgroup Conjecture (SSC). $\pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed surface. (Recently proved by Kahn and Markovic.)

    The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space with an embedded incompressible subsurface.

    Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $\widehat{M}$ with $b_1(\widehat{M})\geq 1$.

    Virtually infinite first Betti number (VIFB). $M$ has finite-sheeted covering spaces $\widehat{M}_k$ with $b_1(\widehat{M}_k)$ arbitrarily large.

    Largeness (L). $\pi_1(M)$ has a finite-index subgroup that surjects a non-abelian free group.

    The Virtually Fibred Conjecture (VFC). $M$ has a finite sheeted cover that is homeomorphic to the mapping torus of a (necessarily pseudo-Anosov) surface automorphism. This is false for graph manifolds. There are fairly easy implications

    $L\Rightarrow VIFB \Rightarrow VPFB \Rightarrow VHC \Rightarrow SSC$.

    Also, a fortiori,

    $VFC\Rightarrow VPFB$.

    Recently, Daniel Wise announced a proof that $VHC\Rightarrow VFC$. His proof also shows that, if $M$ has an embedded geometrically finite subsurface, then we get $L$ and other nice properties.

    This list is similar to the one that Agol links to in the comments. Also, I suppose it's exactly what Daniel Moskovich meant by 'The Virtually Fibred Conjecture, and related problems'. I thought some people might be interested in a little more detail.

    Paul Siegel asks in comments: 'Would it be correct to guess that the "virtually _ conjecture" problems can be translated into a question about the large scale geometry of the fundamental group?'

    Certainly, it's true that most of these can be translated into an assertion about how (some finite-index subgroup of) $\pi_1M$ splits as an amalgamated product, HNN extension or, more generally, as a graph of groups. The equivalence uses the Seifert--van Kampen Theorem in one direction, and something like Proposition 2.3.1 of Culler--Shalen in the other. Rephrased like this, some of the above conjectures turn out as follows.

    The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits.

    Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as an HNN extension.

    Largeness (L). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits as a graph of groups with underlying graph of negative Euler characteristic.

    The Virtually Fibred Conjecture (VFC). $M$ has a finite-sheeted covering space $\widehat{M}$ such that $\pi_1(\widehat{M})$ splits can be written as a semi-direct product

    $\pi_1M\cong \pi_1(\widehat{M}) \cong K\rtimes\mathbb{Z}$

    with $K$ finitely generated. (Here we invoke Stallings' theorem that a 3-manifold whose fundamental group has finitely generated commutator subgroup is fibred.)

    I don't think I know a way to rephrase $VIFB$ in terms of splittings of $\pi_1$.

    Often, when people say 'the large scale geometry of $\pi_1$' they're talking about properties that are invariant under quasi-isometry. I'm really not sure whether these splitting properties (or, more exactly, 'virtually having these splitting properties') are invariant under quasi-isometry. Perhaps something like the work of Mosher--Sageev--Whyte does the trick?
    show/hide this revision's text 3 Fixed typo, added answer to Siegel's comment
    show/hide this revision's text 2 Corrected typos
    show/hide this revision's text 1 [made Community Wiki]