Virtual Haken conjecture

Solved, 2012Topology

Statement

Every compact, orientable, irreducible $3$ -manifold $M$ with infinite fundamental group $\pi_1(M)$ is virtually Haken: it admits a finite-sheeted covering space $\widetilde{M} \to M$ that is a Haken manifold (meaning $\widetilde{M}$ contains a properly embedded, two-sided, incompressible surface of genus at least $1$ , or a properly embedded incompressible disk).

Solved by Ian Agol in 2012 (published in Documenta Mathematica in 2013, with an appendix by Ian Agol, Daniel Groves, and Jason Manning). Following Grigori Perelman's proof of William Thurston's Geometrization Conjecture (2002–2003), the problem reduced to closed hyperbolic $3$ -manifolds. Jeremy Kahn and Vladimir Markovic (2012) proved the Surface Subgroup Conjecture, furnishing ubiquitous quasi-convex surface subgroups in $\pi_1(M)$ , which by the Bergeron–Wise criterion allowed $\pi_1(M)$ to act cocompactly on a $\operatorname{CAT}(0)$ cube complex. Agol completed the program initiated by Daniel Wise and Frédéric Haglund by proving that every cubulated word-hyperbolic group is virtually special, which implies both the Virtual Haken Conjecture and Thurston's Virtual Fibering Conjecture.

Agol's theorem that cubulated hyperbolic groups are virtually special (meaning they have a finite-index subgroup that embeds into a right-angled Artin group) resolved an entire cluster of Thurston's problems simultaneously. Combining Agol's theorem with Daniel Wise's earlier results and his 2008 theorem that fundamental groups of Haken hyperbolic manifolds are virtually RFRS (residually finite rationally solvable), every closed hyperbolic $3$ -manifold $M$ was shown to be virtually fibered over the circle $S^1$ , and its fundamental group $\pi_1(M)$ was shown to be linear over $\mathbb{Z}$ and subgroup separable (LERF).

References

Ian Agol (2013). The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning) · arXiv:1204.2810
Jeremy Kahn, Vladimir Markovic (2012). Immersing almost geodesic surfaces in a closed hyperbolic three manifold · DOI:10.4007/annals.2012.175.3.4 · arXiv:0910.5501
Frédéric Haglund, Daniel T. Wise (2008). Special cube complexes · DOI:10.1007/s00039-008-0643-6

History of the problem

1962Wolfgang Haken introduces normal surface hierarchies for irreducible $3$ -manifolds containing embedded incompressible surfaces (now called Haken manifolds), yielding algorithms to recognize the unknot and homeomorphic Haken manifolds.Wolfgang Haken
1968Friedhelm Waldhausen proves that homotopy equivalent closed Haken manifolds are homeomorphic and asks whether every closed irreducible $3$ -manifold with infinite fundamental group has a finite-sheeted Haken cover.
1982William Thurston publishes his list of $24$ open problems on $3$ -manifolds, highlighting the Virtual Haken Conjecture (Problem $16$ ), the Surface Subgroup Conjecture (Problem $17$ ), and the Virtual Fibering Conjecture (Problem $18$ ).
2003Grigori Perelman completes the proof of Thurston's Geometrization Conjecture via Ricci flow with surgery, reducing the Virtual Haken Conjecture to the case of closed hyperbolic $3$ -manifolds.Grigori Perelman
2012Building on the Kahn–Markovic Surface Subgroup Theorem and Daniel Wise's theory of special $\operatorname{CAT}(0)$ cube complexes, Ian Agol proves that cubulated hyperbolic groups are virtually special, resolving the Virtual Haken Conjecture.