Worked solution: Agol's proof of the Virtual Haken conjecture via cube complexes (2012)
Before Agol's work even starts, Perelman's proof of Thurston's geometrization conjecture (2002-2003) had already cut every closed 3-manifold into standard geometric pieces. Most of those pieces (Seifert fibered spaces, manifolds split by an essential torus) were already known to be virtually Haken by earlier, more elementary arguments.
What was left genuinely open was exactly the pieces carrying a hyperbolic structure: spaces built by gluing straight-sided polyhedra in so that they tile it perfectly, with no group of symmetries acting on the whole picture except the deck transformations themselves.
By Perelman's proof of Thurston's Geometrization Conjecture (2002-2003), every closed, orientable, irreducible 3-manifold decomposes canonically into geometric pieces; the interesting, previously open case of the Virtual Haken conjecture reduces to the piece carrying a hyperbolic structure, i.e. for a discrete, torsion-free group of isometries of hyperbolic 3-space acting freely and cocompactly.
Every other geometric piece in Thurston's classification — spherical, Euclidean, Seifert-fibered, or containing an essential torus — was already known before Agol's work to be virtually Haken by more classical arguments specific to those geometries, so nothing further needs to be said about them here.
So from here on the entire proof concentrates on a single, sharply defined target: closed hyperbolic 3-manifolds, whose fundamental group is word-hyperbolic in the sense of Gromov — exactly the class of groups to which Agol's cube-complex theorem applies.
- Word-hyperbolic group
- A finitely generated group whose Cayley graph, viewed as a metric space, satisfies a thin-triangles condition in the sense of Gromov; such groups behave, at large scale, like discrete groups of isometries of hyperbolic space.