Worked solution: Agol's proof of the Virtual Haken conjecture via cube complexes (2012)
Think of trying to find a flat, rigid disc that you can push through a curved hyperbolic space so that it lies almost perfectly flat everywhere, closing up into a surface without tearing. Kahn and Markovic (2012) proved that inside any closed hyperbolic 3-manifold, such nearly-flat immersed surfaces exist in great abundance, of arbitrarily large genus.
Each such surface contributes a copy of a surface group sitting inside in a well-behaved ("quasi-convex") way — the raw geometric material that the next step will assemble into an algebraic structure Agol's machinery can act on.
Kahn and Markovic (2012) proved the Surface Subgroup Conjecture: every closed hyperbolic 3-manifold contains, for arbitrarily large genera , an immersed surface that is nearly totally geodesic (its second fundamental form is uniformly small), giving a quasi-convex subgroup .
Their proof builds these surfaces probabilistically, gluing together many small, almost-geodesic pairs of pants (three-holed spheres) chosen so that the boundary curves match up with exponentially high probability as the genus grows, a technique with no counterpart in earlier 3-manifold topology.
A single surface only carries limited information about ; the abundance and near-totally-geodesic quality of Kahn-Markovic's surfaces is exactly the raw material Sageev's construction, in the next step, needs to build a rich enough cube complex.
- Quasi-convex subgroup
- A subgroup of a word-hyperbolic group such that geodesics between points of (in the Cayley graph of ) stay within a bounded distance of itself; a well-behaved kind of subgroup that inherits much of the ambient hyperbolic geometry.