Notice that this method didn’t give us just one geodesic loop of length close to R. Any starting point and starting direction can be used in this process, and many of the water droplets will come back near the starting point, so in fact we can generate many such loops. This is a general principle of structure building using exponential mixing.
Exponential mixing “says that whatever structures you find in your manifold, you’ll find in abundance,” Calegari said.
Kahn and Markovic used a similar approach to our loop-building exercise to build “pairs of pants” — surfaces topologically equivalent to a sphere with three holes (a waist hole and two leg holes, so to speak). Pairs of pants are the building blocks of all compact surfaces except the sphere and the torus — for example, gluing together two pairs of pants produces a double torus (see Figure 8).
Given any sufficiently large number R, Kahn and Markovic showed that it is possible to build lots of pairs of pants inside the manifold whose three cuffs each have a length close to R, and that are almost totally geodesic, meaning that each bit of the pants surface looks pretty much flat from the point of view of hyperbolic geometry.
They also showed that at each cuff of a pair of pants, there is another pair of pants emanating from the cuff in roughly the opposite direction. By sewing together these matching pants at the cuffs, Kahn and Markovic produced a large family of compact surfaces that are almost totally geodesic, with some slight buckling at the seams. Surfaces that are almost geodesic are known to be incompressible inside their three-manifold, so Kahn and Markovic’s construction proved the incompressible surface theorem.
Their methods also showed that a three-manifold has not just one incompressible surface, but “a rich structure of almost geodesic surfaces all over the place,” Calegari said.
Kahn and Markovic’s work earned them the 2012 Clay Research Award, presented annually by the Clay Mathematics Institute to recognize major mathematical breakthroughs.
“The techniques of Kahn and Markovic are as compelling as their results, and this body of work will undoubtedly inspire many more threads of inquiry than it ties off,” predicted Jeffrey Brock of Brown University, in an article on Kahn and Markovic’s work in late 2011.
A Hidden Structure
For mathematicians trying to prove the virtual Haken conjecture, Kahn and Markovic’s work created a starting point.
They showed that every manifold is guaranteed to contain an incompressible surface. But this surface may pass through itself, perhaps in many places, instead of being embedded. To get from Kahn and Markovic’s result to the virtual Haken conjecture, mathematicians would have to find a finite cover of the manifold in which, just as in the example of the six-petal and three-petal flowers, the surface lifts to a collection of surfaces that never intersect themselves (though they may intersect each other). If this could be done, each of these would be an embedded, incompressible surface in the cover, meaning the cover would be Haken.
But how, exactly, is such a cover to be found?
“There’s a big gap between Kahn and Markovic’s result and the virtual Haken conjecture,” Dunfield said. “Their finding was important, but at the time it wasn’t so clear whether it would be helpful in getting embedded surfaces.”
Kahn and Markovic’s result caught the attention of Daniel Wise, of McGill University in Montreal. Wise had, in a sense, made a career of figuring out when finite covers remove a topological object’s self-intersections, but he worked in the context of “cube complexes,” objects that are seemingly very different from three-manifolds. Kahn and Markovic’s findings allowed Wise to show other mathematicians that these two contexts are not so very far apart.