Thurston had originally proposed the virtual Haken conjecture in an early attempt to tackle his geometrization conjecture, which he had already proven for Haken three-manifolds. If the virtual Haken conjecture were true, so that every compact three-manifold has a Haken finite cover, it might be possible, Thurston hoped, to use the geometric structure on the cover to build a geometric structure on the original manifold.
Three decades later, well after Perelman proved the geometrization conjecture by very different means, the virtual Haken conjecture and the virtual fibering conjecture remained unsolved. These, and two other related conjectures, were the only questions left unanswered among Thurston’s 23. Computer data strongly suggested that the virtual Haken conjecture was correct: from a computerized list of more than 10,000 hyperbolic three-manifolds, Thurston and Nathan Dunfield, of the University of Illinois at Urbana-Champaign, had managed to find a Haken finite cover for every single one. But computational evidence is not a proof.
“When Thurston proposed it, the virtual Haken conjecture seemed like a small question, but it hung on stubbornly, shining a spotlight on how little we knew about the field,” Minsky said. “It turned out that our ignorance was deep in that direction.”
In 2009, the murky waters surrounding the virtual Haken conjecture started to clear.
That year, Markovic and Jeremy Kahn, then at Stony Brook University and now at Brown, announced the proof of a key step toward proving the virtual Haken conjecture. The result, which we’ll call the “incompressible surface theorem,” states that every compact hyperbolic three-manifold does indeed contain an incompressible surface (which possibly passes through itself instead of being embedded).
Kahn and Markovic’s proof is a prime example of the interplay between three-dimensional topology and geometry: the incompressible surface theorem is a purely topological statement, but to prove it, Kahn and Markovic drew heavily on the wealth of additional structure that hyperbolic geometry provides.
To build surfaces inside a three-manifold, Kahn and Markovic used an attribute of hyperbolic shapes called “exponential mixing.” This means that if you start in any little neighborhood inside your manifold, pick a direction, and imagine that your neighborhood is starting to flow along a river moving roughly in that direction, then your neighborhood will gradually spread out and wind around the three-manifold, reaching every possible location from every possible direction. What’s more, it will spread out very quickly, in a precise “exponential” sense.
This mixing property is unique to hyperbolic three-manifolds and stems roughly from the fact that, unlike in Euclidean space, in hyperbolic space the “straight lines,” or geodesics, curve away from each other. If you pick a small neighborhood in the hyperbolic disk and let it flow in a particular direction, the neighborhood will grow exponentially quickly. Inside a compact three-manifold, a flowing neighborhood will likewise grow exponentially quickly, but since the entire manifold has a finite extent, the neighborhood will end up winding around the manifold again and again, overlapping itself many times. Furthermore — and this is harder to prove — the neighborhood will wind around the manifold evenly, flowing through all spots in the manifold with roughly the same frequency.
Mathematicians have understood this exponential-mixing property for more than 25 years and have thoroughly analyzed the statistics of this “geodesic flow,” figuring out roughly when and how often a given neighborhood will pass by a particular point as the neighborhood flows along. But until Kahn and Markovic tackled the incompressible surface theorem, mathematicians had never successfully harnessed this mixing property in the service of building topological structures in a manifold. (One other mathematician, Lewis Bowen of Texas A&M University, had previously tried to use exponential mixing to build incompressible surfaces in three-manifolds, but his work hit technical obstacles.)
To see how the exponential-mixing property helps to build topological and geometric structures, let’s apply it to a simpler task than building surfaces: building a closed geodesic loop whose length is close to our favorite large number — call it R.
To build our loop, let’s pick any starting point in the manifold and any starting direction, and then imagine turning on a garden hose located in a little neighborhood surrounding that point, and aimed roughly in that direction. The water droplets will stream out along geodesic paths, and as long as R is sufficiently large, the mixing of the flow means that by the time the droplets have traveled a distance R, they will have spread out fairly evenly throughout the whole manifold. In particular, at least one droplet (in fact, many) will have arrived back near the starting point and the starting direction. Then, we can simply build a little bridge connecting that droplet’s geodesic to the starting point, to produce a loop that is almost perfectly geodesic and whose length is very roughly equal to R. It’s not hard to show that by pulling this loop just a bit tighter in the manifold, we can produce a totally geodesic loop.