Low Dimensional Topology

March 8, 2009

Building non-Haken 3-manifolds from geometric pieces.

Filed under: 3-manifolds,Hyperbolic geometry — Jesse Johnson @ 7:52 pm

I asked Ian Biringer to write a few words about his recent work with Juan Souto [1], showing that for any given bound, there are  finitely many “pieces” from which all non-Haken manifolds with bounded injectivity radius can be constructed.  Here’s what he wrote:

The project evolved from some work that Ian Agol did for closed hyperbolic 3-manifolds that have rank = 2 and injectivity radius bounded below by some constant $\epsilon.$ (The rank of a 3-manifold will always here refer to the rank, or minimal number of generators, of its fundamental group.) In any such manifold, there is actually a base point at which the fundamental group can be generated by two loops with length bounded above by some constant depending only on $\epsilon.$ Agol showed that if one takes a Gromov Hausdorff limit of a sequence of such manifolds using the base points above, the limit will be a genus 2 handlebody with a degenerate end. In fact, pulling some compact core of the limit back into the approximating manifolds gives a Heegaard splitting as long as you’re far enough down in the sequence. After some formalism using the fact that there are only countably many closed hyperbolic 3-manifolds, this proves that there are only finitely many closed hyperbolic 3-manifolds with rank = 2 and injectivity radius bounded below that do not have Heegaard genus 2.  (Editor’s note: Ian never wrote down this proof formally.)

In the rank = 2 case, one can sharpen the above work to a good geometric picture of all but finitely many closed hyperbolic 3-manifolds with injectivity radius bounded below. Namely, they all have a Heegaard splitting in which the Heegaard surface has a neighborhood which is a ‘long product region’. This essentially means that it is foliated by surfaces isotopic to the Heegaard surface that have bounded diameter and area; the ‘long’ refers to the distance between the two boundary components of the neighborhood.  Also, this product region can be chosen so that the two complementary components (both homeomorphic to handlebodies) have bounded diameter. Therefore, our manifold looks like a cigar – a long product region in the middle capped off by small handlebodies at both ends. Constructing this geometric picture from the Heegaard splitting is the subject of recent work of Souto, Namazi, Brock and Minsky.

Our project has been to accomplish this for closed hyperbolic 3-manifolds with injectivity radius bounded below by $\epsilon$ and rank = $n$. With the rank increase, our conclusions are necessarily not as strong. For instance, we think that our methods will eventually prove that the Heegaard genus of any such manifold is bounded above by some constant depending only on $n,\epsilon$. We also hope to establish a geometric decomposition similar to the cigar picture above. However, here the situation is more complicated and instead of a single long product region capped off by two handlebodies, we now must use a finite number of bounded diameter ‘building blocks’ connected together with long product regions. Specifically, we have the following conjecture:

Conjecture: Assume that $M$ is a closed hyperbolic 3-manifold with injectivity radius at least $\epsilon$ and rank at most $k$. Then there are disjoint, compact 3-dimensional submanifolds $N_1,\ldots, N_l \subset M$ with the following properties:

1) The number $l$ of these submanifolds is bounded above by some constant depending on $k$. Each $N_i$ has diameter bounded above in terms of $k,\epsilon$, and is homeomorphic to some manifold on a finite list that also depends only on $k,\epsilon$.

2) Every component of $M \setminus \bigcup_i N_i$ is homeomorphic to a product $\Sigma_g \times (0,1)$ for some $g$. The homeomorphism can be chosen so that the level surfaces $\Sigma_g \times \{ t\}$ have both diameter and area bounded above by a constant depending on $\epsilon$. Here, the $N_i$ are the building blocks and the complementary components are the product regions.

We have proven this conjecture and the earlier statement about Heegaard genus when $M$ is non-Haken, and suspect that similar techniques will eventually establish the conjecture in full generality. The technical details become a little bit more daunting, however.