Low Dimensional Topology

February 26, 2010

The quasi-Mapping class group of a Heegaard splitting

Filed under: 3-manifolds,Curve complexes,Heegaard splittings,Mapping class groups — Jesse Johnson @ 9:25 am

Here’s a neat result about mapping class goups of Heegaard splittings that was proved in a recent preprint [1] by Marion Moore and Matt Rathbun:  The mapping class group of a Heegaard splitting is determined by the coarse geometry of the curve complex for the Heegaard surface.   In particular, a Heegaard splitting determines two quasi-convex subsets of the complex of curves for the Heegaard surface and one can define the quasi-mapping class group for a Heegaard splitting in terms of the quasi-isometries of the complex that keep each set within a bounded neighborhood of itself.  Their result shows that (modulo a technicality in genus two) the quasi-Mapping class group of a Heegaard splitting is isomorphic to its mapping class group.

In previous posts, I’ve mentioned the mapping class group of a Heegaard splitting $\Sigma$ of a 3-manifold M: This is the group of automorphisms of M that send $\Sigma$ onto itself, modulo isotopies that keep $\Sigma$ on itself.  The Heegaard surface $\Sigma$ cuts M into two handlebodies, and the set of loops in the curve complex bounding disks in one of these handlebodies is called a handlebody set.  Every automorphism of $\Sigma$ induces an automorphism of the curve complex, and if this automorphism of $\Sigma$ is the restriction to $\Sigma$ of an automorphism of M then the automorphism of the curve complex sends each handlebody set onto either itself or the other handlebody set (if the automorphism “flips” $\Sigma$.  Conversely, every automorphism of the curve complex induces an automorphism of $\Sigma$ [2] and every automorphism that preserves or swaps the handlebody sets extends to the handlebodies, and thus to M.  So, one can define the mapping class group of a Heegaard splitting as the group of isometries/automorphisms of the curve complex for $\Sigma$ that take each handlebody set to itself or to the other handlebody set.  (Well, technically one must take into account the fact that the hyper-elliptic involution of a genus two surface induces the identity map on the curve complex.)

Since it is more common to study the coarse geometry of the curve complex than the exact geometry, it is natural to generalize this definition to quasi-Isometries.  Define the coarse mapping class group of M to be the group of quasi-isometries of the curve complex that send  each handlebody into a $\Delta$ neighborhood of itself or the other handlebody set for some integer $\Delta$, modulo quasi-isometries that move each vertex a bounded distance.   This seems to be the best translation into the coarse language, but it turns out to be not that different from the original. In particular, Rafi-Schleimer [3] have shown (building on a result of Behrstock-Kleiner-Minsky-Mosher and, independently, Hamenstaed [4]) that every quasi-isometry of the curve complex is within a bounded distance from an actual isometry.  (In other words, the curve complex is quasi-isometrically rigid.)  Thus the coarse mapping class group of a Heegaard splitting is isomorphic to the group of isometries of the curve complex that send the handlebody sets into a bounded neighborhood of themselves.

Moore and Rathbun have shown that every isometry of the curve complex that sends a handlebody set into a bounded neighborhood of itself actually sends the handlbody set onto itself, and is therefore induced by an automorphism of the surface that extends to the handlebody.  More precisely, they show that if two handlbody sets are distinct then there is a pseudo-Anosov automorphism of one handlebody whose stable lamination is not a limit of boundaries of disks in the other handlebody, implying that there are vertices in this handlebody set that are arbitrarily far from the other handlbody set.  Therefore the coarse mapping class group of the Heegaard splitting (determined by the coarse geometry of the curve complex) is isomorphic to the regular mapping class group.

Moore and Rathbun’s proof is in fact a modified argument from the main result in the paper, which generalizes a result of Minsky, Moriah and Schleimer [5].  They showed that for any positive integers n, g, there is a knot in the 3-sphere whose complement has a genus g Heegaard splitting of distance at least n. Moore and Rathbun generalized this to show that for every 3-manifold M and for any positive integers g,n with g at least one greater than the minimal Heegaard genus of M, there is a knot in M whose complement has a genus g Heegaard splitting of distance at least n.