Low Dimensional Topology

November 29, 2007

WYSIWYG geometry from Heegaard gluing maps

Recall that WYSIWIG stands for “What you see is what you get”. (I believe this term was first applied to topology by Steve Kerckhoff during his talk at Peter Scott’s birthday conference, but I haven’t been able to track down a reference.) WYSIWYG topology is the idea that there should be a more direct connection between the combinatorics of a topological object and its geometry.

The title of this entry refers a project that was started by Hossein Namazi and Juan Souto (which has since been expanded to include Jeff Brock and Yair Minsky), in which they use the gluing map of a Heegaard splitting to directly calculate (almost) hyperbolic metrics on the ambient manifolds. After waiting a number of years for a preprint to appear on the front, I found out last fall that the preprint had been available the whole time on Namazi’s home page. I will have to get into the habit of looking at peoples web pages for papers that haven’t made it to the front yet.

The first incarnation of the project, a joint paper between Namazi and Souto, considers Heegaard splittings in which the gluing maps are higher and higher powers of pseudo-Anosov surface automorphisms. Given such a Heegaard splitting, they construct a pinched negatively curved metric on the 3-manifold that is pinched less and less for higher powers. The second incarnation, by all four authors (and still in preparation) replaces the high power pseudo-Anosov automorphism with a criteria based on the curve complex, which they call high distance, bounded combinatorics.

Both incarnations begin with a construction related to the model manifold construction developed by Minsky for the ending laminations conjecture. (It is worth noting that Namazi was Minsky’s student.) By constructing what’s called a path hierarchy (due to Masur and Minsky), they construct a metric on a surface cross an interval that looks like a piece of a cusp of a hyperbolic 3-manifold and such that near the boundary of this manifold, the metric is (close to) compatible with a metric constructed on each handlebody. Brock and Souto have a related result (which they announced a few years ago but haven’t written up yet) showing that the distance in the pants complex defined by the Heegaard splitting is (assymptotically) related to the volume of the ambient hyperbolic 3-manifold. My understanding is that they use similar methods (a path hierarchy is very closely related to a path in the pants complex).

It is known that every 3-manifold with a Heegaard splitting of distance three or more is hyperbolic, but the only way to prove this is by showing that every Heegaard splitting of a toroidal of Seifert fibered 3-manifold has distance at most two, and then applying the geometrization conjecture/theorem. There’s no direct/constructive proof. In fact, the difficulty of getting even the results mentioned above shows how far the current knowledge is from getting a constructive proof for lower distance Heegaard splittings. Still, one may hope that the project mentioned above, and its future repercussions may eventually lead to a better understand of how even low distance gluing maps are related to the hyperbolic geometry of the resulting 3-manifold.

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