Why do strongly irreducible Heegaard surfaces look like fibers?

I remember hearing somewhere that strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds "look like" fibers.

I know that Otal's result about short geodesics in hyperbolic mapping tori being unlinked with respect to fibers has an analogue in the setting of strongly irreducible Heegaard surfaces.

Can someone tell me some other similarities between strongly irreducible Heegaard surfaces and fibers in hyperbolic 3-manifolds? Or why someone would say that strongly irreducible Heegaard surfaces look like fibers? Any references would be greatly appreciated.

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1 Answer

Something more basic is true. Strongly irreducible Heegaard splittings act a lot like incompressible surfaces (of which fibers are a special case). Here are two results as evidence.

First, suppose that the three-manifold is equipped with a triangulation. Then Haken showed that incompressible surfaces can be normalized. That is, isotoped to intersect every tetrahedron in a collection of standard disks; normal triangles and normal quads. Rubinstein (and Stocking) showed that strongly irreducible splittings can be "almost normalized"; in every tetrahedron it has normal disks and in at most one tetrahedron there is an almost normal annulus or octagon. Michelle Stocking's thesis is a standard reference for this material. Many people (Rubinstein, Hass, Bachman, Scott, ...) will say in public that the above is a PL version of an analytic truth: incompressible surfaces can be isotoped to be minimal surfaces of index zero while strongly irreducible surfaces can be made minimal of index one.

Second, suppose that $S, T$ are surfaces, with $T$ incompressible. If $S$ is also incompressible then it is an exercise in innermost disks to show that, after isotoping $S$ to meet $T$ minimally, all curves of intersection are essential on both surfaces. If $S$ is instead strongly irreducible then a one-parameter sweep-out argument, followed by an innermost disk argument shows that there is some position of $S$ where all curves of intersection are essential on both surfaces.

Ok, third (I couldn't resist), suppose that the ambient three-manifold is hyperbolic. Minsky's approach to the ending lamination conjecture says that the geometry "around" the incompressible surface or a strongly irreducible splitting can be modelled using "blocks" based on the four-holed sphere or once-holed torus, where the blocks are glued to each other vertically using pairs of pants and horizontally using solid tori (Margulis tubes).

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