# 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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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.