Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds

TL;DR

This paper proves that Haken hyperbolic 3-manifolds have a finite set of strongly irreducible Heegaard surfaces and incompressible surfaces, such that all strongly irreducible Heegaard surfaces are constructed from these via Haken sums.

Contribution

It establishes a finite classification of strongly irreducible Heegaard surfaces in Haken hyperbolic 3-manifolds using Haken sums, extending previous results.

Findings

Finite collection of strongly irreducible Heegaard surfaces and incompressible surfaces

Any strongly irreducible Heegaard surface is a Haken sum of these surfaces

Extension of effective classification results for hyperbolic 3-manifolds

Abstract

Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + \sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.