Strongly irreducible Heegaard splittings of 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.