Surface bundles versus Heegaard splittings

David Bachman, Saul Schleimer
2005 Communications in analysis and geometry  
This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting. As a consequence, if a splitting surface has small genus compared to the translation distance of the monodromy then the splitting is standard. This result gives new
more » ... ation about Heegaard splittings of hyperbolic three-manifolds. Previous work, by Y. Moriah and H. Rubinstein [13], discussed the low genus splittings of negatively-curved manifolds with very short geodesics. Restricting attention to surface bundles and applying Theorems 6.1 and 3.1 gives: This improves results due to H. Rubinstein [15] and M. Lackenby [10] (see Remark 3.5). The two theorems indicate an interesting connection between the combinatorics of the curve complex and the topology of three-manifolds. This accords with other work. Most significantly, Y. Minsky et al. have used the curve complex to prove the Ending Lamination Conjecture. A major step is using a path in the curve complex to give a model of the geometry of a hyperbolic three-manifold. Another example of this connection is found in [6] and [7] . These papers study surface bundles where the fibre is a once punctured torus. Here the Farey graph takes the place of the curve complex. An analysis of ϕ-invariant lines in the Farey graph allows a complete classification of incompressible surfaces in such bundles. As we shall see in the proofs of Theorems 3.1 and 6.1, essential surfaces and strongly irreducible splittings in the mapping torus M (ϕ) yield ϕ-invariant lines in the curve complex of the fibre. It is intriguing to speculate upon axioms for such lines which would, perhaps, lead to classification results for essential surfaces or strongly irreducible splittings. This would be a significant step in the over-all goal of understanding the topology of surface bundles. At the heart of our proof of Theorem 6.1 lies the idea of a "graphic", due to Rubinstein and Scharlemann [16] . The graphic is obtained, as in D. Cooper and M. Scharlemann's paper [5] , by comparing the bundle structure with a given height function and applying Cerf theory. As in their work, our situation requires a delicate analysis of behavior at the vertices of the graphic. The rest of the paper is organized as follows: basic definitions regarding Heegaard splittings, surface bundles, and the curve complex are found in Section 2. With this background we restate the main theorem and corollaries in Section 3. Of main importance is the nature of simple closed curve intersections between a fibre of the bundle and the Heegaard splitting under discussion. This is covered in Section 4, in addition to a preliminary sketch of the proof of Theorem 6.1. The Rubinstein-Scharlemann graphic is discussed in Section 5. Concluding
doi:10.4310/cag.2005.v13.n5.a3 fatcat:2l7bwqmanbbcxc6vawrrisupju