Proximity in the curve complex: boundary reduction and bicompressible surfaces

Abstract

Suppose N is a compressible boundary component of a compact orientable irreducible 3-manifold M and Q is an orientable properly embedded essential surface in M in which each component is incident to N and no component is a disk. Let VN and QN denote respectively the sets of vertices in the curve complex for N represented by boundaries of compressing disks and by boundary components of Q. Theorem: Suppose Q is essential in M, then the distance d(VN, QN) \leq 1 - χ(Q). Hartshorn showed that an incompressible surface in a closed 3-manifold puts a limit on the distance of any Heegaard splitting. The theorem above provides a version of Hartshorn's result for merely compact 3-manifolds. In a similar spirit, here is the main result: Theorem: Suppose Q is connected, separating and compresses on both its sides, but not by way of disjoint disks, then either d(VN, QN) \leq 1 - χ(Q) or Q is obtained from two nested connected boundary-parallel surfaces by a vertical tubing. Forthcoming work with M. Tomova (math.GT/0501140) will show how an augmented version of this theorem leads to the same conclusion as in Hartshorn's theorem, not from an essential surface but from an alternate Heegaard surface. That is, if Q is a Heegaard splitting of a compact M then then no other irreducible Heegaard splitting of M has distance greater than twice the genus of Q.