Boundary curves of surfaces with the 4–plane property

Abstract

Let M be an orientable and irreducible 3-manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4-plane property can realize only finitely many boundary slopes. Moreover, we will show that only finitely many Dehn fillings of M can yield 3-manifolds with nonpositive cubings. This gives the first examples of hyperbolic 3-manifolds that cannot admit any nonpositive cubings.

Heegaard surfaces and measured laminations, I: The Waldhausen conjecture

“…Methods of laminations and branched surfaces have been very useful in solving some seemingly unrelated problems, such as [24,25]. This is the first time that they are used on Heegaard splittings.…”

An algorithm to find vertical tori in small Seifert fiber spaces

“…However, this simple result is not true for immersed surfaces under our definitions of immersed branched surface and carrying above, although every immersed essential surface can also be homotoped into normal form. An analogue of the theorem of Floyd and Oertel for surfaces with the 4-plane property can be found in [12]. Next, we show that this finiteness theorem can be generalized to immersed surfaces with the n-color property.…”

“…For the definitions and notation related to branched surfaces, see [5], [15]. In [2], [12], immersed branched surfaces were introduced to study immersed surfaces with small complexity. Figure 3.1 (see [5], [15], [11] for more details about branched surfaces).…”

“…Then, we give every normal disk in π(γ d ) a color induced from γ d . We still call the corresponding normal disk type in T the disk type d, and we use d × I to denote the product of an interval I and a normal disk of type d. As in [5], [12], we put k (k ≤ n is the number of different colors in γ d ) such products d × I 's in T such that any two disks in the same color are in the same d × I and are transverse to the I -fibers. We can do such construction for every normal disk type in every tetrahedron in T .…”

“…Note that the number of different product regions for a 1-simplex depends on n and the number of tetrahedra incident to this 1-simplex. As in [5], [12], by identify every I -fiber of every product region above to a point, we can construct a singular branched surface and the union of those product regions can be consider as a fibered neighborhood of this singular branched surface. By our construction above, two normal disks in the immersed surface are identified to the same branch sector only if they have the same induced color, and hence the two corresponding planes in the universal cover have the same color and do not intersect.…”