Volume 6, issue 2 (2002)

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Boundary curves of surfaces with the 4–plane property

Tao Li

Geometry & Topology 6 (2002) 609–647
 arXiv: math.GT/0212111
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.

Keywords
3–manifold, immersed surface, nonpositive cubing, 4–plane property, immersed branched surface.
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57M25, 57N10, 57M07