Boundary curves of surfaces with the 4–plane property

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

References

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Publication

Received: 23 March 2001

Revised: 15 March 2002

Accepted: 15 November 2002

Published: 6 December 2002

Proposed: Cameron Gordon

Seconded: Walter Neumann, Michael Freedman

Authors

Tao Li
Department of Mathematics Oklahoma State University Stillwater Oklahoma 74078 USA

Volume 6, issue 2 (2002)

Tao Li