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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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
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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