Geometry & Topology, Vol. 6 (2002) Paper no. 21, pages 609-647.

Boundary curves of surfaces with the 4-plane property

Tao Li

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.

AMS subject classification. Primary: 57M50. Secondary: 57M25, 57N10, 57M07.

DOI: 10.2140/gt.2002.6.609

E-print: arXiv:math.GT/0212111

Submitted to GT on 23 March 2001. (Revised 15 March 2002.) Paper accepted 15 November 2002. Paper published 6 December 2002.

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Tao Li
Department of Mathematics, Oklahoma State University
Stillwater, OK 74078, USA
Email: tli@math.okstate.edu
URL: http://www.math.okstate.edu/~tli
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