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Boundary curves of surfaces with the 4–plane property. (English) Zbl 1021.57008

A surface in a 3-manifold is said to have the 4-plane property if its preimage in the universal cover of the 3-manifold is a union of planes, and among any collection of 4 planes, there is a disjoint pair. In this paper, the author shows that for an orientable and irreducible 3-manifold \(M\) whose boundary is a compressible torus and does not contain any closed nonperipheral embedded incompressible surfaces, the immersed surfaces in \(M\) with the 4-plane property can realize only finitely many boundary slopes. This theorem generalizes a result of A. E. Hatcher [Pac. J. Math. 99, 373-377 (1982; Zbl 0502.57005)] in the theory of incompressible surfaces. However, Hatcher’s theorem is not true for immersed \(\pi _1\)-injective surfaces in general. Aitchison and Rubinstein have shown that if a 3-manifold has a nonpositive cubing, then it contains a surface with the 4-plane property. This paper shows that only finitely many Dehn fillings of \(M\) can yield 3-manifolds with nonpositive cubings. This gives examples of hyperbolic 3-manifolds that do not admit any nonpositive cubings.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M07 Topological methods in group theory

Citations:

Zbl 0502.57005
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References:

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