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Let M be an orientable and irreducible 3-manifold whose boundary is anincompressible torus. Suppose that M does not contain any closed nonperipheralembedded incompressible surfaces. We will show in this paper that the immersedsurfaces in M with the 4-plane property can realize only finitely many boundaryslopes. Moreover, we will show that only finitely many Dehn fillings of M canyield 3-manifolds with nonpositive cubings. This gives the first examples ofhyperbolic 3-manifolds that cannot admit any nonpositive cubings.

Boundary curves of surfaces with the 4–plane property