Boundary curves of surfaces with the 4–plane property

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.