Depth of pleated surfaces in toroidal cusps of hyperbolic $3$–manifolds

Abstract

Let $F$ be a closed essential surface in a hyperbolic $3$–manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth of $F$ in $N$ is bounded above by a constant depending only on the genus of $F$. The result is used to show that an immersed closed essential surface in $M$ which is not coannular to the torus boundary components of $M$ will remain essential in the Dehn filling manifold $M\left(\gamma \right)$ after excluding $C_g$ curves from each torus boundary component of $M$, where $C_g$ is a constant depending only on the genus $g$ of the surface.