Depth of pleated surfaces in toroidal cusps of hyperbolic 3-manifolds

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(gamma) 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.