Volume 9, issue 4 (2009)

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

Ying-Qing Wu

Algebraic & Geometric Topology 9 (2009) 2175–2189
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.

Keywords
pleated surface, hyperbolic manifold, immersed surface, Dehn surgery
Primary: 57N10