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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

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(γ) after excluding Cg curves from each torus boundary component of M, where Cg is a constant depending only on the genus g of the surface.

pleated surface, hyperbolic manifold, immersed surface, Dehn surgery
Mathematical Subject Classification 2000
Primary: 57N10
Received: 10 March 2009
Accepted: 21 September 2009
Published: 21 October 2009
Ying-Qing Wu
Department of Mathematics
University of Iowa
Iowa City, IA 52242