Algebraic & Geometric Topology

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

Ying-Qing Wu

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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.

Article information

Algebr. Geom. Topol., Volume 9, Number 4 (2009), 2175-2189.

Received: 10 March 2009
Accepted: 21 September 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]

pleated surface hyperbolic manifold immersed surface Dehn surgery


Wu, Ying-Qing. Depth of pleated surfaces in toroidal cusps of hyperbolic $3$–manifolds. Algebr. Geom. Topol. 9 (2009), no. 4, 2175--2189. doi:10.2140/agt.2009.9.2175.

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