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

Wu, Ying-Qing

doi:10.2140/agt.2009.9.2175

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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

DOI: 10.2140/agt.2009.9.2175

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 $M (γ)$ 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

Mathematical Subject Classification 2000

Primary: 57N10

References

Publication

Received: 10 March 2009

Accepted: 21 September 2009

Published: 21 October 2009

Authors

Ying-Qing Wu
Department of Mathematics University of Iowa Iowa City, IA 52242

Volume 9, issue 4 (2009)