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25 September 2022
  » arxiv » math.GT/9912049

 Article overview

Immersed surfaces and Dehn surgery
Ying-Qing Wu ;
Date 6 Dec 1999
Subject Geometric Topology MSC-class: 57N10 | math.GT
AbstractLet $F$ be a proper essential immersed surface in a hyperbolic 3-manifold $M$ with boundary disjoint from a torus boundary component $T$ of $M$. Let $alpha$ be the set of coannular slopes of $F$ on $T$. The main theorem of the paper shows that there is a constant $K$ and a finite set of slopes $Lambda$ on $T$, such that if $eta$ is a slope on $T$ with $Delta(eta, alpha_i) > K$ for all $alpha_i$ in $alpha$, and $eta$ is not in $Lambda$, then $F$ remains incompressible after Dehn filling on $T$ along the slope $eta$. In certain sense, this means that $F$ survives most Dehn fillings. The proof uses minimal surface theory, integral of differential forms, and properties of geometrically finite groups. As a consequence of our method, it will also be shown that Freedman tubings of immersed geometrically finite surfaces are essential if the tubes are long enough.
Source arXiv, math.GT/9912049
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