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Hyperbolic manifolds and degenerating handle additions. (English) Zbl 0802.57005
Let $$M$$ be a hyperbolic 3-manifold and let $$\alpha$$ be an essential simple closed curve on $$\partial M$$. Then $$\alpha$$ is called degenerating if the manifold obtained from $$M$$ by adding a 2-handle along $$\alpha$$, if the component of $$\partial M$$ containing $$\alpha$$ is not a torus, or by doing Dehn surgery with meridian $$\alpha$$ if the component of $$\partial M$$ containing $$\alpha$$ is a torus, is not hyperbolic.
Thurston has shown that only finitely many Dehn fillings of a torus component of the boundary of a hyperbolic 3-manifold yield non-hyperbolic manifolds.
Examples show that there can be infinitely many degenerating curves on higher genus boundary components of hyperbolic 3-manifolds $$M$$. Call a degenerating curve $$\alpha$$ basic if it separates its component or if there are no separating degenerating curves which are coplanar with $$\alpha$$ in $$\partial M$$. The main result of the authors states that there is a constant $$k_ g$$ such that the set of basic degenerating curves in a boundary component of genus $$g$$ consists of at most $$k_ g$$ curves.
Reviewer: E.Vogt (Berlin)

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds