Hyperbolic manifolds and degenerating handle additions.

*(English)*Zbl 0802.57005Let \(M\) 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.

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 |