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Hyperbolic manifolds and degenerating handle additions

  • Martin Scharlemann (a1) and Ying-Qing Wu (a1)

Abstract

A 2-handle addition on the boundary of a hyperbolic 3-manifold M is called degenerating if the resulting manifold is not hyperbolic. There are examples that some manifolds admit infinitely many degenerating handle additions. But most of them are not ‘basic’. (See Section 1 for definitions). Our first main theorem shows that there are only finitely many basic degenerating handle additions. We also study the case that one of the handle additions produces a reducible manifold, and another produces a ∂-reducible manifold, showing that in this case either the two attaching curves are disjoint, or they can be isotoped into a once-punctured torus. A byproduct is a combinatorial proof of a similar known result about degenerating hyperbolic structures by Dehn filling.

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Copyright

References

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[1]Bleiler, S. and Hodgson, C., ‘Spherical space forms and Dehn filling’, preprint.
[2]Gordon, C., ‘Boundary slopes of punctured tori in 3-manifolds’, preprint.
[3]Gordon, C., ‘Dehn surgery on knots’, in: Proceedings of the International Congress of Mathematicians, Kyoto 1990 (Springer, Berlin, 1991) pp. 631642.
[4]Gordon, C. and Luecke, J., ‘Reducible manifolds and Dehn surgery’, preprint.
[5]Hempel, J., 3-manifolds, Ann. of Math. Studies 86 (Princeton Univ. Press, Princeton, 1976).
[6]Jaco, W., ‘Lectures on three-manifold topology’, Regional Conference Series in Mathematics 43 (1981).
[7]Myers, R., ‘Excellent 1-manifolds in compact 3-manifolds’, Top. Appl., to appear.
[8]Scharlemann, M., ‘Producing reducible 3-manifolds by surgery on a knot’, Topology 29 (1990), 481500.
[9]Thurston, W., The geometry and topology of 3-manifolds (Lecture notes, Princeton University, 1978).
[10]Wu, Y-Q., ‘Incompressibility of surfaces in surgered 3-manifolds’, Topology 31 (1992), 271280.
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Hyperbolic manifolds and degenerating handle additions

  • Martin Scharlemann (a1) and Ying-Qing Wu (a1)

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