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Hyperbolic manifolds and degenerating handle additions

Part of: Low-dimensional topology Topological manifolds

Published online by Cambridge University Press:  09 April 2009

Martin Scharlemann  and
Ying-Qing Wu
Martin Scharlemann
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA, 93106
Ying-Qing Wu
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA, 93106
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Abstract

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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.

MSC classification

Secondary: 57N10: Topology of general $3$-manifolds 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 1 , August 1993 , pp. 72 - 89
Copyright
Copyright © Australian Mathematical Society 1993

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