J Austral Math Soc Ser A 55 pp72--89, 1993.

Hyperbolic Manifolds and Degenerating Handle Additions

Martin Scharlemann and Ying-Qing Wu

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.

1991 AMS Subject Classification: primary 57N10; secondary 57M50

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Authors

Martin Scharlemann

Department of Mathematics, University of California, Santa Barbara, CA 93016.

Ying-Qing Wu

Department of Mathematics, University of California, Santa Barbara, CA 93016.