HYPERBOLIC MANIFOLDS AND DEGENERATING HANDLE ADDITIONS

A 2-handle addition on the boundary of a hyperbolic 3-manifold M is ca lled degenerating if the resulting manifold is not hyperbolic. There a re examples that some manifolds admit infinitely many degenerating han dle additions. But most of them are not 'basic'. (See Section 1 for de finitions). 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 partial derivative-reducible manifold, showing that in thi s case either the two attaching curves are disjoint, or they can be is otoped into a once-punctured torus. A byproduct is a combinatorial pro of of a similar known result about degenerating hyperbolic structures by Dehn filling.