Universal bounds for hyperbolic Dehn surgery

Abstract

This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic conemanifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.

The shape of hyperbolic Dehn surgery space

“…Once these are established, the arguments to establish uniform bounds closely follow those in [11]. However, the use of manifolds with tubular boundary, as opposed to cone manifolds, leads to subtly different estimates when there are multiple boundary components.…”

“…(Indeed, a geodesic could become nonsimple and change isotopy class.) This was not adequately explained in [11]; for a discussion of this and other subtler issues that arise in the multiple cusp case, the reader may consult Purcell [14].…”

“…Those that don't result in a hyperbolic structure are called exceptional curves. In [11] we showed that there is a universal bound to the number of curves that must be excluded from each boundary component; in particular, when there is one boundary component, there are at most 60 exceptional curves and when there are multiple boundary components, at most 114 curves from each component need to be excluded.…”

“…We now briefly explain how the use of manifolds with tubular boundary allows us to avoid the analytic issues that led to the cone angle restriction in [11].…”

“…The interior of M is known to also admit a complete finite volume hyperbolic metric [11,Lemma 3.8]. Associated to M is a filled hyperbolic manifold M .…”