Non-Wrapping of Hyperbolic Interval Bundles

We demonstrate a condition on the boundary at infinity of a hyperbolic interval bundle N that guarantees that, for any associated geometric limit, there is a compact core for N which embeds under the covering map. The proof involves an analysis of the geometry of torus cusps in a hyperbolic manifold, and techniques of Anderson, Canary and McCullough [AnCM]. Together with results of Holt–Souto [HS] this shows that the locus of non-local-connectivity of the space of once-punctured torus groups is not dense, and describes a relatively open subset of the boundary of the space of once-punctured torus groups consisting of points of non-self-bumping. Received: April 2006, Revision: May 2007, Accepted: December 2007