Effective drilling and filling of tame hyperbolic 3-manifolds

TL;DR

This paper provides effective bounds on metric and geodesic length changes during Dehn fillings of tame hyperbolic 3-manifolds, extending previous finite-volume results to infinite-volume cases.

Contribution

It extends the filling theorem to tame hyperbolic 3-manifolds and develops tools for transferring finite-volume results to infinite-volume contexts.

Findings

Effective bilipschitz bounds for metric changes

Effective bounds on complex length of short geodesics

Extension of the 6-Theorem to infinite-volume manifolds

Abstract

We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into theorems about infinite-volume manifolds. We also prove and apply an infinite-volume version of the 6-Theorem.