Canonical triangulations of Dehn fillings

TL;DR

This paper investigates how the canonical ideal polyhedral decomposition of a cusped hyperbolic 3-manifold changes under Dehn filling, providing predictive methods for the filled manifold's decomposition in many cases.

Contribution

It introduces a way to predict the canonical decomposition of Dehn-filled hyperbolic 3-manifolds from the unfilled manifold's decomposition, especially for generic fillings.

Findings

Canonical decompositions can often be predicted from unfilled manifolds.

The method is exemplified on all fillings of one cusp of the Whitehead link.

Predictive approach applies broadly to hyperbolic Dehn fillings.

Abstract

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold. As an example, we treat all hyperbolic fillings on one cusp of the Whitehead link complement.