Computing Triangulations of Mapping Tori of Surface Homeomorphisms

Abstract

We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. The package is an extension of the software described in [Brinkmann 2000]. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea's triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we obtain a new solution to the restricted conjugacy problem in the mapping class group.

Keywords:

• Mapping tori of surface automorphisms
• pseudo-Anosov automorphisms
• mapping class group
• conjugacy problem