Non-geometric Veering Triangulations

Abstract

Recently, Ian Agol introduced a class of  “veering” ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are “geometric”, i.e. realized in the complete hyperbolic metric with all tetrahedra positively oriented. This article describes a computer program Veering, building on the program Trains by Toby Hall, for generating these triangulations starting from a description of the homeomorphism as a product of Dehn twists. Using this we obtain the first examples of non-geometric veering triangulations; the smallest example we have found is a triangulation with 13 tetrahedra.

2000 AMS Subject Classification::

• Primary 57M50
• Secondary 57Q15, 57M07, 20F10

Keywords:

• veering triangulations
• geometric triangulations
• hyperbolic 3-manifolds
• fibered 3-manifolds
• conjugacy problem for mapping class groups