| Abstract: Agol introduced veering triangulations of mapping tori,
whose combinatorics are canonically associated to the pseudo-Anosov
monodromy. In previous work, Hodgson, Rubinstein, Tillmann and I found
examples of veering triangulations that are not layered and therefore
do not come from Agol's construction. However, non-layered veering triangulations retain many of the good
properties enjoyed by mapping tori. For example, Schleimer and I
constructed a canonical circular ordering of the cusps of the
universal cover of a veering triangulation. Its order completion gives
the veering circle; collapsing a pair of canonically defined
laminations gives a surjection onto the veering sphere. In work in progress, Manning, Schleimer, and I prove that the veering
sphere is the Bowditch boundary of the manifold's fundamental group.
As an application we produce Cannon-Thurston maps for all veering
triangulations. This gives the first examples of Cannon-Thurston maps
that do not come, even virtually, from surface subgroups. |