From veering triangulations to Cannon-Thurston maps

Suppose that M is a complete, cusped, finite-volume oriented hyperbolic three-manifold.  Suppose that V is a transverse veering triangulation on M.  Then we produce a "Cannon-Thurston map" for V: a continuous equivariant surjection \Phi_V from the veering circle of V to the boundary of the universal cover of M.  If M is fibered, and the fiber is carried by V, then V is layered [Landry-Minsky-Taylor].  In this case \Phi_V "agrees" with the classical map [Bowditch, Mj].  If V is not layered then our map is new.

We also give an algorithm to draw approximations of \Phi_V.  In the layered case (where comparisons can be made), our algorithm gives ``better'' pictures for less ``work'' than Thurston's algorithm.

This is joint work with Jason Manning and Henry Segerman.