  The triangulation complexity of a closed orientable 3-manifold is the minimal
number of tetrahedra in any triangulation of the manifold. The main theorem of
the paper gives upper and lower bounds on the triangulation complexity of any
closed orientable hyperbolic 3-manifold that fibres over the circle. We show
that the triangulation complexity of the manifold is equal to the translation
length of the monodromy action on the mapping class group of the fibre, up to a
bounded factor, where the bound depends only on the genus of the fibre.
