The triangulation complexity of fibred 3-manifolds

The triangulation complexity of a closed orientable 3 -manifold M is the minimal number of tetrahedra in any triangulation of M . Our main theorem 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 S , up to a bounded factor, where the bound depends only on the genus of S .