Lower bounds for transversal complexity of torus bundles over the circle

For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S. Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of $M^3$. Usually it is straightforward to give an upper bound for $c(M)$, but obtaining lower bounds remains very difficult. We consider manifolds fibered by tori over the circle, introduce transversal complexity ${\rm tc}(M)$ for such manifolds, and give a lower bound for ${\rm tc}(M)$ in terms of the monodromy of the fiber bundle; this estimate involves a very geometric study of the modular group action on the Farey tesselation of hyperbolic plane. As a byproduct, we construct pseudominimal spines of the manifolds fibered by tori over $S^1$. Finally, we discuss some potential applications of these ideas to other 3-manifolds.