Lower bounds for transversal complexity of torus bundles over the circle

S. S. Anisov

Utrecht University

Abstract: 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.

Key words and phrases: Complexity of 3-manifolds, $T^2$-bundles over $S^1$, Farey tesselation.

MSC: Primary 57M99; Secondary 57M20, 57M50, 57R05, 57R15, 57R22

Received: December 30, 2005

Language: English

DOI: 10.17323/1609-4514-2006-6-1-5-41

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