Tight Contact Structures on Fibred

Abstract

As a first step towards understanding the relationship between foliations and tight contact structures on hyperbolic 3-manifolds, we classify "extremal" tight contact structures on a surface bundle M over the circle with pseudo- Anosov monodromy. More specifically, there is exactly one tight contact structure (up to isotopy) whose Euler class, when evaluated on the fiber, equals the Euler characteristic of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces in M. Indeed, this flexibility can already be seen in surface bundles over the interval, where an analogous classification theorem is also established.