Tight contact structures on hyperbolic three-manifolds

Abstract

Let Σg denote a closed orientable surface of genus g≥2. We consider a certain infinite family of Σg-bundles over circle whose monodromies are taken from some collection of pseudo-Anosov diffeomorphisms. We show the existence of tight contact structure on every closed 3-manifold obtained via rational r-surgery along a section of any member of the family whenever r≠2g−1. Combining with Thurston's hyperbolic Dehn surgery theorem, we obtain infinitely many hyperbolic closed 3-manifolds admitting tight contact structures.