Tight Contact Structures on Laminar Free Hyperbolic Three-Manifolds

Abstract

ABSTRACT. Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Shareshian, and Stein [25], and infinitely many of them do not even admit essential laminations as shown by Fenley

Contact structures, deformations and taut foliations

“…Other examples of tight contact structures on manifolds that admit no Reebless foliations were given by Etgü [8] in the case that the manifold is not Seifert fibered but hyperbolic. However, these examples are neither known to be universally tight, nor is it shown that there are tight contact structures for both orientations.…”