  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, and infinitely many of them do not even admit
essential laminations as shown by Fenley. In this paper, we construct tight
contact structures on a family of 3-manifolds including these examples. These
contact structures are described by contact surgery diagrams and their
tightness is proved using the contact invariant in Heegaard Floer homology.
