R-covered foliations of hyperbolic 3-manifolds

We produce examples of taut foliations of hyperbolic 3-manifolds which are R-covered but not uniform - ie the leaf space of the universal cover is R, but pairs of leaves are not contained in bounded neighborhoods of each other. This answers in the negative a conjecture of Thurston in [7]. We further show that these foliations can be chosen to be C-0 close to foliations by closed surfaces. Our construction underscores the importance of the existence of transverse regulating vector fields and cone fields for R-covered foliations. Finally, we discuss the effect of perturbing arbitrary R-covered foliations.