$\mathbb{R}$–covered foliations of hyperbolic 3-manifolds

Abstract

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. 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.