#### Volume 3, issue 1 (1999)

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

### Danny Calegari

Geometry & Topology 3 (1999) 137–153
 arXiv: math.GT/9808064
##### 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.

##### Keywords
$\mathbb{R}$–covered foliations, slitherings, hyperbolic 3–manifolds, transverse geometry
##### Mathematical Subject Classification
Primary: 57M50, 57R30
Secondary: 53C12