## Geometry & Topology

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

Danny Calegari

#### 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 $C0$ 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.

#### Article information

Source
Geom. Topol., Volume 3, Number 1 (1999), 137-153.

Dates
Revised: 9 April 1999
Accepted: 14 June 1999
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883142

Digital Object Identifier
doi:10.2140/gt.1999.3.137

Mathematical Reviews number (MathSciNet)
MR1695533

Zentralblatt MATH identifier
0924.57014

#### Citation

Calegari, Danny. $\mathbb{R}$–covered foliations of hyperbolic 3-manifolds. Geom. Topol. 3 (1999), no. 1, 137--153. doi:10.2140/gt.1999.3.137. https://projecteuclid.org/euclid.gt/1513883142

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