## Geometry & Topology

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

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

#### Abstract

We produce examples of taut foliations of hyperbolic 3–manifolds which are $\mathbb{R}$–covered but not uniform — ie the leaf space of the universal cover is $\mathbb{R}$, 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 $\mathbb{R}$–covered foliations. Finally, we discuss the effect of perturbing arbitrary $\mathbb{R}$–covered foliations.

#### Article information

**Source**

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

**Dates**

Received: 1 September 1998

Revised: 9 April 1999

Accepted: 14 June 1999

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

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

**Subjects**

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R30: Foliations; geometric theory

Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]

**Keywords**

$\mathbb{R}$–covered foliations slitherings hyperbolic 3–manifolds transverse geometry

#### 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