Geometry & Topology

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

Danny Calegari

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

Article information

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

Received: 1 September 1998
Revised: 9 April 1999
Accepted: 14 June 1999
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R30: Foliations; geometric theory
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]

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


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.

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