#### Geometry & Topology, Vol. 3 (1999)
Paper no. 6, pages 137--153.

## 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 `Three-manifolds, foliations and circles I'
(math.GT/9712268). 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**.
R-covered foliations, slitherings, hyperbolic 3-manifolds, transverse geometry

**AMS subject classification**.
Primary: 57M50, 57R30.
Secondary: 53C12.

**DOI:** 10.2140/gt.1999.3.137

**E-print:** `arXiv:math.GT/9808064`

Submitted to GT on 1 September 1998.
(Revised 9 April 1999.)
Paper accepted 14 June 1999.
Paper published 20 June 1999.

Notes on file formats
Danny Calegari

Department of Mathematics

UC Berkeley

Berkeley, CA 94720

Email: dannyc@math.berkeley.edu

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