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.

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Danny Calegari
Department of Mathematics
UC Berkeley
Berkeley, CA 94720

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