ℝ–covered foliations of hyperbolic 3-manifolds

Calegari, Danny¹

¹ Department of Mathematics, University of California at Berkeley, Berkeley, California 94720, USA

Geometry & topology, Tome 3 (1999) no. 1, pp. 137-153.

Résumé

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 $C^0$ 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.

Détail

DOI : 10.2140/gt.1999.3.137

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

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