ℝ–covered foliations of hyperbolic 3-manifolds

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.

Keywords

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

Mathematical Subject Classification

Primary: 57M50, 57R30

Secondary: 53C12

References

Forward citations

Publication

Received: 1 September 1998

Revised: 9 April 1999

Accepted: 14 June 1999

Published: 20 June 1999

Proposed: David Gabai

Seconded: Walter Neumann, Cameron Gordon

Authors

Danny Calegari
Department of Mathematics University of California at Berkeley Berkeley California 94720 USA

Volume 3, issue 1 (1999)

Danny Calegari