R-covered foliations of hyperbolic 3-manifolds

Details R-covered foliations of hyperbolic 3-manifolds R-covered foliations of hyperbolic 3-manifolds

1998-08-14

arxiv.org/abs/math/9808064v3

Geom. Topol. 3 (1999), 137-153

Mathematics

Geometric Topology

17 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper6.abs.html

Scientific paper

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.

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