Sue Goodman

A condition for the stability of ℝ-covered on foliations of 3-manifolds

We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be R-covered. This condition can be readily verified for many examples. Further, if an R-covered foliation has a compact leaf L, then any transverse loop meeting L lifts to a copy of the leaf space, and the ambient manifold fibers over S1 with L as fiber. The focus in this paper is codimension one, transversely oriented, C foliations of closed 3-manifolds. The property of being R-covered (that is, covered by a trivial product of planes) has been important in the study of foliations, particularly those arising from Anosov flows. Solodov [So] and Barbot [Ba1], [Ba2] have shown that an R-covered Anosov foliation implies that the associated Anosov flow is transitive. Ghys [Gh], in the case of Seifert manifolds, proved that all Anosov foliations are R-covered; Plante [Pl1], in the case that the fundamental group of the ambient 3-manifold is solvable, proved the same. These results are essential in showing the associated Anosov flows are conjugate to standard models—geodesic flows or suspensions of Anosov diffeomorphisms [Pl1, Pl2], [Gh]. Fenley [Fe1, Fe2] has used the hypothesis of R-covered to uncover the rich structure of metric and homotopy properties of the flow lines in many Anosov flows. In general, R-covered foliations are particularly nice since the action of the fundamental group π1(M) of the manifold on the universal cover induces a homomorphism from π1(M) to the group of homeomorphisms of R (where R is the leaf space of the lifted foliation). Taut foliations have been well-studied, especially by Thurston and Gabai. Tautness is the key to Roussarie’s [R] and Thurston’s [T] results on isotoping incompressible tori, and in Thurston’s study of norm-minimizing leaves. Gabai [Ga1, Ga2, Ga3], in turn, used these results, by tautly foliating knot complements, to find the minimal genus spanning surface for a large class of knots and links. In 3-dimensions, an R-covered foliation is easily shown to be taut as long as M 6= S × S (Lemma B in section 3). However, while tautness indicates the absence of dead-end components, it does not imply R-covered as the many non-Rcovered Anosov foliations show. In this paper, we give a sufficient condition for an R-covered foliation to have the property that all foliations sufficiently close to it in the C metric are also R-covered. This dates back to a question posed by W. Thurston in 1976. A key element of the proof lies in finding a property of a branched surface which carries only foliations Received by the editors September 3, 1996 and, in revised form, April 18, 1998. 2000 Mathematics Subject Classification. Primary 57M12, 57M20, 57N10, 57R30.