Regulating flows, topology of foliations and rigidity

Abstract

Abstract.A flow transverse to a foliation is regulating if, in the universal cover, an arbitrary orbit of the flow intersects every leaf of the lifted foliation. This implies that the foliation is R-covered, that is, its leaf space in the universal cover is homeomorphic to the reals. We analyse the converse of this implication to study the topology of the leaf space of certain foliations. We prove that if a pseudo-Anosov flow is transverse to an R-covered foliation and the flow is not an R-covered Anosov flow, then the flow is regulating for the foliation. Using this we show that several interesting classes of foliations are not R-covered. Finally we show a rigidity result: if an R-covered Anosov flow is transverse to a foliation but is not regulating, then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow.

Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations

“…First of all this implies that Φ is regulating for F, for otherwise the aforementioned result from [Fe9] shows that Φ is an R-covered Anosov flow − in particular Λ s has Hausdorff leaf space. Also by theorem 3.6 the fact that Λ s has non Hausdorff leaf space implies that there are closed orbits α, β of Φ so that α is freely homotopic to the inverse of β.…”

“…We also say that Φ * does not regulate F . In [Fe9] we analysed a similar situation and proved the following: if Υ is a pseudo-Anosov flow transverse to an R-covered foliation and Υ is not regulating, then Υ is an R-covered Anosov flow. The same arguments work with an almost pseudo-Anosov flow transverse to an R-covered foliation.…”

R-covered branched surfaces

“…Consequently, the boundary of each generating disk yields a smooth embedded copy of S 1 in the branch set. For example, Now, one can construct non-R-covered foliations arbitrarily close to the foliation F ( [Ca], [Fe5]) so, as discussed in Section 2, W cannot be stably R-covered. Indeed, if we choose λ to be the essential loop indicated by the dashed curve in Figure 3.7 and let α be the embedded arc indicated by the dotted line, it is straightforward to check that there are no corresponding curves η and τ as in the hypotheses of Theorem 3.2.…”

“…However, when the foliation has no compact leaves, very little is known. One of the few things that is known is that foliations transverse to a non-Rcovered pseudo Anosov flow φ are R-covered precisely when φ is regulating; there are, however, many examples of R-covered Anosov flows transverse to R-covered foliations that are not regulating [Fe5].…”

“…In fact, Calegari has described a way in which one could use shearing to perturb a foliation of a hyperbolic surface bundle over a circle to possibly obtain an arbitrarily close non-R-covered foliation [Ca]. While Calegari does not prove whether a non-R-covered foliation can result from such a shearing, his observations indicate that one might, and his example seems similar to one of Fenley's examples [Fe5] proving nonstability of the R-covered property.…”