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

Abstract

Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if F is a Reebless, finite depth foliation in a hyperbolic manifold, then it has the continuous extension property. Such finite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classes of foliations, including a large class of foliations where all leaves are dense and infinitely many examples with one sided branching. One key tool is a detailed understanding of asymptotic properties of almost pseudo-Anosov singular 1-dimensional foliations in the leaves of F lifted to the universal cover.Hence any orientable, hyperbolic 3-manifold with non finite second homology has such a foliation with the continuous extension property. Conjecturally any closed, hyperbolic 3-manifold has a finite cover with positive first Betti number. This would imply there is always a foliation with the continuous extension property in a finite cover. The proof of corollary A is simple given previous results: If necessary take a double cover and assume that F is transversely oriented. Then Mosher and Gabai proved that such F is almost transverse to a pseudo-Anosov flow Φ [Mo5]. We proved, jointly with Mosher that these flows are quasigeodesic [Fe-Mo]. The main theorem then implies corollary A. The result in [Fe-Mo] is only for finite depth foliations: the proof depends heavily on the existence of a compact leaf, M being hyperbolic and the direct association with a hierarchy. By Thurston's geometrization theorem [Th1, Th2, Mor] it suffices to assume that M is atoroidal.Corollary B − There are infinitely many foliations with all leaves dense which have the continuous extension property. Many of these have one sided branching.

Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry

“…The case of finite depth foliations was resolved very recently in [28] using work of Mosher and the author [49; 50; 29]. For general foliations with two sided branching, Calegari [12] constructed a very full lamination transverse to F , like the stable/unstable foliation of a flow.…”

Pseudo-Anosov flows in toroidal manifolds

“…It was proved in [Fe7] that any complementary component of L covers M . This implies that given x in A(h), there is g in π 1 (M ) with g(x) in B, which is disjoint from A(h).…”

“…The first author has done extensive work [Fe3,Fe7] in the closed atoroidal case (where in fact by Perelman's work [Pe1,Pe2,Pe3], M is hyperbolic). The questions addressed in that analysis were more of a geometric nature: Exactly when is the flow is quasigeodesic?…”

“…In the applications to the topology of 3-manifolds there is a requirement that p is at least 3, which is the convention here as well. Pseudo-Anosov flows have been used very successfully to analyse the topology and geometry of 3-manifolds [Mo1,Mo2,Mo3,Fe3,Fe7,Fe8] and they are much more common than Anosov flows [Fr,RSS,Fe6]. They are much more flexible because for instance they survive most Dehn surgeries on closed orbits [Fr], see also section 8 for new examples.…”

“…Notice that there is an ongoing broad study of pseudo-Anosov flows in closed, hyperbolic manifolds by the second author [Fe3,Fe7,Fe8], which is mostly orthogonal to this article. In our situation classical 3-dimensional topology will play a much bigger role.…”