Sergio R. Fenley

Foliations with good geometry

The goal of this article is to show that there is a large class of closed hyperbolic 3-manifolds admitting codimension one foliations with good large scale geometric properties. We obtain results in two directions. First there is an internal result: A possibly singular foliation in a manifold is quasi-isometric if, when lifted to the universal cover, distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover. This means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3-manifolds can never be quasi-isometric. In this article we produce a large class of singular quasi-isometric, codimension one foliations in closed hyperbolic 3-manifolds. The foliations are stable and unstable foliations of pseudo-Anosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3-manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let g be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then g is not quasi-isometric, but suppose that g is transverse to a quasigeodesic pseudo-Anosov flow with quasi-isometric stable and unstable foliations which are given by the internal result. We then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis. The main tools used to prove these results are first a link between geometric properties of stable/unstable foliations of pseudo-Anosov flows and the topology of these foliations in the tuniversal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliations in the universal cover. Reebless codimension one foliations are extremely useful for understanding the topology of 3-manifolds. For instance they imply that the manifold is irreducible [Ro], its universal cover is homeomorphic to R3 [Pa], leaves are 7rl-injective [No] and transversals are never null homotopic [No]. Hence they reflect topological properties of the manifold. As for which manifolds have Reebless foliations, Gabai [Gal, Ga2, Ga3] proved that any compact, oriented, irreducible 3-manifold with nonzero first Betti number has many Reebless finite depth foliations. Roughly, a

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

Given a general pseudo-Anosov flow in a closed three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We construct a natural compactification of this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, we then show: The ideal circle of the orbit space has a natural quotient space which is a sphere. This sphere is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold. The main result is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This gives an entirely new proof that the fundamental group of a closed, atoroidal 3‐manifold which fibers over the circle is Gromov hyperbolic. In addition with further geometric analysis, the main result also implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and that the stable/unstable foliations of these flows are quasi-isometric foliations. Finally we apply these results to (nonsingular) foliations: if a foliation is R‐covered or with one sided branching in an aspherical, atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity. 37C85, 37D20, 53C23, 57R30; 58D19, 37D50, 57M50

Laminar free hyperbolic 3-manifolds

The purpose of the article is to prove that there are infinitely many closed hyperbolic 3-manifolds which do not admit essential laminations. The manifolds are obtained by Dehn surgery on torus bundles over the circle. This gives a definitive negative answer to a fundamental question posed by Gabai and Oertel when they introduced essential laminations. The proof is obtained by analysing group actions on on trees and showing that certain 3-manifold groups only have trivial actions on trees. There are corollaries concerning the existence of Reebless foliations and pseudo-Anosov flows.

Quasi-Fuchsian Seifert surfaces

Abstract. Let

Pseudo-Anosov flows in toroidal manifolds

K \subset S^3

Pseudo-Anosov Flows and Incompressible Tori

be a non fibered knot with hyperbolic complement. Given a Seifert surface of minimal genus for

Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

K

Regulating flows, topology of foliations and rigidity

, we prove that it corresponds to a quasi-Fuchsian group, by showing it has no accidental parabolics. In particular, this shows that any lift of the Seifert surface to the universal cover is a quasi-disk and its limit set is a quasi-circle in the sphere at infinity.

Harmonic functions on ℝ-covered foliations

We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3‐manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a suspension Anosov flow. Then we study the interaction of a general pseudo-Anosov flow with possible Seifert fibered pieces in the torus decomposition: if the fiber is associated with a periodic orbit of the flow, we show that there is a standard and very simple form for the flow in the piece using Birkhoff annuli. This form is strongly connected with the topology of the Seifert piece. We also construct a large new class of examples in many graph manifolds, which is extremely general and flexible. We construct other new classes of examples, some of which are generalized pseudo-Anosov flows which have one-prong singularities and which show that the above results in Seifert fibered and solvable manifolds do not apply to one-prong pseudo-Anosov flows. Finally we also analyse immersed and embedded incompressible tori in optimal position with respect to a pseudo-Anosov flow. 37D20, 37D50; 57M60, 57R30 The goal of this article is to start a systematic study of pseudo-Anosov flows in toroidal 3‐manifolds. More specifically we analyse such flows in closed manifolds which are not hyperbolic or in pieces of the torus decomposition which are not hyperbolic and we obtain substantial results in Seifert fibered pieces. We also produce many new examples of pseudo-Anosov flows, including a large new class in graph manifolds and we study optimal position of tori with respect to arbitrary pseudo-Anosov flows. The study of hyperbolic flows in toroidal manifolds was initiated by Ghys [29], who analysed Anosov flows in 3‐dimensional circle bundles. Ghys showed that up to finite covers, the flow is topologically equivalent to the geodesic flow in the unit tangent bundle of a hyperbolic surface. This was later strengthened by the first author who showed that this holds if the manifold is Seifert fibered [2]. In the mid 70s

Local and global properties of limit sets of foliations of quasigeodesic Anosov flows

We study incompressible tori in 3-manifolds supporting pseudo-Anosov flows and more generally Z ⊕ Z subgroups of the fundamental group of such a manifold. If no element in this subgroup can be represented by a closed orbit of the pseudo-Anosov flow, we prove that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism of the torus. In particular it is non singular and is an Anosov flow. It follows that either a pseudo-Anosov flow is topologically conjugate to a suspension Anosov flow, or any immersed incompressible torus can be realized as a free homotopy from a closed orbit of the flow to itself. The key tool is an analysis of group actions on non-Hausdorff trees, also known as R-order trees – we produce an invariant axis in the free action case. An application of these results is the following: suppose the manifold has an R-covered foliation transverse to a pseudo-Anosov flow. If the flow is not an R-covered Anosov flow, then it follows that the manifold is atoroidal.