pseudo-Anosov Flow Research Articles

From loom spaces to veering triangulations

We introduce loom spaces , a generalisation of both the leaf spaces associated to pseudo-Anosov flows and the link spaces associated to veering triangulations. Following work of Guéritaud, we prove that there is a locally veering triangulation canonically associated to every loom space, and that the realisation of this triangulation is homeomorphic to \mathbb{R}^{3} .

Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity

AbstractWe introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.

$\mathbb{R}$-covered foliations and transverse pseudo-Anosov flows in atoroidal pieces

We study the transverse geometric behavior of 2-dimensional foliations in 3-manifolds. We show that an \mathbb{R} -covered, transversely orientable foliation with Gromov hyperbolic leaves in a closed 3-manifold admits a regulating, transverse pseudo-Anosov flow (in the appropriate sense) in each atoroidal piece of the manifold. The flow is a blow up of a one prong pseudo-Anosov flow. In addition we show that there is a regulating flow for the whole foliation. We also determine how deck transformations act on the universal circle of the foliation.

Flows, growth rates, and the veering polynomial

AbstractFor a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$ -manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $ . We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $ , and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M. Our work can be used to study the flow $\varphi $ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$ -manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.

Free Seifert pieces of pseudo-Anosov flows

We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In a previous paper [Ba-Fe1] we described the structure of a pseudo-Anosov flow restricted to a periodic piece up to isotopy along the flow. In the present paper we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. In the proof we introduce almost k-convergence groups and prove a convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we carefully define what is a hyperbolic blow up.

Quasigeodesic pseudo-Anosov flows in hyperbolic 3-manifolds and connections with large scale geometry

In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover. We prove that such flows are quasigeodesic if and only if there is an upper bound, depending only on the flow, on the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow. The main ingredient is a proof that, under the boundedness condition, the fundamental group of the manifold acts as a uniform convergence group on a flow ideal boundary of the universal cover. We also construct a flow ideal compactification of the universal cover, and prove that it is equivariantly homeomorphic to the Gromov compactification. This implies the quasigeodesic behavior of the flow. The flow ideal boundary and flow ideal compactification are constructed using only the structure of the flow.

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

In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds.Geom. Topol. 17(2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of$M$mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.

Rigidity of pseudo-Anosov flows transverse to $\mathbb{R}$-covered foliations

A foliation is \mathbb{R} -covered if the leaf space of the lifted foliation to the universal cover is homeomorphic to the set of real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to a fixed \mathbb{R} -covered foliation. If there are two transverse pseudo-Anosov flows, then the foliation is weakly conjugate to the stable foliation of an \mathbb{R} -covered Anosov flow. The proof uses the universal circle for \mathbb{R} -covered foliations.

Pseudo-Anosov flows in toroidal manifolds

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.

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 three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify 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 and 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 implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and we show that the stable/unstable foliations of these flows are quasi-isometric. Finally we apply these results to foliations: if a foliation is R-covered or with one sided branching in an 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.

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

Let $\mathcal{F}$ be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that $\mathcalF$ 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 $\mathcal{F}$ is a Reebless, infinite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such infinite 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 extremely useful tool is a detailed understanding of the topological structure and asymptotic properties of the 1-dimensional foliations in the leaves of e $\mathcal{F}$ induced by the stable and unstable foliations of the flow.

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.

Uniform 1-cochains and genuine laminations

We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting a transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are the straightening of the coarse laminations defined by Calegari, by using minimal surface techniques. Moreover, when we collapse these laminations, we get a topological pseudo-Anosov flow, as defined by Mosher.

Regulating flows, topology of foliations and rigidity

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.

Laminations and groups of homeomorphisms of the circle

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that π1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so π1(M) is isomorphic to a subgroup of Homeo(S 1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S 1. As a corollary, the Weeks manifold does not admit a tight essential lamination with solid torus guts, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston’s universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem.

Pseudo-Anosov Flows and Incompressible Tori

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.

Foliations with One-Sided Branching

We generalize the main results from the author's paper in Geom. Topol. 4 (2000), 457–515 and from Thurston's eprint math.GT/9712268 to taut foliations with one-sided branching. First constructed by Meigniez, these foliations occupy an intermediate position between ℝ-covered foliations and arbitrary taut foliations of 3-manifolds. We show that for a taut foliation $$F$$ with one-sided branching of an atoroidal 3-manifold M, one can construct a pair of genuine laminations Λ± of M transverse to $$F$$ with solid torus complementary regions which bind every leaf of $$F$$ in a geodesic lamination. These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston (unpublished), which maps monotonely and π1(M)-equivariantly to each of the circles at infinity of the leaves of $$\tilde F$$ , and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of $$\tilde F$$ . In particular, let $$\tilde F$$ denote the pulled-back foliation of the universal cover, and co-orient $$\tilde F$$ so that the leaf space branches in the negative direction. Then for any pair of leaves of $$\tilde F$$ with μλ, the leaf λ is asymptotic to μ in a dense set of directions at infinity. This is a macroscopic version of an infinitesimal result from Thurston and gives much more drastic control over the topology and geometry of $$F$$ , than is achieved by him. The pair of laminations Λ± can be used to produce a pseudo-Anosov flow transverse to $$F$$ which is regulating in the nonbranching direction. Rigidity results for Λ± in the ℝ-covered case are extended to the case of one-sided branching. In particular, an ℝ-covered foliation can only be deformed to a foliation with one-sided branching along one of the two laminations canonically associated to the ℝ-coveredfoliation constructed in Geom. Topol. 4 (2000), 457–515, and these laminations become exactly the laminations Λ± for the new branched foliation. Other corollaries include that the ambient manifold is δ-hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.

Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows

We analyse the topological and geometrical behavior of foliations on 3-manifolds. We consider the transverse structure of an R-covered foliation in a 3-manifold, where R-covered means that in the universal cover the leaf space of the foliation is Hausdorff. If the manifold is aspherical we prove that either there is an incompressible torus in the manifold; or there is a transverse pseudo-Anosov flow. It follows that manifolds with R-covered foliations satisfy the weak hyperbolization conjecture.

The geometry of ℝ–covered foliations

We study R-covered foliations of 3-manifolds from the point of view of their transverse geometry. For an R-covered foliation in an atoroidal 3-manifold M, we show that M-tilde can be partially compactified by a canonical cylinder S^1_univ x R on which pi_1(M) acts by elements of Homeo(S^1) x Homeo(R), where the S^1 factor is canonically identified with the circle at infinity of each leaf of F-tilde. We construct a pair of very full genuine laminations transverse to each other and to F, which bind every leaf of F. This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for F, analogous to Thurston's structure theorem for surface bundles over a circle with pseudo-Anosov monodromy. A corollary of the existence of this structure is that the underlying manifold M is homotopy rigid in the sense that a self-homeomorphism homotopic to the identity is isotopic to the identity. Furthermore, the product structures at infinity are rigid under deformations of the foliation F through R-covered foliations, in the sense that the representations of pi_1(M) in Homeo((S^1_univ)_t) are all conjugate for a family parameterized by t. Another corollary is that the ambient manifold has word-hyperbolic fundamental group. Finally we speculate on connections between these results and a program to prove the geometrization conjecture for tautly foliated 3-manifolds.

Geometry and topology of ℝ-covered foliations

An R \mathbb {R} -covered foliation is a special type of taut foliation on a 3 3 -manifold: one for which holonomy is defined for all transversals and all time. The universal cover of a manifold M M with such a foliation can be partially compactified by a cylinder at infinity, somewhat analogous to the sphere at infinity of a hyperbolic manifold. The action of π 1 ( M ) \pi _1(M) on this cylinder decomposes into a product by elements of Homeo ( S 1 ) × Homeo ( R ) \text {Homeo}(S^1)\times \text {Homeo}(\mathbb {R}) . The action on the S 1 S^1 factor of this cylinder is rigid under deformations of the foliation through R \mathbb {R} -covered foliations. Such a foliation admits a pair of transverse genuine laminations whose complementary regions are solid tori with finitely many boundary leaves, which can be blown down to give a transverse regulating pseudo-Anosov flow. These results all fit in an essential way into Thurston’s program to geometrize manifolds admitting taut foliations.