My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobius-like groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress that I know of on a conjectural program I formulated a few years ago.

One of the main results of the paper is to show that every quasigeodesic flow on a closed hyperbolic 3-manifold either has a closed orbit, or the fundamental group of the manifold admits an action on a circle with some very peculiar properties, namely that it is Mobius-like but not Mobius. The problem of giving necessary and sufficient conditions on a vector field on a 3-manifold to guarantee the existence of a closed orbit is a long and interesting one, and the introduction to the paper gives a brief sketch of this history as follows:

In 1950, Seifert asked whether every nonsingular flow on the 3-sphere has a closed orbit. Schweitzer gave a counterexample in 1974 and showed more generally that every homotopy class of nonsingular flows on a 3-manifold contains a C^1 representative with no closed orbits. Schweitzer’s examples were generalized considerably and it is known that the flows can be taken to be smooth or volume-preserving.

On the other hand, Taubes’ 2007 proof of the 3-dimensional Weinstein conjecture shows that flows satisfying certain geometric constraints must have closed orbits. Explicitly, Taubes showed that every Reeb vector field on a closed 3-manifold has a closed orbit. Reeb flows are geodesible, i.e. there is a Riemannian metric in which the flowlines are geodesics. Complementary to this result, though by different methods, Rechtman showed in 2010 that the only geodesible real analytic flows on closed 3-manifolds that contain no closed orbits are on torus bundles over the circle with reducible monodromy.

Geodesibility is a local condition, and furthermore one that is not stable under perturbations. By contrast, a nonsingular flow is said to be quasigeodesic if the flowlines of the flow pulled back to the universal cover are quasigeodesics. This is a macroscopic condition, and when the ambient 3-manifold is hyperbolic it is a stable condition under C^0 perturbations; this stability is for global topological reasons and not because the flow itself is structurally stable (which it will not typically be).

Calegari conjectured in 2006 that quasigeodesic flows on closed hyperbolic 3- manifolds should all have closed orbits, and moreover that every homotopy class of quasigeodesic flow should contain a pseudo-Anosov representative that is unique up to isotopy. Pseudo-Anosov flows are hyperbolic and therefore structurally stable, so this conjecture implies that one should be able to deduce the existence of closed orbits from the dynamics of the fundamental group on the orbit space in the universal cover.

Our paper is devoted to fleshing out some aspects of Calegari’s conjectural program. We are able to find conditions that guarantee the existence of a closed orbit for a quasigeodesic flow on a closed hyperbolic 3-manifold expressed in terms of the action of the fundamental group on an associated “universal circle”.

To go more deeply into this, let me start with some basic definitions. We are concerned always with a closed hyperbolic 3-manifold M with a 1-dimensional foliation X (the leaves of the foliation are the flowlines of the flow). The universal cover of M is isometric to hyperbolic 3-space, and the foliation lifts to a 1-dimensional foliation \widetilde{X} of the universal cover. To say that the flow (foliation) is quasigeodesic is to say that the leaves of \widetilde{X} are quasigeodesics in hyperbolic 3-space.

The fact that the flowlines are quasigeodesics easily implies that the leaf space P of \widetilde{X} (i.e. the quotient of hyperbolic 3-space by the equivalence relation that collapses every leaf to a point) is Hausdorff; since it is simply-connected and noncompact, it is homeomorphic to the plane. Notice that the plane P comes together with an action by the fundamental group \pi_1(M); a closed orbit of the flow corresponds precisely to a fixed point for some nontrivial element of \pi_1(M).

Now, every oriented quasigeodesic in hyperbolic 3-space is asymptotic to two distinct points in the sphere at infinity. It follows that we can define two equivariant endpoint maps e^{\pm}:P \to S^2_\infty. The point preimages of S^2_\infty under e^+ (say) decompose the plane into closed, connected sets. It turns out that each of these sets is unbounded and therefore has a nonempty collection of ends. The nice thing about a collection of disjoint, closed, connected, unbounded subsets of the plane is that the set of ends of such subsets can be circularly ordered in a canonical way, and one therefore obtains a natural action of \pi_1(M) on a circularly ordered set, which can be bootstrapped to a (faithful) action of \pi_1(M) on a so-called universal circle (S^1_u)^+ by homeomorphisms. This much of the story is contained in my 2006 paper.

Some hyperbolic 3-manifolds have fundamental groups which do not act faithfully on a circle; from this one deduces that there are hyperbolic 3-manifolds with no quasigeodesic flow, which answered a long-standing question of Thurston. We’ll return to this question in a minute.

Now, from the discussion above, we see that the existence of a quasigeodesic flow gives rise to a natural action of \pi_1(M) on a plane P and a circle (S^1_u)^+. It is natural to wonder if (and in fact I conjectured that) there is a natural topology on P \cup (S^1_u)^+, compatible with the \pi_1(M) actions, for which the union is homeomorphic to a closed disk. This is the first main theorem Steven proves:

Compactification Theorem (Frankel): There is a natural compactification \overline{P} of P homeomorphic to the closed disc so that \partial P = (S^1_u)^+. The action of \pi_1(M) on P extends to \overline{P} and restricts to the universal circle action on \partial P.

The proof of this is quite deep and involved. One of the main difficulties is that a priori, the point preimages under the endpoint maps e^\pm are arbitrary closed subsets of the plane, so dealing with their separation properties is very involved. Steven develops his theory in quite some generality. An unbounded decomposition of the plane is a partition of the plane into unbounded continua; Steven’s main theorem is that any such decomposition with uncountably many elements gives rise to a canonical compactification of the plane, homeomorphic to the disk. Applying this to the special case arising in the context of a quasigeodesic flow, gives the Compactification Theorem above.

The story can be repeated with e^- in place of e^+, and one gets another universal circle compactifying P. In fact, one can work with both e^+ and e^- simultaneously, and obtain a “master” compactification, obtained by adding a “master” universal circle S^1_u, with canonical monotone surjections to the positive and negative universal circles constructed as above. One must deal with generalized unbounded decompositions to achieve this result; this is Theorem 7.9 in Steven’s paper. Using this, one can build a “universal sphere” S^2_u from two copies of P glued together along S^1_u. From the construction, the following conjecture seems quite plausible:

Conjecture 1: The maps e^\pm:P \to S^2_\infty extend to a monotone map E:S^2_u \to S^2_\infty.

Note that since the image of S^1_u under such a hypothetical map should be both closed and invariant, it should be equal to all of S^2_u; i.e. it would be a group-invariant Peano curve. Examples of such curves arise very naturally by the Cannon-Thurston construction associated to surface bundles. A proof of Conjecture 1 would give a new proof (and considerable generalization) of the Cannon-Thurston theorem. The connection between quasigeodesic flows and surface bundles is the simple fact that any 1-dimensional foliation of a hyperbolic 3-manifold transverse to the surfaces of a surface fibration is quasigeodesic, and the universal circle in this case should be the circle at infinity of the universal cover of a surface fiber.

Let’s return to the question of closed orbits. Now, any homeomorphism of a closed disk has a fixed point, by the Brouwer fixed point theorem. So one deduces either that X has a closed leaf, or that every nontrivial element of \pi_1(M) has at least one fixed point in S^1_u. To make more progress, one must understand the relationship between the dynamics of \pi_1(M) on S^1_u and the dynamics on S^2_\infty. A positive answer to Conjecture 1 above would simplify things, but even without it, Steven is able to get a great deal of traction.

Let’s consider the following definition:

Definition: A group of homeomorphisms of the circle is Mobius-like if every element is conjugate to an element of PSL(2,\mathbb{R}). It is rotationless if every element is conjugate to a hyperbolic or parabolic element. It is Mobius if the entire group is conjugate into PSL(2,\mathbb{R}).

With this definition, Steven’s next main theorem is the following:

Mobius-like Theorem (Frankel): Let X be a quasigeodesic flow on a closed hyperbolic 3- manifold M. Suppose that the action of \pi_1(M) on the universal circle S^1_u is not a rotationless Mobius-like group. Then X has a closed orbit.

This is nicely complemented by:

Conjugacy Theorem (Frankel): Let X be a quasigeodesic flow on a closed hyperbolic 3- manifold M. Then the action of \pi_1(M) on S^1_u is not conjugate into PSL(2,\mathbb{R}).

Steven conjectures that the action of \pi_1(M) on S^1_u should never be Mobius-like; this would imply that every quasigeodesic flow on a hyperbolic 3-manifold should have a closed orbit.

While we’re being speculative, let’s imagine how far such a program could go. Quasigeodesicity persists under C^0 perturbations, even though a quasigeodesic flow need not be structurally stable (for example, it could contain a solid torus foliated by closed orbits). We can create closed orbits by a small perturbation, and these give rise to fixed points in P for the perturbed actions. The connected preimage under e^+ containing the fixed point must itself be fixed, and so must its set of ends. If this set is finite, some power fixes the ends pointwise; a similar picture holds for e^-, and we should obtain a collection of fixed points in the master circle S^1_u which one expects to have alternating source-sink dynamics. In this way, we expect to be able to produce a pair of invariant stable/unstable laminations. These should give rise in turn to a pseudo-Anosov quasigeodesic flow, whose closed orbits should correspond to Nielsen classes of closed orbits of the original flow X. Hence (conjecturally), not only should X have one closed orbit, it should have infinitely many! Explicitly:

Conjecture 2: Let X be a quasigeodesic flow on a hyperbolic 3-manifold M. Then X should be homotopic to a pseudo-Anosov quasigeodesic flow Y whose closed orbits should be in bijection to free homotopy classes of closed orbits of X.

The relationship between X and Y should be like the relationship between a surface homeomorphism, and its pseudo-Anosov representative. Interestingly enough, the “stable/unstable laminations” we would like to find are already actually known to exist; they are constructed in Theorem B of my paper. What is missing is the interpretation of these laminations as the residue on the universal circle of a pair of stable/unstable laminations of the flow space of a homotopic flow.

How canonical should Y be? As far as I know, there is no known obstruction to the following conjecture:

Conjecture 3: Every connected component of the space of quasigeodesic flows on a hyperbolic 3-manifold should contain a unique pseudo-Anosov quasigeodesic flow, up to isotopy.

Well, this picture is all very nice, if true. But it raises the significant problem of constructing quasigeodesic flows, or understanding exactly which hyperbolic 3-manifolds do or don’t have them. As remarked above, the existence of a quasigeodesic flow implies that the fundamental group is circularly orderable, and therefore that some finite index subgroup is left orderable. In fact, if M is an integral homology sphere, the fundamental group is circularly orderable if and only if it is left orderable. The condition of left orderability is quite interesting in its own right; there are many known examples of hyperbolic 3-manifolds whose fundamental groups are not left orderable (e.g. double branched covers of alternating knots in the 3-sphere), and some people are trying to connect up this condition to the concept of a (Heegaard Floer Homology) L-space.

But I prefer to be a bit more optimistic, and look at a quasigeodesic flow as a potentially quite flexible structure. Suppose N is a hyperbolic 3-manifold with a cusp. Such a 3-manifold has nontrivial 2-dimensional (relative) homology, and combined work of Fenley-Gabai-Mosher shows that it admits a pseudo-Anosov flow, which persists (and is quasigeodesic) in “most” Dehn fillings (see e.g. Fenley-Mosher or my foliations book for a discussion of this). Now, if we have a hyperbolic 3-manifold M with an embedded geodesic \gamma with a sufficiently thick embedded tube around it, we know M-\gamma is hyperbolic, and has such a nice flow. We can try to extend this flow over M by spinning it around \gamma. It is plausible that the resulting flow on M should be quasigeodesic: far from \gamma, it should be quasigeodesic because the geometry should be close to the geometry of M-\gamma, and quasigeodesity is stable. Close to \gamma, it should be quasigeodesic, because it wraps around and around \gamma. Anyway, I think it is worth making another conjecture:

Conjecture 4: For any t there is a T(t) so that if M is a hyperbolic 3-manifold with an embedded geodesic \gamma of length t contained in an embedded tube of radius at least T(t), then M admits a quasigeodesic flow.

If M is an arbitrary hyperbolic 3-manifold, one can find a finite cover \hat{M} satisfying the hypotheses of this conjecture, by using the fact that cyclic groups in hyperbolic 3-manifold groups are subgroup separable.

Some elements of this program are more approachable than others, but Steven’s work definitely represents a big step forward.