<![CDATA[Filling geodesics and hyperbolic complements]]>

<![CDATA[Filling geodesics and hyperbolic complements]]>Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact (i.e. they preserve a contact form — that is, a 1-form $\alpha$ for which $\alpha \wedge d\alpha$ is a volume form). Their preprint gives some very interesting new constructions of such flows, obtained by surgery along a Legendrian knot (one tangent to the kernel of the contact form) which is transverse to the stable/unstable foliations of the Anosov flow.

This idea of using surgery to modify Anosov flows goes back at least to Fried, Goodman and Handel-Thurston in the early 80’s. In the 90’s Fenley analyzed the geometry and topology of the resulting manifolds obtained by surgery, obtaining strong results. This was in the days before Perelman, when perhaps the main goal of 3-manifold topology was to prove Thurston’s Geometrization Conjecture, and one of the main avenues of attack was to prove the conjecture under the hypothesis of some extra structure, for example the existence of a certain kind of foliation or flow. Barbot showed that the stable/unstable foliations of contact Anosov flows are ${\bf R}$ -covered (i.e. their leaf space in the universal cover is Hausdorff, and therefore homeomorphic to the real line); meanwhile, Fenley and I obtained essentially a strong classification theorem for manifolds with ${\bf R}$ -covered foliations, showing (before Perelman!) that they satisfy a weak version of the geometrization conjecture (that their fundamental groups either contain a ${\bf Z}^2$ or are word-hyperbolic), and that they contain quasigeodesic pseudo-Anosov flows. Thus they can be understood and analyzed in many ways, and we have an essentially complete picture of their geometry and topology.

So we have known for quite some time that Anosov flows are quite flexible, and there are many known constructions. By contrast, the only known contact Anosov flows were very special — essentially the only general example available before this paper was the geodesic flow on a Riemannian (or Finsler) manifold of negative curvature.

The key example is a hyperbolic surface $S$ . A geodesic $\gamma$ in $S$ lifts to a knot $K$ in the unit tangent bundle $UTS$ , by associating to each $p \in \gamma$ the unit vector in $T_pS$ perpendicular to $\gamma'(p)$ and on the positive side. This knot is Legendrian, and is transverse to both the stable and unstable foliations, so Foulon-Hasselblatt show that one can do surgery on it to produce interesting new contact Anosov flows on new manifolds. If $UTS-K$ is hyperbolic, the result of a sufficiently big surgery will be a hyperbolic manifold. Consequently, Foulon-Hasselblatt raise the natural question of what conditions on $\gamma$ ensure that $UTS-K$ is hyperbolic.

In fact, one obvious necessary condition is that $\gamma$ should be filling in $S$ , that is, the complementary regions to $S-\gamma$ should be polygons. For, otherwise, an essential embedded loop $\alpha$ in $S-\gamma$ suspends to an essential embedded (non-boundary parallel) torus in $UTS-K$ , which is an obstruction to hyperbolicity. In fact, Foulon-Hasselblatt ask explicitly whether this filling condition is sufficient.

Anyway, when I read this, I immediately felt that this should be the only obstruction. I have been out of 3-manifold theory for a while, but the statement seemed vaguely familiar, and I’m reasonably confident that this fact is somewhere in the literature (though who knows; I’d be grateful to any reader that can point me to a specific reference). It is also vaguely reminiscent of the well-known theorem of Menasco that if $L$ is a nonsplit prime alternating link which is not a torus link, then $S^3-L$ is hyperbolic. On the other hand, it turns out to be simple enough to prove directly, so the purpose of this blog post (apart from to break my record of only blogging in odd numbered years) is to give a short proof of this fact.

Let’s be a bit more precise. Let $S$ be a closed, oriented hyperbolic surface, let $\Gamma$ be a finite union of immersed, oriented, primitive geodesics in $S$ . Assume that no two components of $\Gamma$ are the same geodesic with opposite orientation.

Associated to $\Gamma$ is the link $L(\Gamma)$ (or just $L$ if $\Gamma$ is understood) in the unit tangent bundle $UTS$ consisting of unit vectors $v$ based at points $p$ of (the image of) $\Gamma$ for which the ordered pair $v,\Gamma'(p)$ make an oriented orthonormal basis for $T_pS$ . Note that it is just as easy to let $S$ and $\Gamma$ be unoriented, and work in the projective unit tangent bundle instead of $UTS$ .

Theorem. If $\Gamma$ is filling in $S$ then $UTS-L$ is hyperbolic.

Note that we could just as well take $L$ to be the set of tangent vectors to $\Gamma$ , since this is isotopic (as a link) to $L$ .

Proof: Thurston famously showed (also see here for a detailed proof) that a 3-manifold with boundary is hyperbolic if and only if it is irreducible (i.e. every embedded sphere bounds a ball), has infinite fundamental group, and contains no essential embedded torus which is not parallel to a boundary component. Since $UTS$ is irreducible, and no component of $L$ is contained in a ball, so is $UTS-L$ . Furthermore, $\pi_1(UTS-L)$ surjects onto $\pi_1(UTS)$ which is infinite. It follows that either $UTS-L$ is hyperbolic, or it contains an embedded essential non-boundary parallel torus $T$ . We show that no such $T$ can exist.

The inclusion $i:UTS-L \to UTS$ induces a map on fundamental groups. We let $G = i_*\pi_1(T)$ . This is a free abelian group (because $\pi_1(UTS)$ is torsion free) of rank at most 2. The proof reduces to a case-by-case analysis depending on the rank of $G$ .

Case ${\rm Rank}(G)=2$ . In this case, since $T$ is embedded in $UTS-L$ , the map on fundamental groups is injective, and therefore (by the classification of essential embedded tori in Seifert fibered spaces) $T$ is vertical — i.e. it is the union of circle fibers over an embedded essential loop $\alpha$ in $S$ . Since $\Gamma$ is filling, some $\gamma \subset \Gamma$ has nontrivial (geometric) intersection number with $\gamma$ . Evidently, the lift $\gamma^\perp \subset L$ intersects $T$ .

Case ${\rm Rank}(G)=0$ . In this case, $G={\rm id}$ and $T$ lifts to an embedded torus in the universal cover $\widetilde{UTS}$ . The flowlines of the geodesic flow make $\widetilde{UTS}$ into a topological product ${\bf R}^2 \times {\bf R}$ , and $\tilde{L}$ sits in this as $X\times {\bf R}$ for some discrete set $X\subset {\bf R}^2$ . Hence any embedded torus in the complement is compressible; this shows that $T$ was already compressible in $UTS-L$ .

Case ${\rm Rank}(G)=1$ . In this case $G$ is cyclic and is equal to $\langle g\rangle$ for some $g\in G$ . A component of the preimage $\tilde{T} \subset \widetilde{UTS}$ is proper, homeomorphic to a cylinder, and of uniformly bounded thickness (i.e. it is foliated by circles of uniformly bounded diameter). To see this, first foliate $T$ by circles in the homotopy class of the kernel of $i_*$ , and lift the foliation to $\tilde{T}$ . Note also that the foliation is $\langle g\rangle$ -equivariant. The idea of the proof is now straightforward: $\tilde{T}$ must separate some of the components of $\tilde{L}$ from others. Since it has bounded thickness, it can only separate finitely many. But if $T$ is not boundary parallel, it must separate at least two. These two must be a finite Hausdorff distance apart in $\widetilde{UTS}$ ; this will readily imply that they are both lifts of the same geodesic $\gamma$ in $\Gamma$ , which will give a contradiction. We now flesh out this argument.

Let’s consider the geometry of $\widetilde{UTS}$ . There is a canonical 1-Lipschitz projection $\pi:\widetilde{UTS} \to {\bf H}^2$ with fibers homeomorphic to ${\bf R}$ (these are lifted flowlines of the circle flow on $UTS$ ). Moreover, there is a foliation of $\widetilde{UTS}$ by stable leaves of the geodesic flow (on $UTS$ ). If we pick one stable leaf $\lambda$ and let $z\in \pi_1(UTS)$ denote the generator of the center, then the “slab” $E$ between $\lambda$ and $z(\lambda)$ (a fundamental domain for the deck action of $\langle z \rangle$ on $\widetilde{UTS}$ ) is quasi-isometric to ${\bf H}^2$ , and every component of $\tilde{L}$ in this slab is quasi-isometric to a geodesic in ${\bf H}^2$ .

Now consider how $\tilde{T}$ sits in $\widetilde{UTS}$ . Since $T$ is essential in $UTS-L$ , it follows that $\tilde{T}$ is essential in $\widetilde{UTS} - \tilde{L}$ ; the latter is homeomorphic to ${\bf R}^2-X \times {\bf R}$ for some discrete set $X$ , so $\tilde{T}$ must be properly isotopic to a proper annulus of the form $\beta \times {\bf R}$ for some embedded $\beta$ in ${\bf R}^2-X$ (warning: this product structure is purely topological, and not quasi-metrical). Since $T$ is assumed not to be boundary parallel, $\beta$ must enclose at least two distinct points of $X$ , corresponding to components $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ of $\tilde{L}$ ; i.e. $\tilde{T}$ must separate these components from other components. Recall that $\tilde{T}$ is foliated ( $\langle g\rangle$ -equivariantly if we like) by circles $S^1_t$ of uniformly bounded diameter. Each such circle bounds a disk $D_t$ in $\widetilde{UTS}$ of uniformly bounded diameter, and moreover each such disk intersects $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ in at least two points. The set of such disks $D_t$ is proper; in particular, it follows that $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ contain a pair of proper sequences of points which are a uniformly bounded distance apart; in particular, $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ are a finite Hausdorff distance apart. It follows that the $\tilde{\gamma}_i$ have the same projection to ${\bf H}^2$ under $\pi$ , and therefore they are lifts of the same component $\gamma$ of $\Gamma$ .

But now one readily obtains a contradiction. Pick an arc $\beta$ disjoint from $\tilde{T}$ from one component to the other. This can be chosen to project to a closed essential loop in $UTS$ , and we deduce that $\tilde{T}$ encloses infinitely many lifts of $\gamma$ once it encloses two. But this set of lifts is discrete, so only finitely many have uniformly bounded Hausdorff distance from any one of them. This contradiction completes the proof. qed.

This method of proof is a little bit specific and possibly not as simple as possible, but I believe it does generalize to the case that $\Gamma$ is a filling union of round circles in $S$ — i.e. curves of constant geodesic curvature (which might vary from component to component). Some of these circles (those with extrinsic curvature $<1$ ) will be essential in $S$ and some (those with extrinsic curvature $>1$ ) will be inessential; but their lifts to $UTS$ will all be essential, and if the union is filling in $S$ , the complement of the lifts will be atoroidal. The main issues to deal with are to show that the preimage in $\widetilde{UTS}$ looks like a collection of straight lines in ${\bf R}^3$ (i.e. it is globally unknotted) and to deal with the fact that there are now distinct circles whose lifts are a finite Hausdorff distance apart (any two circles with extrinsic curvature $>1$ ). Casson-Jungreis‘s criterion for unknottedness could be used to deal with the first issue, I think.

This is now no longer relevant to contact Anosov surgery, but rather to “regulating surgery”, of the kind considered in another paper I wrote some time ago. (Idle remark: this was the third paper I ever wrote, and I was very pleased with myself for disproving a conjecture of Thurston. Naturally the silence from the mathematical world was deafening. Finally a review appeared on MathSciNet, and I thought: finally I’ll get some feedback! Needless to say I found the actual review a little disappointing . . . C’est la vie)

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