Taut Foliations, Left-Orders, and Pseudo-Anosov Mapping Tori

TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI

JONATHAN ZUNG

Abstract. For a large class of 3-manifolds with taut foliations, we construct an action of π1(M) on R by orientation preserving homeomorphisms which captures the transverse geometry of the leaves. This action is complementary to Thurston’s universal circle. Applications include the left-orderability of the fundamental groups of every non-trivial surgery on the ﬁgure eight knot. Our techniques also apply to at least 2598 manifolds representing 44.7% of the non-L-space rational homology spheres in the Hodgson-Weeks census of small closed hyperbolic 3-manifolds.

1. Introduction A taut foliation on a 3-manifold M is a valuable structure. A taut foliation may be used to certify genus minimizing surfaces or to certify nontriviality of transverse loops [Rou74, Thu86, Gab83, Nov]. The existence of a taut foliation puts various constraints on π1(M). For example, Thurston showed that if M carries a taut foliation, then π1(M) is circularly orderable, i.e. it acts on S1 by homeomorphisms [Thu97, CD03]. In another direction, Kronheimer and Mrowka made a connection to Floer theory via contact and symplectic geometry, showing that a taut foliation gives rise to a nontrivial class in monopole Floer homology [KM97, KM07]. Ozsv´athand Szab´o established parallel results in the setting of Heegaard Floer homology [OS04]. The L-space conjecture is a proposed sharpening of the connections outlined above [BGW11]. It posits that the following are equivalent for an orientable, irreducible rational homology sphere M: arXiv:2006.07706v2 [math.GT] 1 Sep 2021 (1) M has a co-orientable taut foliation

(2) π1(M) is left-orderable (i.e. π1(M) acts faithfully on R by orientation preserving homeomorphisms).

(3) M is not an L-space (i.e. its Heegaard Floer homology HFd(M) 2 satisfies the strict inequality rank(HFd(M)) > |H (M; Z)|) [OS03]. Technology for deciding conditions (1) and (3) is well developed; for example, Dunfield verified the equivalence of (1) and (3) for 99.8% of the manifolds in his census of ∼300, 000 small hyperbolic rational homology spheres [Dun19]. 1 2 JONATHAN ZUNG

Techniques for deciding (2) are harder to come by. For every non-left- orderable group, there is a ﬁnite length certiﬁcate proving its non-left- orderability. On the other hand, left-orderability is not known to be decidable. Here are a few practical methods for proving the left-orderability of 3-manifold group:

• Try to lift PSL(2, R) representations of π1(M) to PSL(2^, R) which 2 acts on the the universal cover of the circle at inﬁnity in H [EHN81, CD18].

• If M has a taut foliation, try to lift Thurston’s action of π1(M) on 1 S to an action on R. This works whenever the Euler class of the plane ﬁeld tangent to the foliation vanishes [CD03, Section 7] [BH19, Section 5].

• We say that a foliation F is R-covered if the leaf space of the lift of F to the universal cover of M is homeomorphic to R (e.g. in the case of the foliation of a fibered 3-manifold by fiber surfaces). Since π1(M) always acts on the leaf space, we get an action on R. The third technique is appealing since it directly uses the transverse geometry of the foliation, but is limited in generality since most taut foliations are not R-covered. In this paper, we demonstrate a method for improving the third technique to work for more general taut foliations. We study a family of 3- manifolds with taut foliations which are not R-covered, but whose leaf spaces admit a map to R such that the action of π1 descends to an action on R. The question of the existence of taut foliations and left-orderings compatible in this sense was first raised by Thurston [Cal02, Section 8.1][Thu97]. We prove the following:

Theorem 1. Let Σ be an orientable closed surface and ϕ: Σ → Σ a pseudo- Anosov map with orientable invariant foliations. Suppose further that ϕ preserves the orientation of these foliations. Let Mϕ be mapping torus of Σ. Let Mϕ(p; q) be the result of non-zero surgery along any collection closed orbits of ϕ. If the surgery slopes all have the same sign, then Mϕ(p; q) has left-orderable fundamental group.

Here we take the zero slope, also known as the degeneracy slope, to be the one which crosses no prongs. See Convention 1 for a full explanation of our slope conventions.

Theorem 2. With the assumptions of Theorem 1, there is a taut foliation F on Mϕ(p; q). Let L be the leaf space of Fe. Then there is a continuous, monotone map f : L → R so that the action of π1(Mϕ(p; q)) on L descends to a nontrivial action on R. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 3

Corollary 1. For any n ≥ 1, any non-trivial surgery on the n-fold cyclic branched cover of the ﬁgure-eight knot has left orderable fundamental group.

Previously, orderability for surgeries on the figure eight knot was known only for slopes in [−4, 4] ∪ Z. The range (−4, 4) was treated by the representation theoretic approach of [BGW11], while the toroidal exceptional surgeries { 4, −4 } were resolved in [CLW13] using a gluing theorem for amalgamations of left-orderable groups. It has been known to those in the field that Fenley’s work on R-covered Anosov flows applies to show that integer surgeries on the figure eight knot are left-orderable [Fen98]. Hu recently gave another approach to the case of integer slopes by showing that certain taut foliations on these manifolds have vanishing Euler class [Hu19]. In Section 2, we set up notation and construct taut foliations on the manifolds of Theorem 1. In Section 3, we analyze the branching in the leaf spaces of these foliations.

Our approach to defining the map f : L → R is to glue together certain branches of the leaf space L. This point of view is outlined in Section 4. More formally, in Section 5 we define an R-bundle with structure group + partial Homeo (R) and a flat partial connection J . We complete this bundle by adding a point at infinity to each fiber. The resulting S1 bundle has an honest flat connection J . Moreover, this S1 bundle has vanishing Euler class and hence lifts to a flat R bundle. Finally, in Section 6 we report on computations showing that manifolds satisfying the hypotheses of Theorem 1 are abundant in the Hodgson-Weeks census. Acknowledgements. The author would like to thank David Gabai and Sergio Fenley for several helpful discussions about this work. Nathan Dunfield and Mark Bell generously shared data from their census of monodromies of small hyperbolic manifolds. The author is indebted to Peter Ozsváthfor his consistent encouragement and guidance during this project.

2. Foliations on surgeries on pseudo-Anosov mapping tori In this section, we describe the construction of taut foliations on our class of 3-manifolds. One might colloquially describe the construction as “stuﬃng the guts of the suspension of the ϕ-invariant lamination with monkey saddles”. We give a more detailed description and set up notation below. Let Σ be a closed orientable surface. Let ϕ: Σ → Σ be a pseudo-Anosov s u map such that its stable and unstable foliations, denoted F |Σ and F |Σ, are orientable. This implies that each singularity of ϕ has an even number of s u prongs. Suppose further that ϕ preserves the orientations of F |Σ and F |Σ.

Let Mϕ denote the mapping torus of ϕ, and let K1,...,Kn ⊂ Mϕ be the suspensions of any n periodic orbits of ϕ. For ease of exposition, we will 4 JONATHAN ZUNG always assume that the suspensions of the singularities of ϕ are included in K1,...,Kn.

Let Λ|Σ be the stable invariant lamination of ϕ produced by splitting open s s u F |Σ along the prongs at each singularity. Let F , F , and Λ be the s u suspensions of F |Σ, F |Σ, and Λ|Σ in Mϕ. The orientability constraints are equivalent to the orientability of F s.

Example 1. The figure eight knot complement fibers over the circle with a genus 1 fiber and pseudo-Anosov monodromy. We can choose coordinates on 2 2 1 the fiber T \ (0, 0) so that the monodromy is ( 1 1 ). Since this matrix has distinct positive real eigenvalues, the monodromy preserves the orientation of the invariant foliations as desired.

Proof of Corollary 1. The nth cyclic branched cover of the ﬁgure eight knot 2 1 n has monodromy ( 1 1 ) . With our slope conventions, surgery along the zero slope yields the nth cyclic branched cover. By Theorem 1, surgery along any non-zero slope yields a manifold with left-orderable fundamental group.

Example 2. We can generate examples with a given ﬁber genus and pre- scribed singularities by enumerating periodic splitting sequences of train tracks, as in [PH16]. One of the lowest volume examples appearing in the genus 2, 1-singularity enumeration is the 1-cusped hyperbolic manifold m038.

Example 3. The (−2, 3, 7) pretzel knot is fibered with a genus 5 fiber and pseudo-Anosov monodromy. In this case, the monodromy ϕ has a single 18- prong singularity at the boundary of the fiber, so the invariant foliations are orientable. However, ϕ reverses this orientation. The branched double cover of the (−2, 3, 7) pretzel knot does preserve the orientation of the invariant foliations, and so satisfies the given condition.

Convention 1 (Slope conventions). For each i, let mi be the period of the orbit Ki. The mi singularities in Ki ∩ Σ have the same number, denoted ki, of prongs. We assume throughout that ki is even. Deﬁne ωi so that 2πωi is the counterclockwise angle by which ϕmi rotates one of these singularities. (We are thinking of each singularity is a cone point with angle 2πki).

We describe slopes in a slightly nonstandard way. A slope of (pi; qi) corresponds with a curve in ∂N(Ki) deﬁned as follows. Choose a point near m q Ki ∩ Σ and ﬂow it along ϕ i i . Typically, the resulting curve will not close up since ϕmi may rotate singularities. We close it by appending a path in Σ which walks around the relevant singularity by a clockwise angle of 2πpi. Not all such pairs represent slopes; a pair (pi; qi) corresponds with a closed curve if and only pi = ωiqi mod ki. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 5

1.000000?1.000000?1.000000?

1.302775?1.302775?1.302775? 0.580691?0.580691?0.580691? 0.419308?0.419308?0.419308?

1.722083?1.722083?1.722083? 0.722083?0.722083?0.722083?

0.419308?0.419308?0.419308? 1.302775?1.302775?1.302775?

1.000000?1.000000?1.000000?

2.243488?2.243488?2.243488? 1.243488?1.243488?1.243488? 2.243488?2.243488?2.243488?

1.000000?1.000000?1.000000?

Figure 1. This figure shows a train track carrying an invariant lamination for the monodromy of the fibered, 1-cusped 3-manifold m038. The fiber surface has genus 2 with one puncture at the vertices of the octagon. The complement of the train track is a punctured ideal hexagon. The figure was generated using Mark Bell’s program flipper [Bel13].

We always assume qi ≥ 0. We say that a slope is positive, resp. negative when pi is positive, resp. negative. We say that the slope is ∞ when qi = 0. The ∞ slope corresponds with the ﬁber slope and is declared to be both positive and negative. The zero slope or the degeneracy slope is (pi; qi) = (0; ki/ gcd(ωi, ki)). In an abuse of notation, the zero slope and the inﬁnity slope may intersect more than once.

Let Mϕ(p; q) denote the manifold obtained by slope (pi; qi) surgery along Ki, and let Ki(p; q) denote the core of the Dehn ﬁlling of Ki in Mϕ(p; q). 6 JONATHAN ZUNG

Construction 1. Given a sutured solid torus D such that the sutures are parallel with non-meridional slope, there is a foliation of D by planes compatible with the sutures. Recall that the sutures on a sutured manifold divide its boundary into (possibly disconnected) positive and negative subsurfaces. We can construct the desired foliation beginning with the obvious product foliation of the solid torus by disks, and then combing the edges of the disks to expose their positive sides in the positive regions of ∂D and their negative sides in the negative regions of ∂D. This is called a foliation by a stack of monkey saddles. See Fig. 2.

Figure 2. On the left is the standard foliation of D2 × S1, where S1 is cut open. In the middle, we alternately comb the edges of the disks to expose their positive and negative sides. Grey shows the positive sides of the leaves, while white shows the negative sides. On the right, we show the limiting conﬁguration which has 4 annular leaves at the boundary which we call walls and inﬁnitely tall saddle-like leaves (homeomorphic to planes) on the interior. The interior saddle-like leaves accumulate on the walls. We can vary the number of legs of the saddle’s rider or the gluing of the top and bottom of the picture to get sutures of any desired non-meridional slope.

Construction 2. Let (pi; qi) be any choice of slopes with pi 6= 0. Then the manifold Mϕ(p; q) carries a taut foliation constructed as follows. Let ◦ s Mϕ = Mϕ \{ Ki }. Let F be the codimension 1 weak stable foliation of ◦ s the suspension ﬂow of ϕ on Mϕ. Split open F along its prongs to obtain ◦ a lamination Λ on Mϕ ⊂ Mϕ(p; q). The complement Mϕ(p; q) \ Λ is a collection of ideal polygon bundles over S1. With our conventions (see Convention 1), these polygons are 2pi-gons since a closed curve of slope (pi; qi) decomposes into an arc of slope (0; qi), which intersects no prongs, and an arc of slope (pi; 0), which wraps around an angle of 2πpi and therefore intersects 2pi prongs. Fill in these bundles with the foliations constructed in Construction 1. The resulting foliation is taut because it contains no compact leaves; in fact, all leaves are either cylinders or planes.

Example 4. When pi = 1 for all i, the complementary regions can be blown down without without inserting any leaves at all. The resulting 3-manifold TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 7 carries an Anosov flow. Fried shows that every transitive Anosov flow with orientable invariant foliations is obtained by this construction [Fri83]. The transitivity condition is not severe; every Anosov flow on a hyperbolic 3- manifold is transitive. His construction might require the slopes to have different signs, so our results will not hold for all of these manifolds.

Let F denote the taut foliation constructed in Construction 2. Let G denote the guts of Λ, i.e. the compact subspace of M \ Λ obtained by chopping off the ends of the ideal polygon bundles. Let I denote the interstitial region, i.e. the part of M \ Λ we just cut off. Topologically, I is a disjoint union of I-bundles over half-infinite cylinders. We have Mϕ(p; q) = Λ ∪ G ∪ I. Note that the distinct components of G have disjoint closures.

It will also be convenient to work with another decomposition Mϕ(p; q) = Λ0 ∪ G0. We obtain Λ0 by blowing down I and deﬁne G0 to be the closure 0 0 of Mϕ(p; q) \ Λ . Some of the leaves of Λ are branched surfaces instead of surfaces, so Λ0 is not a lamination, but a branched lamination. The advantage of Λ0 over Λ is that there is a Solv metric supported on Λ0.

Figure 3. Blowing down I. The shaded region on the left is G, and the shaded region on the right is G0.

The following lemma will be useful later:

◦ Lemma 1. Let Σ be a ﬁber surface in the ﬁbered manifold Mϕ(p; q) \G. ◦ Suppose γ is a path in Σ ∩ (Mϕ(p; q) \G) with endpoints on two walls of G. 0 Suppose that γ makes Ntan tangencies with F. Then there is an arc γ in ◦ 0 0 Σ ∩ (Mϕ(p; q) \G ) with endpoints on the corresponding two walls of G and in the same homotopy class rel. endpoints as γ such that γ0 makes at most 0 Ntan tangencies with Λ . (Here we are identifying relative homotopy classes ◦ ◦ ◦ 0 ◦ 0 of paths in (Σ ∩(Mϕ(p; q)\G), Σ ∩∂G) and (Σ ∩(Mϕ(p; q)\∂G ), Σ ∩∂G ) in the obvious way.) The converse also holds.

Proof. First, homotope γ so that all of its tangencies with F and all of its self-intersections occur on the interior of I. This may be done without introducing any new tangencies. Let Iε be the subset of I of thickness less 8 JONATHAN ZUNG than ε. We begin by blowing down Iε for some ε > 0. Let Fε and Λε be the images of F and Λ under this blowdown.

If we choose ε small enough, then γ intersects F ∩ Iε transversely. Moreover, if we choose ε small enough, then the blowdown changes the tangent plane ﬁeld of F by a uniformly small amount. Thus, we can arrange that that γ has Ntan tangencies with Fε.

Now, homotope γ so that all of its tangencies with Fε and its self intersections occur inside Λε. At this stage, γ \ Λε is a collection C of disjoint segments transverse to Fε. We now blow down the rest of I so that each of the segments in C blows down to a point. The curve γ then blows down to a curve γ0 with the desired properties. The converse is easier; Λ is obtained by splitting open the prongs in Λ0, and 0 this may be done without introducing new tangencies with γ .

3. Structure of branching in L

Let L be the leaf space of the lift Fe of F to M^ϕ(p; q). In this section, we roughly prove that “all branching in L happens in the saddle regions”. These results are not logically required for the proof of the main theorem (and indeed do not hold in full generality), but are interesting in their own right and provide motivation for subsequent constructions. In this section, we make the following assumption:

Assumption 1. All the monkey saddles used in the construction of F have at least four sides.

In the case where all the monkey saddles have two sides, the resulting manifolds carry Anosov ﬂows. The branching in their invariant foliations was analyzed in [Fen98]. Recall that for a taut foliation, the leaf space of the universal cover is a simply connected (possibly non-Hausdorﬀ) 1-manifold. Each leaf of the universal cover is homeomorphic to a plane, and its stabilizer under the action of π1 by deck transformations is the fundamental group of the projected leaf in Mϕ(p; q). A branch locus is maximal set of at least two points in L that are pairwise non-separable. [Cal07, Chapter 4]

Proposition 1. Assume Assumption 1. For each orbit Ki there are two + − branch loci called Bi and Bi (unique up to covering transformations) each of which is ﬁnite and has pi points corresponding with coherent lifts of the positively or negatively oriented walls of the ﬁlling saddle region. Moreover, these are all of the branch loci of Fe up to covering transformations. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 9

Proof. The core of a saddle region is a curve transverse to F and is therefore non-contractible. Therefore, the S1 worth of saddle-like leaves in a saddle region lifts to an R worth of saddle-like leaves (plus translates thereof) in Fe. This R worth of leaves limits to lifts of the positive (resp. negative) walls in the +∞ (resp −∞) direction, so the positively (resp. negatively) oriented walls of a saddle region form a branch locus. We choose one of the translates + − of this branch locus and call it Bi (resp. Bi ). To show that there is no branching elsewhere, it suffices to show that any curve can be “pulled tight” relative to F so that it is transverse to F except for controlled intervals in G. Roughly speaking, our strategy will be to pull γ tight relative to the natural Solv metric on Λ0. We say that an arc in G with endpoints on the interior of the walls in ∂G is inessential if it can be compressed into a wall of ∂G, and essential otherwise. An essential arc with endpoints in oppositely oriented walls can be homotoped rel. boundary to be transverse to F, while an essential arc with endpoints in similarly oriented walls can be homotoped rel. boundary to have a single tangency with F. An essential arc of the latter type lifts to a short curve ± in L that connects two points in Bi . It suffices to show that any curve γ with endpoints on leaves λ1, λ2 can be homotoped relative to its endpoints so that either (1) γ is transverse to F except for finitely many tangencies, each of which is contained in an essential subarc of γ in G or

(2) λ1 = λ2 and γ lies inside λ1. We call such a curve efficient. Moreover, we can without loss of generality assume that λ1 and λ2 are walls of G. For, suppose that λe1 and λe2 are adjacent, non-separated leaves of Fe in a branch locus not equal to one of ± the Bi . Then there is a family of leaves λet limiting to both λf1 and λf2 from, say, above in L. Any path γe ⊂ M^ϕ(p; q) from λe1 to λf2 descends to a path γ in Mϕ(p; q) that cannot be homotoped to become efficient. Since F is taut, we can augment the beginning of γ with a descending transversal from λ1 to a wall of G. Similarly, we can augment the end of γ with a descending transversal from λ2 to a wall of G. This resulting path is also not homotopic to an efficient one, but has endpoints on walls of G.

Given any arc γ with endpoints in leaves λ1, λ2 which are walls of G, we may homotope it so that it intersects ∂G transversely on the interior of the walls. Deﬁne

• Ntan as the number of tangencies between γ and F in Mϕ(p; q) \G.

• Ness as the number of components of γ ∩ G that are essential. 10 JONATHAN ZUNG

• Niness as the number of components of γ ∩ G that are inessential.

Now choose γ in its relative homotopy class to minimize Ness + 1.01Ntan + 1.02Niness. With this choice, we claim that Ntan = Niness = 0.

Suppose Niness > 0. Then we can compress an inessential arc in G. This decreases Niness by 1 at the cost of increasing Ntan by one, violating the minimality assumption.

Now suppose that Ntan > 0. Then there is a component γ0 of γ∩(Mϕ(p; q)\G) ◦ containing a tangency with F. Let Σ be the fiber of the fibering Mϕ(p; q) \ 1 ◦ G → S which contains γ0(0). We can arrange that γ0 lies inside of Σ by ◦ using a homotopy of γ which pushes γ0 down into Σ along the flowlines of the suspension of ϕ. This possibly slides γ0(1) along its wall of G, but does not change the objective function. Using Lemma 1, we may replace γ0 with 0 an arc γ0 having the properties that 0 0 (1) γ0 is in the support of Λ and 0 (2) γ0 is homotopic to γ0 rel. endpoints and

(3) the number of tangencies of γ0 with F is equal to the number of 0 0 tangencies of γ0 with Λ . There is a natural Euclidean metric on Λ0 ∩ Σ◦ inherited from the pseudo- 0 Anosov structure of ϕ. Pull γ0 tight relative to this metric. Since the leaves of Λ0 ∩ Σ◦ are geodesics with respect to this metric, the number of tangencies 0 0 between γ0 and Λ does not increase during tightening. We have a couple cases: 0 0 (1) γ0 tightens to a geodesic and is not contained in a single leaf of Λ . 0 0 Since the leaves of Λ ∩ Σ are geodesic, γ0 cannot have any tangencies 0 with leaves of Λ . Thus, using Lemma 1 we can also homotope γ0 to be transverse to F and decrease Ntan. 0 0 (2) γ0 tightens to an arc contained in a wall of ∂G . Note that distinct components of the guts have distinct boundary leaves, so this arc is really contained in a single wall of some component of G0. If γ0 = γ, then by the correspondence in Lemma 1 we have successfully compressed γ into a leaf and we’re done. Otherwise, using the correspondence in Lemma 1, we can compress γ0 into G which reduces Ntan by 1. If γ0 shares an endpoint with γ, then Ness +Niness remains constant. See Fig. 4b. Otherwise, Ness + Niness decreases by one (although each could individually increase). Therefore, the objective function decreases. See Fig. 4c for an example in which Ntan decreases from 1 to 0 and Ness + Niness decreases from 2 to 1. 0 0 0 (3) γ0 approaches G during tightening. Then there is a subarc of γ0 which wraps around ∂G0, making contact along a subarc of Euclidean angle > π. This arc must have a tangency to Λ0. Push π of the TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 11

corresponding subarc of γ0 into G so as to decrease Ntan by exactly one. See Fig. 4d. We claim that the new arc created is essential so that Ness increases by one. This fact crucially uses the assumption that the saddle region is an ideal n-gon bundle for some n ≥ 4. A pair of walls of such a saddle region which are separated by two prongs must have distinct lifts in the universal cover of the solid torus saddle region. Therefore, the endpoints of the newly created arc in G lie in diﬀerent walls in this Z-cover and so the new arc is essential. In total, the objective function has decreased.

(a) (b)

Figure 4. Various tightening moves

It follows that the minimizer γ has Ntan = Niness = 0, so every tangency between γ and F occurs inside an essential arc in G. Finally, we pull each component of γ ∩ G tight so as to contain either 0 or 1 tangencies with F. The arc γ is now efficient. Remark 1. Although Construction 2 works more generally for manifolds with a pseudo-Anosov flow having orientable invariant foliations, Proposition 1 does not hold. For example, there are many manifolds supporting Anosov flows whose invariant foliations have branching [Fen98]. The existence of a Solv metric on Mϕ \G and the presence of “negative curvature” in G from Assumption 1 are crucial.

+ − Proposition 2. Assume Assumption 1. The stabilizers of Bi and Bi in ± π1(Mϕ(p; q)) are both equal to the inﬁnite cyclic group Stab(Bi ) generated by 12 JONATHAN ZUNG

± a conjugate to the Dehn ﬁlling core Ki(p; q). Stab(Bi ) preserves a circular + order on the points of Bi .

Proof. Let R be the relevant saddle region. Observe that Stab(R) = + − Stab(Bi ) = Stab(Bi ). Every leaf in R is homeomorphic to a plane, and so has trivial stabilizer. Therefore, there is at most one element of Stab(R) mapping between lifts λ1, λ2 ∈ Fe of a leaf λ ∈ R. There is always a power of Ki(p; q) which accomplishes this transformation, so Stab(R) consists only of such elements. The circular order on walls is the order in which they appear as sides of a ﬁber of the ideal polygon bundle structure on R.

Let us now recall a standard fact about periodic orbits of ϕ.

Lemma 2. The non-planar leaves of Λ are all cylindrical. Moreover, they are in correspondence with primitive periodic orbits of ϕ. This correspondence is k-to-one for the leaves intersecting ∂G, where k is the number of walls of the incident component of G. It is one-to-one for all other leaves.

Proof. Since ϕ preserves no closed curve, the intersection of any leaf in F s with Σ is either a prong (homeomorphic to [0, ∞)) or a copy of R. Leaves of F s with nontrivial topology are suspensions of those leaves of F s ∩ Σ which are preserved by some power of ϕ, and hence are homeomorphic to cylinders. Since ϕ is pseudo-Anosov, its action contracts lengths in each leaf of F s ∩ Σ. A contraction on [0, ∞) or R has a unique fixed point, so each cylindrical leaf of F s contains exactly one primitive orbit of the suspension flow of ϕ. Conversely, the suspension of a primitive periodic orbit of ϕ is either contained in a unique leaf of F s or is a singular orbit contained in two or more prongs. We obtain Λ from F s by splitting open the leaves of F s which contain orbits in { Ki }. This has the effect of doubling such leaves and then gluing any leaves of the form [0, ∞) × S1 in pairs; the conclusion of the lemma follows for the leaves intersecting ∂G. All other leaves are preserved in the passage s from F to Λ.

± Proposition 3. Assume Assumption 1. No element of Stab(Bi ) stabilizes + − any point in the Hausdorﬃﬁcation of L aside from Bi and Bi .

± Proof. Stab(Bi ) is generated by a loop freely homotopic to the Dehn surgery core Ki(p; q). The stabilizer of a cylindrical leaf in Λ is represented by a loop freely homotopic to the suspension of a periodic point of ϕ. We must show that there are no nontrivial free homotopies among loops Ki(p; q) (or powers thereof) or suspensions periodic orbits of ϕ (or powers thereof). Call the set of these curves O. For simplicity, we replace each Ki(p; q) with a power thereof that is homotopic to one living inside a wall of ∂G. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 13

First, we rule out a nontrivial homotopy between two curves in O that stays inside Mϕ(p; q) \G. Now Mϕ(p; q) \G embeds in the ﬁbered manifold Mϕ so that the elements of O are suspensions of periodic orbits of ϕ. Since ϕ is pseudo-Anosov, these suspensions are never homotopic to one another.

Now we must rule out a nontrivial homotopy between curves in O that might pass through the guts G. The idea is that the saddle regions contains lots of negative curvature, but an annulus giving rise to the purported homotopy has Euler characteristic zero and therefore can’t cut across any saddle region. To be more precise, let γ and η be two elements of O. Suppose γ is homotopic to η in Mϕ(p; q). Then γ and η cobound an annulus A. Replacing γ and η by γk and ηk if necessary, we can assume that γ and η both lie in leaves of F. Homotope this annulus to become immersed. By Roussarie-Thurston, we can homotope the annulus so that the induced foliation has no critical points [Rou74, Thu86]. We can also make A transverse to ∂G and the corners of ∂G. Let C = A ∩ ∂G. Choose A minimizing the number of components of C.

Case 1 C contains an innermost loop α that is inessential in A. Let D be the disk in A bounded by α. The boundary of D is a 2n-gon with sides alternating between arcs transverse to the foliation and arcs tangent to the foliation. Here, 2n is the number of corners of ∂G that ∂D intersects. Moreover, the angles between adjacent arcs are all convex or all concave, depending on whether int D lies inside or outside G. See Fig. 5a for a picture of the convex case with n = 4. The foliation on D has no critical points, so actually we must have 2n = 4χ(D) = 4. Since every saddle region is an ideal polygon bundle with ﬁber having at least four sides, a meridian in ∂G crosses at least eight corners. Therefore, α is not a meridian or a multiple of a meridian. Furthermore, α cannot be homotopic to a multiple of a core of G since α is zero in π1(Mϕ(p; q)). Therefore, α must be inessential in ∂G. We can then compress D into ∂G (while maintaining Roussarie-Thurston general position) and eliminate the curve α from C, contradicting our minimality assumption.

Case 2 C contains only loops that are essential in A. Let α be any element of C. As in the ﬁrst case, α ∩ A alternates between arcs between transverse and tangent to F, and the angles between adjacent arcs all either convex or concave depending on which side of α is on the interior of G. Let n be the number of alternations between transverse and tangent arcs in α. See Fig. 5b for the case n = 4. Let R be the annulus cobounded by γ and α. Since F ∩ R has no singularities, we have n = 2χ(R) = 0. Thus, α actually intersects no corners of ∂G and is either entirely transverse to F (i.e. is contained in a suture of ∂G) or contained in a wall of G. Moreover, α is not contractible in ∂G since α is homotopic in Mϕ(p; q) to γ which is even non-contractible 14 JONATHAN ZUNG

(a) (b)

Figure 5. The annulus A is shown along with the induced foliation in a neighbourhood of α in two disallowed conﬁgu- rations. If α crosses the sutures in ∂G as shown, there is no way to extend the foliation to the rest of A without critical points, contradicting Roussarie-Thurston.

in Mϕ(p; q). Therefore, α is a closed curve parallel to the sutures in some component of ∂G. The arguments above hold for any α ∈ C. The curves in C cut A into a collection of annuli which we label A1 ...AM . Each Ak is either an annulus living in Mϕ(p; q) \G or an annulus in G. In the ﬁrst alternative, Ak represents a free homotopy in Mϕ(p; q) \G; as noted at the beginning of the proof, the two boundary components of such an Ak must represent the same element of O. In the latter alternative, we can trivially make the same conclusion. Therefore, γ and η represent the same element of O as desired.

4. Gluing branches In this section, we give some motivation for the constructions in Section 5. We ∼ wish to define on L an equivalence relation, denoted ∼, such that (L/ ∼) = R. We first describe this gluing process by an analogy. Consider an infinite rooted binary tree T , each of whose vertices has a distinguished left and right child. We think of T not combinatorially, but as a topological space. Let v0 be the root vertex of T . Given a vertex v ∈ T , let vL represent its left child and vR represent its right child. We use fractional powers of R to 1 denote points that are on edges of the tree; for example, vR 2 is the midpoint k α of the path [v, vR]. Then any point on the tree is of the form v0{L, R} L k α or v0{L, R} R for some k ≥ 0, 0 ≤ α < 1. At each vertex v, glue the infinite paths [v, vLLL . . . ) to [v, vRRR . . . ) together by the obvious map [0, ∞) → [0, ∞) which preserves depths of vertices. See Fig. 6. This has the effect of collapsing the entire tree down to a copy of [0, ∞). TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 15

Figure 6. The red paths are of the form [v, vLLL . . . ) and the blue paths are of the form [v, vRRR . . . ). Points which are equivalent with respect to the gluing are marked with the same number.

To illustrate what could go wrong, let us choose a slightly different gluing for which the quotient is not [0, ∞). Consider what happens if instead at each vertex v ∈ T we glue the infinite descending paths [v, vLRRR . . . ) and [v, vRLRRR . . . ) by a homeomorphism which sends [v, vL] to [v, vRL] by a dilation by a factor of two, and sends [vL, vLRRR . . . ) to [vRL, vRLRRR . . . ) n by an isometry. See Fig. 7. The points v0R L are identified for all n ≥ 0. Therefore, [v0, v0RRR... ) is not properly embedded in the quotient space. It turns out that the quotient space is not [0, ∞) but a tree with infinite valence at each vertex. For example, the points v0LL and v0RLL are not comparable in the partial order induced on the quotient. For points a, b ∈ T , let [a] and [b] denote their images in the quotient. Write a ≥ b if a is an ancestor of b in T , and [a] ≥ [b] if [a] is an ancestor of [b] in the quotient. The interested reader is invited to show the following: (1) Each equivalence class under the gluing has a canonical representative k s s s of the form v0{L, R} LLR , v0LR , or v0R for some s ≥ 0. (2) Suppose a and b are canonical representatives of their equivalence s classes. Write a = rW R for some word W in L and R, and s ∈ R maximal. Then [a] ≥ [b] if and only if one of the following holds: (a) a ≥ b n (b) W is the empty word and v0WR L ≥ b for some integer n ≤ s 0 0 k (c) W = W L and v0W R LL ≥ b for some integer k ≥ 0 We wish to perform a similar gluing on the leaf space L. The leaf space should be thought of as a kind of tree, but possibly with a dense set of branching points. There are two new features in this case. First, there may be branching in both the upward and downward directions. Second, we must do this gluing in a π1-equivariant way so that the action of π1 will descend to the quotient L/ ∼. The gluings we performed in the case of the binary tree are equivariant with respect to the semigroup of isometric self-maps of the binary tree which preserve the left and right children at each node. 16 JONATHAN ZUNG

Figure 7. Two paths that are to be glued together, [v0, v0LRRR . . . ) and [v0, v0RLRRR . . . ), are shown in red and blue respectively. Dotted lines are drawn through the equivalence classes 2, 20, and 200. These classes map to non- separable points in the quotient. Their canonical representatives are v0LL, v0RLL, and v0RRLL. For any s ≥ 0, the s 1 point v0LR (marked with the label 2 − 2s ) is a common ancestor of all three of these points in the quotient.

More care will be required in L since the stabilizer of a branch point may be nontrivial. ± Let bi be a generator for Stab(Bi ). Let ai be the smallest positive integer ai such that bi stabilizes Bi pointwise. In other words, ai is the integer such ai that bi is freely homotopic to the essential loop in a wall of G. We ﬁx the TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 17

ai orientation of bi by asking that the pushoﬀ of bi to the outside of G has positive intersection number with Σ. With this choice, the holonomy of F on ai the outside of G along a curve freely homotopic to bi is always repelling. + For each x ∈ Bi (i.e. a lift of a wall of a saddle region), let (x, ∞) ⊂ L be the lift to L of the transverse loop obtained by pushing the essential + loop in x slightly outside the saddle region corresponding to Bi . Another way to think of (x, ∞) is as the path in L correponding with the set of leaves of Fe intersecting a prong of the unstable invariant foliation F u. Let + [x, ∞) = { x } ∪ (x, ∞). The paths [x, ∞) | x ∈ Bi will play the role of the paths [v, vRRR . . . ) or [v, vLLL . . . ) in the case of the binary tree.

ai Lemma 3. [x, ∞) is a bi invariant path from x to ∞ in L. Furthermore, bai acts on [x, ∞) by an expanding dilation ﬁxing x.

Proof. Since (x, ∞) is a lift of a loop homotopic to ai times the core of the ai corresponding saddle region, [x, ∞) is bi invariant. If [x, ∞) had a greatest ai lower bound, then the greatest lower bound would be ﬁxed by bi . By Proposition 3, this never happens. Therefore, [x, ∞) is properly embedded in L. Since the holonomy around the essential loop in x is repelling on the ai outside of G and bi has no ﬁxed points in [x, ∞) besides x, it must be that ai bi acts by a homeomorphism conjugate to a dilation with stretch factor > 1.

Call [x, ∞) the invariant path at x. We are now ready to give a ﬁrst attempt at deﬁning the equivalence relation ∼ on L.

+ − Construction 3. For each point x in the branch locus Bi (resp. Bi ), construct the upward (resp. downward) oriented path [x, ∞) ⊂ L from x ai to +∞ (resp. −∞) which is invariant under b as in Lemma 3. Now bi S i acts on + [x, ∞). We shall now glue together the various paths in the x∈Bi + set [x, ∞) | x ∈ Bi . Up to reparameterizing [x, ∞), we can assume that ai ai + bi acts on each [x, ∞) by dilation by a factor 2 . For each x ∈ Bi , each 1 m y ∈ [x, ∞) and each m ∈ Z, declare y ∼ 2m bi y. This has the eﬀect of gluing + + together the paths { [x, ∞) | x ∈ O } for O ⊂ Bi an orbit of the Stab(Bi ) + action on Bi . See Fig. 8. The action of bi descends to the quotient L/ ∼. In the quotient, bi acts on [x, ∞)/ ∼ as a dilation by a factor of 2.

Having glued together [xi, ∞) and [xj∞) for xi, xj in the same orbit of the + Stab(Bi ) action, we now wish to glue those paths when xi and xj are in diﬀerent orbits in any bi equivariant way. This is possible since bi acts in the same way on [xi, ∞)/ ∼ and [xj, ∞)/ ∼. In total, we have glued together + + ± |Bi | semi-inﬁnite paths from Bi to ∞ together in a Stab(Bi ) equivariant − ± way. Perform this gluing procedure for Bi as well as all translates of Bi in a π1-equivariant way. 18 JONATHAN ZUNG

S Figure 8. On the left, the action of bi on x∈O[x, ∞) is − shown for the case ai = 3. Here, O = { x1, x2, x3 } ⊂ Bi . The action of bi descends to the quotient by ∼. This action on the quotient is shown on the right.

Note that if x ∼ y, then gx ∼ gy for any g ∈ π1(Mϕ(p; q)). Therefore, Construction 3 gives a π1-equivariant equivalence relation ∼ on leaves. Un- fortunately, L/ ∼ is not homeomorphic to R due to a phenomenon like that of Fig. 7. In Section 5, we present a coarsening of this equivalence relation which ﬁnishes the job.

5. The flat connection Jb ◦ We prove the main theorems in this section. Recall that Mϕ = Mϕ \{ Ki }. Our first goal is to construct a trivial S1-bundle (with structure group + 1 ◦ Homeo (S )) on Mϕ with a flat connection J whose monodromy around the filling curves on the boundary components is trivial. This S1 bundle will lift to an R-bundle with a flat connection Jb having the same property. 5.1. Preliminaries on connections and partial connections. Given a bundle F → E −→π B, a connection H is a choice of a homeomorphism −1 −1 Hγ : π (γ(0)) → π (γ(1)) for each piecewise smooth path γ : [0, 1] → B. These homeomorphisms are required to be independent of the parameterization of γ and to satisfy functoriality conditions with respect concatenation of paths. We refer to Hγ as parallel transport along γ with respect to H. The functoriality conditions are:

(1) Hλ∗γ = Hλ ◦ Hγ where λ and γ are concatenable paths in B and λ ∗ γ is their concatenation. −1 (2) Hγ−1∗γ = id, where γ is γ traversed backwards. (3) parallel transport along a trivial path is the identity map. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 19

Said in a different way, let C be the category whose objects are topological spaces homeomorphic to F and whose morphisms are homeomorphisms. An F -bundle over B assigns to each point x ∈ B the fiber over x which is an object in C. A connection is an extension of this assignment to a functor from the groupoid of paths in B to C. This perspective will be useful because we will sometimes specify a connection by defining it on a set of generators for the groupoid of paths in B. We might also replace the groupoid of paths in B with a slightly larger but equivalent groupoid.

Remark 2. For the purposes of this paper, we will not need to ask that Hγ is close to the identity when γ is a short path. Thus, our deﬁnition of a connection makes sense even in the absence of a local product structure on the bundle. This is useful because we will actually use a connection to deﬁne the local product structure.

A connection is called flat if Hγ = id whenever γ is contractible. A flat connection gives rise to a homomorphism of π1(B) into Homeo(F ) defined by [γ] 7→ Hγ for [γ] ∈ π1(B).

A section of π is called flat with respect to H or H-flat if Hγ(s(γ(0))) = s(γ(1)) for all paths γ : [0, 1] → B. When H is understood, we will suppress mentioning it and simply say that a section is flat.

We may also refer to sections of π over a curve γ : [0, 1] → B; this simply means a section of the pullback bundle over [0, 1]. We will usually express such a section as a map s: [0, 1] → E. A ﬂat section over a path is also called a parallel section. For any path γ : [0, 1] → B and x ∈ π−1(γ(0)), there is a unique parallel section sx over γ satisfying sx(0) = x. It is traced out by parallel transport of x along γ. In equations, this means s (t) = H (x) x γ|[0,t] where γ |[0,t] is the restriction of γ to [0, t]. A partial connection is similar to a connection except that the homeomor- −1 phisms Hγ may not be deﬁned on all of π (γ(0)). A partial connection H is a choice of a homeomorphism Hγ for each path γ : [0, 1] → B from some (topological) subspace of π−1(γ(0)) to some subspace of π−1(γ(1)). The subspaces may depend on the path γ. The homeomorphisms are again required to be independent of the parameterization of γ and functorial with respect to composition of paths:

−1 (1) Hλ∗γ(x) = Hλ(Hγ(x)) for all x ∈ π (γ(0)) at which the right side is deﬁned.

−1 (2) Hγ−1∗γ(x) = x for all x ∈ π (γ(0)) at which Hγ is defined. (3) Parallel transport along a trivial path is defined on the entire fiber π−1(γ(0)) and is equal to the identity map. 20 JONATHAN ZUNG

A partial connection is called ﬂat if for each point x ∈ B and each compact set W ⊂ π−1(x), there exists a neighbourhood U of x such that for every contractible path γ : [0, 1] → U, Hγ is deﬁned on all of W and agrees with the identity map.

Remark 3. In contrast with the case of flat connections, the monodromy of a flat partial connection around a long, contractible loop need not agree with the identity map on its domain of definition.

A section is called flat with respect to a partial connection H if Hγ(s(γ(0))) = s(γ(1)) for all paths γ : [0, 1] → B along which Hγ is defined on s(γ(0)). Define flat or parallel sections over paths analogously to the case of connections. Given a path γ : [0, 1] → B and a point x ∈ π−1(γ(0)), one may attempt to define a parallel section s over γ by s (t) = H (x). How- x x γ|[0,t] ever, the right side may fail to be defined for large t. In this case, let n o t (x) = sup t | H (x) is defined . Then we say that the parallel max γ|[0,t] section sx blows up at time tmax (x). A flat partial connection is roughly the same thing as a foliated bundle, though we will not use that language since the total space of our bundle may not apriori have a topology making it a manifold.

Example 5. A partial connection H on the bundle (0, 1) → (0, 1) × R → R may be defined as follows. For any γ parameterizing a curve in R that is monotonically increasing or decreasing, Hγ is a homeomorphism between an open subinterval of the fiber over γ(0) and an open subinterval of the fiber over γ(1). We define Hγ(t) = t + γ(0) − γ(1) for any t satisfying t ∈ (0, 1) and t + γ(0) − γ(1) ∈ (0, 1). For γ which is not monotonically increasing or decreasing, Hγ is defined by composition of monotonic paths. In this case, the range and domain of Hγ will be smaller than that specified in the formula. See Fig. 9.

Figure 9. The total space for the bundle is shown with the ﬁber direction vertical. Parallel sections for the partial connection H are shown in red. Each parallel section is deﬁned only for a limited time. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 21

partial ◦ ◦ 5.2. The partial connection J . Let Σ be a typical fiber of Mϕ, ◦ i.e. Σ punctured at its intersections with the Ki’s. Mϕ comes equipped with transverse stable and unstable codimension 1 foliations F s and F u. It also has a natural incomplete Solv metric g locally expressible as dt2 +λtdx2 +λ−tdy2, where x and y are local coordinates on Σ◦, λ > 1 is the stretch factor of ϕ, and t is the coordinate transverse to Σ◦. By the orientability constraints, we may consistently establish cardinal directions north, south, east, and west. We take F s ∩ Σ◦ to be east-west and F u ∩ Σ◦ to be north-south. For us, a rectangle will always mean a rectangle in a fiber surface free of singularities on its interior with top and bottom parallel to the east-west direction and left and right sides parallel to the north-south direction. Recall that we s ◦ constructed F by splitting open F . Let Σp be the fiber surface containing a point p. A generalized leaf of F u ∩ Σ◦ is one of: (1) a leaf of F u ∩ Σ◦ which is not a prong, or (2) the concatenation of two prongs of F u ∩ Σ◦ which meet at the same singularity and make an angle of π (i.e. a limit of leaves of F u ∩ Σ◦ approaching the singularity). We may similarly define a generalized leaf of F u. The concatenation of two s ◦ prongs P1 and P2 of F ∩ Σ incident with the same singularity q formally ◦ requires an extra dummy point since q itself is not a point in Mϕ. We usually ∗ ∗ call the dummy point q so that the generalized leaf is P1 ∪ { q } ∪ P2. If there is more than one generalized leaf in play, we will use q∗∗, q∗∗∗, etc. to denote their dummy points. A point on a prong of F u ∩ Σ◦ is contained in exactly two generalized leaves, but it would be better if each point were contained in exactly one generalized leaf. Thus, we will formally double each point p on a prong P of F u ∩ Σ◦ into two points pwest and peast. The points pwest and peast should be regarded as ◦ points in Mϕ infinitesimally perturbed to the west and east of P respectively. Let Z be the resulting (non-Hausdorff, in fact non-T0) space containing a pair of topologically indistinguishable points for each point on a prong of F u. In Z, each point is contained in exactly one generalized leaf. One can construct a path in Z from pwest to peast whose image is the two point set ◦ { pwest, peast }; such a path should be regarded as an infinitesimal path in Mϕ crossing P from pwest to peast. Let Eπ = {(x, y) |x, y ∈ Z and u y is on the generalized leaf of F ∩ Σx which contains x}.

Let π : Eπ → Z be the projection map π : (x, y) 7→ x. Then π deﬁnes an R ◦ bundle over Z. Intuitively, one should think of π as the R-bundle over Mϕ ◦ u ◦ whose ﬁber over a point p ∈ Mϕ is the leaf of F ∩ Σp containing p. 22 JONATHAN ZUNG

Figure 10. The unstable foliation near a singularity with ki = 4. The ﬁbers over pwest and peast are each unions of two prongs.

Deﬁne the auxiliary map θ : Eπ → Z by (x, y) 7→ y. In practice, we visualize points in E via their images under θ.

Note that there is no obvious local product topology on Eπ since the leaves of F u diverge around every singular point. We will eventually define a topology by using a flat connection in Section 5.4. In the rest of this section, when there is no chance of confusion, we will ◦ conflate Z and Mϕ. This means that we will think of pwest and peast as ◦ ◦ points in Mϕ, think of π as an R-bundle over Mϕ, and think of θ as a map ◦ from Eπ to Mϕ. We will also think of the path from pwest to peast in Z as ◦ ◦ an infinitesimal path in Mϕ with distinct endpoints. Since Mϕ embeds in ◦ Z, a connection on Z restricts to a connection on Mϕ. Thus, we only gain generality by working in Z. Now we shall define a partial connection J partial on the bundle π. We initially define J partial on three types of short paths which generate a dense set of ◦ ◦ −1 −1 paths in Mϕ. For a curve γ : [0, 1] → Mϕ, let hγ : θ◦π (γ(0)) → θ◦π (γ(1)) s partial be the holonomy of F along γ. Whenever hγ exists, we define Jγ so that partial θ ◦ Jγ = hγ ◦ θ. (1) However, J partial will also be defined on some curves γ along which holonomy does not exist. Type 1 Suppose γ is a path in a generalized leaf µ of F u. Then we define partial s ∼ 2 s J using the holonomy of F . In µe = R , the leaves of Ff ∩ µe and the lifts to µe of the θ-images of fibers of π form two transverse TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 23

codimension-1 foliations. This pair of foliations is topologically 2 conjugate to the horizontal and vertical foliations of R , so hγ is fully −1 partial defined on θ ◦ π (γ(0)). We may then define Jγ as in Eq. (1). ◦ Type 2 Suppose R is a rectangle in Σp for some p. Let γ be a horizontal path from the east to the west side of R. In this case, hγ identifies the east and west sides of the rectangle by the obvious isometry, so we define J partial as in Eq. (1). (If an endpoint of γ lies on a prong, partial then we make the same definition of Jγ regardless of whether the endpoint is infinitesimally perturbed to the east or the west. We also allow the north or south side of the rectangle to contain a singularity.) u Type 3 Let p be a point on a prong P of F ∩Σp and let γ be the infinitesimal path from pwest to peast. Let q be the singularity terminating P . Let Peast and Pwest be the prongs at q adjacent to P so that −1 ∗ θ ◦ π (pwest) = P ∪ { q } ∪ Pwest and −1 ∗∗ θ ◦ π (peast) = P ∪ { q } ∪ Peast. −1 −1 Briefly conflating θ ◦ π (pwest) with π (pwest), let us define partial ∗ ∗∗ Jγ : P ∪ { q } ∪ Pwest → P ∪ { q } ∪ Peast separately in two subcases: partial ∗ Type 3a If p lies to the south of q, then declare that Jγ sends q to ∗∗ q , acts as the identity on P , and stretches Pwest by a factor m q /p of λ i i i , where (pi; qi) is the surgery slope at the relevant singularity and mi is the ϕ-period of the singularity. Here, λ is the stretch factor of ϕ and the relevant metric is the singular ◦ Euclidean metric induced on Σfp by g. Type 3b If p lies to the north of q, then make the same definition except that we use the inverse stretch factor λ−miqi/pi instead.

Note that hγ is defined on P , but not on Pwest. Thus, we had some freedom to choose the dilation factors in type 3a and 3b. Our choice is designed to make Proposition 5 work. ◦ 0 Any path in Mϕ may be C approximated by a concatenation of paths included in the definition above, so we can define J partial for any path γ in ◦ Mϕ by composition of parallel transport maps. This definition is independent of the approximation used by Proposition 4.

Remark 4. The deﬁnition of J partial along Type 3 curves is motivated by Construction 3. The images in L of Pwest and Peast are paths the form [x, ∞) constructed in Section 4. The failure of J partial to be a connection is related to the failure of L/ ∼ to be a line. 24 JONATHAN ZUNG

◦ −1 Proposition 4. Given any point p ∈ Mϕ and a compact subset W of π (p), the monodromy of J partial around sufficiently small contractible loops based at p formed by concatenating type 1, 2, and 3 curves is defined on W and equal to the identity. It follows that J partial is flat.

Proof. Without loss of generality, assume p ∈ int θ(W ). If p does not lie on a prong, then one can find a tall, skinny rectangle R containing a neighbourhood ◦ of θ(W ). Thicken this rectangle by ε in the direction transverse to Σp. Now holonomy of transversals to F s in R × (−ε, ε) exists along all curves in a neighbourhood p, so the required monodromies are trivial in this case. ◦ Suppose instead that p = pwest lies on a prong P ⊂ Σp. This time, choose a tall, skinny rectangle Rwest such that its east side contains θ(W ). Choose an- partial other tall, skinny rectangle Reast such that its west side contains θ(Jγ W ), where γ is the type 3 curve from pwest to peast. Let U be an ε-thickening of ◦ Rwest ∪ Reast in the direction transverse to Σp. By construction, p lies on the interior of U. We need to show that monodromy around a contractible loop in U based at p is trivial on W . The null-homotopy of such a loop may be decomposed into disks whose boundaries are “commutators” of type i and type j arcs, i.e. quadrilaterals whose sides alternate between type i and type j curves. Moreover, we can arrange that the only commutators crossing P × (−ε, ε) are commutators of type 3 and type 1 arcs. Any commutator contained in Rwest × (−ε, ε) or Reast × (−ε, ε) has trivial monodromy, so it only remains to check that the holonomy around a commutator of a type 1 arc transverse to Σ◦ and a type 3 arc vanishes. Parallel transport along a type 1 arc which ◦ moves a distance α in the direction transverse to Σp acts as a dilation by factor λα (relative to the induced Euclidean metric on fibers). On the other hand, parallel transport along a type 3 curve acts as a piecewise dilation. These two maps commute, as desired. Finally, it’s easy to check that parallel transport along any path γ ⊂ U starting at p is defined on W .

Proposition 5. The monodromy of J partial around a filling curve on the ◦ boundary of Mϕ is trivial. To be precise, let p be a point on an unstable ◦ prong incident to a boundary component of Mϕ. For any compact subspace W ⊂ π−1(p), the monodromy of J partial around a small enough loop γ based at p and homotopic to the Dehn filling meridian is defined on W and equal to the identity.

Proof. The boundary curve in N(Ki) can be written as the suspension of m q a small perturbation of a point in Σ ∩ Ki under the map ϕ i i and an arc ◦ in Σ travelling a clockwise angle of 2πpi around the singularity. Parallel transport along the ﬁrst arc stretches distances by a factor of λmiqi . Roughly speaking, the second arc contains pi subarcs of type 3b and pi subarcs which TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 25 are (the inverse of) type 3a. Together, J partial-parallel transport along these arcs produces a dilation of a factor of (λ−miqi/pi )pi . Thus, the composite monodromy is dilation by a factor λmiqi (λ−miqi/pi )pi = 1.

Figure 11

In more detail, let us analyze parallel transport around a short arc travelling an angle of 2π around the singularity. Fig. 11 shows the foliation F u ∩ Σ◦ near a singularity along with an arc γ travelling an angle of 2π around the singularity. The arc γ is chosen so that γ(0), γ(1/2) and γ(1) lie on prongs of the stable foliation F s ∩ Σ◦ (not shown). The θ-images of half ﬁbers of −1 π are labelled P1 ...P6, so that, for example, θ ◦ π (γ(0)) = P1 ∪ P2 and γ(0) = P1 ∩ P2. We will abuse notation and use P1 to refer to both a half u ◦ −1 leaf F ∩ Σ and its θ-preimage in π (γ(0)), and similarly for P2 ...P6.

Let R1, R2, and R3 be rectangles bounded on the east and west sides by the u ◦ Pi’s and the prongs of F ∩ Σ as shown shaded in Fig. 11. We might not be able to make the rectangles arbitrarily long in the north south direction due to hitting other singularities. However, if γ(0), γ(1/2), and γ(1) hug very close to the singularity, these rectangles can be chosen to be as long in the north-south direction as desired.

Restrict attention to points in the Pi’s such that the θ-images of their parallel transports along γ stay inside R1 ∪ R2 ∪ R3. Call the set of such points N. For example, N ∩ (P1 ∪ P2) is an interval containing γ(0).

Now γ|[0,1/2] decomposes as a type 2 curve in R1, the inverse of a type 3a partial curve crossing the prong, and a type 2 curve in R2. Therefore, J maps γ|[0,1/2] P2 ∩ N to P3 ∩ N by an isometry and maps P1 ∩ N to P4 ∩ N by a stretch −miqi/pi by a factor λ . Similarly, γ|[1/2,1] decomposes as a type 2 curve in 26 JONATHAN ZUNG

R2, a type 3b curve, and a type 2 curve in R3. Thus, γ |[1/2,1] maps P4 ∩ N to P5 ∩ N by an isometry and P3 ∩ N to P6 ∩ N by a stretch by a factor m q /p partial λ i i i . The composite Jγ maps (P1 ∪ P2) ∩ N to (P5 ∪ P6) ∩ N by a dilation of λ−qimi/pi . This justifies and makes precise the claim made in the first paragraph of the proof. Finally, if we take γ(0), γ(1), and γ(1/2) to hug close to the singularity, then R1,R2 and R3 may be made as long as needed, and in turn N may be made as large as desired. By the accounting of dilation factors from the first paragraph, the monodromy of γ is the identity on as large a subspace of π−1(p) as desired.

5.3. Blowup time for parallel sections. Let γ : [0, ∞) → Σ◦ parameter- ize an eastward ray in a leaf of Σ◦ ∩ F s. Given a point x ∈ π−1(γ(0)), recall from Section 5.1 that one may attempt to contruct a parallel section sx : [0, ∞) → Eπ with sx(0) = x. However, sx typically blows up in ﬁnite time and can only be deﬁned over [0, tmax (x)) for some tmax (x) > 0. We will abuse notation and sometimes think of sx as a function of t, whose value ◦ is the signed north-south distance in Σf from γ(t) to θ ◦ sx(t). Similarly, we will think of x as a real number. In this section, we show that tmax is locally −1 a homeomorphism from π (γ(0)) to R.

Recall that we defined λ so that λ > 1. We also assumed that pi all have the m q /p same sign. Without loss of generality, we assume pi ≥ 0. Let αi = λ i i i . m q /p Note that αi ≥ 1. Let αmax = maxi λ i i i . If αmax = 1, then Mϕ(p; q) is fibered and there is no blowup of sections. So we assume that αmax > 1. m q /p Now we can define αmin = min λ i i i | qi > 0 . We call a singularity magnifying if the associated coefficient satisfies αi > 1. A ragged rectangle is the region in Σf◦ swept out by a interval in F s ∩ Σ◦ under the time t northward flow along F u ∩ Σ◦ (i.e. the map that sends each point t units due north). It looks like a rectangle with some vertical slits cut into the northern edge.

It is best to visualize all our constructions via the developing map D : Σf◦ → 2 R which is a local isometry away from singularities and a branched cover at singularities. We can arrange that D(F^s ∩ Σ◦) and D(F^u ∩ Σ◦) are the 2 horizontal, resp. vertical foliations of R . Thus, the θ-images of ﬁbers of π also correspond to vertical lines in this picture. Ragged rectangles project via D to honest rectangles. We can also arrange that the image of γ is the positive x axis. −1 Given x ∈ π (γ(0)), the section sx(t) may be constructed geometrically. If x > 0, push the vertical segment [γ(0), θ(x)] eastward along γ. The endpoint sweeps out the θ image of the section sx(t). See Fig. 12. Whenever the segment hits a singularity in orbit i at distance q from γ(t) with 0 < q < sx(t), TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 27 the segment continues on the other side with length q + αi(sx(t) − q). If x < 0, then the same procedure works except that the new length of the segment is q + (1/αi)(sx(t) − q).

Figure 12. Constructing sx(t) by pushing a ﬁber of π past a singularity, as seen in the image of the developing map.

When x < 0, the section sx(t) stays bounded for all time because γ can be expressed as a composite of type 2 and type 3b curves, neither of which increases |sx(t)|. On the other hand, when x > 0, sx(t) grows with t and possibly blows up. Lemma 4 below guarantees that we can expect many singularities to be encountered during this procedure.

Lemma 4. There exists a positive constant κ such that for every ε > 0, there exists a large enough Aε so that every ragged rectangle of area A > Aε has number of magnifying singularities in the range [(κ − ε)A, (κ + ε)A].

Proof. First, we may use the action of ϕ to turn any ragged rectangle of area A into a new ragged rectangle of width O(1) and height O(A) having the same number of magnifying singularities. Now the lemma follows the ergodicity of the translation flow on flat surfaces. Masur’s criterion states that the translation flow on a flat surface is ergodic whenever the corresponding flow et 0 on Teichmüllerspace given by multiplying the metric by stays in 0 e−t some compact set [Zor06, Sec. 3.7]. A pseudo-Anosov map corresponds with a closed orbit under this flow. Therefore, Masur’s criterion is fulfilled.

We now deﬁne A∗ to be a constant large enough that every ragged rectangle of area A∗ contains a magnifying singularity.

Lemma 5. There exists a constant C independent of γ such that whenever x > S > 0, sx(t) blows up in time tmax (x) < C/S. 28 JONATHAN ZUNG

Figure 13. In this example, Σ◦ = T 2, ϕ is the monodromy of the figure eight knot, and the surgery coefficient is ◦ (p1; q1) = (5, 1). This picture shows D(Σf) with the 2-pronged singularities of ϕ drawn as dots. Sections sx(t) are shown in colours for various choices of x. These sections have been projected to Σ◦ by θ. Whenever x ∈ π−1(γ(0)) satisfies x < 0, the section sx(t) exists for all time. When x > 0, sx(t) blows up in finite time.

Proof. By Lemma 4, a prong from a magnifying singularity intersects the ◦ S 2 image of γ in Σ at north-south distance < 2 in time t < A∗ · S . Therefore,

2A S S s ∗ > + α (2) S 2 min 2 1 + α = min S (3) 2 Repeating the argument to find more nearby singularities, we find that n −i! i 2A∗ X 1 + αmin 1 + αmin s > S (4) S 2 2 i=1 as long as the left side is defined. Taking n → ∞, we find that

−1 2A∗ 2 tmax (x) < 1 − . (5) S 1 + αmin The right side is of the form C/S as desired. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 29

Lemma 6. There exists a constant c independent of γ such that whenever 0 ≤ x < S, sx(t) exists for time c/S.

Proof. By Lemma 4, we can choose a constant Aκ so that every ragged rectangle of area A > Aκ has fewer than 2κA singularities.

2κAκ Consider a ragged rectangle R of area Aκ, with height (αmax ) S and 2κAκ width c/S := Aκ/ (αmax ) S . This ragged rectangle has fewer than 2κAκ singularities. Since |sx(t)| grows by at most αmax each time it encounters a singularity, θ ◦ sx(t) cannot exit through the top of R. Therefore, sx(t) exists for time c/S.

−1 Proposition 6. tmax : π (γ(0)) → (0, ∞) is surjective.

Proof. We ﬁrst show that the image of tmax is dense. For any t0 and any −1 ε > 0, Lemma 5 guarantees that if x ∈ π (γ(t0)) is large enough, the parallel section over γ passing through x will blow up before t = t0 + ε. This section exists for all t < t0, so tmax (s(0)) ∈ (t0, t0 + ε).

Since tmax is non-decreasing and has dense image, it is surjective.

−1 Proposition 7. tmax : π (γ(0)) → (0, ∞) is injective.

Proof. Choose x, y ∈ π−1(γ(0)). Assume 0 < x < y. We want to show that sy explodes strictly before sx.

Case 1 sy(t)/sx(t) is unbounded. Choose t so that sy(t)/sx(t) > C/c, where C and c are the constants from Lemma 5 and Lemma 6. Then by those two lemmas, sy explodes strictly before sx.

Case 2 sy(t)/sx(t) stays bounded. Let { tj } be the sequence of times at which sx grows, and let bj be the height of the singularity encountered at time tj. At time tj, we have

s (t + ε) − s (t + ε) αt · (sy(tj − ε) − sx(tj − ε)) y j x j = j (6) sx(tj + ε) bj + αtj · (sx(tj − ε) − bj)

where αtj is the magniﬁcation factor at the singularity encountered at time tj and ε is small enough. If bj/sx(tj) is constant size, then (sy(t) − sx(t))/sx(t) grows by a constant factor. Therefore, we must have limj→∞ bj/sx(tj) = 0.

Let N ∈ N and ε > 0 be constants to be determined. Choose n large enough that bi/sx(ti) < ε for all i > n. By Lemma 5, there exists m > n such that N−1 N+1 (αmax ) sx(tn) < sx(tm) < (αmax ) sx(tn) (7) 30 JONATHAN ZUNG

and C tm − tn < (8) sx(tn)

Now consider the ragged rectangle R with base γ([tn, tm]) and height εsx(tm). We have that

Area(R) = εsx(tm)(tm − tn) (9)

C N+1 < εsx(tn) (αmax ) (10) sx(tn) N+1 = Cε(αmax ) . (11)

Since bj/sx(tj) < ε for every j ∈ [n, m], the ragged rectangle contains every singularity which the section sx encounters between time tn and tm. Since sx can grow by a factor of at most αmax each time it encounters a singularity, there must be at least N singularities in R. Taking N large enough and then ε small enough, we ﬁnd a ragged rectangle of arbitrarily small area containing many singularities, contradicting Lemma 4.

5.4. The full connection J . The uniqueness statements for blowup proven in the previous section suggest that we add a point at infinity to each fiber 1 Π ◦ of π, obtaining an new bundle S → EΠ −→ Mϕ. Via the embedding of fibers of π into fibers of Π, J partial is naturally a partial connection on Π as well. We now define an honest connection J on Π which extends J partial partial partial (i.e. for all γ ⊂ Mϕ(p; q), we have Jγ = Jγ on the domain of Jγ in Π−1(γ(0))).

partial −1 On a type 1 curve γ, the domain of Jγ is all of π (γ(0)). We define partial J in the only sensible way which extends J , asking that Jγ sends the point at infinity in Π−1(γ(0)) to the point at infinity in Π−1(γ(1)). It remains to define J on curves oriented east-west, i.e. those considered in ◦ Section 5.3. Given γ : [0, t] → Mϕ parameterizing an eastward interval in a ◦ s −1 leaf of Σ ∩F and a point x ∈ Π (γ(0)), let s0 be the parallel section over γ partial traced out by parallel transport of x along γ using J . Suppose s0 blows up towards +∞ at time tmax (x). (When x = ∞, we say that tmax (x) = 0, and when x < 0 we have tmax (x) = ∞.) By Proposition 6 and Proposition 7, there exists a unique parallel section s1 defined on [tmax (x), ∞) that blows up towards −∞ as t approaches tmax (x) from the right. We now define J in cases:  s (t) if t < t (x)  0 max Jγ(x) = s1(t) if t > tmax (x) (12)  ∞ if t = tmax (x) See Fig. 14a. TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 31

(a) We continue the example from Fig. 13. Here we show the θ-image of a parallel section for J over a curve γ oriented east-west. The section is piecewise constant. It starts out equal to s0, wraps around ∞ at t = tmax (x), and continues as s1.

(b) A parallel section for J over a line of negative slope. The section sx intersects the ∞ section inﬁnitely many times.

Figure 14

Proposition 8. J is ﬂat.

◦ Proof. Consider three types of short arcs in Mϕ: eastward arcs, northward arcs, and transverse arcs (i.e. arcs contained in ﬂowlines of the suspension ◦ 0 ﬂow of ϕ). Every curve in Mϕ can be C approximated by such curves. Recall that by a commutator of, say, eastward and northward arcs, we mean a quadrilateral whose sides alternate between eastward and northward arcs. As in Proposition 4, it is enough to show that the monodromy of J around all commutators is trivial. 32 JONATHAN ZUNG

It is easy to check that the monodromy around a commutator of eastward and northward arcs vanishes (i.e. the monodromy around the boundary of a rectangle is trivial) and that the commutator of northward and transverse arcs vanishes (both are type 1 curves). Finally, consider the commutator of a transverse arc of length ε and an eastward arc γ. Observe that the entire construction of Jγ for an eastward arc γ is equivariant with respect to dilations in the north-south or east-west directions, and in particular, a dilation by λε in the north-south direction and λ−ε in the east-west direction where ε. This is equivalent to the vanishing of the desired monodromy.

We can now rectify the lack of a good topology on EΠ. A ﬂat connection gives a local product structure on EΠ, so we endow EΠ with the corresponding local product topology.

Π has a special section, called the ∞-section and denoted r∞, whose value at any point is the point at infinity in the corresponding fiber. There is another section, called the zero-section and denoted r0, defined by the property

θ(r0(p)) = p ◦ for all p ∈ Mϕ. Note that neither of these sections is ﬂat with respect to J . Nevertheless, these sections are continuous.

Lemma 7. The ∞-section and the zero-section are both continuous sections of Π.

Proof. Lemma 6 gives a quantitative bound on how fast ﬂat sections can explode to ∞ in the east and west directions. In the north-south and transverse directions, ﬂat sections do not blow up. It follows that the graph of the ∞-section is closed and r∞ is continuous. ◦ Now let’s check continuity of the zero-section near a point p ∈ Mϕ. Take a ◦ s u neighbourhood U of a point p ∈ Mϕ which F and F foliate as products. The set 0 U = { y ∈ Eπ | θ(y) ∈ U, π(y) ∈ U } s is a neighbourhood of r0(p). Since holonomy of F exists for all transverse arcs in U, the topology on U 0 is just the standard product topology. The graph of the zero section is clearly closed in this topology.

Proposition 9. The monodromy of J around a ﬁlling curve on the boundary is trivial. Moreover, a parallel section over such a ﬁlling curve has zero intersection number with the ∞-section.

Proof. Since J is flat, we are free to compute the monodromy using a curve hugging very close to the boundary. Since J agrees with J partial when they are both defined, Proposition 5 tells us that the monodromy is the identity TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 33 for all points in the fiber not equal to ∞. Then it must be equal to the identity at ∞ as well. For the second statement, it suffices to check that just one parallel section of J has zero intersection number with the ∞-section. Simply take any parallel section of J partial along the meridian; such a section has no intersections with the ∞-section.

Proposition 9 guarantees that (Π, J ) extends to a bundle over Mϕ(p; q) with a flat connection. Moreover, the ∞-section extends to a section over 1 Mϕ(p; q). Thus, the Euler class of Π vanishes and the S -bundle with flat connection (Π, J ) unrolls to an R-bundle with flat connection which we call (Πb, Jb).

+ Proof of Theorem 1. Jb gives a homomorphism from π1(M) into Homeo (R). This map is nontrivial since the monodromy around an essential loop in one of the walls of the monkey saddles acts nontrivially; indeed, even the monodromy of J partial around such a curve is a dilation. Theorem 1.1 of [BRW02] states that for fundamental groups of irreducible 3-manifolds, the existence of any nontrivial map to a left-orderable group is equivalent to the existence of a left-ordering.

5.5. The ﬁber of Πb and the leaf space.

◦ ◦ Proof of Theorem 2. Let Mϕ be the cover of Mϕ associated with our Dehn ◦ ﬁllings. Mϕ embeds in M^ϕ(p; q). We use bars to denote the lifts of objects ◦ s ◦ s in Mϕ (e.g. F ) to objects in Mϕ (e.g. F ). Let K be the the leaf space of F s, where prongs incident with the same singularity are considered the same leaf. Let f1 : L → K be the monotone, π1(Mϕ(p; q))-equivariant map which crushes each interval in L corresponding with a lift of a saddle region down to a point.

The zero-section of Π lifts to a section of Πb since it never intersects the ∞-section. Call this lifted section rb0. ◦ ◦ Select a basepoint p ∈ Mϕ and a lift p ∈ Mϕ. We can deﬁne a monotone, π1(Mϕ(p; q))-equivariant map −1 f2 : K → Πb (p) as follows. For any leaf λ of F s, choose a point q on λ and a path γ : [0, 1] → ◦ ◦ Mϕ from p to q. Let γ be the projection of γ to Mϕ. We declare that

f2(λ) = Jbγ(rb0(q)).

Since Jb extends to a flat bundle over Mϕ(p; q), this definition is independent of the choice of γ in its relative homotopy class in Mϕ(p; q). Since the 34 JONATHAN ZUNG zero-section is flat on λ, this definition is independent of the choice of q. Finally, it’s easy to check that two prongs of F s corresponding with the same point in K have the same value of f2.

Let’s check π1-equivariance. Given g ∈ π1(Mϕ(p; q)), represent g by a loop ◦ γ2 in Mϕ. Then f (gλ) = J (r (q)) 2 bγ2∗γ b0 = J ◦ J (r (q)) bγ2 bγ b0 = gf2(λ)

Next, we check that f2 is a homeomorphism on short arcs transverse to s F s. Indeed, any short enough transversal to F terminating at q may be ◦ represented by a curve γ3 : [0, 1] → Mϕ such that γ3(1) = q and γ3 is the −1 θ-image of a curve γ4 in π (q). Then γ3 is a type 1 curve and we have for any t ∈ [0, 1], f (γ (t)) = J ◦ J (r (γ (t))) 2 3 bγ bγ3|[t,1] b0 3

= Jbγ(γ4(t)).

It follows that f2 is continuous and monotone. Thus, we conclude that the composition f = f2 ◦ f1 is a continuous, monotone, π1(Mϕ(p; q))-equivariant map. The π1(Mϕ(p; q)) action on the image is nontrivial as noted in the proof of Theorem 1.

Remark 5. The map f can be visualized quite cleanly in the setting of Fig. 14b. Extend γ linearly in both directions. The vertical red lines (plus their completions at ∞) are θ-images of fibers of Π. Lift these fibers to half-open subintervals of fibers of Πb. Choose the lifts so that they intersect the zero-section in Πb. Now parallel transport these subintervals to Πb −1(γ(0)) along γ. The transported intervals are disjoint and cover Πb −1(γ(0)), so the vertical lines (plus their completions at ∞) contain a representative from each point preimage of f. Therefore, the quotient of the leaf space we have constructed can be seen as the concatenation of all of the vertical red lines in Fig. 14b. This picture is analogous to that of the step map in the setting of skew-Anosov flows.

6. Computations Building on work of Dunfield and Bell, we were able to find 2598 manifolds in the Hodgson-Weeks census which can be constructed by a surgery satisfying the hypotheses of Theorem 1. This represents about 44.7% of the 5801 non-L- spaces in the Hodgson-Weeks census [HW94, Dun19]. Dunfield and Bell found monodromies for many of the fibered, orientable 1-cusped manifolds that can be triangulated with at most 9 tetrahedra [BN]. Using Bell’s program flipper, they were able to find invariant laminations for about 25,700 of TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 35 them [Bel13]. About 800 of these have orientable invariant laminations and monodromy preserving these orientations. The first few such examples with genus ≥ 2 are listed in Table 1.

Name Genus m038 2 m120 3 s090 4 v0224 5 m221 3 t00448 6 o9 00896 7 s173 4 v0248 6 m289 2 t00682 4 m305 2 s296 2 m310 3 t00707 3

Table 1. The ﬁrst few 1-cusped ﬁbered manifolds with genus ≥ 2 and monodromy satisfying the conditions of Theorem 1.

We used flipper to drill these manifolds along their pseudo-Anosov singularities and produce ideal triangulations of the resulting many-cusped manifolds. Using SnapPy, we performed surgeries with small coefficients on these manifolds satisfying the constraints of Theorem 1 and identified the resulting manifolds in the Hodgson-Weeks census [CDGW]. We found 2598 manifolds in the census, the first of which are shown in Table 2. Dunfield obtained orderability results for many manifolds in the Hodgson- Weeks census either by constructing a taut foliation with vanishing Euler class or by constructing a PSL(2, R) representation that lifts to a PSLg (2, R) representation [Dun19]. Table 3 shows the overlap in applicability between these methods and ours.

7. Remarks and questions

• For which taut foliations does there exist a π1-equivariant, order preserving map from the leaf space of the universal cover to R? As a ﬁrst step, we suggest the following conjecture:

Conjecture 1. Theorem 2 holds without the condition that the surgery slopes have the same sign. 36 JONATHAN ZUNG

Name Underlying ﬁbered manifold Volume m003(-2,3) m004 0.981 m004(6,1) m004 1.284 m004(1,2) m004 1.398 m003(-3,4) v0650 1.414 m009(4,1) m023 1.414 m004(3,2) m004 1.440 m004(7,1) m004 1.463 m004(5,2) m004 1.529 m015(5,1) t03310 1.757 m009(5,1) m009 1.831 m009(-5,1) m009 1.831 m011(1,3) v1577 1.831 m009(1,2) m009 1.843 m007(-5,1) o9 31045 1.843 m006(-5,1) m009 1.941

Table 2. The ﬁrst few (closed) manifolds in the Hodgson- Weeks census to which Theorem 1 applies.

Theorem 1 applies? Yes No Taut foliation or PSL(2, R) Yes 1795 1730 rep with Euler class 0? No 803 1473

Table 3. The overlap in applicability between our method and previously used methods for proving orderability on the non-L-space rational homology spheres in the Hodgson-Weeks census.

This would greatly expand the scope of the results in this paper; for example, by [Fri83] it would include every 3-manifold carrying a transitive pseudo- Anosov ﬂow with orientable invariant foliations. The diﬃculty is that parallel sections now typically blow up towards both +∞ and −∞.

• Can the map f : L → R be upgraded to a strictly monotone map? We expect that f factors as

µ η L R Πb −1(p) (13)

R R TAUT FOLIATIONS, LEFT-ORDERS, AND PSEUDO-ANOSOV MAPPING TORI 37 where µ is locally a homeomorphism onto its image, η is monotonic, and R is yet to be defined. The composition f = η ◦ µ collapses the leaves of each saddle region in L to a point. Therefore, R should be obtained from Πb −1(p) by blowing up the f-image of a leaf in a saddle region to a closed interval. The difficulty is that different saddle regions could conceivably map to the same point under f. • What is the best possible analytic quality of the representations we have constructed? • A generic pseudo-Anosov map will violate the orientability constraints of Theorem 1. For surgery coefficients satisfying an appropriate parity condition, we expect that the methods of this paper may be extended to give an action of π1 on R by possibly orientation-reversing homeomorphisms. • What can be said about fillings along the degeneracy slope? Experiments suggest that these are L-spaces, and hence do not carry taut foliations.

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Department of Mathematics, Princeton Email address: [email protected]