Fibered faces, veering triangulations, and the arc complex

# Fibered faces, veering triangulations, and the arc complex

Yair N. Minsky Department of Mathematics
Yale University
and  Samuel J. Taylor Department of Mathematics
Temple University
July 27, 2019
###### Abstract.

We study the connections between subsurface projections in curve and arc complexes in fibered 3-manifolds and Agol’s veering triangulation. The main theme is that large-distance subsurfaces in fibers are associated to large simplicial regions in the veering triangulation, and this correspondence holds uniformly for all fibers in a given fibered face of the Thurston norm.

This work was partially supported by NSF grants DMS-1311844 and DMS-1400498.

## 1. Introduction

Let be a 3-manifold fibering over the circle with fiber and pseudo-Anosov monodromy . The stable/unstable laminations of give rise to a function on the essential subsurfaces of ,

 Y↦dY(λ+,λ−),

where denotes distance in the curve and arc complex of between the lifts of to the cover of homeomorphic to . This distance function plays an important role in the geometry of the mapping class group of [MM00, BKMM12, MS13], and in the hyperbolic geometry of the manifold [Min10, BCM12].

In this paper we study the function when is fixed and the fibration is varied. The fibrations of a given manifold are organized by the faces of the unit ball of Thurston’s norm on , where each fibered face has the property that every irreducible integral class in the open cone represents a fiber. There is a pseudo-Anosov flow which is transverse to every fiber represented by , and whose stable/unstable laminations intersect each fiber to give the laminations associated to its monodromy. With this we note that the projection distance can be defined for any subsurface of any fiber in . We use to denote this quantity.

Our main results give explicit connections between and the veering triangulation of , introduced by Agol [Ago11] and refined by Guéritaud [Gué15], with the main feature being that when satisfies explicit lower bounds, a thickening of is realized as an embedded subcomplex of the veering triangulation. In this way, the “complexity” of the monodromy is visible directly in the triangulation in a way that is independent of the choice of fiber in the face . This is in contrast with the results of [BCM12] in which the estimates relating to the hyperbolic geometry of are heavily dependent on the genus of the fiber.

The results are cleanest in the setting of a fully-punctured fiber, that is when the singularities of the monodromy are assumed to be punctures of the surface (one can obtain such examples by starting with any and puncturing the singularities and their flow orbits). All fibers in a face are fully-punctured when any one is, and in this case we say that is a fully-punctured face. In this setting is a cusped manifold and the veering triangulation is an ideal triangulation of .

We obtain bounds on that hold for in any fiber of a given fibered face:

###### Theorem 1.1 (Bounding projections over a fibered face).

Let be a hyperbolic 3-manifold with fully-punctured fibered face and veering triangulation . For any essential subsurface of any fiber of ,

 α⋅(dW(Λ−,Λ+)−β)<|τ|,

where is the number of tetrahedra in , and when is an annulus and and when is not an annulus.

Note that this means that for each subsurface , no matter which fiber lies in. Further, the complexity of any subsurface of any fiber of with is also bounded in terms of alone.

In addition, given one fiber with a collection of subsurfaces of large , we obtain control over the appearance of high-distance subsurfaces in all other fibers:

###### Theorem 1.2 (Subsurface dichotomy).

Let be a hyperbolic 3-manifold with fully-punctured fibered face and suppose that and are each fibers in . If is a subsurface of , then either is isotopic along the flow to a subsurface of , or

 3|χ(S)|≥dW(Λ−,Λ+)−β,

where if is an annulus and otherwise.

One can apply this theorem with taken to be the smallest-complexity fiber in . In this case there is some finite list of “large” subsurfaces of , and for all other fibers and all subsurfaces with sufficiently large, is already accounted for on this finite list.

For a sample application of Theorem 1.2, let be an essential annulus with core curve in a fiber of and suppose that for some . (We note that it is easy to construct explicit examples of with as large as one wishes by starting with a pseudo-Anosov homeomorphism of with large twisting about the curve .) If is trivial in , then Theorem 1.2 (or more precisely Corollary 6.7) implies that is actually isotopic to a simple closed curve in every fiber in the open cone containing . When is nontrivial in it determines a codimension- hyperplane in consisting of cohomology classes which vanish on . For each fiber of either is contained in , in which case is isotopic to a simple closed curve in as before, or lies outside of and . We remark that the second alternative is non-vacuous so long as has rank at least 2.

The general (non-fully-punctured) setting is also approachable with our techniques, but a number of complications arise and the connection to the veering triangulation of the fully-punctured manifold is much less explicit. An extension of the results in this paper to the general setting will be the subject of a subsequent paper.

### Pockets in the veering triangulation

When is a subsurface of a fiber in and , we show (Theorem 5.3) that is realized simplicially in the veering triangulation lifted to the cover . If is even larger then this realization can be thickened to a “pocket”, which is a simplicial region bounded by two isotopic copies of . With sufficiently large assumptions this pocket can be made to embed in as well, and this is our main tool for connecting arc complexes to the veering triangulation and establishing Theorems 1.1 and 1.2:

###### Theorem 1.3.

Suppose is a subsurface of a fiber with , where if is nonannular and if is an annulus. Then there is an embedded simplicial pocket in isotopic to a thickening of , and with .

In this statement, and refer to the triangulations of the top and bottom surfaces of the pockets, regarded as simplices in the curve and arc complex . Also, denotes the smallest -distance between an arc of and an arc of .

The veering triangulation in fact recovers a number of aspects of the geometry of curve and arc complexes in a fairly concrete way. As an illustration we prove

###### Theorem 1.4.

In the fully punctured setting, the arcs of the veering triangulation form a geodesically connected subset of the curve and arc graph, in the sense that any two points in are connected by a geodesic that lies in .

### Hierarchies of pockets

One is naturally led to generalize Theorem 1.3 from a result embedding one pocket at a time to a description of all pockets at once. Indeed Proposition 6.5 tells us that whenever subsurfaces and of have large enough projection distances and are not nested, they have associated pockets and which are disjoint in . These facts, taken together with Theorem 1.4, strongly suggest that the veering triangulation encodes the hierarchy of curve complex geodesics between as introduced by Masur-Minsky in [MM00]. We expect that, using a version of Theorem 1.4 that applies to subsurfaces and adapting the notion of “tight geodesic” from [MM00], one can carry out a hierarchy-like construction within the veering triangulation and recover much of the structure found in [MM00], with more concrete control, at least in the fully-punctured setting. We plan to explore this approach in future work.

### Related and motivating work

The theme of using fibered 3-manifolds to study infinite families of monodromy maps is deeply explored in McMullen [McM00] and Farb-Leininger-Margalit [FLM11], where the focus is on Teichmüller translation distance.

Distance inequalities analogous to Theorem 1.2, in the setting of Heegaard splittings rather than surface bundles, appear in Hartshorn [Har02], and then more fully in Scharlemann-Tomova [ST06]. Bachman-Schleimer [BS05] use Heegaard surfaces to give bounds on the curve-complex translation distance of the monodromy of a fibering. All of these bounds apply to entire surfaces and not to subsurface projections. In Johnson-Minsky-Moriah [JMM10], subsurface projections are considered in the setting of Heegaard splittings. A basic difficulty in these papers which we do not encounter here is the compressibility of the Heegaard surfaces, which makes it tricky to control essential intersections. On the other hand, unlike the surfaces and handlebodies that are used to obtain control in the Heegaard setting, the foliations we consider here are infinite objects, and the connection between them and finite arc systems in the surface is a priori dependent on the fiber complexity. The veering triangulation provides a finite object that captures this connection in a more uniform way.

The totally-geodesic statement of Theorem 1.4 should be compared to Theorem 1.2 of Tang-Webb [TW15], in which Teichmüller disks give rise to quasi-convex sets in curve complexes. While the results of Tang-Webb are more general, they are coarse, and it is interesting that in our setting a tighter statement holds. Finally, we note that work by several authors has focused on geometric aspects of the veering triangulation, including [HRST11, FG13, HIS16].

### Summary of the paper

In Section 2 we set some notation and give Guéritaud’s construction of the veering triangulation. We also recall basic facts about curve and arc complexes, subsurface projections and Thurston’s norm on homology. We spend some time in this section describing the flat geometry of a punctured surface with an integrable holomorphic quadratic differential, and in particular giving an explicit description of the circle at infinity of its universal cover (Proposition 2.2). While this is a fairly familiar picture, some delicate issues arise because of the incompleteness of the metric at the punctures.

In Section 3 we study sections of the veering triangulations, which are simplicial surfaces isotopic to in the cover , and transverse to the suspension flow of the monodromy. These can be thought of as triangulations of the surface using only edges coming from the veering triangulation. We prove Lemma 3.2 which says that a partial triangulation of using only edges from can always be extended to a full section, and Proposition 3.3 which says that any two extensions of a partial triangulation are connected by a sequence of “tetrahedron moves”. This is what allows us to define and study the “pockets” that arise between any two sections.

In Section 4 we define a simple but useful construction, rectangle and triangle hulls, which map saddle connections in our surface to unions of edges of the veering triangulation. An immediate consequence of the properties of these hulls is a proof of LABEL:th:totalgeodesic.

In Section 5 we apply the flat geometry developed in Section 2 to control the convex hulls of subsurfaces of the fiber, and then use Section 4 to construct what we call -hulls, which are representatives of the homotopy class of a subsurface that are simplicial with respect to the veering triangulations. Theorem 5.3 states that quite mild assumptions on imply that the -hull of has embedded interior. The idea here is that any pinching point of the -hull is crossed by leaves of and that intersect each other very little. The main results of both Section 4 and Section 5 apply in a general setting and do not require that the surface be fully-punctured.

In Section 6 we put these ideas together to prove our main theorems for fibered manifolds with a fully-punctured fibered face. In Proposition 6.2 we describe the maximal pocket associated to a subsurface with sufficiently large (greater than 2, for nonannular ). We then introduce the notion of an isolated pocket, which is a subpocket of the maximal pocket that has good embedding properties in the manifold . The existence and embedding properties of these pockets are established in Lemma 6.4 and Proposition 6.5, which together allow us to prove Theorem 1.3.

From here, a simple counting argument gives Theorem 1.1: the size of the embedded isolated pockets is bounded from below in terms of and , and from above by the total number of veering tetrahedra.

To obtain Theorem 1.2, we use the pocket embedding results to show that, if is a subsurface of one fiber and essentially intersects another fiber , then must cross every level surface of the isolated pocket of , and hence the complexity of gives an upper bound for . To complete the proof we need to show that, if does not essentially cross , it must be isotopic to an embedded (and not merely immersed) subsurface of . This is handled by Lemma 6.6, which may be of independent interest. It gives a uniform upper bound for when corresponds to a finitely generated subgroup of , unless covers an embedded subsurface.

### Acknowledgments

The authors are grateful to Ian Agol and François Guéritaud for explaining their work to us. We also thank Tarik Aougab, Jeff Brock, and Dave Futer for helpful conversations and William Worden for pointing out some typos in an earlier draft. Finally, we thank the referee for a thorough reading of our paper and comments which improved its readability.

## 2. Background

The following notation will hold throughout the paper. Let be a closed Riemann surface with an integrable meromorphic quadratic differential . We remind the reader that may have poles of order . We denote the vertical and horizontal foliations of by and respectively. Let be a finite subset of that includes the poles of if any, and let . Let denote the union of with the set of zeros of . We require further that has no horizontal or vertical saddle connections, that is no leaves of that connect two points of . This situation holds in particular if are the stable/unstable foliations of a pseudo-Anosov map , which will often be the case for us. If (i.e. contains all zeros of ) we say is fully-punctured.

Let denote the metric completion of the universal cover of , and note that there is an infinite branched covering , infinitley branched over the points of . The preimage of is the set of completion points. The space is a complete CAT space with the metric induced by .

### 2.1. Veering triangulations

In this section let . The veering triangulation, originally defined by Agol in [Ago11] in the case where corresponds to a pseudo-Anosov , is an ideal layered triangulation of which projects to a triangulation of the mapping torus of . The definition we give here is due to Guéritaud [Gué15]. (Agol’s “veering” property itself will not actually play a role in this paper, so we will not give its definition).

A singularity-free rectangle in is an embedded rectangle whose edges consist of leaf segments of the lifts of and whose interior contains no singularities of . If is a maximal singularity-free rectangle in then it must contain a singularity on each edge. Note that there cannot be more than one singularity on an edge since have no saddle connections. We associate to an ideal tetrahedron whose vertices are , as in Figure 1. This tetrahedron comes equipped with a “flattening” map into as pictured.

The tetrahedron comes with a natural orientation, inherited from the orientation of using the convention that the edge connecting the horizontal boundaries of the rectangle lies above the edge connecting the vertical boundaries. This orientation is indicated in Figure 1.

The union of all these ideal tetrahedra, with faces identified whenever they map to the same triangle in , is Guéritaud’s construction of the veering triangulation of .

###### Theorem 2.1.

[Gué15] Suppose that is fully-punctured. The complex of tetrahedra associated to maximal rectangles of is an ideal triangulation of , and the maps of tetrahedra to their defining rectangles piece together to a fibration . The action of on lifts simplicially to , and equivariantly with respect to . The quotient is a triangulation of .

If corresponds to a pseudo-Anosov then the action of on lifts simplicially and -equivariantly to . The quotient is a triangulation of the mapping torus . The fibers of descend to flow lines for the suspension flow of .

We will frequently abuse notation and use to refer to the triangulation both in and in its covers.

We note that a saddle connection of is an edge of if and only if spans a singularity-free rectangle in . See Figure 2.

If and are two crossing -edges spanning rectangles and , note that crosses from top to bottom, or from left to right – any other configuration would contradict the singularity-free property of the rectangles (Figure 3). If denotes the absolute value of the slope of with respect to , we can see that crosses from top to bottom if and only if crosses and . We say that is more vertical than and also write . We will see that corresponds to lying higher than in the uppward flow direction.

Indeed we can see already that the relation is transitive, since if and then the rectangle of is forced to intersect the rectangle of .

We conclude with a brief description of the local structure of around an edge : The rectangle spanned by can be extended horizontally to define a tetrahedron lying below in the flow direction (Figure 2), and vertically to define a tetrahedron lying above in the flow direction. Call these and as in Figure 4. Between these, on each side of , is a sequence of tetrahedra so that two successive tetrahedra in the sequence share a triangular face adjacent to . We find this sequence by starting with one of the two top faces of , extending its spanning rectangle vertically until it hits a singularity, and calling the tetrahedron whose projection is inscribed in the new rectangle. If the new singularity belongs to we are done , otherwise we repeat from the top face of containing to find , and continue in this manner. Figure 4 illustrates this structure on one side of an edge . Repeating on the other side, note that the link of the edge is a circle, as expected.

### 2.2. Arc and curve complexes

The arc and curve complex for a compact surface is usually defined as follows: its vertices are essential homotopy classes of embedded circles and properly embedded arcs , where “essential” means not homotopic to a point or into the boundary [MM00]. We must be clear about the meaning of homotopy classes here, for the case of arcs: If is not an annulus, homotopies of arcs are assumed to be homotopies of maps of pairs. When is an annulus the homotopies are also required to fix the endpoints. Simplices of , in all cases, correspond to tuples of vertices which can be simultaneously realized by maps that are disjoint on their interiors. We endow with the simplicial distance on its -skeleton.

It will be useful, in the non-annular case, to observe that the following definition is equivalent: Instead of maps of closed intervals consider proper embeddings into the interior of , with equivalence arising from proper homotopy. This definition is independent of the compactification of . The natural isomorphism between these two versions of is induced by a straightening construction in a collar neighborhood of the boundary.

If is an essential subsurface (meaning the inclusion of is -injective and is not homotopic to a point or to an end of ), we have subsurface projections which are defined for simplices that intersect essentially. Namely, after lifting to the cover associated to (i.e. the cover to which lifts homeomorphically and for which ), we obtain a collection of properly embedded disjoint essential arcs and curves, which determine a simplex of . We let be the union of these vertices [MM00]. We make a similar definition for a lamination that intersects essentially, except that we include not just the leaves of but all leaves that one can add in the complement of which accumulate on . This is natural when we realize as a measured foliation (as we do in most of the paper), and need to include generalized leaves, which are leaves that are allowed to pass through singularities. Note that the diameter of in is at most 2.

Note that when is an annulus these arcs have natural endpoints coming from the standard compactification of by a circle at infinity. We remark that does not depend on any choice of hyperbolic metric on .

When is not an annulus and and are in minimal position, we can also identify with the isotopy classes of components of .

These definitions naturally extend to immersed surfaces arising from covers of . Let be a finitely generated subgroup of . Then the corresponding cover has a compact core – a compact subsurface such that is a collection of boundary parallel annuli. For curves or laminations of , we have lifts to and define .

Throughout this paper, when are two laminations or arc/curve systems, we denote by the minimal distance between their images in , that is

 dY(λ,λ′)=min{dY(l,l′):l∈πY(λ),l′∈πY(λ′)}.

To denote the maximal distance between and in we write

 diamY(λ,λ′)=diamA(Y)(πY(λ)∪πY(λ′)).

### 2.3. Flat geometry

In this section we return to the singular Euclidean geometry of and describe a circle at infinity for the flat metric induced by on the universal cover . We identify with after fixing a reference hyperbolic metric on . Because of incompleteness of the flat metric at the punctures , the connection between the circle we will describe and the usual circle at infinity for requires a bit of care. A related discussion appears in Guéritaud [Gué15], although he deals explicitly only with the fully-punctured case. With this picture of the circle at infinity we will be able to describe in terms of -geodesic representatives, and to describe a -convex hull for essential subsurfaces of . In this section we do not assume that is fully-punctured.

The completion points in correspond to parabolic fixed points for in , and we abuse notation slightly by identifying with this subset of .

A complete -geodesic ray is either a geodesic ray of infinite length, or a finite-length geodesic segment that terminates in . A complete -geodesic line is a geodesic which is split by any point into two complete -geodesic rays. Our goal in this section is to describe a circle at infinity that corresponds to endpoints of these rays.

###### Proposition 2.2.

There is a compactification of on which acts by homeomorphisms, with the following properties:

1. There is a -equivariant homeomorphism , extending the identification of with and taking to the corresponding parabolic fixed points in .

2. If is a complete -geodesic line in then its image in is an embedded arc with endpoints on and interior points in . Conversely, every pair of distinct points in are the endpoints of a complete -geodesic line. The termination point in of a complete -geodesic ray is in if and only if it has finite length.

3. The -geodesic line connecting distinct is either unique, or there is a family of parallel geodesics making up an infinite Euclidean strip.

One of the tricky points of this picture is that -geodesic rays and lines may meet points of the boundary not just at their endpoints.

###### Proof.

When and is a closed surface, is quasi-isometric to and the proposition holds for the standard Gromov compactification. We assume from now on that .

We begin by setting and endowing it with the topology obtained by taking, for each , horoballs based at as a neighborhood basis for .

###### Lemma 2.3.

The natural identification of with extends to a homeomorphism from to .

###### Proof.

First note that is discrete as both a subspace of and of . Hence, it suffices to show that a sequence of points in converges to a point in if and only if it converges to in . This follows from the fact that the horoball neighborhoods of descend to cusp neighborhoods in which form a neighborhood basis for the puncture that is equivalent to the neighborhood basis of -metric balls. ∎

Our strategy now is to form the Freudenthal space of and equivalently , which appends a space of ends. This space will be compact but not Hausdorff, and after a mild quotient we will obtain the desired compactification which can be identified with . Simple properties of this construction will then allow us to obtain the geometric conclusions in part (2) of the proposition.

Let be the space of ends of , that is the inverse limit of the system of path components of complements of compact sets in . The Freudenthal space is the union endowed with the toplogy generated by using path components of complements of compacta to describe neighborhood bases for the ends. Because is not locally compact, is not guaranteed to be compact, and we have to take a bit of care to describe it.

The construction can of course be repeated for , and the homeomorphism of Lemma 2.3 gives rise to a homeomorphism . Let us work in now, where we can describe the ends concretely using the following observations:

Every compact set meets in a finite set (since is discrete in ), and such a is contained in an embedded closed disk which also meets at . (This is not hard to see but does require attention to deal correctly with the horoball neighborhood bases). The components of determine a partition of , which in fact depends only on the set and not on (if is another disk meeting at , then is contained in a third disk , and this common refinement of the neighborhoods gives the same partition). Thus we have a more manageble (countable) inverse system of neighborhoods in , and with this description it is not hard to see that is a Cantor set.

For each there are two distinguished ends defined as follows: For each finite subset with at least two points one of which is , the two partition terms adjacent to in the circle (or equivalently, in the boundary of any meeting in ) define neighborhoods in , and this pair of neighborhood systems determines and respectively.

One can also see that (and ) and do not admit disjoint neighborhoods, and this is why is not Hausdorff. We are therefore led to define the quotient space

 β(^H2)=Fr(^H2)/∼,

where we make the identifications , for each .

We can make the same definitions in , obtaining

 β(^X)=Fr(^X)/∼,

which we rename . Since the definitions are purely in terms of the topology of the spaces and , the homeomorphism of Lemma 2.3 extends to a homeomorphism .

Part (1) of Proposition 2.2 follows once we establish that the identity map of extends to a homeomorphism

 β(^H2)≅¯¯¯¯¯¯H2.

This is not hard to see once we observe that the disks used above to define neighborhood systems can be chosen to be ideal hyperbolic polygons. Their halfspace complements serve as neighborhood systems for points of . A sequence converges in to a point if it is eventually contained in any union of a horoball centered at p and two half-planes adjacent to on opposite sides. This is modeled exactly by the equivalence relation .

For part (2), let be a fundamental domain for in , which may be chosen to be a disk with vertices at points of , and of finite -diameter. Translates of can be glued to build a sequence of nested disks exhausting , each of which meets in a finite set of vertices, and whose boundary is composed of arcs of bounded diameter between successive vertices.

A complete -geodesic ray either has finite length and terminates in a point of , or has infinite length in which case it leaves every compact set of , and visits each point of at most once. Thus it must terminate in a point of in the Freudenthal space. We claim that this point cannot be or for . If terminates in , then for each disk ( large) it must pass through the edge of adjacent to on the side associated to . Any two such consecutive edges meet in at one of finitely many angles (images of corners of ), and hence the accumulated angle between edges goes to with . If we replace these edges by their -geodesic representatives, the angles still go to . This means that contains infinitely many disjoint subsegments whose endpoints are a bounded distance from , but this contradicts the assumption that is a geodesic ray.

The image of in the quotient therefore terminates in a point of when it has finite length, and a point in otherwise. The same is true for both ends of a complete -geodesic line , and we note that both ends of cannot land on the same point because then we would have a sequence of segments of length going to with both endpoints of on the same edge or on two consecutive edges of , a contradiction to the fact that is a geodesic and the arcs in have bounded -length.

Now let be two distinct points in . Assume first that both are not in . Then for large enough , they are in separate components of the complement of . If we let and be sequences in , then eventually and are in the same components of the complement of as and , respectively. The geodesic from to must therefore pass through the corresponding boundary segments of and in particular through , so we can extract a convergent subsequence as . Letting and diagonalizing we obtain a limiting geodesic which terminates in as desired. If or the same argument works except that we can take or . This establishes part (2).

Now let and be two -geodesics terminating in and . If and are in then since the metric is CAT(0). If then both and pass through infinitely many segments of on their way to . Since these segments have uniformly bounded lengths, and remain a bounded distance apart. If then again CAT(0) implies that , and if then and must cobound an infinite flat strip. This establishes part (3). ∎

With Proposition 2.2 in hand we can consider each complete -geodesic line in as an arc in the closed disk , which by the Jordan curve theorem separates the disk into at least components. Each component is an open disk whose closure meets in a subarc of one of the complementary arcs of the endpoints of . We call the union of disks whose closures meet one of these complementary arcs of the endpoints of an open side of . The closure of each open side in is then a connected union of closed disks, attached to each other along the points of that meets on the circle. We call the closure of the open side of in the side . Note that , and if and are the two sides of , then . See Figure 5.

With this picture we can state the following:

###### Corollary 2.4.

Let be disjoint arcs in with well-defined, distinct endpoints on and let be -geodesic lines with the same endpoints as and , respectively. Then is contained in a single side of .

###### Proof.

Letting and be the arcs of minus the endpoints of , the endpoints of must lie in one of them, say , since and are disjoint.

Since and are geodesics in the CAT space , their intersection is connected. If their intersection is empty, then the corollary is clear. Otherwise, is one or two arcs, each with one endpoint on and the other on . It follows that is on one open side of , and the corollary follows. ∎

### Subsurfaces and projections in the flat metric

Let be an essential compact subsurface, and let be the associated cover of . (Here we have identified with the deck transformations of and fixed within its conjugacy class.) For any lamination in , we want to show that the projection can be realized by subsegments of the -geodesic representative of . Recall that is not necessarily fully-punctured.

We say a boundary component of is puncture-parallel if it bounds a disk in that contains a single point of . We denote the corresponding subset of by and refer to them as the punctures of . Let denote the subset of punctures of which are encircled by the boundary components of the lift of to . In terms of the completed space , is exactly the set of completion points which have finite total angle. Let denote the union of the puncture-parallel components of and let denote the rest. Observe that the components of are in natural bijection with and set .

Identifying with , let be the limit set of , , and the set of parabolic fixed points of . Let denote the compactification of given by , adding a point for each puncture-parallel end of , and a circle for each of the other ends. Now given a lamination (or foliation) , realized geodesically in the hyperbolic metric on , its lift to extends to properly embedded arcs in , of which the ones that are essential give .

Proposition 2.2 allows us to perform the same construction with the -geodesic representative of . Note that the leaves we obtain may meet points of in their interior, but a slight perturbation produces properly embedded lines in which are properly isotopic to the leaves coming from .

If is an annulus the same construction works, with the observation that the ends of cannot be puncture-parallel and hence is a closed annulus and the leaves have well-defined endpoints in its boundary. We have proved:

###### Lemma 2.5.

Let be an essential subsurface. If is a proper arc or lamination in then the lifts of its -geodesic representatives to , after discarding inessential components, give representatives of .

### q-convex hulls

We will need a flat-geometry analogue of the hyperbolic convex hull. The main idea is simple – pull the boundary of the regular convex hull tight using -geodesics. The only difficulty comes from the fact that these geodesics can pass through parabolic fixed points, and fail to be disjoint from each other, so the resulting object may fail to be an embedded surface. Our discussion is similar to Section of Rafi [Raf05], but the discussion there requires adjustments to handle correctly the incompleteness at punctures.

As above, identify with . Let be a closed set and let be the convex hull of in . We define as follows.

Assume first that has at least 3 points. Each boundary geodesic of has the same endpoints as a (biinfinite) -geodesic . By part (3) of Proposition 2.2, is unique unless it is part of a parallel family of geodesics, making a Euclidean strip.

The plane is divided by into two sides as in the discussion before Corollary 2.4, and one of the sides, which we call , meets in a subset of the complement of . Recall that is either a disk or a string of disks attached along puncture points. If is one of a parallel family of geodesics, we include this family in . After deleting from the interiors of for all in (which are disjoint by Corollary 2.4), we obtain , the -convex hull.

If has 2 points then is the closed Euclidean strip formed by the union of -geodesics joining those two points.

Now fixing a subsurface we can define a -convex hull for the cover , by taking a quotient of the -convex hull of the limit set of . This quotient, which we will denote by , lies in the completion . Because may not be homeomorphic to , we pay explicit attention to a marking map between and its hull.

Let be the lift of the inclusion map to the cover.

###### Lemma 2.6.

The lift is homotopic to a map whose image is the -hull such that

1. The homotopy from to has the property that for all .

2. Each component of is taken by to the corresponding completion point of .

3. If is an annulus then the image of is either a maximal flat cylinder in or the unique geodesic representative of the core of in .

4. If is not an annulus then each component of is taken by to a -geodesic representative in . If there is a flat cylinder in the homotopy class of then the interior of the cylinder is disjoint from .

5. There is a deformation retraction . For each component of , the preimage intersects in either an open annulus or a union of open disks joined in a cycle along points in their closures.

6. If the interior is a disk then is a homeomorphism from to its image.

###### Proof.

Let and let denote the limit set of . As usual, can be identified with . After isotopy we may assume is this identification.

First assume that is not an annulus. Form as above, and for a boundary geodesic of define and its side as in the discussion above. The quotient of is a geodesic representative of a component of , and the quotient of the open side in is either an open annulus or a union of open disks joined in a cycle along points in their completion. The -geodesic may pass through points of , so that there is a homotopy from to rel endpoints which stays in until the last instant.

We may equivariantly deform the identity to a map , which takes each to : since is contractible, it suffices to give a -invariant triangulation of and define the homotopy successively on the skeleta. This homotopy descends to a map from to , and can be chosen so that the puncture-parallel boundary components map to the corresponding points of . This gives the desired map and establishes properties (1-4).

Using the description of the sides , we may equivariantly retract to , giving rise to the retraction of part (5).

Finally, if the interior of is a disk, then its quotient is a surface. Our homotopy yields a homotopy-equivalence of to this surface which preserves peripheral structure and can therefore be deformed rel boundary to a homeomorphism. We let be this homeomorphism, giving part .

When is a (nonperipheral) annulus, is a pair of points and we recall from above that is either a flat strip in which descends to a flat cylinder in , or it is a single geodesic. The proof in the annular case now proceeds exactly as above. ∎

Let be the composition of with the (branched) covering and set . Note that this will be a 1-complex of saddle connections and not necessarily a homeomorphic image of .

### 2.4. Fibered faces of the Thurston norm

A fibration of a finite-volume hyperbolic 3-manifold over the circle comes with the following structure: there is an integral cohomology class in represented by , which is the Poincaré dual of the fiber . There is a representation of as a quotient where and is called the monodromy map. This map is pseudo-Anosov and has stable and unstable (singular) measured foliations and on . Finally there is the suspension flow inherited from the natural action on , and suspensions of which are flow-invariant 2-dimensional foliations of . All these objects are defined up to isotopy.

The fibrations of are organized by the Thurston norm on [Thu86] (see also [CC00]). This norm has a polyhedral unit ball with the following properties:

1. Every cohomology class dual to a fiber is in the cone over a top-dimensional open face of .

2. If contains a cohomology class dual to a fiber then every irreducible integral class in is dual to a fiber. is called a fibered face and its irreducible integral classes are called fibered classes.

3. For a fibered class with associated fiber , .

In particular if and is fibered then there are infinitely many fibrations, with fibers of arbitrarily large complexity. We will abuse terminology a bit by saying that a fiber (rather than its Poincaré dual) is in .

The fibered faces also organize the suspension flows and the stable/unstable foliations: If is a fibered face then there is a single flow and a single pair of foliations whose leaves are invariant by , such that every fibration associated to may be isotoped so that its suspension flow is up to a reparameterization, and the foliations for the monodromy of its fiber are . These results were proven by Fried [Fri82]; see also McMullen [McM00].

### Veering triangulation of a fibered face

A key fact for us is that the veering triangulation of the manifold depends only on the fibered face and not on a particular fiber. This was known to Agol for his original construction (see sketch in [Ago12]), but Guéritaud’s construction makes it almost immediate.

###### Proposition 2.7 (Invariance of τ).

Let be a hyperbolic 3-manifold with fully-punctured fibered face . Let and be fibers of each contained in and let and be the corresponding veering triangulations of . Then, after an isotopy preserving transversality to the suspension flow, .

###### Proof.

The suspension flow associated to lifts to the universal cover , and any fiber in is covered by a copy of its universal cover in which meets every flow line transversely, exactly once. Thus we may identify with the leaf space of this flow. The lifts of the suspended laminations project to the leaf space where they are identified with the lifts of to .

The foliated rectangles used in the construction of from on depend only on the (unmeasured) foliations . Thus the abstract cell structure of depends only on the fibered face and not on the fiber. The map from each tetrahedron to its rectangle does depend a bit on the fiber, as we choose -geodesics for the edges (and the metric depends on the fiber); but the edges are always mapped to arcs in the rectangle that are transverse to both foliations. It follows that there is a transversality-preserving isotopy between the triangulations associated to any two fibers. ∎

### Fibers and projections

We next turn to a few lemmas relating subsurface projections over the various fibers in a fixed face of the Thurston norm ball.

###### Lemma 2.8.

If is a fibered face for and is an infinite covering where is a fiber in and is finitely generated, then the projection distance depends only on and the conjugacy class of the subgroup (and not on ).

Note that need not correspond to an embedded subsurface of .

###### Proof.

As in the proof of Proposition 2.7, can be identified with the leaf space of the flow in . The action of on descends to , and thus the cover is identified with the quotient and the lifts of to are identified with the images of in . Thus the projection can be obtained without reference to the fiber . ∎

This lemma justifies the notation used in the introduction.

We will also require the following lemma, where we allow maps homotopic to fibers which are not necessarily embeddings.

###### Lemma 2.9.

Let be a fiber of . Let be a compact surface and let be a map which is homotopic to the inclusion. Suppose that is inessential in , i.e. each component of the intersection is homotopic into the ends of . Then the image of is contained in .

###### Proof.

Let be the cohomology class dual to . Since meets inessentially, every loop in can be pushed off of so vanishes on . But the kernel of in is exactly , so the image of is in . ∎

## 3. Sections and pockets of the veering triangulation

In this section the surface is fully-punctured. A section of the veering triangulation is an embedding which is simplicial with respect to an ideal triangulation of , and is a section of the fibration (hence transverse to the vertical flow). By simplicial we mean that the map takes simplices to simplices. The edges of are saddle connections of that are also edges of (i.e. those which span singularity-free rectangles), and indeed any triangulation by -edges gives rise to a section. We will abuse terminology a bit by letting denote both the triangulation and the section.

A diagonal flip between sections is an isotopy that pushes through a single tetrahedron of , either above it or below it. Equivalently, if is a maximal rectangle and its associated tetrahedron, the bottom two faces of might appear in , in which case would be obtained by replacing these with the top two faces. This is an upward flip, and the opposite is a downward flip. We will refer to the transition as both a diagonal flip/exchange and a tetrahedron move, depending on the perspective.

An edge of can be flipped downward exactly when it is the tallest edge, with respect to , among the edges in either of the two triangles adjacent to it. This makes the top edge of a tetrahedron (i.e. the diagonal of a quadrilateral that connects the horizontal sides of the corresponding rectangle). Similarly it can be flipped upward when it is the widest edge among its neighbors. See Figure 7.

In particular it follows that every section has to admit both an upward and downward flip – simply find the tallest edge and the widest edge.

However it is not a priori obvious that a section even exists. Guéritaud gives an argument for this and more:

###### Lemma 3.1 ([Gué15]).

There is a sequence of sections separated by upward diagonal flips, which sweeps through the entire manifold . Moreover, when covers the manifold , this sequence is invariant by the deck translation .

We remark that Agol had previously proven a version of Lemma 3.1 with his original definition of the veering triangulation [Ago11].

For an alternative proof that sections exist, see the second proof of Lemma 3.2. We remark that Lemma 3.1 does not give a complete picture of all possible sections of . In this section we will establish a bit more structure.

For a subcomplex , denote by the collection of sections of containing the edges of . A necessary condition for to be nonempty is that is an embedded complex in composed of -simplices. We will continue to blur the distinction between and .

Our first result states that the necessary condition is sufficient:

###### Lemma 3.2 (Extension lemma).

Suppose that is a collection of -edges in with pairwise disjoint interiors. Then is nonempty.

The second states that is always connected by tetrahedron moves. This includes in particular the case of , the set of all sections.

###### Proposition 3.3 (Connectivity).

If is a collection of -edges in with pairwise disjoint interiors, then is connected via tetrahedron moves.

### Finding flippable edges

Let be a section and let be an edge of , which is not an edge of . Any edge of crossing must do so from top to bottom () or left to right (), as in Section 2.1, and we further note that all edges of that cross do it consistently, all top-bottom or all left-right, since they are disjoint from each other.

###### Lemma 3.4.

Let be a section and suppose that an edge of is crossed by an edge of . If , then there is an edge of crossing which is downward flippable. Similarly if then there is an edge of crossing which is upward flippable.

###### Proof.

Assuming the crossings of are top to bottom, let be the edge crossing that has largest height with respect to . Let be a triangle of on either side of and let be its tallest edge. Drawing the rectangle in which is inscribed (Figure 8) one sees that , the rectangle of , is forced to cross it from left to right. Hence, the edge must also cross . Therefore, by choice of . It follows that is a downward flippable edge. ∎

### Pockets

Let and be two sections and their intersection, as a subcomplex in . Because both sections are embedded copies of transverse to the suspension flow, their union divides into two unbounded regions and some number of bounded regions. Each bounded region is a union of tetrahedra bounded by two isotopic subsurfaces of and , which correspond to a component of the complement of in . The isotopy is obtained by following the flow, and if it takes the subsurface of upward to the subsurface of we say that lies above in . We call a pocket over , and sometimes write . We call the base of the pocket .

###### Lemma 3.5.

With notation as above, lies above in the pocket if and only if, for every edge of in and edge of in , if and cross then .

Note that, for each edge of in there is in fact an edge of in which crosses , since both and are triangulations, with no common edges in .

###### Proof.

Suppose that lies above in and let be an edge of in ; hence, it is in the top boundary of . Let be the tetrahedron of for which is the top edge. Via the local picture around (see Section 2.1 and Figure 4), we see that lies locally below . Its interior is of course disjoint from and (and the whole - skeleton), hence it is inside . Let be the bottom edge of . Note . If is in , stop (with ). Otherwise it is in the interior of , and we can repeat with the tetrahedron for which is the top edge. We get a sequence of steps terminating in some in , which must be in the boundary of , and conclude (by the transitivity of as in Section 2.1). Now from the paragraph before Lemma 3.4, the same slope relation holds for every edge of crossing , hence giving the first implication of the lemma. For the other direction, exchange the roles of and in the proof. ∎

### Connectedness of T(k)

We can now prove Proposition 3.3.

###### Proof.

Let us consider , in . Let be one of the pockets, and suppose lies above in . Lemma 3.5 together with Lemma 3.4 implies that