# Increasing the number of fibered faces
of

arithmetic hyperbolic 3-manifolds

###### Abstract.

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston’s Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried’s dynamical characterization of the fibered faces. The origin of the basic fibration is the modular elliptic curve , which admits multiplication by the ring of integers of . We first base change the holomorphic differential on to a cusp form on over , and then transfer over to a quaternion algebra ramified only at the primes above ; the fundamental group of is a quotient of the principal congruence subgroup of of level . To analyze the topological properties of , we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.

## 1. Introduction

The most mysterious variant of the circle of questions surrounding Waldhausen’s Virtual Haken Conjecture [Waldhausen68] is:

###### 1.1 Virtual Fibration Conjecture (Thurston).

If is a finite-volume hyperbolic 3-manifold, then has a finite cover which fibers over the circle, i.e. is a surface bundle over the circle.

This is a very natural question, equivalent to asking whether contains a geometrically infinite surface group. However, compared to the other forms of the Virtual Haken Conjecture, there are relatively few non-trivial examples where it is known to hold, especially in the case of closed manifolds (but see [Reid:bundles, Leininger2002, Walsh2005, Button2005] and especially [Agol2007]). Moreover, there are indications that fibering over the circle is, in suitable senses, a rare property compared, for example, to simply having non-trivial first cohomology [DunfieldDThurston, Masters2005].

Despite this, we show here that certain manifolds satisfy Conjecture 1.1 in a very strong way, in that they have finite covers which fiber over the circle in many distinct ways. For a 3-manifold , the set of classes in which can be represented by fibrations over the circle are organized by the Thurston norm on . The unit ball in this norm is a finite polytope where certain top-dimensional faces, called the fibered faces, correspond to those cohomology classes coming from fibrations (see Section 2 for details). The number of fibered faces thus measures the number of fundamentally different ways that can fiber over the circle. We will sometimes abusively refer to these faces of the Thurston norm ball as “the fibered faces of ”.

If is a finite covering map, the induced map takes each fibered face of to one in ; if is strictly larger than , then it may (but need not) have additional fibered faces. The qualitative form of our main result is:

###### 1.2 Theorem.

There exists a closed hyperbolic 3-manifold which has a sequence of finite covers so that the number of fibered faces of the Thurston norm ball of goes to infinity.

Moreover, we prove a quantitative refinement of this result (Theorem 1.4) which bounds from below the number of fibered faces of in terms of the degree of the cover. While it is the closed case of Conjecture 1.1 that is most interesting, we also give an example of a non-compact finite-volume hyperbolic 3-manifold with the same property (Theorem LABEL:thm_whitehead) using a simple topological argument.

The example manifold of Theorem 1.2 is arithmetic, and the proof uses detailed number-theoretic information about it, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried’s dynamical characterization of the fibered faces. To state the geometric part of the theorem, we need to introduce the Hecke operators (see Section 3.1 for details). Suppose is a closed hyperbolic 3-manifold, and we have a pair of finite covering maps ; when is arithmetic there are many such pairs of covering maps because the commensurator of in is very large. The associated Hecke operator is the endomorphism of defined by , where is the transfer map. The simplest form of our main geometric lemma is the following:

###### 1.3 Lemma.

Let be a closed hyperbolic 3-manifold, and a pair of finite covering maps. If for some coming from a fibration over the circle, then and lie in distinct fibered faces.

We prove this lemma in Section 3. Then in Section LABEL:sec_example, we give a manifold with an infinite tower of covers to which Lemma 1.3 applies at each step, thus proving Theorem 1.2. In fact, we show the following refined quantitative version:

###### 1.4 Theorem.

There is a closed hyperbolic 3-manifold of arithmetic type, with an infinite family of finite covers of degree , where the number of fibered faces of satisfies

In particular, for any , there is a constant such that

Note that this bound for is slower than any positive power of , but is faster than any (positive) power of . For context, the Betti number is bounded above by (a constant times) the degree , and bounded below (see Proposition LABEL:prop_lower_bound), for any , by (a constant times) for large enough (relative to ), while is at least as large as (a constant times) . In our non-compact example of Theorem LABEL:thm_whitehead, the number of faces grows exponentially in the degree while grows linearly.

We now describe the basic ideas behind the construction of the manifolds in Theorem 1.4. For an arithmetic hyperbolic 3-manifold , one has a Hecke operator as above associated to each prime ideal of the field of definition. The key to applying Lemma 1.3 repeatedly is to have a fibered class which is killed by infinitely many such Hecke operators. One can produce cohomology classes which are killed by infinitely many Hecke operators using the special class of CM forms. Fortunately, there is a manifold of manageable size whose cohomology contains a CM form coming from base change of an automorphic form associated to a certain elliptic curve with complex multiplication, and that class turns out to fiber over the circle!

### 1.1. Outline of the arithmetic construction

Here is a sketch of how the manifolds of Theorem 1.4 are built from arithmetic; for details, see Section LABEL:sec_example. Let be the elliptic curve over defined by , which has conductor and admits complex multiplication by the ring of integers of the imaginary quadratic field . It corresponds to a holomorphic cusp form of weight for the congruence subgroup of acting on the upper half plane , given by with , where for every prime , the eigenvalue of under the Hecke operator is . The differential is invariant under , and hence defines a holomorphic -form on , and by the cuspidality of , this differential extends to the natural (cusp) compactification of . Put , in which the prime splits as , with . Let denote the base change of to , which is a cusp form on the hyperbolic -space of “weight ” and level for the group . One can associate to a cuspidal automorphic representation of with trivial central character and conductor ; here denotes the adèle ring of . By a basic property of base change, we have, for every degree prime of above a rational prime unramified in , the Hecke eigenvalue of equals the Hecke eigenvalue of , namely .

Let be the quaternion division algebra over which is ramified exactly at the primes and . Fortunately for us, the local components of at and are both supercuspidal (see Lemma 5.3); this has to do with the fact that does not acquire good reduction over an abelian extension of , this fact being controlled, thanks to a useful criterion of D. Rohrlich [Rohr], by the valuation (at and ) of the discriminant of . (After writing this paper, we learned of an earlier proof of the supercuspidality at in [GL], where the argument is somewhat different.) Therefore, by the Jacquet-Langlands correspondence, there is an associated cusp form of weight on relative to a congruence subgroup of the units in a maximal order , such that for every unramified prime , the Hecke eigenvalues of and of coincide. Moreover, for , we can read off the conductor and the dimension of the associated representation of from the local correspondence (see Lemma 5.5). It follows that , which is co-compact, is the principal congruence subgroup of level . Our base manifold in Theorem 1.4 is . While is 3-dimensional, we show that the new subspace of is 2-dimensional and, as a module under the Hecke algebra of correspondences, is isotypic of type , the cohomology class defined by . A difficult computation shows that fibers over the circle, and moreover, this can be associated with a class of type .

Now consider the set of rational primes which are inert in , but are split in as ; it has density . Then, if we put

the Hecke operators act by zero on . Each in gives a covering of of degree which defines the associated Hecke operator. Lemma 1.3 allows us, since , to conclude that the two natural transforms of define cohomology classes of which lie on two different fibered faces. If we order according to the rational primes defining them, then we inductively build a covering of degree such that there are at least distinct fibered faces on . Using the density of , we get the lower bound for given in Theorem 1.4 in terms of .

To understand why we chose to look at this particular example, it may be helpful to note the following. Suppose is an indefinite quaternion algebra over , and a congruence subgroup of with associated Shimura curve over , which is a compact Riemann surface. For any imaginary quadratic field such that is still a division algebra, we may consider the hyperbolic -manifold , for a congruence subgroup of . When , the surface embeds in , and since it is totally geodesic, the cohomology class defined by cannot give rise to any fibering of over the circle [ThurstonFibered]. This suggests that we will fail to construct cohomology classes on with the desired fibering property if we transfer to a cusp form on which is the base change of an elliptic modular cusp form of weight which will transfer to such a . It is not an accident that we chose our example above where the of interest does not transfer to any indefinite quaternion algebra over .

Finally, it should be noted that if one starts with a non-CM elliptic curve over , then one knows, by a theorem of Elkies, that there are infinitely many primes for which is zero [Elkies1987]. But for our method to work we would also need an example where this property holds for an infinite set of primes which split in a suitable imaginary quadratic field . Even then it would only give a qualitative result as would have density zero. Our quantitative result (Theorem 1.4) depends on the density of the corresponding in the CM case being .

### 1.2. Moral

For an arithmetic hyperbolic 3-manifold, the commensurator of its fundamental group is very large, in fact dense in . Recently, there has been much important work which exploits this density geometrically, see [LackenbyLongReid2006, CooperLongReid2006, Venkataramana2006, Agol2006]. On the number-theoretic side, the theory of automorphic forms tells us a great deal about the cohomology of arithmetic hyperbolic 3-manifolds as it relates to virtual Haken type questions (see e.g. [Clozel1987, DunfieldFCalegari2006]). To the best of our knowledge, our work here is the first time that more refined automorphic information has been combined with geometric arguments and yields, for example, a geometric/topological interpretation of the vanishing of the Hecke eigenvalues. Thus we hope for deeper connections between these two areas in the future. In particular, it would be very interesting to answer the following:

###### 1.5 Question.

Is there an automorphic criterion which implies that certain cohomology classes of arithmetic hyperbolic 3-manifold give fibrations over the circle?

### 1.3. Practical methods for computing the Thurston norm

The example manifold of Theorem 1.4 is quite complicated from a topological point of view; its hyperbolic volume is about 100 and triangulations of it need some 130 tetrahedra. Despite this, we were able to compute its Thurston norm and check that it fibers over the circle, which is necessary for the proof of Theorem 1.4. To do this, we used new methods for both these tasks. While loosely based on normal surfaces, these techniques eschew guaranteed termination in favor of quick results. The basic idea is to consider only normal surfaces representing elements of that are “obvious” with respect to the triangulation, and then randomize the triangulation until a minimal norm surface is found. These same techniques have been useful in many other examples and are of independent interest. See Sections LABEL:subsec_ex_thurston and LABEL:subsec-fibering-M for a complete description of our method, which can often determine whether a manifold made up of several hundred tetrahedra fibers. Computing the Thurston norm is more subtle, and in the case of our particular , we had to heavily exploit its symmetries, but we also suggest a general method, as of yet untested, for attacking this.

### 1.4. Improvements

In subsequent work, Long and Reid have considerably strengthened Lemma 1.3 and its extension Theorem LABEL:cong_cover_thm by showing that the hypothesis on the Hecke operators can be dispensed with:

###### 1.6 Theorem (Long and Reid [LongReid2007]).

Let be a closed arithmetic hyperbolic 3-manifold. If fibers over the circle, then has finite covers whose Thurston norm balls have arbitrarily many fibered faces.

In addition to the work of Fried [Fried1979] on which Lemma 1.3 hinges, the proof of Theorem 1.6 uses work of Cooper, Long, and Reid on suspension pseudo-Anosov flows [CooperLongReid1994], as strengthened by Masters [Masters2006]. In Section LABEL:sec_long_reid we give a simplified proof of Theorem 1.6 using only Fried’s theorem; a different, but equally concise, simplification was given by Agol [Agol2007]. In any case, the soft nature of its proof mean that Theorem 1.6 cannot be used to prove quantitative results such as Theorem 1.4. However, Theorem 1.6 does have the advantage that it is much easier to apply.

In a major breakthrough, Agol has just shown there are infinitely many commensurability classes of arithmetic hyperbolic 3-manifolds which fiber over the circle [Agol2007]. Combining with Theorem 1.6, this means the qualitative behavior of Theorem 1.2 actually occurs in an infinite number of examples, providing further evidence for Conjecture 1.1.

In Theorems 1.4 and 1.6, the distinct fibered faces are all equivalent under the isometry group of the cover manifolds; indeed this is intrinsic to the method. A natural question is whether one can find a tower of covers where the fibered faces fall into arbitrarily many classes modulo isometries. In the case of manifolds with cusps, Theorem LABEL:thm_whitehead gives such examples since the number of faces grows exponentially in the degree of the cover, whereas the size of the isometry group of a hyperbolic manifold is bounded linearly in the volume.

### 1.5. Paper outline

In Section 2, we review the basics of the Thurston norm and Fried’s dynamical characterization of the fibered faces. In Section 3, we discuss Hecke operators and congruence covers in the context of arithmetic hyperbolic 3-manifolds, and then prove Lemma 1.3 and its generalization Theorem LABEL:cong_cover_thm which underpins Theorem 1.4. We give the precise description of the manifold used in Theorem 1.4 in Section LABEL:sec_example. We then analyze the automorphic and topological properties of this manifold in Sections LABEL:sec-automorphic and LABEL:sec-topological-props respectively. In Section LABEL:sec_proof_of_main_thm we assemble the pieces and prove Theorem 1.4. Section LABEL:sec-whitehead gives our example of this phenomenon in the case of hyperbolic 3-manifolds with cusps. Finally, Section LABEL:sec_long_reid gives our simplified proof of Theorem 1.6.

### 1.6. Acknowledgments

Dunfield was partially supported by a Sloan Fellowship and US NSF grants DMS-0405491 and DMS-0707136, and some of this work was done while he was at Caltech. Ramakrishnan was partially supported by US NSF grants DMS-0402044 and DMS-0701089. We thank Philippe Michel for a useful discussion concerning the quantitative version of our main result (Theorem 1.4), and Danny Calegari for suggesting that we look at families of link complements in the cusped case, which led us to Theorem LABEL:thm_whitehead. We also thank Darren Long and Alan Reid for sending us an early version of [LongReid2007], and Benedict Gross for informing us of an earlier proof of Lemma 5.3 in [GL]. Finally, we thank the authors of the software packages [SnapPea, Snap, Cremona1984, MAGMA214, SAGE], without which this paper would not have been possible, as well as the referee for carefully reading the paper and suggesting improvements in exposition.

## 2. Fibered faces of the Thurston norm ball

Let be a closed orientable hyperbolic 3-manifold. When a cohomology class can be represented by a fibration , we say that it fibers. In this section, we review the work of Thurston and Fried on the structure of the set of fibered classes in . It is not hard to see that fibers if and only if it can be represented by a nowhere vanishing 1-form: a fibration gives rise to such a form by pulling back the standard orientation 1-form on , and conversely such a form can be integrated (since its periods are integers) into a map to which is a fibration. If we pass to real coefficients, the latter condition clearly defines an open subset of ; a fundamental result of Thurston shows that must have the following very restricted form. This set is prescribed by the Thurston norm, which measures the simplest surface that represents the Poincaré dual of a cohomology class.

More precisely, for define its Thurston norm by

where is the Euler characteristic of , and we further require that has no 2-sphere components. Thurston showed that this gives a norm on which extends continuously to one on ; see [ThurstonNorm] for details of the assertions made in this paragraph. Moreover, the unit ball of this norm is a bounded convex polytope, i.e. the convex hull of finitely many points. Moreover, there are top-dimensional faces of , called the fibered faces, so that fibers if and only if it lies in the cone over the interior of a fibered face, that is, the ray from the origin through intersects the interior of such a face. We will say that such a fibered lies in the corresponding fibered face. Finally, as the map preserves fibering, the fibered faces come in pairs interchanged by this symmetry of the Thurston norm ball. When determining the number of fibered faces, we often count in terms of these fibered face pairs, and say that fibered classes and lie in genuinely distinct fibered faces if both and are not in the fibered face of .

### 2.1. Behavior of fibered faces under covers

Now suppose is a finite covering map. The natural map is an embedding, and a deep theorem of Gabai shows that preserves the Thurston norm [Gabai83, Cor. 6.18]. Equivalently, if we denote the unit Thurston norm balls of and by and respectively, we have . By work of Stallings, a class represents a fibration if and only if does [Stallings62]. In particular, if fibers but does not, then is larger than . Thus each fibered face of gives rise to a distinct fibered face of ; if has additional cohomology, we can hope that has new fibered faces, but the interiors of these must be disjoint from .

### 2.2. Fried’s work

We now turn to Fried’s dynamical characterization of when two fibrations lie in the same fibered face, which is in terms of a certain flow that is transverse to the fibers of the fibration . Suppose is a self-homeomorphism of a closed surface of genus at least 2, and consider the mapping torus with monodromy :

Thurston proved that is hyperbolic if and only if is what is called pseudo-Anosov [ThurstonFibered, Otal96]. The latter means that is isotopic to a homeomorphism which preserves a pair of foliations of in a controlled way. Henceforth, we always assume that has been isotoped to such a preferred representative. Now, has a natural suspension flow which is transverse to the circle fibers, where a point moves at unit speed in the –direction. We will call the transverse pseudo-Anosov flow.

Conversely, given coming from a fibration, the monodromy of the bundle structure is well-defined up to isotopy. Thus there is a corresponding transverse pseudo-Anosov flow , which is well-defined up to isotopy. Fried’s first result is:

###### 2.1 Theorem (Fried [Fried1979]).

Let be a closed orientable hyperbolic 3-manifold. Then two fibrations of over the circle lie over the same fibered face if and only if the corresponding transverse pseudo-Anosov flows are isotopic.

Fried also provided the following characterization of those lying over a fibered face. Let be the flow associated to a particular fibered face , and let be the set of homology directions of , namely the set of all accumulation points of the homology classes of long, nearly closed orbits of . Fried showed that the dual cone to ,

is precisely the cone on the interior of the fibered face [Fried1979, Thm. 7].

One kind of element of is the homology class of a closed flowline of . There are always closed flowlines, for instance coming from the orbits of the finitely many singular points of the invariant foliations, which are permuted among themselves by the monodromy. The particular consequence of Fried’s work that we will use here, which is immediate from our discussion, is:

###### 2.2 Lemma (Fried).

Let be a closed orientable hyperbolic 3-manifold, with fibered classes . Let be a closed orbit of the transverse pseudo-Anosov flow for . If , then and lie over genuinely distinct fibered faces.

## 3. The geometric theorem

We begin this section by proving Lemma 1.3, which contains the key geometric idea of this paper: a fibered cohomology class which is annihilated by a Hecke operator gives rise to two genuinely distinct fibered faces in the corresponding cover. We then give a detailed review of Hecke operators for arithmetic hyperbolic 3-manifolds, and end by proving the complete version of our geometric result (Theorem LABEL:cong_cover_thm) which is needed to prove Theorem 1.4.

### 3.1. Hecke operators

From a geometric point of view, a Hecke operator is the map on cohomology induced from the following setup. Suppose is a topological space, and we have a pair of finite covering maps . The map on singular chains which takes a singular simplex to the sum of its inverse images under induces transfer homomorphisms and which run in opposite directions to the usual maps that induces on (co)homology (see e.g. [HatcherBook, §3.G] for details). The Hecke operator of this pair of covering maps is the endomorphism of defined by , where the pullback map and is the transfer map.

###### 3.1 Remark.

In this paper, will always be a 3-manifold and we will be interested in the Hecke operator on . While the argument below is given purely in terms of cohomology, the geometrically minded reader may prefer to contemplate the Poincaré dual group . There, the Hecke operator on is the composite , where is the transfer map. Such Hecke operators commute with the Poincaré duality isomorphism , and so it makes no difference whether one takes the homological or cohomological point of view. The action of on a class is particularly simple to think of geometrically: If an embedded surface represents , then the immersed surface represents .

### 3.2. Main geometric idea

When is arithmetic, there are many manifolds which cover it in distinct ways. Before getting into this, let us give the central topological idea of Theorem 1.2 in its simplest form from the introduction:

###### 1.3 Lemma.

Let be a closed hyperbolic 3-manifold, and suppose comes from a fibration of over the circle. Further assume that are a pair of finite covering maps. If then and lie in genuinely distinct fibered faces.

###### Proof.

We will apply Lemma 2.2 to justify our claim. Downstairs in , let be a closed orbit of the pseudo-Anosov flow associated to the fibration . Then is a closed orbit of the flow associated to the fibration . To calculate , note that for any one has and hence

as required to apply the lemma. ∎

In the case of a tower of covers of , the following strengthening of the previous lemma will be needed to work inductively:

###### 3.2 Lemma.

Let be a closed hyperbolic 3-manifold, and be a pair of finite covering maps. Suppose lie in genuinely distinct fibered faces. If for all , then has at least pairs of fibered faces. More precisely, lie in genuinely distinct fibered faces.

###### Proof.

By the discussion in Section 2.1, it is clear that the lie in distinct fibered faces, as do the . Thus it remains to distinguish the face of from that of . As before, let be a closed orbit of the pseudo-Anosov flow associated to , so that is an orbit of the flow associated to . Then

and so Lemma 2.2 applies as needed. ∎

### 3.3. Sources of Hecke operators

Now we turn to the source of such multiple covering maps. For a hyperbolic 3-manifold , let be its fundamental group, thought of as a lattice in . The commensurator of is the subgroup

When is arithmetic, is dense in , and Margulis showed that the converse is true as well; indeed, if is not arithmetic, then has finite index in .

Regardless, can be associated to a Hecke operator as follows. Let , and be the corresponding closed hyperbolic 3-manifold. Consider the finite covering map induced by the inclusion . We will define a second such covering map by considering , and analogously . Now, as , the action of on induces an isometry giving us the picture