Geodesics and compression bodies

Geodesics and compression bodies

Abstract.

We consider hyperbolic structures on the compression body with genus 2 positive boundary and genus 1 negative boundary. Note that deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus. We call this arc the core tunnel of . We conjecture that, in any geometrically finite structure on , the core tunnel is isotopic to a geodesic. By considering Ford domains, we show this conjecture holds for many geometrically finite structures. Additionally, we give an algorithm to compute the Ford domain of such a manifold, and a procedure which has been implemented to visualize many of these Ford domains. Our computer implementation gives further evidence for the conjecture.

1. Introduction

For a hyperbolic manifold with torus boundary component , every homotopically nontrivial arc in with endpoints on is homotopic to a geodesic. However, it seems to be a difficult problem to identify arcs in which are isotopic to a geodesic, given only a topological description of .

One place this problem arises is in the study of unknotting tunnels. An unknotting tunnel for a 3–manifold with torus boundary components is defined to be an arc from to such that is a handlebody. Manifolds (other than a solid torus) that admit unknotting tunnels are tunnel number one manifolds. Adams asked whether the unknotting tunnel of a hyperbolic tunnel number one manifold is always isotopic to a geodesic [adams:tunnels]. This has been shown to be the case for many classes of hyperbolic tunnel number one manifolds ([adams-reid], [sakuma-weeks]). Recently, Cooper, Futer, and Purcell showed that the conjecture is true for a generic manifold, in an appropriate sense of generic [cfp:tunnels]. The original question still remains open, however.

The purpose of this paper is to present and motivate a related question. Any tunnel number one manifold is built by attaching a compression body to a handlebody, and the unknotting tunnel corresponds to an arc in the compression body. We call the core tunnel of . Given Adams’ question on whether an unknotting tunnel is isotopic to a geodesic, it seems natural to ask whether the arc is isotopic to a geodesic under a complete hyperbolic structure on .

The compression body admits many complete hyperbolic structures. Here, we examine those that are geometrically finite, and show that for many such structures, the core tunnel is isotopic to a geodesic. In order to investigate such structures, we develop algorithms to find the Ford domains for geometrically finite structures on . We present one algorithm that is guaranteed to find the Ford domain in finite time and terminate, but which is impractical in practice, and a procedure which has been implemented for the computer, which will find the Ford domain and terminate for large families of geometrically finite structures, and which we conjecture will always find the Ford domain.

Computer investigation and the theorems proven for families of geometrically finite hyperbolic structures lead us to the following conjecture.

Conjecture 1.1.

Let be a compression body with a torus, and a genus two surface. Suppose is given a geometrically finite hyperbolic structure. Then the core tunnel of is isotopic to a geodesic.

In fact, we conjecture something stronger. We conjecture that the core tunnel is not only isotopic to a geodesic, but always dual to a face of the Ford domain. This is Conjecture 5.11, explained in Section 5.

The techniques of this paper can be seen as an extension of work of Jørgensen [jorgensen], who found Ford domains of geometrically finite structures on , where is a once–punctured torus. Jørgensen’s work was extended and expanded by others, including Akiyoshi, Sakuma, Wada, and Yamashita [aswy, aswy:book]. Wada implemented an algorithm to determine Ford domains of these manifolds [wada:opti].

A complete understanding of the geometry of compression bodies, for example through a study of Ford domains, could lead to many interesting applications, since compression bodies are building blocks of more complicated manifolds via Heegaard splitting techniques. With Cooper, we have already applied some of the ideas in this paper to build tunnel number one manifolds with arbitrarily long unknotting tunnels [clp:length].

1.1. Acknowledgements

Both authors were supported by the Leverhulme trust. Lackenby was supported by an EPSRC Advanced Research Fellowship. Purcell was supported by NSF grants and the Alfred P. Sloan foundation.

2. Background and preliminary material

In this section we review terminology and results used throughout the paper. Our intent is to make this paper as self–contained as possible, and also to emphasize relations between the geometry and topology of compression bodies.

First, we review definitions and results on compression bodies, which are the manifolds we study. Next, we review what it means for these manifolds to admit a geometrically finite hyperbolic structure. We then recall the definition of a Ford domain, since we will be using Ford domains to examine geometrically finite hyperbolic structures on compression bodies. We also give a few definitions relevant to Ford domains, such as visible isometric spheres, Ford spines, and complexes dual to Ford spines. Ford domains of geometrically finite manifolds are finite sided polyhedra; thus we can often identify a Ford domain using the Poincaré polyhedron theorem. Finally, we review this theorem and some of its relevant consequences.

2.1. Compression bodies

The manifolds we study in this paper are compression bodies with negative boundary a single torus, and positive boundary a genus surface.

Recall that a compression body is either a handlebody, or the result of taking the product of a closed, oriented (possibly disconnected) surface and the interval , and attaching 1–handles to such that the result is connected. The negative boundary is and is denoted . When is a handlebody, . The positive boundary is , and is denoted .

Let be the compression body for which is a torus and is a genus surface. We will call this the –compression body, where the numbers refer to the genus of the boundary components. Note the –compression body is formed by taking a torus crossed with and attaching a single 1–handle to . The 1–handle retracts to a single arc, the core of the 1–handle.

Let be the union of the core of the 1–handle with two vertical arcs in attached to its endpoints. Thus, is a properly embedded arc in , and is a regular neighborhood of . We refer to as the core tunnel of . See Figure 1, which first appeared in [clp:length].

Figure 1. The –compression body. The core tunnel is the thick line shown, with endpoints on the torus boundary.

The fundamental group of a –compression body is isomorphic to . We will denote the generators of the factor by , , and we will denote the generator of the second factor by .

2.2. Hyperbolic structures

We are interested in the isotopy class of the arc when we put a complete hyperbolic structure on the interior of the –compression body . We obtain such a structure by taking a discrete, faithful representation and considering the manifold .

Definition 2.1.

A discrete subgroup is geometrically finite if admits a finite–sided, convex fundamental domain. In this case, we will also say that the manifold is geometrically finite.

The following gives a useful fact about geometrically finite groups in .

Theorem 2.2 (Bowditch, Proposition 5.7 [bowditch]).

If a subgroup is geometrically finite, then every convex fundamental domain for has finitely many faces.

Definition 2.3.

A discrete subgroup is minimally parabolic if it has no rank one parabolic subgroups.

Thus for a discrete, faithful representation , the image will be minimally parabolic if for all , the element is parabolic if and only if is conjugate to an element of the fundamental group of a torus boundary component of .

Definition 2.4.

A discrete, faithful representation is a minimally parabolic geometrically finite uniformization of if is minimally parabolic and geometrically finite, and is homeomorphic to the interior of .

2.3. Isometric spheres and Ford domains

To examine structures on , we examine paths of Ford domains. This is similar to the technique of Jørgensen [jorgensen], developed and expanded by Akiyoshi, Sakuma, Wada, and Yamashita [aswy:book], to study hyperbolic structures on punctured torus bundles. Much of the basic material on Ford domains which we review here can also be found in [aswy:book].

Throughout this subsection, let be a hyperbolic manifold with a single rank 2 cusp, for example, the –compression body. In the upper half space model for , assume the point at infinity in projects to the cusp. Let be any horosphere about infinity. Let denote the subgroup that fixes . By assumption, .

Definition 2.5.

For any , will be a horosphere centered at a point of , where we view the boundary at infinity of to be . Define the set to be the set of points in equidistant from and . Then is the isometric sphere of .

Note that is well–defined even if and overlap. It will be a Euclidean hemisphere orthogonal to the boundary of .

The following is well known, and follows from standard calculations. We include a proof for completeness.

Lemma 2.6.

If

then the center of the Euclidean hemisphere is . Its Euclidean radius is .

Proof.

The fact that the center is is clear.

Consider the geodesic running from to . It consists of points of the form in . It will meet the horosphere about infinity at some height , and the horosphere at some height . The radius of the isometric sphere is the height of the point equidistant from points and .

Note that , and hence is given by the height of , which can be computed to be . Thus . Then the point equidistant from and is the point of height . ∎

Definition 2.7.

Let denote the open half ball bounded by , and define to be the set

Note is invariant under , which acts by Euclidean translations on . We call the equivariant Ford domain.

When bounds a horoball that projects to an embedded horoball neighborhood about the rank 2 cusp of , is the set of points in which are at least as close to as to any of its translates under . Provided is discrete, such an embedded horoball neighborhood of the cusp always exists, by the Margulis lemma.

Definition 2.8.

A vertical fundamental domain for is a fundamental domain for the action of cut out by finitely many vertical geodesic planes in .

Definition 2.9.

A Ford domain of is the intersection of with a vertical fundamental domain for the action of .

A Ford domain is not canonical because the choice of fundamental domain for is not canonical. However, the equivariant Ford domain in is canonical, and for purposes of this paper, is often more useful than the actual Ford domain.

Note that Ford domains are convex fundamental domains (cf. [aswy:book, Proposition A.1.2]). Thus we have the following corollary of Bowditch’s Theorem 2.2.

Corollary 2.10.

is geometrically finite if and only if a Ford domain for has a finite number of faces.

Example 2.11.

Let be any complex number such that , and let and in be linearly independent over with , . Let be the representation defined by

(Recall that and denote the generators of the factor of , and denotes an additional generator of .)

By Lemma 2.6, has center , radius , and has center , radius . Since , will not meet . By choice of , , all translates of and under are disjoint.

We will see in Lemma 2.27 that gives a minimally parabolic geometrically finite uniformization of , and that for this example, consists of the exterior of (open) half–spaces and , bounded by and , respectively, as well as translates of these two isometric spheres under . Thus we will show that the Ford domain for this example is as shown in Figure 2. Before proving this fact, we need additional definitions and lemmas. We use this example to illustrate these definitions and lemmas.

Figure 2. Left: Schematic picture of the Ford domain of Example 2.11. Right: Three dimensional view of in , for , , and .

2.4. Visible faces and Ford domains

Let be a hyperbolic manifold with a single rank two cusp, and let denote a maximal rank two parabolic subgroup, which we may assume fixes the point at infinity in . Notice that , the equivariant Ford domain of , has a natural cell structure.

Definition 2.12.

Let . We say is visible if there exists a 2–dimensional cell of the cell structure on contained in .

Similarly, we say the intersection of isometric spheres is visible if there exists a cell of contained in of the same dimension as .

Thus in Example 2.11, we claim that the only visible isometric spheres are , , and the translates of these under . There are no visible edges for this example.

There is an alternate definition of visible, Lemma 2.13. Let be a horosphere about infinity that bounds a horoball which is embedded under the projection to .

Lemma 2.13.

For , is visible if and only if there exists an open set such that is not empty, and for every and every , the hyperbolic distances satisfy

Similarly, if is not empty, then it is visible if and only if there exists an open such that is not empty, and for every and every ,

Proof.

An isometric sphere, or interesction of isometric spheres, is visible if and only if it contains a cell of of the same dimension. This will happen if and only if there is some open set in which intersects the isometric sphere, or intersections of isometric spheres, in the cell of in . The result follows now by definition of : a point is in if and only if it is not contained in any open half space , , if and only if . ∎

We can say something even stronger for isometric spheres:

Lemma 2.14.

For discrete, the following are equivalent.

  1. The isometric sphere is visible.

  2. There exists an open set such that is not empty and for any and any ,

  3. is not contained in .

Proof.

If (2) holds, then Lemma 2.13 implies is visible. Conversely, suppose is visible. Let be as in Lemma 2.13, so that for all , and all , . Suppose there is some such that for all we have equality: . Then the isometric spheres and must agree on an open subset, hence they must agree everywhere. In particular, their centers must agree: .

Now, notice that is the subgroup of fixing , since fixes if and only if fixes infinity, so lies in . Next note that since , fixes . So . Thus . We have shown (1) if and only if (2).

Finally, (2) clearly implies (3). If is not visible, then for any , either , which implies , or is in a cell of with dimension at most . In this case, for some . Thus (3) implies (1). ∎

Notice that in the above proof, we showed that if two isometric spheres and agree, then . It is clear that if , then .

We now present two results on visible faces of the Ford domain. Again these are well known, but we include proofs for completeness.

Lemma 2.15.

Let be a discrete, torsion free subgroup of with a rank two parabolic subgroup fixing the point at infinity, and let . Then is visible if and only if is visible. Moreover, takes isometrically to , sending the half space bounded by to the exterior of the half space .

Proof.

Let be a horosphere about infinity in that bounds a horoball which projects to an embedded neighborhood of the cusp of .

First, note that under , is mapped isometrically to , since takes to , and to , and hence takes to the set of points equidistant from these two horospheres. This is the isometric sphere . Note the half space , which contains , must be mapped to the exterior of , which contains , as claimed.

Suppose is visible. Then there exists an open set , with not empty, so that for every in , and for every , .

Now consider the action of on this picture. The set is open in , and for all , we have , for some , so the distance , for all . So is visible.

To finish, apply the same proof to . ∎

Lemma 2.16.

Gluing isometric spheres corresponding to and of Example 2.11 gives a manifold homeomorphic to the interior of the –compression body .

Proof.

In the example, first glue sides of the vertical fundamental domain via the parabolic transformations fixing infinity. The result is homeomorphic to the cross product of a torus and an open interval . Next glue the face to via . The result is topologically equivalent to attaching a 1–handle, yielding a manifold homeomorphic to . ∎

Lemma 2.17.

Let be a discrete, torsion free subgroup of with a rank two parabolic subgroup fixing the point at infinity. Suppose , with and visible, and suppose is visible. Then is visible, and maps the visible portion of isometrically to the visible portion of . In addition, there must be some visible isometric sphere , not equal to , such that .

Notice that in Lemma 2.17, may be equal to , but is not necessarily so. In fact, may not be visible, such as in the case that there is a quadrilateral dual to . We discuss dual faces later.

Proof.

Let be a horosphere about infinity which bounds a horoball that projects to an embedded neighborhood of the cusp of . Suppose is visible. By Lemma 2.13, there exists an open set such that for all , and all , the hyperbolic distance is less than or equal to the hyperbolic distance . Since , we also have .

Apply to this picture. We obtain:

for all . Thus for all in the intersection of the open set and , satisfies the inequality of Lemma 2.13, and so is visible. Since this works for any such open set , and the 1–cell of contained in may be covered with such open sets, maps visible portions isometrically.

Finally, since is visible, it contains a 1–dimensional cell of . There must be two 2–dimensional cells of bordering . One of these is contained in , using the fact that is visible and Lemma 2.15. The other must be contained in some (possibly, but not necessarily ), and so this is visible. ∎

The first part of Lemma 2.17 is a portion of what Akiyoshi, Sakuma, Wada, and Yamashita call the chain rule for isometric circles [aswy:book, Lemma 4.1.2].

Additionally, we present a result that allows us to identify geometrically finite uniformizations that are minimally parabolic.

Lemma 2.18.

Suppose is a geometrically finite uniformization. Suppose none of the visible isometric spheres of the Ford domain of are visibly tangent on their boundaries. Then is minimally parabolic.

By visibly tangent, we mean the following. Set , and assume a neighborhood of infinity in projects to the rank two cusp of , with fixing infinity in . For any , the isometric sphere has boundary that is a circle on the boundary at infinity of . This circle bounds an open disk in . Two isometric spheres and are visibly tangent if their corresponding disks and are tangent on , and for any other , the point of tangency is not contained in the open disk .

Proof.

Suppose is not minimally parabolic. Then it must have a rank 1 cusp. Apply an isometry to so that the point at infinity projects to this rank 1 cusp. The Ford domain becomes a region meeting this cusp, with finitely many faces. Take a horosphere about infinity sufficiently small that the intersection of the horosphere with gives a subset of Euclidean space with sides identified by elements of , conjugated appropriately.

The side identifications of this subset of Euclidean space, given by the side identifications of , generate the fundamental group of the cusp. But this is a rank 1 cusp, hence its fundamental group is . Therefore, the side identification is given by a single Euclidean translation. The Ford domain intersects this horosphere in an infinite strip, and the side identification glues the strip into an annulus. Note this implies two faces of are tangent at infinity.

Now apply an isometry, taking us back to our usual view of , with the point at infinity projecting to the rank 2 cusp of the –compression body . The two faces of tangent at infinity are taken to two isometric spheres of the Ford domain, tangent at a visible point on the boundary at infinity. ∎

We will see that the converse to Lemma 2.18 is not true. There exist examples of geometrically finite representations for which two visible isometric spheres are visibly tangent, and yet the representation is still minimally parabolic. Such an example is given, for example, in Example 4.1, with .

Remark 2.19.

In Example 2.11, we claimed that the only visible isometric spheres are those of , , and their translates under . Since none of these isometric spheres are visibly tangent, provided the claim is true, Lemma 2.18 will imply that this representation is minimally parabolic.

2.5. The Ford spine

Let be discrete and geometrically finite. When we glue the Ford domain into the manifold , the faces of the Ford domain will be glued together in pairs to form .

Definition 2.20.

The Ford spine of is defined to be the image of the faces, edges, and 0–cells of under the covering .

A spine usually refers to a subset of the manifold for which there is a retraction of the manifold. Using that definition, the Ford spine is not strictly a spine. However, the union of the Ford spine and the non-toroidal boundary components will be a spine for a manifold with a single rank 2 cusp.

To make that last sentence precise, recall that given a geometrically finite uniformization , the domain of discontinuity is the complement of the limit set of in the boundary at infinity . See, for example, Marden [marden-book, section 2.4].

Lemma 2.21.

Let be a minimally parabolic geometrically finite uniformization of a 3–manifold with a single rank 2 cusp. Then the manifold retracts onto the union of the Ford spine and the boundary at infinity .

Proof.

Let be a horosphere about infinity in that bounds a horoball which projects to an embedded horoball neighborhood of the cusp of . Let be any point in . The nearest point on to lies on a vertical line running from to infinity. These vertical lines give a foliation of . All such lines have one endpoint on infinity, and the other endpoint on or an isometric sphere of . We obtain our retraction by mapping the point to the endpoint of its associated vertical line, then quotienting out by the action of . ∎

To any face of the Ford spine, we obtain an associated collection of visible elements of : those whose isometric sphere projects to (or more carefully, a subset of their isometric sphere projects to the face ).

Definition 2.22.

We will say that an element of corresponds to a face of the Ford spine of if is visible and (the visible subset of) projects to . In this case, we also say corresponds to . Notice the correspondence is not unique: if corresponds to , then so does and for any words .

Remark 2.23.

Consider again the unifomization of given in Example 2.11. We will see that the Ford domain of this example has faces coming from a vertical fundamental domain and the two isometric spheres and . Hence the Ford spine of this manifold consists of a single face, corresponding to .

2.6. Poincaré polyhedron theorem

We need a tool to identify the Ford domain of a hyperbolic manifold. This tool will be Lemma 2.26. The proof of that lemma uses the Poincaré polyhedron theorem, which we use repeatedly in this paper. Those results we use most frequently are presented in this subsection. Our primary reference is Epstein and Petronio [epstein-petronio], which contains a version of the Poincaré theorem that does not require finite polyhedra.

The setup for the following theorems is the same. We begin with a finite number of elements of , , as well as a parabolic subgroup of , fixing the point at infinity. Let be a polyhedron cut out by isometric spheres corresponding to and , as well as either:

  1. all isometric spheres given by translations of and under , or

  2. a vertical fundamental domain for the action of .

An example of the former would be an equivariant Ford domain, . An example of the latter would be a Ford domain. Note that in both cases, we allow to contain an open neighborhood of a point on the boundary at infinity of , so it will not necessarily have finite volume.

Let be the object obtained from by gluing isometric spheres corresponding to and via the isometry , for all , and then, if applicable, gluing faces of the vertical fundamental domain by parabolic isometries in .

Theorem 2.24 (Poincaré polyhedron theorem, weaker version).

For , as above, if is a smooth hyperbolic manifold, then

  • the group generated by face pairings is discrete,

  • .

Proof.

The result will follow essentially from [epstein-petronio, Theorem 5.5]. First we check the conditions of this theorem. Since is a smooth hyperbolic manifold, the condition Pairing, requiring faces to meet isometrically, will hold. Similarly, the condition Cyclic must hold, requiring the monodromy around an edge in the identification to be the identity, and sums of dihedral angles to be . Condition Connected is automatically true for a single polyhedron (rather than a collection of polyhedra). Finally, note that since we have a finite number of original isometric spheres corresponding to and their inverses, and translation by an element in moves an isometric sphere a fixed positive distance, any isometric sphere of can meet only finitely many other isometric spheres. This is sufficient to imply condition Locally finite.

We need to show the universal cover of is complete. Since is a smooth hyperbolic manifold and is complete, will be complete if and only if the link of its ideal vertex inherits a Euclidean structure coming from horospherical cross sections to , by [thurston:book, Theorem 3.4.23]. In the case that is cut out only by isometric spheres and their translates under , there is nothing to show. In the case that is cut out by a vertical fundamental domain, we know the holonomy of the link of this vertex is given by the group , which is a rank 2 subgroup of fixing the point at infinity. Thus it acts on a horosphere about infinity by Euclidean isometries, and so is indeed complete. It follows that is complete.

Thus all the conditions for [epstein-petronio, Theorem 5.5] hold, and the developing map is a covering map, with covering transformations generated by . It follows that is discrete, and . ∎

Theorem 2.25 (Poincaré polyhedron theorem).

For , as above, and the group generated by face pairings, suppose each face pairing maps a face of isometrically to another face of , and that for each edge of , i.e. for each equivalence class of intersections of isometric spheres under the equivalence given by the gluing, the sum of dihedral angles about is , and the monodromy around the edge is the identity. Then

  • is a smooth hyperbolic manifold with , and

  • is discrete.

Proof.

Again this follows from various results in [epstein-petronio]. Because faces of are mapped isometrically, we have the condition Pairing. The fact that dihedral angles sum to and the monodromy is the identity implies condition Cyclic. Again because isometric spheres can meet only finitely many others in , we have condition Locally finite, and because we have a single polyhedron, we have condition Connected. When we send to via the developing map, we may find a horosphere about infinity disjoint from the isometric spheres forming faces of . In the case that is cut out by a vertical fundamental domain, since preserves this horosphere and acts on it by Euclidean transformations, in the terminology of Epstein and Petronio, the universal cover of the boundary of has a consistent horosphere. This is true automatically if is not cut out by a vertical fundamental domain. Then by [epstein-petronio, Theorem 6.3], the universal cover of is complete. Now Poincaré’s Theorem [epstein-petronio, Theorem 5.5] implies the developing map is a covering map, hence is a smooth, complete hyperbolic manifold with a discrete group. ∎

Our first application of Poincaré’s theorem is the following lemma, which helps us identify Ford domains.

Lemma 2.26.

Let be a subgroup of with a rank 2 parabolic subgroup fixing the point at infinity.

Suppose the isometric spheres corresponding to a finite set of elements of , as well as their translates under , cut out a region so that the quotient under face pairings and the group yields a smooth hyperbolic manifold with fundamental group . Then is discrete and geometrically finite, and must be the equivariant Ford domain of .

Similarly, suppose the isometric spheres corresponding to a finite set of elements of , as well as a vertical fundamental domain for , cut out a polyhedron , so that face pairings given by the isometries corresponding to isometric spheres and to elements of yield a smooth hyperbolic manifold with fundamental group . Then is discrete and geometrically finite, and must be a Ford domain of .

Proof.

In both cases, Theorem 2.24 immediately implies that is discrete. The fact that is geometrically finite follows directly from the definition.

In the case of the polyhedron , suppose is not a Ford domain. Since the Ford domain is only well–defined up to choice of fundamental region for , there is a Ford domain with the same choice of vertical fundamental domain for as for . Since is not a Ford domain, and do not coincide. Because both are cut out by isometric spheres corresponding to elements of , there must be a visible face that cuts out the domain that does not agree with any of those that cut out the domain . Hence is a strict subset of , and there is some point in which lies in the interior of , but does not lie in the Ford domain.

Now consider the covering map . This map glues both and into the manifold , since both are fundamental regions for the manifold. Now consider applied to . Because lies in the interior of , and is a fundamental domain, there is no other point of mapped to . On the other hand, does not lie in the Ford domain . Thus there is some preimage of under which does lie in . But is a subset of . Hence we have in such that . This contradiction finishes the proof in the case of the polyhedron .

The proof for is nearly identical. Again if is not the equivariant Ford domain , then there is an additional visible face of besides those that cut out , and again there is some point in which lies in the interior of , but does not lie in . Again the covering map glues and into the manifold , and again since a point lies in but not in , we have some in such that . Again this is a contradiction. ∎

We may now complete the proof that the Ford domain of the representation of Example 2.11 is as shown in Figure 2.

Lemma 2.27.

Let be the representation given in Example 2.11. Then gives a minimally parabolic geometrically finite uniformization of , and a Ford domain is given by the intersection of a vertical fundamental domain for with the half–spaces exterior to the two isometric spheres and .

Proof.

We have seen that , , and the translates of these isometric spheres under are all disjoint. Select a vertical fundamental domain for which contains the isometric spheres and . This is possible by choice of and , particularly because the translation lengths and are greater than .

Let be the region in the interior of the vertical fundamental domain, exterior to the half–spaces and bounded by and , respectively. Then when we identify vertical sides of via elements of , and identify and via , the object we obtain is a smooth hyperbolic manifold, by Theorem 2.25, since has no edges. Lemma 2.26 now implies that is a Ford domain for , and that is geometrically finite. Lemma 2.18 implies is minimally parabolic. Finally, Lemma 2.16 shows is homeomorphic to the interior of , so this is indeed a uniformization of . ∎

We conclude this section by stating a lemma that will help us identify representations which are not discrete. It is essentially the Shimizu–Leutbecher lemma [maskit, Proposition II.C.5].

Lemma 2.28.

Let be a discrete, torsion free subgroup of such that has a rank two cusp. Suppose that the point at infinity projects to this cusp, and let be its stabilizer in . Then for all , the isometric sphere of has radius at most the minimal (Euclidean) translation length of all non-trivial elements in .

3. Algorithm to compute Ford domains

We will use Ford domains to study geometrically finite minimally parabolic uniformizations of the –compression body. To facilitate this study, we have developed algorithms to construct Ford domains. In this section, we present an algorithm which is guaranteed to construct the Ford domain, but is impractical. We also present a practical procedure which we have implemented, which we conjecture will always construct the Ford domain of the –compression body.

3.1. An initial algorithm

Let be a discrete, geometrically finite subgroup of such that is homeomorphic to the interior of the –compression body. We will assume that is given by an explicit set of matrix generators. We now present an (impractical) algorithm to find the Ford domain of . Assume without loss of generality that in the universal cover , the point at infinity is fixed by the rank 2 cusp subgroup, .

Algorithm 3.1.

Enumerate all elements of the group: . Again we assume that each is given as a matrix with explicit entries. Step through the list of group elements. At the -th step:

  1. Draw isometric spheres corresponding to and .

  2. If these isometric spheres are visible over other previously drawn isometric spheres (corresponding to and their inverses), check if the object obtained by gluing pairs of currently visible, previously drawn isometric spheres via the corresponding isometries satisfies the hypotheses of Theorem 2.25.

  3. If it does satisfy these hypotheses, then by the Poincaré polyhedron theorem, Theorem 2.24, the fundamental group of the manifold is generated by isometries corresponding to face identifications. Therefore, if we can write the generators of as words in the isometries of these faces, we will be done, by Lemma 2.26. Put this manifold into a list of manifolds built by repeating the previous two steps.

  4. For each manifold in the list of manifolds built by steps (1) and (2), we have an enumeration of words in the group elements generated by gluing isometries of faces: .

    1. For each generator of , step through the first words of to see if equals one of these words.

    2. If each can be written as a word in one of the first elements of , we are done. The Ford domain is given by the isometric spheres which are the faces of this manifold.

Note that in step (2), if we find that isometric spheres glue to give a manifold, it does not necessarily follow that this manifold is our original compression body. For example, we may have found a non-trivial cover of the original compression body. Therefore, steps (3) and (4) are required.

Since Ford domains of geometrically finite hyperbolic manifolds have a finite number of faces, after a finite number of steps, Algorithm 3.1 will have drawn all isometric spheres corresponding to visible faces. Since identifying a finite number of generators as words in a finite number of generators given by face pairings can be done in a finite number of steps, after a finite number of steps the algorithm will terminate.

3.2. A practical procedure

The algorithm above is impractical for computer implementation. In this section we present a practical procedure, which will generate the Ford domain and terminate in many cases for a –compression body. We conjecture it will terminate for all cases.

We have implemented this procedure, and used the images it produced to analyze behavior of paths of Ford domains. The computer images of this paper were generated by this program.

Procedure 3.2.

Let , be parabolic, fixing a common point at infinity in . Let be loxodromic, such that .

Conjugate such that

We will hold two lists: The list of elements to draw, , and the list of elements that have been drawn . These are ordered lists.

Initialization. Replace and if necessary, so that the lattice generated by and has generators of shortest length.

Replace if necessary so that is within the parallelogram with vertices at , , , and .

Add and to the list of elements to draw, .

Loop. While the list is non-empty, do the following.

  1. Remove the first element of , call it . Consider the isometric sphere of . Check against elements of . If is no longer visible, discard and start over with the next element of . If is still visible, draw the isometric sphere determined by to the screen. Add to the end of the list .

    Now also draw isometric spheres of each element of the form , where , lie in , with chosen so that we draw only those translates of which are contained in the region of the screen.

  2. For each in the list of drawn elements , find integers , such that the center of is nearest the center of .

    For each isometric sphere of the form , with , in , check if that isometric sphere and intersect visibly. That is, check if they intersect and, if so, if the edge of their intersection is visible from infinity. (In the case of , no need to check for intersections of and the isometric sphere of the newly added last element of .)

    We claim that if intersects any translate of under , then that translate will have the form where , are in . See Lemma 3.3 below.

  3. If and do intersect visibly, then the isometric sphere of the element should be drawn, where , so that . Step through the lists and to ensure the isometric sphere hasn’t been drawn already, and is not yet slated to be drawn (to avoid adding the same sequence of faces repeatedly – note there are more time effective ways of ensuring the same thing). If is not in either list, then add , and to the end of the list to be drawn.

Lemma 3.3.

Suppose and are parabolic fixing the point at infinity, chosen as above such that has the shortest translation length in the group , and such that has the shortest translation length of all parabolics independent from . Suppose and are loxodromic such that the group is discrete. Choose integers such that the center of is nearer the center of than the center of any other translate of under . Then if intersects any translate of , that translate must be of the form for .

Proof.

Apply an isometry to so that translates by exactly along the real axis in . Note that after this isometry, by Lemma 2.28, all isometric spheres have radius at most . Hence if two intersect, the distance between their centers is less than . Let denote the center of . We may apply another isometry of so that in . Finally, since is the shortest translation independent of , must translate to be within the hyperbolic triangle on with vertices , , .

Since the center of , denote it by , is closer to than to any of the translates of under , the real coordinate of in must have absolute value at most . Similarly, the difference in imaginary coordinates of and is at least , for otherwise the square of the distance between and some lattice point of the form is at most . Finally, we may assume the imaginary coordinate of is positive, by symmetry of the lattice.

Suppose meets , where one of or is greater than . Then the distance between and on is at most . On the other hand, if , then the difference between the imaginary coordinates of and is at least , which is a contradiction. So suppose and . Then the difference in real coordinates of and is at least , which is again a contradiction. ∎

Theorem 3.4.

Suppose each of the spheres drawn by Procedure 3.2 is a face of the Ford domain of a geometrically finite uniformization of the –compression body . Then the procedure draws (at least one translate under of) all visible isometric spheres, and the procedure terminates.

Proof.

The fact that the procedure terminates follows from Corollary 2.10: there are only finitely many visible faces, and each face the procedure draws is visible.

The fact that the procedure draws all visible isometric spheres of the Ford domain will follow from Lemma 2.26 and the Poincaré polyhedron theorem, as follows.

First, suppose the faces corresponding to and are visible, and they do not intersect each other or any other faces. Then the procedure terminates after drawing these faces and a few translates under . Because there are no edges of intersection, the argument of Lemma 2.27 implies that the only visible face of the Ford domain corresponds to (and ), and in this case we are done.

So suppose two isometric spheres drawn by the procedure intersect. Say isometric spheres and intersect. Then the procedure will draw . Since the procedure only draws visible isometric spheres, must be visible. By Lemma 2.17, it intersects in an edge which is mapped isometrically to the edge of . Changing roles of and in the same lemma, the isometric sphere must be visible, and is mapped isometrically to .

Now notice that the faces of the Ford domain corresponding to the pairs and , and , and and are the only faces that meet the edge class of (up to translation by ). This can be seen by noting that takes and to and , respectively. Then apply . This sends and to and , respectively. Finally apply , which sends and to and , respectively. Thus the monodromy is given by . As for dihedral angles around this edge class, because the monodromy is the identity, the sum of the dihedral angles must be a multiple of . Since there are only three faces in the edge class, and the dihedral angle between any two faces is less than , the sum of the dihedral angles around the edge must be exactly