Small Hyperbolic 3-Manifolds With Geodesic Boundary

We classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps. The manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). And there is a single cusped manifold, that we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), or of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp, and one having two cusps. Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web. MSC (2000): 57M50 (primary), 57M20, 57M27 (secondary). This paper is devoted to the class of all orientable finite-volume hyperbolic 3manifolds having non-empty compact totally geodesic boundary and admitting a minimal ideal triangulation with either three or four but no fewer tetrahedra. We describe the theoretical background and experimental results of a computer program that has enabled us to classify all such manifolds. (The case of manifolds obtained from two tetrahedra was previously dealt with in [8]). We also provide an overall discussion of the most important features of all these manifolds, namely of: • their volumes; • the shape of their canonical Kojima decomposition; 1 • the presence of cusps. These geometric invariants have all been determined by our computer program. The complete list of manifolds in SnapPea format and detailed information on the invariants is available from [19]. 1 Preliminaries and statements We consider in this paper the class H of orientable 3-manifolds M having compact non-empty boundary ∂M and admitting a complete finite-volume hyperbolic metric with respect to which ∂M is totally geodesic. It is a well-known fact [9] that such an M is the union of a compact portion and some cusps based on tori, so it has a natural compactification obtained by adding some tori. The elements of H are regarded up to homeomorphism, or equivalently isometry (by Mostow’s rigidity). Candidate hyperbolic manifolds Let us now introduce the class H̃ of 3-manifolds M such that: • M is orientable, compact, boundary-irreducible and acylindrical; • ∂M consists of some tori (possibly none of them) and at least one surface of negative Euler characteristic. The basic theory of hyperbolic manifolds implies that, up to identifying a manifold with its natural compactification, the inclusion H ⊂ H̃ holds. We note that, by Thurston’s hyperbolization, an element of H̃ actually lies in H if and only if it is atoroidal. However we do not require atoroidality in the definition of H̃, for a reason that will be mentioned later in this section and explained in detail in Section 2. Let ∆ denote the standard tetrahedron, and let ∆∗ be ∆ minus open stars of its vertices. Let M be a compact 3-manifold with ∂M 6= ∅. An ideal triangulation of M is a realization of M as a gluing of a finite number of copies of ∆∗, induced by a simplicial face-pairing of the corresponding ∆’s. We denote by Cn the class of all orientable manifolds admitting an ideal triangulation with n, but no fewer, tetrahedra, and we set: Hn = H∩ Cn, H̃n = H̃ ∩ Cn. We can now quickly explain why we did not include atoroidality in the definition of H̃. The point is that there is a general notion [12] of complexity c(M) for a compact 3-manifold M , and c(M) coincides with the minimal number of tetrahedra in an ideal triangulation precisely when M is boundary-irreducible and acylindrical. This property makes it feasible to enumerate the elements of H̃n. 2 To summarize our definitions, we can interpret Hn as the set of 3-manifolds which have complexity n and are hyperbolic with non-empty compact geodesic boundary, while H̃n is the set of complexity-n manifolds which are only “candidate hyperbolic.” Enumeration strategy The general strategy of our classification result is then as follows: • We employ the technology of standard spines [12] (and more particularly ographs [1]), together with certain minimality tests (see Section 2 below), to produce for n = 3, 4 a list of triangulations with n tetrahedra such that every element of H̃n is represented by some triangulation in the list. Note that the same element of H̃n is represented by several distinct triangulations. Moreover, there could a priori be in the list triangulations representing manifolds of complexity lower than n, but the result of the classification itself actually shows that our minimality tests are sophisticated enough to ensure this does not happen; • We write and solve the hyperbolicity equations (see [5] and Section 3 below) for all the triangulations, finding solutions in the vast majority of cases (all of them for n = 3); • We compute the tilts (see [5, 15] and Section 3 again) of each of the geometric triangulations thus found, whence determining whether the triangulation (or maybe a partial assembling of the tetrahedra of the triangulation) gives Kojima’s canonical decomposition; when it does not, we modify the triangulation according to the strategy described in [5], eventually finding the canonical decomposition in all cases; • We compare the canonical decompositions to each other, thus finding precisely which pairs of triangulations in the list represent identical manifolds; we then build a list of distinct hyperbolic manifolds, which coincides with Hn because of the next point; • We prove that when the hyperbolicity equations have no solution then indeed the manifold is not a member of Hn, because it contains an incompressible torus (this is shown in Section 2). Even if the next point is not really part of the classification strategy, we single it out as an important one: • We compute the volume of all the elements of Hn using the geometric triangulations already found and the formulae from [16]. 3 One-edged triangulations Before turning to the description of our discoveries, we must mention another point. Let us denote by Σg the orientable surface of genus g, and by K(M) the blocks of the canonical Kojima decomposition of M ∈ H. We have introduced in [6] the class Mn of orientable manifolds having an ideal triangulation with n tetrahedra and a single edge, and we have shown that for n ≥ 2 and M ∈ Mn: • M is hyperbolic with geodesic boundary Σn; • M has a unique ideal triangulation with n tetrahedra, which coincides with K(M); moreover c(M) = n and Mn = {M ∈ H̃n : ∂M = Σn}; • the volume of M depends only on n and can be computed explicitly. These facts imply in particular that Mn is contained in Hn. Results We can now state our main results, recalling first [8] that H1 = ∅ and H2 = M2 has eight elements, and pointing out that all the values of volumes in our statements are approximate, not exact ones. More accurate approximations are available on the web [19]. We also emphasize that our results indeed have an experimental nature, but we have checked by hand a number of cases and always found perfect agreement with the results found by the computer. Results in complexity 3 We have discovered that: • H3 coincides with H̃3 and has 151 elements; • M3 consists of 74 elements of volume 10.428602; • all the 77 elements of H3 \M3 have boundary Σ2, and one of them also has one cusp. Moreover the elements M of H3 \M3 split as follows: • 73 compact M ’s with K(M) consisting of three tetrahedra; vol(M) attains on them 15 different values, ranges from 7.107592 to 8.513926, and has maximal multiplicity nine, with distribution according to number of manifolds as shown in Table 3 (see the Appendix); • three compact M ’s with K(M) consisting of four tetrahedra; they all have the same volume 7.758268; • one non-compact M ; it has a single toric cusp, K(M) consists of three tetrahedra, and vol(M) = 7.797637. 4 Figure 1: The cusped manifold having complexity three and non-empty boundary is the complement of a knot in the genus-two handlebody. The cusped element of H3 turns out to be a very interesting manifold. In [7] we have analyzed all the Dehn fillings of its toric cusp, improving previously known bounds on the distance between non-hyperbolic fillings. In particular, we have shown that there are fillings giving the genus-2 handlebody, so the manifold in question is a knot complement, as shown in Fig. 1. Results in complexity 4 We have discovered that: • H4 has 5033 elements, and H̃4 and has 6 more; • 5002 elements of H4 are compact; more precisely: – 2340 have boundary Σ4 ( i.e. they belong to M4); – 2034 have boundary Σ3; – 628 have boundary Σ2; • 31 elements of H4 have cusps; more precisely: – 12 have one cusp and boundary Σ3; – 18 have one cusp and boundary Σ2; – one has two cusps and boundary Σ2. More detailed information about the volume and the shape of the canonical Kojima decomposition of these manifolds is described in Tables 1 and 2. In these tables each box corresponds to the manifolds M having a prescribed boundary and type of K(M). The first information we provide (in boldface) within the box is the number of distinct suchM ’s. When all theM ’s in the box have the same volume, we indicate its value. Otherwise we indicate the minimum, the maximum, the number of different values, and the maximal multiplicity of the values of the volume function, 5 Σ4 Σ3 Σ2 4 tetra 234