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Article / Book Information
題目(和文)

双曲体積，ファイバーの通約性，そして例外的手術．理論と計算，双方向からの研究

Title(English)

Hyperbolic Volume, Fibered Commensurability, and Exceptional
Surgeries, Theory v.s. Computation

著者(和文)

正井秀俊

Author(English)

Hidetoshi Masai

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第9421号,授与年月日:2014年3月26日,
学位の種別:課程博士,
審査員:小島 定吉,渡辺 治,梅原雅顕,西畑伸也,寺嶋Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第9421号,
Conferred date:2014/3/26,
Degree Type:Course doctor,Examiner:,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis

種別(和文)

要約

Type(English)

OutlinePowered by T2R2 (Tokyo Institute Research Repository)

郁二

論文の要約
平成２５年度博士論文

Hyperbolic Volume, Fibered Commensurability,and Exceptional Surgeries, Theory v.s. Computation

Hidetoshi MASAI
正井秀俊

Department of Mathematical and Computing sciences
Tokyo Institute of Technology

Supervisor Sadayoshi Kojima

Outline of the paper
A hyperbolic 3-manifold is a Riemannian manifold of constant sectional curvature −1.
We mainly consider hyperbolic 3-manifolds of ﬁnite volume. By the Mostow-Prasad
rigidity theorem, two such manifolds are isometric if and only if they are homeomorphic.
Hence we may study hyperbolic geometry via topology and vice versa. Further, thanks
to the Mostow-Prasad rigidity, computers can be very useful tools for the study ofhyperbolic 3-manifolds. In this outline, we explain what we study in the thesis. Here,
we lay stress especially on how we used computer.

1 Theoretical study
The ﬁrst main topic of this thesis is the hyperbolic volumes, which is one of the mostfundamental geometric invariants of hyperbolic manifolds. We also consider ﬁberedcommensurability, a notion introduced in [5].

1.1 On volume preserving moves on graphs with parabolic meridians [16].
We discuss cutting operation on hyperbolic 3-manifolds along thrice punctured spheres.
If we cut a hyperbolic link complement along thrice punctured spheres, we get a complement of hyperbolic graph with parabolic meridians. By using this relationship between
links and graphs, we give a method to compute the best possible upper bounds for volumes of hyperbolic link complements in terms of the twist numbers. Here, twist number
is deﬁned for a link diagram as the number of twists, a maximal collection of bigon
regions. We compute approximate values of the best possible upper bounds for the casewhere twist numbers are less than 10. To compute those values we used two programs
called plantri [3] and Orb [9]. We ﬁrst enumerate planar graphs by plantri and compute the (approximate) volumes of them by Orb. Those (approximate) values of upperbounds are better than the known bounds due to Agol-Thurston [14].

1.2 On the number of hyperbolic 3-manifolds of a given volume (joint workwith Craig Hodgson) [10].
We discuss the number N (v) of hyperbolic 3-manifolds of volume v. By a work ofJørgensen and Thurston (c.f. [8]), the number N (v) is known to be ﬁnite for all v ∈ R.
We ﬁrst investigate censuses on SnapPea and computed (approximate) volumes. Thenwe drew graphs (see ﬁgure 1). Those graphs suggest that• sup N (w) grows exponentially as v goes to inﬁnity, andw≤v• there are inﬁnitely many v with N (v) = 1.
We show that the experiments above are actually the case.
Theorem 1. For each n ≥ 3, there are at least 2n /(2n) diﬀerent hyperbolic link com-

5045Frequency

403530252015105001234Volume5678Fig. 1 Frequencies of volumes of hyperbolic 3-manifolds in the closed and cusped censuses.

plements of volume 4nV8 .
Theorem 2. There is an inﬁnite sequence of hyperbolic Dehn ﬁllings M (ai , bi ) on theﬁgure eight knot complement M = m004 such that the manifolds M (ai , bi ) are determined by their volumes, amongst all ﬁnite volume orientable hyperbolic 3-manifolds.
Here, “m004” is the SnapPea’s notation of the ﬁgure eight knot complement.

1.3 Equidecomposability, volume formulae and orthospectra (joint work withGreg McShane)[18].
We compare two formulae by Bridgeman-Kahn [2] and Calegari [4] that compute thevolume of a given compact hyperbolic n-manifold M with totally geodesic boundary
by using the data called orthospectrum. The orthospectrum of M is the set of lengthof orthogeodesics, geodesics perpendicular to the boundary ∂M at both ends. The twoidentities are derived by means of diﬀerent decompositions. They deﬁned decompositionsof unit tangent bundle T1 (M ) such that each piece corresponds to an orthogeodesic. Let
us denote by B(l) (resp. C(l)) the piece deﬁned by Bridgeman-Kahn (resp. Calegari)that corresponds to an orthogeodesic of length l. They also showed that the geometry of each piece is determined by the length of corresponding orthogeodesic. Since
vol2n−1 (T1 (M )) = voln (M )voln−1 (Sn−1 ) (Sm is the unit sphere of dimension m), thesum of the volumes of B(l) or C(l), where l runs over the orthospectrum, is equal tovoln (M ) times voln−1 (Sn−1 ), a function of n − 1. Here, volm denotes the volume of anm-dimensional orientable manifold. Thus by associating the volume of B(l) or C(l) toeach element l in the orthospectrum of M , we can compute the volume of M . Since bothformulae hold for every such hyperbolic manifolds, it is natural to ask if the volume ofB(l) and C(l) coincide or not. For n = 2, Calegari showed that vol3 (B(l)) = vol3 (C(l))[4]. Further, for n = 3, we numerically computed the volumes of B(l) and C(l), andobserved that they are very close for many l ∈ R. In §3, we verify that the volumes ofB(l) and C(l) coincide for any n ≥ 2.
Theorem 3. For all n ≥ 2,
vol2n−1 (B(l)) = vol2n−1 (C(l)).
Both Bridgeman-Kahn and Calegari derived formulae for the volume of B(l) and C(l),which involve integrals and are complicated. We derived a simple formula of the volumeof C(l) (and hence, B(l)) for the case where n = 3.
Theorem 4.
vol5 (C(l)) =2π(l + 1).
e2l − 1

1.4 On commensurability of ﬁbrations on a hyperbolic 3-manifold [17].
We consider ﬁbered commensurability of ﬁbrations on a hyperbolic 3-manifold. The
notion of ﬁbered commensurability is introduced by Calegari-Sun-Wang [5]. Note thatfor a given ﬁbration on an orientable 3-manifold, we have an associated pair (F, φ) of
the ﬁbre surface F and the monodromy φ : F → F . Roughly speaking, two ﬁbrations(on possibly distinct manifolds) corresponding to (F1 , φ) and (F2 , φ) are said to becommensurable if there is a ﬁnite covering Fe of both F1 and F2 , and the lifts φei of φikkf1 1 = φ
f2 2 for some k1 , k2 ∈ Z\{0}. Two ﬁbrations on a 3-manifold(i = 1, 2) satisfying φ
M are said to be symmetric if there is a homeomorphism ϕ : M → M that maps one
to the other. We give a necessary condition for manifolds to have non-symmetric but
commensurable ﬁbrations.
Theorem 5. Suppose that M is a hyperbolic 3-manifold that does not have hidden symmetries. Then, any pair of ﬁbrations of M is either symmetric or non-commensurable,but not both.
Here, a hidden symmetry of M is an element of the commensurator of Γ which is notin the normalizer of Γ, where Γ < PSL(2, C) is the image of a holonomy representationof π1 (M ). On the other hand, we also show that there exist manifolds with many
commensurable ﬁbrations.
Theorem 6. For any n ∈ N, there exists a hyperbolic 3-manifolds with at least nmutually non-symmetric but commensurable ﬁbrations.

2 Computational study
In the theoretical part, we mostly used computer to have questions that quite likely to betrue, and veriﬁed them theoretically. In most of those questions, inﬁnitely many values

or manifolds are involved, and hence we cannot prove them naively by using computer.
On the other hand, once we have a theory that shows “ﬁniteness”, then computers canbe powerful tools to prove a theorem. In the second part of this thesis, we develop
computer programs that perform rigorous proofs.

2.1 Veriﬁed computations for hyperbolic 3-manifolds (joint work with Neil
Hoﬀman, Kazuhiro Ichihara, Masahide Kashiwagi, Shin’ichi Oishi, andAkitoshi Takayasu)[13].
We developed the package hikmot that enables us to prove the hyperbolicity of a given
triangulated 3-manifold. The program hikmot can fail verifying the hyperbolicity eventhough a given manifold is actually hyperbolic, and hence it can NOT prove the nonhyperbolicity. However, hikmot usually works and in fact we can prove hyperbolicity of
all manifolds in several censuses on SnapPy [7].
Theorem 7. All the manifolds in OrientableCuspedCensus [6] are hyperbolic.
Theorem 8. All the manifolds in OrientableClosedCensus [11] are hyperbolic.
Since to prove the hyperbolicity, we deal with equations with complex variables, socalled gluing equations, the notion of interval arithmetic plays a key role. On computer,
we cannot deal with all real numbers or complex numbers. We usually use ﬂoating
point arithmetic, that is an approximate computation. Therefore, in principle, it cannot
prove any mathematical theorem concerning real values or complex values. The intervalarithmetic has been introduced in order to solve this problem [20, 21, 19]. Insteadof dealing with approximate value, in interval arithmetic, we compute intervals thatcontain rigorous values. In principle, this cannot prove any equality, however, it canprove inequalities. To solve a gluing equation, we use Newton’s method. If we apply
Newton’s method by using ﬂoating point arithmetic, then we can only get approximate
solutions and more critically, this cannot verify the convergence of Newton’s method.
There are several theorems that give suﬃcient conditions, which are inequalities, forthe convergence of Newton’s method. Putting those theorems and interval arithmetic
together, we can prove that there is a solution for a gluing equation and hence thehyperbolicity of a manifold.

2.2 Exceptional surgeries on alternating knots (joint work with KazuhiroIchihara)[12].
We use hikmot together with a python code based on the codes in [15] to classify all
the exceptional surgeries along alternating knots. Thurston [22] has proved that for agiven hyperbolic knot, all but ﬁnitely many surgeries along the knot give hyperbolic
manifolds. We call those ﬁnitely many surgeries that give non-hyperbolic manifolds
exceptional. The codes in [15] are based on SnapPea. The code ﬁrst applies so-called
the 6-theorem due to Agol [1] and Lackenby [14]. The 6-theorem shows that all surgery
slopes of geometric lengths greater than 6 give rise to hyperbolic manifolds by Dehnsurgery. Then by using SnapPea, the code tries to ﬁnd hyperbolic metrics on manifolds
obtained by Dehn surgery along slopes of geometric length less than or equal to 6. We
modiﬁed the codes [15] so that each computation is done by interval arithmetic. (Toapply the 6-theorem we need to verify an inequality). For slopes of geometric length lessthan or equal to 6, we use hikmot to prove the hyperbolicity of the surgered manifolds.
The codes might return slopes which are non-exceptional, however, never miss slopesof exceptional surgeries. Thanks to several known works to classify the exceptional
surgeries along alternating knots, it suﬃces to prove that about 30000 links with acondition on the slopes for certain components do not admit exceptional surgeries. By
applying our codes to those links, we see that none of such links admit any exceptional
surgeries that satisfy the condition. In average, it takes about one hour to verify nonexistence of exceptional surgeries for each link. Since 30,000 hours ≈ 3.4 years, we used
TSUBAME, the supercomputer of Tokyo Tech. Thus with a big aid of computer, wehave a classiﬁcation of the exceptional surgeries along the alternating knots.

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