数理解析研究所講究録
第 1571 巻 2007 年 5-17

5Ford domains of a certain hyperbolic
3-manifold whose boundary consists of a pairof once-punctured tori, II
大阪市立大学数学研究所 秋吉宏尚 (Hirotaka Akiyoshi)
Osaka City University Advanced Mathematical Institute1

Introduction

This is a sequel to [3]. Our initial problem is the following.
Problem 1.1. Characterize the combinatorial structures of the Ford domains for hyperbolic structures on a 3-manifold which has a pair of punctured
tori as boundary.
The Jorgensen theory tells in detail the combinatorial structures of theFord domains of hyperbolic structures on punctured torus bundles. We expect to understand in detail the hyperbolic structures on manifolds with
non-fiber surfaces from the combinatorial structures of Ford domains. Problem 1.1 is the first step to the attempt to fill in the box with “???” in thefollowing table, which tells the relation between analytic and combinatorialaspects of Thurston’s Hyperbolization Theorem for Haken manifolds.

A family of 3-manifolds with a pair of punctured tori as boundary2(resp. ).
We denote the one-holed torus (resp. once-punctured torus) byof
Let be an essential simple closed curve on the level surface
the exterior of , i.e.,
the product manifold $T_{0}\cross[-1,1]$ , and denote by$T$$T_{0}$$T_{0}\cross\{0\}$$\gamma$$M_{0}$$\gamma$6, where–Int
is a regular neighborhood of .
For each sign
, we denote the one-holed torusby ,and denote the natural homeomorphism between
by :
and
.
We define the slopes ( $=free$ homotopy classes) and in
as follows.
is the meridian slope of , i.e.,
is represented by an essential simple
closed curve which bounds a disk in, and is the slope represented by
the intersection ofand the annulus. Thengenerates$M_{0}=T_{0}\cross[-1\rangle 1]$$N(\gamma)$$N(\gamma)$$\gamma$$\epsilon=\pm$$T_{0}\cross\{\epsilon 1\}\subset\partial M_{0}$$T_{0}$$\iota^{\epsilon}$$T_{0}^{\epsilon}$$\lambda$

$\mu$

$\mu$

$T_{0}arrow T_{0}^{\epsilon}$$\partial N(\gamma)$$\mu$$\gamma$$N(\gamma)$$\lambda$$\gamma^{t}\cross[0,1]$$\partial N(\gamma)$$H_{1}(\partial N(\gamma))$$T_{0}^{\epsilon}$$\{\mu, \lambda\}$.

For a pair of coprime integers $(p, q)$ , let $M(p, q)$ be the result of Dehn
filling onwith slope $p\mu+q\lambda$ , i.e., the manifold obtained frombygluing the solid torus by an orientation-reversing homeomorphism
so that the meridian of is identified with a simple closed. We regard
curve onof slope
as a submanifold of $M(p, q)$
by using the canonical embedding. In particular, $M(\pm 1,0)$ is canonically
homeomorphic to the product $T_{0}\cross[-1,1]$ .
$M_{0}$$M_{0}$$V$$\partial Varrow$$V$$\partial N(\gamma)\subset\partial M_{0}$$p\mu+q\lambda$$\partial N(\gamma)$$M_{0}$

Proposition 2.1. For any pair of coprime integers $(p, q),$ $M(p, q)$ is homeomorphic to the handlebody of genus 2.

Set $P=\partial T_{0}\cross[-1,1]$ . In contrast to Proposition 2.1, the pair $(M(p, q),P)$
does not necessarily admit a product structure.
is incompressible in $M(p, q)$ if and only
Proposition 2.2. The surfaces
if $(p, q)\neq(0, \pm 1)$ . In this case, it follows that $(M(p, q),$ $P$ ) is an atoroidal
Haken pared manifold in the sense of [12].
$T_{0}^{\pm}$

By the Thurston’s Hyperbolization Theorem for Haken pared manifolds(cf. [12, Theorem 1.43]), we obtain the following corollary.

Corollary 2.3. For any pair of coprime integers $(p)q)\neq(0, \pm 1),$ $M(p, q)$
admits a complete geometrically finite hyperbolic structure with the parabolic
locus $P$ .

For the rest of this paper, $(p, q)$ denotes a pair of coprime integers distinct
from $(0, \pm 1)$ .

the space of geometricallyDefinition 2.4. We shall denote byfinite (marked) hyperbolic structures on the pared manifold $(M(p, q),$ $P$ ) withthe parabolic locus .
$\mathcal{M}\mathcal{P}(p, q)$$P$

is not empty. The following proposition folBy Corollary 2.3,
lows from Marden’s isomorphism theorem.
$\mathcal{M}\mathcal{P}(p, q)$

is isomorphic to the square
Proposition 2.5. The spaceof the Teichmuller space of the punctured torus.
$\mathcal{M}\mathcal{P}(p, q)$$\mathcal{T}$$\mathcal{T}\cross \mathcal{T}$7

3

Punctured torus groups

We fix a generator pairby .
$\alpha\beta\alpha^{-1}\beta^{-1}$$(\alpha, \beta)$of$\pi_{1}(T)$and denote the commutator$[\alpha, \beta]=$$\kappa$

Definition 3.1. A representation : $\pi_{1}(T)arrow PSL(2, \mathbb{C})$ is a (marked)
punctured torus group if it is a faithful discrete representation which maps
to a parabolic element. Two punctured torus groups are said to be equivalent
if they are conjugate to each other. Denote the set of equivalence classes ofpunctured torus groups. The set
is identified with a subset of the-representation space of. The topology on
is induced$\rho$$\kappa$$by\overline{Q\mathcal{F}}$$\overline{\mathcal{Q}\mathcal{F}}$$PSL(2, \mathbb{C})$$\pi_{1}(T)$$\overline{\mathcal{Q}\mathcal{F}}$from this identification.

For any
is homeomorphic to Int $(T_{0}\cross(-1,1))$ . For
each end of $T_{0}\cross(-1,1)$ , one can define the end invariant of , denoted by, as follows. If there is asimply connected component of
the domain of discontinuity ofcorresponding to the end, then its
quotient determines amarked conformal structure on , which is defined tobe the end invariant. If the subset of domain of discontinuity of$\rho\in\overline{Q\mathcal{F}},$$\mathbb{H}^{3}/p(\pi_{1}(T))$$\rho$$\lambda^{\epsilon}(\rho)(\epsilon\in \{-, +\})$$\rho(\pi_{1}(T))$$T$$\lambda^{\epsilon}(\rho)\in \mathcal{T}$corresponding to the end is the disjoint union of countable familyof round disks, then its quotient determines amarked conformal structure
on $T$ with node, which is defined to be the end invariant.
If the subset of domain of discontinuity ofcorresponding to theend is empty, then one can define the end invariant
byusing asequence of simple closed geodesics whlch exits the end. Hereiscompactified via Thurston’s compacticfication. Thenand(resp.
).
is canonically identified with(resp.
and(See [14] for details.)$\rho(\pi_{1}(T))$$\lambda^{\epsilon}(\rho)\in\partial_{\mathbb{Q}}\mathcal{T}$$\rho(\pi_{1}(T))$$\lambda^{\epsilon}(\rho)\in\partial \mathcal{T}-\partial_{\mathbb{Q}}\mathcal{T}$$\mathcal{T}$$\mathcal{T}$$\overline{\mathbb{H}^{2}}$$\mathbb{H}^{2}$$\partial_{\mathbb{Q}}\mathcal{T})$Definition 3.2. The end invariant map
defined by$\lambda$$\lambda(\rho)=(\lambda^{-}(\rho), \lambda^{+}(\rho))(\rho\in\overline{\mathcal{Q}\mathcal{F}})$:.

$\mathcal{T}\cup\partial \mathcal{T}$$\partial_{\mathbb{Q}}\mathbb{H}^{2}=\hat{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}$$\overline{\mathcal{Q}\mathcal{F}}arrow \mathbb{H}^{2}\cross \mathbb{H}^{2}-diag(\partial \mathbb{H}^{2})$is:Theorem 3.3 (Minsky [14]). The end invari ant map
is a bijection and its inverse is a continuous map. $In$.
particular,
is equal to the closure of the quasifuchsian space$\lambda=(\lambda^{-}, \lambda^{+})$$\overline{\mathcal{Q}\mathcal{F}}arrow$$\overline{\mathbb{H}^{2}}\cross\overline{\mathbb{H}^{2}}-diag(\partial \mathbb{H}^{2})$$\overline{Q\mathcal{F}}$$\mathcal{Q}\mathcal{F}$: $\pi_{1}(M(p, q))arrow PSL(2, \mathbb{C})$ be, let
the holonomy representation. Then let : $\pi_{1}(T)arrow PSL(2, \mathbb{C})\cross PSL(2, \mathbb{C})$. The following proposition follows
be the composition
from Proposition 2.2 and the covering theorem [7].

Given an element$\sigma\in \mathcal{M}\mathcal{P}(p, q)$$p_{\sigma}\wedge$$\iota_{\sigma}$$\iota_{\sigma}=(\rho_{\sigma}\wedge 0\iota_{*}^{-\wedge}\rho_{\sigma}0\iota_{*}^{+})$

Proposition 3.4. The correspondence.
into$\mathcal{M}\mathcal{P}(p, q)$$\mathcal{Q}\mathcal{F}\cross \mathcal{Q}\mathcal{F}$$\sigma\mapsto\iota_{\sigma}$

induces an embedding of8

Figure 1: Ford domain of a cyclic Kleinian group

4

Ford domain

In what follows, we use the upper half space model for$\mathbb{H}^{3}$.

Definition 4.1. For an element of $PSL(2,\mathbb{C})$ which does not stabilize ,the isometric hemisphere,, of is the set of points inwhere acts
as an isometry with respect to the canonical Euclidean metric on the upper
half space. We denote the exterior ofby.
$\infty$$\gamma$$Ih(\gamma)$$\mathbb{H}^{3}$$\gamma$$\gamma$$Ih(\gamma)$$Eh(\gamma)$

Definition 4.2. For a Kleinian group , the Ford domain,, of isdefined bywhere
is the stabilizer ofin F.
For any hyperbolic structure on
or $M(p, q)$ , the Ford domain for isdefined to be the Ford domain of the image of a holonomy representation forwhich sends the peripheral elementof the fundamental group$Ph(\Gamma)$$\Gamma$$Ph( \Gamma)=\bigcap_{\gamma\in\Gamma-\Gamma_{\infty}}Eh(\gamma))$$\sigma$$\infty$$M_{0}$$\sigma$$[\partial T_{0}\cross\{0\}]$$\sigma$to$\Gamma_{\infty}$$\Gamma$$\{\begin{array}{ll}1 20 l\end{array}\}$.

Example 4.3. The Ford domainof the cyclic Kleinian group,$\gamma\in
PSL(2,\mathbb{C})$generated by a loxodromic element
which does not stabilize(face’$Ph(\Gamma)$, is as depicted in Figures 1 and 2. Everyofis supported by anisometric hemisphere; there are 8 faces in this example. The characterizationof combinatorial structures of the Ford domains of cyclic Kleinian groups is
given by Jorgensen [10] (cf. [8]).
$Ph(\langle\gamma\rangle)$$\langle\gamma\rangle$$\infty$Let
be the Farey tessellation of.
The combinatorial structure of the Ford domain offoris$\nu=(\nu^{-},\nu^{+})$
characterized by the extension of Jorgensen’s side parameter:. (See [5, Section 4] for detail.)$\mathcal{D}=\{\gamma\langle\infty,0,1\rangle|\gamma\in PSL(2, \mathbb{Z})\}$$\mathbb{H}^{2}$$\rho(\pi_{1}(T))$$\overline{Q\mathcal{F}}arrow \mathbb{H}^{2}\cross \mathbb{H}^{2}-diag(\partial \mathbb{H}^{2})$$\rho\in\overline{\mathcal{Q}\mathcal{F}}$9Figure 2: Ford domain of a cyclic Kleinian group and its dual

Figure 3: Ford domain of a quasifuchsian punctured torus group
Here we briefly review the idea of the characterization. The combinatorial
structures of Ford domains are characterized by using EPH-decomposition
introduced in [4].
beDefinition 4.4. For a hyperbolic structure on
or $M(p,q)$ , let
the subcomplex of the EPH-decomposition for consisting of the Euclidean
consisting of the facets whose
facets. Let
be the subcomplex ofvertices correspond to the parabolic locus .
$M_{0}$$\sigma$$\Delta_{E}(\sigma)$$\sigma$$\Delta_{E,0}(\sigma)$$\triangle_{E}(\sigma)$$P$

is
By the observation in [4, Section 10], it can be proved that
dual to the Ford domain for .
, a topological ideal polyhedral
For any pointcomplex Trg(v) is defined in [5, Section 5]. Then, for any(see [5, Theorem
is isotopic to $Rg(\nu(\rho))$ in the convex core of5.1]). Figure 3 illustrates the Ford domain of a generic quasifuchsian puncand atured torus group, and Figure 4 illustrates the intersection of
sufficiently small horosphere centered at , which is the fixed point of theparabollc subgroup corresponding to by normalization.
$\triangle_{E,0}(\sigma)$$\sigma$$\nu\in\overline{\mathbb{H}^{2}}\cross\overline{\mathbb{H}^{2}}-diag(\partial \mathbb{H}^{2})$$\rho\in\overline{\mathcal{Q}\mathcal{F}},$$\Delta_{E,0}(\rho)$$\mathbb{H}^{3}/\rho(\pi_{1}(T))$$\Delta_{E}(p)$$\infty$$P$

10

Figure 4: Dual of Ford domain

5

Ford domains for hyperbolic structures in$\mathcal{M}\mathcal{P}(p, q)$

To answer Problem 1.1 for the pared manifold $(M(p, q),$ $P$ ) with a coprime
integers $(p, q)\neq(O, \pm 1)$ , we will follow the following program.
(1) Construct a geometrically finite hyperbolic structure, , on the pared
manifoldwith the parabolic locus, whereis the regular neighborhood inof the union of$\sigma_{0}$$(M_{0}, P\cup\partial N(\gamma)\cup N(\alpha^{\pm}))$$N(\alpha^{\pm})$$P\cup\partial N(\gamma)\cup$$N(\alpha^{\pm})$$T_{0}^{\pm}$two simple closed curves$\alpha^{\pm}\subset T_{0}^{\pm}$.

(2) Construct a geometrically finite hyperbolic structure,, on thepared manifold $(M(p, q),$inby hyperbolic Dehn
filling on the structure .
$\sigma(p, q)$$P\cup N(\alpha^{\pm}))$$\partial \mathcal{M}\mathcal{P}(p, q)$$\sigma_{0}$(3) By using the (geometric continuity” argument, which is used in the Jorgensen theory, characterize the combinatorial structures of Ford domainsof the structures in.
$\mathcal{M}\mathcal{P}(p, q)$

See Figures 5 and 6, which illustrates the Ford domains for$\sigma_{0}$and$\sigma(3,5)$.

For the step (1), the desired hyperbolic structure, , is obtained fromtwo copies of the manifold of the double cusp group(oo, 1/2) by gluing
along a pair of boundary components of their convex cores.
$\sigma_{0}$$\lambda^{-1}$

Definition 5.1. For eachwith $0<s<1$ , let

be the complex obtained from the two copies of the complex $Rg(\infty, s)$ by gluing them together
along the edge with slope(see Figure 7).
$s\in\hat{\mathbb{Q}}$$\Delta_{0}^{s}$$\infty$

The following proposition follows from [5, Theorem 9.1].

11

Figure 5: Ford domain for

Figure 6: Ford domain for$\sigma_{0}$$\sigma(3,5)$

Figure 7: The link of the ideal vertex of$\Delta_{0}^{1/2}$

12

Figure 8: The link of the ideal vertex ofcase $(p, q)=(3,5)$
Proposition 5.2. The complex.
complex$\Delta_{E,0}(\sigma_{0})$$\triangle^{1/2}(p, q)$; this figure illustrates theis combinatorially equivalent to the$\Delta_{0}^{1/2}$

For the step (2), the following Proposition 5.4 is proved by studying theFord domains after hyperbolic Dehn filling. See [2] for an outline, in which
the definition of layered sohd torus is not correct; it should be modified asfollows. The following construction is parallel to the construction of the
topological ideal triangulation of the two bridge link complement introducedin [15]. Let
be the triangle of with vertices $0,1/2$ and 1/3. For any
coprime integers $(p, q)$ , let be the geodesic inwith endpoints $p/q$ and$s^{+}\in\{0,1/2,1/3\}$ which intersects the interior of. Let be the triangleof with vertex $p/q$ whose interior intersects . Let
be the triangle whichshares an edge withand does not contain $p/q$ . Let
be the vertex ofwhich is not contained in . We introduce the equlvalence $relation\sim_{s^{-}}$
on the boundary component of $Rg(s^{-}, s^{+})$ corresponding tofollowing$Trg(s^{-},s^{+})/\sim_{s^{-}}$[15, Section II.2]. Let $V(p, q)$ be the quotient space. Then$V(p, q)$ is homeomorphic to the solid torus with apoint on the boundary
anddo not share an edge. We can see also thatremoved whenever
$V(p,
q)$
the meridian ofhas slope $p/q$ .
$\sigma^{+}$$\mathcal{D}$$\mathbb{H}^{2}$$l$$\sigma^{+}$$l$$\mathcal{D}$$\sigma^{-}$$\sigma^{-,*}$$s^{-}$$\sigma^{-}$$\sigma^{-,*}$$\sigma^{-}$$\sigma^{-,*}$$\sigma^{+}$$\sigma^{-,*}$

be the double cover of $V(p,q)$ . For eachDefinition 5.3. Lettowith $0<s<1$ , let
be the complex obtained by gluingwith edges of slopes $(0,1/3,1/2)$ and theso that the triangle ofwith edges of slopes $(\infty, 0,1)$ are identified. (See Figure 8.
triangle of
See also Figures 10 and 11.)$\overline{V}\cdot(p, q)$$s\in \mathbb{Q}$$\uparrow(p,q)\sim_{V}$$\Delta^{s}(p, q)$$\partial\tilde{V}(p, q)$$\triangle_{0}^{s}$$\partial\triangle_{0}^{s}$

Proposition 5.4. For all but finite coprime integers $(p, q)$ , the complex(see Figure 8).
is combinatorially equivalent to$\Delta_{E,0}(\sigma(p, q))$$\Delta^{1/2}(p)q)$

13

Figure 9: Ford domain for a hyperbolic structure in
for $(p, q)=(3,5)$$J_{sym}^{thick}\subseteq \mathcal{M}\mathcal{P}_{S\}},m(p, q)$consisting of the elements
be the subspace of
whose image
inby the map defined in Proposition 2.5 satisfies.
is contained inthatis the mirror image of . Then
The symmetry of this kind seems to be useful to carry out the “geometriccontinuity’) argument.

Let$\mathcal{M}\mathcal{P}(p, q)$$\mathcal{M}\mathcal{P}_{sym}(p\rangle q)$$(\lambda^{-},\cdot\lambda^{+})$$\lambda^{+}$$\mathcal{T}\cross \mathcal{T}$$\sigma(p, q)$$\lambda^{-}$$\partial \mathcal{M}\mathcal{P}_{sym}(p, q)$

Question 5.5. Is the combinatorial structure of the Ford domain for every
characterized by a way similar to that
hyperbolic structure in
given Fn Proposition 5.4?
$\mathcal{M}\mathcal{P}_{s\}},m(p)q)$

is
is thick good if
We say a hyperbolic structurewith $0<s<1$ , andcombinatorially equivalent tofor some
consisting of the thick good structures bydenote the subset of. Figure 9 illustrates the Ford domain for a hyperbolic structure confor $(p, q)=(3,5)$ . We can draw a conjectural
tained infor $(p, q)=(3,5)$ as Figure 12. In the Pgure, the hyperpicture of, where is a lift to abolic structures are parametrized byrepresentation of the holonomy representation for the structure. A point in
the plane is colored gray if the corresponding representation determines anembedding into of a simplicial complex which is supposed to be the dual of
the Ford domain and if the radii of the isometric hemispheres corresponding
to the vertices of the complex do not exceed 1. The condition on the radiiof isometric hemispheres is necessary for the corresponding holonomy representation to be discrete. Those points are colored by two different colors$\sigma\in \mathcal{M}\mathcal{P}_{sym}(p)q)$$\triangle^{s}(p, q)$$\triangle_{E}(\sigma)$$s\in \mathbb{Q}$$\mathcal{M}\mathcal{P}_{sym}(p, q)$$\mathcal{J}_{sym}^{thick}$$\mathcal{J}_{sym}^{thick}\subset \mathcal{M}\mathcal{P}_{sym}(p, q)$$\mathcal{J}_{sym}^{thick}$$Tr\rho(\gamma)$$\mathbb{C}$$\rho$$SL(2, \mathbb{C})-$14

Figure 10: Idea of construction of the dual complex (1): The dual complex isbased on two copies of the dual complex for a quasifuchsian punctured torus
group

15

Figure 11: Idea of construction of the dual complex (2): By drilling and fillingwith the dual complex for a cyclic Kleinian group, we obtain the desiredcomplexaccording to the change of combinatorial structures of the Ford domains.
Question 5.6. How is the Ford domain for a hyperbolic structure incharacterized?

$\mathcal{M}\mathcal{P}(p,q)-$$\mathcal{J}_{sym}^{thick}$

References[1] H. Akiyoshi, “On the Ford domains of once-punctured torus groups”,
Hyperbolic spaces and related topics, RIMS, Kyoto, Kokyuroku 1104
(1999), 109-121.

decompositions of cusped hyperbolic 3manifolds obtained by Dehn fillings”, Perspectives of Hyperbolic Spaces,
RIMS, Kyoto, Kokyuroku 1329, 121-132, (2003).

[2] H. Akiyoshi,(Canonical[3] H. Akiyoshi, “Ford domains of a certain hyperbolic 3-manifold whoseboundary consists of a pair of once-punctured tori”, Complex Analysis
and Geometry of Hyperbolic Spaces, RIMS, Kyoto, Kokyuroku 1518,
63-75, (2006).

16

Figure 12: Conjectural picture of$f_{sym}$

for $(p, q)=(3,5)$[4] H. Akiyoshi and M. Sakuma, (Comparing two convex hull constructions
for cusped hyperbolic $manifold_{S}$ Kleinian groups and hyperbolic 3manifolds (Warwick, 2001), 209-246, London Math. Soc. Lecture Note
Ser., 299, Cambridge Univ. Press, Cambridge, (2003).
)’[5] H. Akiyoshi, M. Sakuma, M. Wada and Y. Yamashita, “Jorgensen’s
picture of punctured torus groups and its refinement“, Kleinian groups
and hyperbolic 3-manifolds (Warwick, 2001), 247-273, London Math.
Soc. Lecture Note Ser., 299, Cambridge Univ. Press, Cambridge, (2003).
[6] H. Akiyoshi, M. Sakuma, M. Wada, and Y. Yamashita, “Punctured
torus groups and 2-bridge knot groups (I), preprint.

[7] R. Canary, (A covering theorem for hyperbolic 3-manifolds and its applications”, Topology, 35 (1996), 751-778.
[8] T. A. Drumm and J. A. Poritz, “Ford and Dirichlet domains for cyclic
andsubgroups of $PSL(2, C)$ action on
Conformal Geometryand Dynamics, An Electronic Journal of the A.M.S. Vol. 3 (1999), 116$\mathbb{H}_{\mathbb{R}}^{3}$$\partial \mathbb{H}_{R)}^{3}$

150.

[9] W. Floyd and A. Hatcher, “Incompressible surfaces in punctured torus
bundles”, Topology Appl., 13 (1982), 263-282.

[10] T. Jorgensen, “On cyclic groups of Mobius transformations“, Math.
Scand., 33 (1973), 250-260.
[11] T. Jorgensen, “On pairs of punctured tori“, in Kleinian Groups andHyperbolic 3-Manifolds, Y. Komori, V. Markovic &C. Series (Eds.),
London Mathematical Society Lecture Notes 299, Cambridge University
Press, (2003).

17[12] M. Kapovich, Hyperbolic manifolds and discrete groups, Progress inMathematics 183, Birkhauser Boston, Inc., Boston, MA, (2001).
[13] M. Lackenby, “The canonical decomposition of once-punctured torus
bundles”, Comment. Math. Helv., 78 (2003), no. 2, 363-384.

[14] Y. Minsky, “The classification of punctured torus groups”, Ann.
Math., 149 (1999), 559-626.

of[15] M. Sakuma and J. Weeks, “Examples of canonical decompositions ofhyperbolic link complements”, Japan. J. Math. (N.S.) 21 (1995), no. 2,

393-439.
Osaka City University Advanced Mathematical Institute,
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japane-mail: akiyoshi@sci.osaka-cu.ac.jp