数理解析研究所講究録1104 巻 1999 年 122-132

122

A canonical cellular decomposition of theTeichm\"uller space of compact surfaces withboundary
Akira USHIJIMA’ 牛島顕

Department of Mathematics, Graduate School of Science,
Osaka University, 1-1 Machikaneyama-cho, Toyonaka,
Osaka 560-0043, Japan. ecip. osaka-u. . jp
-mail address:$E$$\mathrm{s}\mathrm{m}\mathrm{v}\mathrm{O}\mathrm{O}\mathrm{l}\mathrm{u}\mathrm{a}\omega_{\mathrm{e}}\mathrm{x}$$\mathrm{a}\mathrm{c}$Abstract
This article is a summary of [Usl].
Using the Euclidean decomposition of the hyperbolic surface, R. C.
Penner gave a canonical cellular decomposition of the decorated Teichm\"uller
space of punctured surfaces, which is invariant by the action of the mapping class group. Adapting his method, we give a canonical cellular decomposition of the Teichn\"uller space of compact orientable surfaces with
non-empty boundary.

Introduction

1

This article is a summary of [Usl].
’
TeR. C. Penner introduced in [Pe] a method for dividing theichm\"uller space of
surfaces by “natural” cells. Here, “decorated”’
means thatmeans that each puncture is given some “weight,)’ andthe decomposition is invariant by the action of the mapping class group. In
his method, the Euclidean decomposition of punctured surfaces with weight
an important role (see [EP]). Since then, it has been tried to extend his
construction to the Teichm\"uller space of other kinds cf surfaces (see Table 1).
S. Kojima introduced in [Ko] a canonical method to decompose compact
hyperbolic manifolds with non-empty totally geodesic boundary into truncatedpolyhedra. In this paper, using this decomposition and Penner’s method, wegive a canonical cellular decomposition of the Teichn\"uller space of compact
orientable surfaces with non-empty boundary (see Theorem 2.1).
$‘(\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d})$$\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}^{}\mathrm{d}$$‘(\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l})$$\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}_{\mathrm{S}}$*The author is partially$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{e}}\mathrm{d}$by JSPS Research Fellowships for Young Scientists.

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On the other hand, for the decomposition, M. N\"a\"at\"anen obtained a cellular
decomposition of the Teichm\"uller space of closed surfaces with a distinguishedpoint in [N\"a]. In her study, the decomposition of such surfaces introduced in[NP] plays a role of Euclidean decomposition in Penner’s work.
Table 1: Cellular decompositions of Teichm\"uller spaces by the canonical decomposition of the surface

canon. decomp.

surfacecusped srfc.
srfc. with a point. srfc. with bdry.

[EP]
[NP]
[Ko]

$\mathrm{c}\mathrm{p}\mathrm{t}$

$Tei$

.

$sp$

.

application
Penner et. al.

[Pe][N\"a][Usl].

$[\mathrm{N}\mathrm{N}.]$$.\mathrm{e}\mathrm{t}\mathrm{c}$.

Main theorem

2Let
be a compact orientable surface obtained
closed orientable surface
genus
by removing the interior of disjoint closed disks on the surface.
ofMoreover we assume 2 $g-2+r>0$ . This assumption means that
admitsa complete hyperbolic structure. Now we denote by
the Teichm\"uller spaceof. By the assumption as above, we regard
as the set of hyperbolic(with each boundary component being totally geodesic) up
structures on, the set of diffeomorphisms acting on, which is homotopic
toto the identity relative to the boundary. Each element ofdetermines a
group
consisting of the orientation-preserving
marked discrete subgroup of theplane.
hyperbolicof
Sb we denote by
the element oftheisometries$:=$, namely. We denote by
the mapping class group of, where.
is the set of diffeomorphisms acting on
For detailed definitions, please see, for example, [Ra, Th].
. Then, as we saw the above, it gives a hyperbolicof
Fix an elementwith each boundary component being totally geodesic. Now
structure on
the set of points ineach of which admits at least two distinct shortest paths
to the boundary consists the graph (see the left of Figure 1). We call this graphobtained by the dual of the cut
the cut locus, and the decomposition of(see the center ofwith respect tolocus the canonical decomposition of
Figure 1). It is known that the decomposition is actually cellular, that is, eachpiece obtained by the decomposition is homeomorphic to a disk. For detailed
definitions, please see [Ko].
with the following two conditions: each arc
be a set of arcs inLetis a disjointly embedded simple arc connecting boundaries (maybe the same
boundary), and the closure of each complementary region of arcs is a hexagon.
a truncated triangulation of
We call such region a truncated tniangle, and$F_{g,r}$$\mathrm{h}\mathrm{o}\mathrm{m}$

$r$

$g$

$F_{g,r}$$\mathcal{T}_{g,r}$$F_{g,r}$$T_{g,r}$$F_{g,r}$$F_{g,r}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0}F_{g,r}$$\mathcal{T}_{g,r}$$\mathrm{H}^{2}$$\Gamma_{m}$$F_{g,r}$$\mathrm{M}\mathrm{C}_{g,r}$$\mathcal{T}_{g,r}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}F_{g,r}/\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}_{0^{F_{g,r}}}$$F_{g,r}$$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}F_{g_{)}r}$$\Gamma_{m}$$T_{g,r}$$F_{g,r}$$F_{g,r}$$F_{g,r}$$F_{g,r}$$\triangle$$\mathrm{M}\mathrm{C}_{g,r}$$\Gamma_{fn}$$F_{g,r}$$\triangle$

124$F_{g,r}$. Euler characteristic considerations show that there are 6 $g-6+3r$ arcs
. We call the cellular decomposition ofobtained by deleting several
arcs (maybe empty) ffom a truncated triangulation a truncated cellular decom(see the right of Figure 1). Of course if we delete much arcs
position of
from , then the decomposition is not even cellular. For a truncated cellular, we denote byeach ofthe set of points indecompositionof
whose canonical decomposition coincides (topologically) to . By the definition
throughof the canonical decomposition, it is easy to see that the union of-invariant decomposition of
all truncated cellular decompositions gives an. Furthermore we can prove the following theorem, the main theorem of thisarticle:in$\triangle$$F_{g,r}$$F_{g,r}$$\triangle$$\triangle$$C\circ(\triangle)$$F_{g,r}$$T_{g,r}$$\triangle$$C\circ(\triangle)$$\mathrm{M}\mathrm{C}_{g,r}$$\mathcal{T}_{g,r}$, Theorem 6.6]) If
is a truncated cellular decom.
The setis an open cell of dimension
is a
is a truncated cellular decomposition ofTheorem 2.1 (,position of$\triangle$$[\mathrm{U}\mathrm{s}1$$F_{g,r}$$\{C\circ(\triangle)|\triangle$$C\circ(\triangle)$$\neq\triangle$$\mathrm{M}\mathrm{C}_{g,r^{-}}inva\dot{n}ant$$F_{g,r,\backslash }\}$

cellular decomposition of
itself. Furthermore, the isotropy group ofleavinginis isomorphic to the (finite) group of mapping classes ofinvariant.
$C(\triangle)$$\mathcal{T}_{g,r}$$F_{g,r}$$\mathrm{M}\mathrm{C}_{g,r}$$\triangle$

Figure 1: A canonical decomposition of3$F_{0,3}$

Proof of the main theorem

is a truncated triangulation, not
In this section, under the assumption thatis homeoa truncated cellular decomposition, we explain the proof thatmorphic to an open ball of dimension $q:=6g-6+3r$ by the following four
steps.
$\triangle$$C\circ(\triangle)$

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3.1

$s$

-length coordinate

Let $\triangle=(c_{1}, c_{2,\ldots,q}c)$ be a truncated triangulation of. For any element, we define the -length of relative toofas follows:$F_{g,r}$$\Gamma_{m}$$s$$\mathcal{T}_{g,r}$$\Gamma_{m}$$\mathrm{q}$$s(c_{i}; \mathrm{r}_{m}):=\sqrt{2}\cosh\frac{d_{i}}{2}\in \mathrm{R}_{s}$,where means the hyperbolic distance of the geodesic , andUsing -lengths, we define the mappingfromtoas follows:$d_{i}$$\mathrm{R}_{s}:=\{t\in \mathrm{R}|t>\sqrt{\mathit{2}}\}$$c_{i}$$s$$S_{\triangle}$$\mathcal{T}_{g,r}$$S_{\triangle}(\Gamma_{m}):=(s(c_{1};\Gamma_{m}), s(C2;\Gamma_{m}),$$\ldots,$$\mathrm{R}_{s}^{q}$$S(C_{q};\Gamma m))\in \mathrm{R}_{\mathit{8}}^{q}$.

For this mapping we have the following theorem:, Theorem 4.1]) If
Theorem 3.1 (thenis a homeomorphism.
$[\mathrm{U}\mathrm{s}1$$F_{g,rf}$$\triangle$

is a truncated triangulation$of\square$$S_{\triangle}$.histheorem, for a fixed$\Sigma\in \mathrm{R}_{s}^{q}$, the pair$(\triangle, \Sigma)$$\mathcal{T}_{g,r}\mathrm{B}\mathrm{y}\mathrm{t}$Definition 3.2 (short) Fix a point$(\triangle, \Sigma)$of$\mathcal{T}_{g,r}$

is rega.rded as a point of.

in , we denote by
the -length of . Then we saythat $(e, \Sigma(e))$ is short if
holds, wheremeans the hyperbolic distance between and (see Figure 2).

(1) For any arc$e$$\triangle$$\Sigma(e)$

$s$

$e$

$\Sigma(e)<\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}_{\mathrm{H}}(\partial_{1}, \partial_{2})$$\partial_{1}$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}_{\mathrm{H}}(\partial_{1}, \partial_{2})$$\partial_{2}$Figure 2: $(e, \Sigma(e))$ is short
By the definition of the
proposition:

decomposition, we have the following$\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{i}_{\mathrm{C}\mathrm{a}}.1$

Proposition 3.3 (cf. [Usl, Theorem 6:1]) For an eleme$(\triangle, \Sigma)$is short if and only if$(\triangle, \Sigma)\in$$C\circ(\triangle)$.

.

$nt(\triangle, \Sigma)\in T_{g,r}$,$\square$This proposition implies that it is important to obtain an efficient tool to decide
the given arc is short or not, and the following -length coordinate is the one.
$h$

.

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3.2

$h$

-length coordinate

is decomposed into
For a truncated triangulation , the boundary of$:=$
the set of such segments. Let
several segments. We denote by$\{t\in \mathrm{R}|t>0\}$ .
, and we define the -length ofFix
foras follows (see also Figure 3):$\triangle$$F_{g,r}$$B_{\triangle}$$\mathrm{R}_{+}$$h$$\Sigma\in \mathrm{R}_{s}^{q}$$E\in B_{\triangle}$$\Gamma_{m}=(\triangle, \Sigma)$$h(E, \Gamma_{m}):=\frac{\Sigma(e)}{\Sigma(a)\Sigma(b)}$Figure 3:$h$.

-length of $E$

by transforming thetofromNow we define the mapping-length coordinate into the -length coordinate.
. It is easy to see that $\Sigma(e)^{-2}=$
by
We here observe the image of$h(A, \Gamma_{m})h(B, \Gamma m)$ holds under the situation of Figure 3. But it also holds that$\Sigma(e)^{-2}=h(C, \Gamma m)h(D, \Gamma m)$ .
is demanded the
So the element offollowing condition at every edge of the truncated triangulation:$I_{\triangle}$$\mathrm{R}_{+}^{B_{\Delta}}\approx \mathrm{R}_{+}^{2q}$$\mathrm{R}_{s}^{q}$

$h$

$s$

$\mathcal{T}_{g,r}$$I_{\triangle}$$I_{\triangle}(\mathcal{T}_{g_{)}r})$$h(A, \Gamma_{m})h(B, \Gamma m)=h(C, \Gamma_{m})h(D, \Gamma m)$.

We call this equation the coupling equation. Furthermore, since -lengths are, we also demand the following condition:
greater than$s$$\sqrt{\mathit{2}}$$(0<)h(A, \Gamma m)h(B,\Gamma m)<\frac{1}{\mathit{2}}$.

We call this inequality the coupling inequality. On the other hand, we can easily
satisfying the two conditions denoted above are also
see that elements in. Thus we obtain the following theorem:
elements in$\mathrm{R}_{+}^{2q}$$I_{\triangle}(\mathcal{T}_{g,r})$, Proposition 4.4]) The mapping
is an embedding
Theorem 3.4 (is characterized by the
Explicitly,
intoofequations and the coupling inequalities.
$I_{\triangle}$$[\mathrm{U}\mathrm{s}1$$\mathrm{R}_{+}^{B_{\Delta}}$$I_{\triangle}(\tau_{\mathit{9}},r)\subset \mathrm{R}_{+}^{B_{\Delta}}$$\mathcal{T}_{g,r}$$couplin_{\square }g$127

Using the -length coordinate, we can easily see whether the given edge isshort or not.
$h$

, Theorem 6.1]) Under the situation of Figure 3,
Proposition 3.5 ($(e, \Sigma(e))$ is short
if and only if the inequality $h(A,\Gamma_{m})+h(B,\Gamma_{m})+h(c,\Gamma_{m})+\square$$h(D,\Gamma_{m})>h(E,\Gamma_{m})+h(F, \Gamma_{m})$ holds.
$[\mathrm{U}\mathrm{s}1$Note
We can extend Proposition 3.5 to the following one:
Proposition 3.6 Under the situation of Figure(resp.
$>)$
if and only if $h(A,\Gamma_{m})+h(B, \Gamma_{m})+h(C, \Gamma_{m})+h(D, \Gamma_{m})=h(E, \Gamma_{m})+\square$(resp. ) holds.
$\mathit{3}_{f}\Sigma(e)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}_{\mathrm{H}}(E, F)$$h(F, \Gamma_{m})$$<$These two propositions are kinds of so-called “tilt proposition.” In [Us2], we
study a generalization of the tilt proposition.

3.3

Changing bases

As we saw the preceding subsection, Though the -length coordinate gives aneffective formula to decide whether the given edge is short or not. But the space
is twice the dimension of. So we extract the information of the short from.
the half-dimensional space of, we define a pair of vectors
For each edge
inandas the following figure:$h$$\mathcal{T}_{g,r}$$\mathrm{R}^{2q}$$e\in\triangle$$B_{e}$$B_{e}$$C_{e}$$\mathrm{R}^{\mathcal{B}_{\Delta}}\approx \mathrm{R}^{2q}$$C_{e}$

Figure 4: vectors$B_{e}$

and$C_{e}$

does not give any effects to the inequality in Proposition 3.5, we caneasily obtain the following proposition:

Since$C_{e}$

Proposition 3.7 (is a basis of$[\mathrm{U}\mathrm{s}1$, Lemma 6.3])$\mathrm{R}^{B_{\Delta}}\approx \mathrm{R}^{2q}$. Namely(1) The set of vectors$\{B_{e}, c_{e}\}_{6}\in\triangle$$\mathrm{R}^{B_{\Delta}}\cong\langle B_{e}\rangle_{e\in\triangle}\oplus\langle C_{e}\rangle_{e\in\triangle}$

.

128

(2) Suppose$(\triangle, \Sigma)$

3.4$I_{\triangle}(( \triangle, \Sigma))=\sum_{e\in\triangle}x_{ee}$$B+ \sum_{e\in\triangle}yeC_{e}$for some

is short if and only if $x_{e}>0$ for every$e\in\triangle$$x_{e},y_{e},$ $\in$

R. Then.

$\square$

The core of the proof

We define subsets of$X^{\mathrm{o}}$$v_{\triangle()}\circ\triangle$$g_{\triangle}\circ(\triangle)$

$:=$

$:=$

$:=$

$\mathrm{R}_{+}^{B_{\Delta}}$

as follows:$\{\sum_{e\in\triangle}xeBe\in \mathrm{R}_{+}^{B}\Delta$$x_{e}>0\}$,$\{z\in \mathrm{R}_{+^{\Delta}}B|\mathrm{a}\mathrm{n}\mathrm{d}(z\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{S}\triangle, I^{-}1(_{Z}\triangle)\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{C}\mathrm{o})\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{p}1\mathrm{i}\mathrm{n}_{\mathrm{h}}\mathrm{g}_{\mathrm{o}\mathrm{r}}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S},$$\{z\in D\mathrm{o}_{\triangle}(\triangle)|z$satisfies the coupling

$\}$

,

$\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}\}$by an immediate consequence of TheoWe note thatrem 3.1, Proposition 3.3 and Theorem 3.4.
Now the following theorem holds:$I_{\triangle}\circ S_{\triangle}(C(\circ\triangle))=\mathcal{G}_{\triangle}\circ(\triangle)$, Theorem 6.4] or [Pe, Theorem 5.4]) The projection
Theorem 3.8 (to .
induces a homeomo rphism from$[\mathrm{U}\mathrm{s}1$$D_{\triangle}\circ(\triangle)$$x^{\mathrm{o}}$$\square$$\Pi_{\triangle}$

By the definition of the coupling inequality, we can easily prove thatand the open unit ball inis homeomorphic to the intersection ofcentered at the origin. Thus we obtain the following theorem, which is the goalof this section:$\mathcal{G}_{\triangle}\circ(\triangle)$$D\mathrm{o}_{\triangle}(\triangle)$, Theorem 6.5]) The set
Theorem 3.9 (open ball of dimension $q=\mathit{6}g-\mathit{6}+3r$ .
$[\mathrm{U}\mathrm{s}1$A$\mathrm{R}^{B_{\Delta}}$$C\circ(\triangle)$

is $h_{omeom}o\mathit{7}phiC$ to

ExamplesWe see the decomposition of$\mathcal{T}_{0,3}$

in page 8, and$\mathcal{T}_{1,1}$

in pages 9 and 10.

$an\square$

129

Example 1.

$\mathcal{T}_{0,3}$$x:= \sqrt{2}\cosh\frac{d_{X}}{\mathit{2}}$,$y:= \sqrt{\mathit{2}}\cosh\frac{d_{Y}}{\mathit{2}}$,$z:= \sqrt{\mathit{2}}\cosh\frac{d_{Z}}{\mathit{2}}$

.

130

Example 2.

$\mathcal{T}_{1,1}$$x:= \sqrt{2}\cosh\frac{d_{X}}{2}$,$y:= \sqrt{2}\cosh\frac{d_{Y}}{\mathit{2}}$,$z:= \sqrt{2}\cosh\frac{d_{Z}}{2}$

.

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$[egg2]$$[egg1]$$[egg3]$$[egg5]$$[egg7]$$[egg9]$$[egg4]$$[egg6]$$[egg8]$

${ }$

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References[EP] D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, Journal of Differential Geometry 27 (1988),

67-80.
[Ko] Sadayoshi Kojima, Polyhedral decomposition of hyperbolic manifolds withboundary, On the Geometric Structure of Manifolds, edited by Dong Pyo
Chi, Proceedings of Workshops in Pure Mathematics, Volume 10, Part III(1990), 37-57.
[N\"a] Marjatta N\"a\"at\"anen, A cellular parameterization for dosed surfaces with adistinguished point, Annales Academiae Scientiarum Fennicae Series A. I.
Mathematica 18 (1993), 45-64.

[NN] Toshihiro Nakanishi and Marjatta N\"a\"at\"anen, The Teichm\"uller space of apunctured surface represented as a real algebraic space, Michigan Mathematical Journal 42 (1995), 235-258.

[NP] M. N\"a\"at\"anen and R. C. Penner, The convex hull construction for compact
surfaces and the Dirichlet polygon, Bulletin of the London MathematicalSociety 23 (1991), 568-574.
[Pe] R. C. Penner, The decorated Teichm\"uller space of punctured surfaces, Communications in Mathematical Physics 113 (1987), 299-339.
[Ra] John G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Textsof Mathematics, 149, Springer-Verlag, 1994.
[Th] William P. Thurston, Three Dimensional Geometw and Topology, Princeton Mathematical Series, 35, Princeton University Press, 1997.

[Usl] Akira Ushijima, A canonical cellular decomposition of the Teichm\"uller
space of compact surfaces with boundary, Communications in Mathematical
Physics 201 (1999), 305-326.
[Us2] Akira Ushijima, The tilt formula for weighted simplices in hyperbolic
space, preprint.