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数理解析研究所講究録
第2062巻 2018年 133-138

A coordinate system for the Teichmüller space of a compact
surface and a rational represesentation of the mapping class group
Toshihiro Nakanishi

Shimane University§1. Teichmüller spaces and mapping class groups. Let S=S_{g,n} denotec_{n} . Wea compact oriented surface of genus g with n boundary curves \mathrm{c}_{1}>0assume that 2g-2+n. The fundamental group $\Gamma$_{g,n}$\pi$_{1}(S) has the=presentation:\langle a_{1}, b_{1} ,\cdots, a_{g}, b_{g}, c_{1} ,\cdots, c_{n} :(\displaystyle\prod_{j=1}^{g}[a_{j},b_{j}])c_{1}\cdots\mathrm{c}_{n}=1\rangle,aba^{-1}b^{-1} is the commutator of a and b , and we denote also
where [a, b]
by c_{j} the homotopy class of c_{j} . Let L(L_{1}, L_{n}) \in \mathb {R}_{\geq 0}^{n} and $\Gamma$_{g,n}(L) be
the Teichmüller space of isotopy classes of complete hyperbolic metrics on==

the interior I(S) of’S with the length of the geodesic isotopic to c_{j} is L_{j} forn (c_{j} corresponds to a puncture if L_{j} =0. ) Let C=C_{g,n} denote theset of isotopy classes of unoriented closed curves in I(S) . Each $\gamma$\in C defines areal analytic function on $\Gamma$_{g,n}(L) called the geodesic length function associated

j=1,to $\gamma$ : For each X\in$\Gamma$_{g,n}(L)P_{ $\gamma$}(X)= the length of the geodesic representation in $\gamma$ on X.
We also define $\tau$_{ $\gamma$}(X)=2\cosh(\ell_{ $\gamma$}(X)/2) . X defines a Fuchsian representation$\chi$ of $\Gamma$_{g,n} into PSL(2, \mathbb{R}) up to conjugacy and we have$\tau$_{ $\gamma$}(X)=|\mathrm{t}\mathrm{r} $\chi$( $\gamma$)|.
We call $\tau$_{$\gam a$} the trace function associated to $\gamma$ . We can identify X \in $\Gamma$_{g,n}(L)with the simultaneous conjugacy class \mathcal{G}(X) of a tuple of matrices in SL(2, \mathbb{R})(A_{1}, B_{1}, A_{g}, B_{g}, C_{1}, C_{n})=( $\chi$(a_{1}), $\chi$(b_{1}), $\chi$(a_{g}), $\chi$(b_{g}), $\chi$(c_{1}), $\chi$(c_{ $\eta$}))with \mathrm{t}\mathrm{r}A_{j}>0, \mathrm{t}\mathrm{r}B_{j} > 0 (j = 1, g) and \mathrm{t}\mathrm{r}C_{j}(j=1, n) , and hence identify $\Gamma$_{g,n}(L) with=-2\cosh(L_{j}/2)

T_{g,n} (\ell_{1}, l_{n})=\{\mathcal{G}(X):X\in$\Gamma$_{g,n}(L)\}.

=-l_{j} < 0134The Teichmüller space $\Gamma$_{g,n}(L) is homeomorphic to \mathbb{R}^{d} , where d=6g-6+2n.

Let \mathcal{M}C_{g,n} denote the mapping class group of the surface S=S_{g,n} . Eachelement [f] of \mathcal{M}C_{g_{)}n} is the isotopy class of an orientation preserving diffeomor‐phism f : S\rightarrow S preserving each boundary curve setwise. \mathcal{M}C_{g,n} acts on theTeiichmüller space $\Gamma$_{g,n}(L) . If X= (S, $\sigma$) \in$\Gamma$_{g,n}(L) , where $\sigma$ is a hyperbolic
metric on S , then [f](X) is the isotopy class of (S, f^{*} $\sigma$) . This group induces a
subgroup of outer automorphisms of the surface group $\Gamma$_{g,n}.
The first statement of the following theorem is proved by Schmutz, Oku‐mura, Feng Luo and others. For a proof of the full statement, see [8].
Theorem 1 There are simple closed curves $\gamma$_{1}$\gamma$_{d+1}on I(S) such that$\Phi$:$\Gamma$_{g,n}(L)\rightar ow \mathbb{R}^{d+1}defined by $\Phi$(X)=($\tau$_{$\gamma$_{1}}(X), $\tau$_{$\gamma$_{d+1}}(X)) is an embedding. Moreover, the map‐ping class group \mathcal{M}C_{g,n} acts on $\Phi$($\Gamma$_{g,n}(L)) as a group of rational transfor‐
mations in the coordinates x_{1}
x_{d+\mathrm{i}} of \mathbb{R}^{d+1} and l_{1},\ldots,\ell_{n} over the rational
number field.
§2. Finite subgroups of the mapping class group of genus 2 surface.
For the rest of this note, $\Gamma$_{g} means the Teichmüller space of the closed surface ofgenus g . By the Nielsen‐Kerckhoff realization theorem [5], each finite subgroup
G of \mathcal{M}C\mathcal{G}_{g}\mathcal{M}C\mathcal{G}_{g,0} acts on a Riemann surface R of genus g as a groupof conformal automorphisms. For each $\varphi$\in \mathcal{M}C\mathcal{G}_{g} , let $\varphi$_{*} denote the rationaltransformation acting on $\Phi$($\Gamma$_{g}) obtained by Theorem 1. Let x_{0} = $\Phi$(X_{0}) bex_{0} for some m > 0 , then $\varphi$ is an
an arbitrary point of $\Phi$($\Gamma$_{g}) . If $\varphi$_{*}^{m}(x_{0})isotopy class of a conformal automorphism (including the identity map) on the==Riemann surface X_{0} and we can conclude that $\varphi$ is elliptic or it has a finiteorder. Since the order of an elliptic element is at most a number Pg depending

only on g ( \leq 84(g-1) by Riemann‐Hurwitz formula), we can detect whether
an element of \mathcal{M}C_{g} is ellptic or not by showing some $\varphi$_{*}^{m} (1 \leq m\leq P_{g}) fixes
x_{0}.

Let G be a finite subgroup of \mathcal{M}C_{g} and assume that all elements of G fix
a Riemann surface R of genus g . If the genus of the factor surface R/G is hp_{n} withand the covering map $\pi$ : R\rightarrow R/G is branched over n points p_{1}branching orders m_{j} with m_{1}\leq m_{2}\leq\cdots\leq m_{n} , then (h;m_{1}, m_{n}) is the typeof the orbifold R/G . In stead of (h, m_{1}, m_{n}) , we often write (h;$\nu$_{1}^{r}1, $\nu$_{p}^{r_{p}})($\nu$_{1}<\cdots<$\nu$_{p}) if \mathrm{u}_{j} appears r_{j} times in (m_{1}, m_{n}) .

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The mapping class group \mathcal{M}C\mathcal{G}_{2} of a closed orientable surface of genus 2
is generated by Dehn twists $\omega$_{1}, $\omega$_{2}, $\omega$_{3}, $\omega$_{4} and $\omega$_{5} with the following defining

relations (see [1, p.184]):
if |i-j| \geq 2, 1\leq i, j\leq 5$\omega$_{i}$\omega$_{i+1}$\omega$_{i}=$\omega$_{i+1}$\omega$_{i}$\omega$_{i+1} (1\leq i\leq 4)$\omega$_{i}$\omega$_{j}=$\omega$_{j}$\omega$_{i}($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{6}=1($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1})^{2}=1$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1} and $\omega$_{i} commute for i=1 , 2, 3, 4, 5

(1)
(2)
(3)
(4)
(5)

In [2] S. A. Broughton classified completely the finite subgroups of \mathcal{M}C\mathcal{G}_{2},up to topological equivalence. After a lengthy calculations, Nakamura and$\omega$_{5} for thethe author found explicit expressions by the Dehn twists $\omega$_{1},generator‐systems in Broughton’s list.

Theorem 2 ([9]). A non‐trivial finite subgroup of \mathcal{M}C\mathcal{G}_{2} of a closed ori‐
entable surface of genus 2 is conjugate with one of the groups in the tablebelow.

The table shows the group G_{*} corresponding to (2,*) in [2] with generators, $\omega$_{5} , the order |G_{*}| and the orbifold type.

expressed in $\omega$_{1} ,(2.a) G_{a}=\{x : x^{2}=1\rangle\cong \mathbb{Z}_{2}, x=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1} , 2, (0;2^{6})(2.b) G_{b}=\langle x : x^{2}=1\rangle\cong \mathbb{Z}_{2}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3} , 2, (1;2^{2}) .
(2.c) G_{c}=\langle x : x^{3}=1\rangle\cong \mathbb{Z}_{3}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{2} , 3, (0;3^{4}) .

(2.e) G_{e}=\langle x : x^{4}=1\rangle\cong \mathbb{Z}_{4}, x=($\omega$_{1}$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4})^{2} , 4, (0;2^{2},4^{2}) .

[x, y] 1\} \cong \mathbb{Z}_{2} \times \mathbb{Z}_{2}, x=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1}.
y=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3}, 4, (0;2^{5})(2. f) G_{f}=\langle x : x^{2}=y^{2}==.

(2.h) G_{h}=\langle x : x^{5}=1\rangle\cong \mathbb{Z}_{5}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4})^{2} , 5, (0;5^{3}) .

(2.i) Gi=(x : x^{6}=1\}\cong \mathbb{Z}_{6}, x=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5},6 , (0,3,62).
(2.\mathrm{k}.1) G_{k1}=\langle x:x^{6}=1\rangle\cong \mathbb{Z}_{6}, x=$\omega$_{1}$\omega$_{2}$\omega$_{5}^{-1}$\omega$_{4}^{-1} , 6, (0,2^{2},3^{2}) .

136\langle x, y : x^{2}

1, xyx^{-1}y=($\omega$_{1}$\omega$_{2}$\omega$_{5}^{-1}$\omega$_{4}^{-1})^{2} , 6, (0,2^{2},3^{2}) .

(2.\mathrm{k}.2) G_{k2}

=

=

y^{3}

=

=

y^{-1}\rangle \cong D_{3}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3},(2.1) G_{l}=\langle x:x^{8}=1\}\cong \mathbb{Z}_{8}, x=$\omega$_{1}$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4} , 8, (0;2,8,8).
(2.m) G_{m}=\langle x, y : x^{4}=y^{4}=1, x^{2}=y^{2}, xyx^{-1}=y^{-1}\rangle\cong\tilde{D}_{2}, x=($\omega$_{1}$\omega$_{2}$\omega$_{1}$\omega$_{3}$\omega$_{4})^{2},y=($\omega$_{2}$\omega$_{3}$\omega$_{5}$\omega$_{4}$\omega$_{3})^{2}, 8, (0;4,4,4)(2.n) G_{n}=\langle x, y : x^{2}

=

y^{4}

=

.

1, xyx^{-1}y=($\omega$_{1}$\omega$_{2}$\omega$_{4}$\omega$_{3}$\omega$_{2})^{2}, 8, (0,2^{3},4)=y^{-1}\rangle \cong D_{4}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3},.

(2.0) G_{o}=\langle x:x^{10}=1\rangle\cong \mathbb{Z}_{10}, x=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4} , 10, (0,2,5,10) .
(2.p) G_{p}=\langle x, y:x^{2}=y^{6}=[x, y]=1\rangle\cong \mathbb{Z}_{2}\times \mathbb{Z}_{6}, x=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1},y=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5} , 12, (2, 6, 6).
\{x : x^{4}

1, xyx^{-1}y^{3}y=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{2}, 12, (0,3,4^{2})(2. r) G_{r}(2.s) G_{s}==\langle x, y : x^{2}

=

=

=

=

y^{6}

=

1, xyx^{-1}y=$\omega$_{1}$\omega$_{2}$\omega$_{5}^{-1}$\omega$_{4}^{-1}, 12, (0,2^{3},3)=\{x, y : x^{2}

=

=

=

($\omega$_{1}$\omega$_{2}$\omega$_{4}$\omega$_{3}$\omega$_{2})^{2},y^{-1}\rangle \cong D_{6}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3},y^{3}\rangle \cong D_{2,8,3}, x=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3},=.

1, xyx^{-1}y^{8}y=$\omega$_{1}$\omega$_{2}$\omega$_{4}$\omega$_{3}$\omega$_{2} , 16, (0,2,4,8).

(2.u) G_{u}y^{-1}\rangle \cong D_{4,3,-1}, x.

=:\langle
X,xyx^{-1}=y,xzx^{-1}=zy,xwx^{-1}=w^{-1}x^{2}=y^{2}=z^{2}=w^{3}=[y,z]=[y, w]=[z, w]=1 \rangle\mathbb{Z}_{2}\ltimes(\mathbb{Z}_{2}\times \mathbb{Z}_{2}\times \mathbb{Z}_{3}) x=($\omega$_{1}$\omega$_{2}$\omega$_{1}$\omega$_{4}^{-1}$\omega$_{5}^{-1}$\omega$_{4}^{-1})($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3}, y=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1},(2.w) G_{w}\congy, Z, w=,z=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{3}, \mathrm{w}=($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5})^{4} , 24, (0,2,4,6).

(2.x) G_{x}=\langle x, y : x^{3}=y^{4}=1, xy^{2}=y^{2}x , (xy)3 =1\rangle\cong SL_{2}(3) ,x=($\omega$_{2}$\omega$_{1}$\omega$_{4}^{-1}$\omega$_{5}^{-1})($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1}) , y=($\omega$_{1}$\omega$_{2}$\omega$_{1}$\omega$_{3}$\omega$_{4})^{2}, 24, (0,3^{2},4)\langle x,
GL_{2}(3)(2.aa) G_{xx}=y, u :u^{2}=xy^{-1}x^{-1}y^{2}uyu^{-1}=x^{-1}yxx^{3}=y^{4}=(xy)^{3}=1,xy^{2}=y^{2}x,uxu^{-1}=y^{-1}x^{-1}y,\rangle\cong,

x= ($\omega$_{2}$\omega$_{1}$\omega$_{4}^{-1}$\omega$_{5}^{-1}$\omega$_{4}^{-1})($\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}^{2}$\omega$_{4}$\omega$_{3}$\omega$_{2}$\omega$_{1}) , y= ($\omega$_{1}$\omega$_{2}$\omega$_{1}$\omega$_{3}$\omega$_{4})^{2}, u=$\omega$_{2}$\omega$_{3}$\omega$_{5}$\omega$_{4}$\omega$_{3}, 48, (0,2,3,8)

For general g >Dehn twists$\omega$_{0}, $\omega$_{1}1,

the mapping class group \mathcal{M}C_{g} is generated 2g+ 1$\omega$_{2g}called Humphries generators (See Theorem 4.14

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and Figure 4.5 in [3]) such that the same relations as in (1) and (2) hold and$\zeta$^{2g+2}=$\eta$^{4g+2}=1 , where, with an additional Dehn twist $\omega$_{2g+1} about a curve
c_{2g+1}=m_{g} in Figure 4.5 in [3],$\zeta$=$\omega$_{1}$\omega$_{2}

.

.

.

$\eta$=$\omega$_{1}$\omega$_{2}$\omega$_{2g+1},

.

.

.

$\omega$_{2g}.

We have by (1) and (2)$\omega$_{2} $\zeta$ = $\omega$_{1}$\omega$_{2}$\omega$_{1}($\omega$_{3} . . . $\omega$_{2g+1})= ($\omega$_{1}$\omega$_{2} . . . $\omega$_{2g+1})$\omega$_{1}= $\zeta \omega$_{1}and likewisefor i=1,$\omega$_{i+1} $\zeta$= $\zeta \omega$_{i}

2g .

(6)

By using this we have also that$\omega$_{1} $\zeta$ = $\zeta \zeta$^{-1}$\omega$_{1} $\zeta$==$\zeta \omega$_{2g+1}^{-1}$\omega$_{2g}^{-1}$\zeta \omega$_{2g+1}^{-1}$\omega$_{2g}^{-1}

.

.

.

.

.

.

.

.

$\omega$_{2}^{-1} $\zeta$$\omega$_{3}^{-1} $\zeta \omega$_{1}^{-1}:=$\zeta$^{2}$\omega$_{2g}^{-1}$\omega$_{2g-1}^{-1}.

$\omega$_{1}^{-1}=$\zeta$^{2}$\eta$^{-1}and hence $\omega$_{1}=$\zeta$^{2}$\eta$^{-1}$\zeta$^{-1} . Then by (6)$\omega$_{2}=$\zeta$^{3}$\eta$^{-1}$\zeta$^{-2}, $\omega$_{3}=$\zeta$^{4}$\eta$^{-1}$\zeta$^{-3},,$\omega$_{2g+1}=$\zeta$^{2g+2}$\eta$^{-1}$\zeta$^{-2g-1}=$\eta$^{-1}$\zeta$^{-2g-1}

If g=2 , then c_{0}=c_{5} and hence we obtain Korkmaz’s theorem [6] for g=2.
Theorem 3 The mapping class group \mathcal{M}C_{2} is generated by $\zeta$=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4}$\omega$_{5}
and $\eta$=$\omega$_{1}$\omega$_{2}$\omega$_{3}$\omega$_{4} satisfying $\zeta$^{6}=$\eta$^{10}=1.
Hirose obtained expressions by Dehn twists of all torsions in the mapping classgroup \mathcal{M}C_{g} with g\leq 4 in [4].
This note is based on work with Gou Nakamura, Aichi Institute of Technology.

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References[1] Birman, J. S., The Braids, Links and Mapping Class Groups, Ann. ofMath. Studies 82, Princeton Univ. Press, 1974.

[2] Broughton, A. S, Classifying finite group actions on surfaces of low genus,
Journal of Pure and Applied Algebra, 69 (1990), 233‐270.
[3] Farb, B. and D. Margalit, A Primer on Mapping Clas\mathcal{S} Groups. Pronceton
University Press, 2011.

[4] Hirose, S., Presentation of periodic maps on oriented closed surfaces ofgenera up to 4, Osaka J. Math., 47 (20109, 385‐421.
[5] Kerckhoff, S. P., The Nielsen realization problem, Ann. of Math., 117(1983), 235−265[6] Korkmaz, M., Generating the surface mapping class group by two ele‐ments, Trans. Amer. Math. Soc., 357, 3299‐3310.

[7] Feng Luo, Geodesic length functions and Teichmüller spaces, J. Differen‐tial Geom., 48 (1998), 275‐317.
[8] G. Nakamura and T. Nakanishi, Parametrizations of Teichmüller spaces
by trace functions and action of mapping class groups, Conform. Geom.

Dyn., 20 (2016), 25‐42.
[9] G. Nakamura and T. Nakanishi, Presentation of finite subgroups of map‐ping class group of genus 2 surface by Dehn‐Lickorish‐Humphries genera‐
tors, Preprint.
DEPARTMENT OF MATHEMATICS, SHIMANE UNIVERSITY, MATSUE, 690‐
8504, JAPANE‐mail addres \mathcal{S} : tosihiro@riko.shimane −u.ac.jp