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Author(s)
Citation
Issue Date
URL
Trace fields of genus 3 surfaces with regular fundamental
polygons (Perspectives of Hyperbolic Spaces II)
Nakamura, Gou
数理解析研究所講究録 (2004), 1387: 170-178
2004-07
http://hdl.handle.net/2433/25798
Right
Type
Textversion
Departmental Bulletin Paper
publisher
Kyoto University
数理解析研究所講究録 1387 巻 2004 年 170-178
170
Ikace fields of genus 3 surfaces with regular fundamental polygons
豪 (
Nakamura)
Aichi Institute of Technology
愛知工業大学
中村
$\mathrm{G}\mathrm{o}\mathrm{u}$
1. Introduction
Let
be a Fuchsian group. The trace field
of is the field generated over Q by the
traces of elements in . In [5] M. Naatanen and T. Kuusalo determined the trace fields of all Fuchsian
groups of signature (2; 0) with aregular polygon ae afundamental polygon. In the present paper we shall
consider the trace fields br the case of signature (3,0) analogously.
$\Gamma\subset \mathrm{S}\mathrm{L}(2,\mathrm{R})$
$\Gamma$
$\mathrm{t}\mathrm{r}(\Gamma)$
$\Gamma$
2. Regular fundamental polygons and trace fields
By Euler’s formula we see that there are 4 regular polygons to be a compact surface of genus three.
1. 30-gon with each angle
mgon
-gon with ea&anQe
each angle
angle
3. llgon with each ange
4.
angle
with each ange
2.
$12\ovalbox{\tt\small REJECT} \mathrm{n}$
$2\pi/3$
$\pi/2$
,
$2\pi/7$
$\pi/6$
,
,
.
By using a computer we can show the side pairing patterns for each polygon..
Theorem 1. There exist $927$
, 112 for 14-g0n
side pairing patterns for 30-g0n, 297 for 20-g0n,
927 side-pairing
14-gon and 82 for
20
12-gon up to mimr
images.
mirror inagoe.
$\infty \mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}$
$\mathrm{s}\mathrm{i}\mathrm{d}\triangleright \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{i}\mathrm{n}\mathrm{g}$
$\mathrm{p}\mathrm{l}\mathrm{y}\Re \mathrm{n}$
$\alpha \mathrm{i}\#$
$\mathrm{g}\mathrm{o}\mathrm{n}$
$12-\Re \mathrm{n}$
The following is mentioned for the case of $(2,0)$ in [5].
Lemma 2. Let be a Fuchsian group of signature (3; 0) with a regular
as a fundamental polygon
$(n =6,7,10,15)$ . Then is a subgroup of the triangle group
of type $(2, 2n/(n-5)$ , $2n)$ .
$\Gamma$
$2n-\Re \mathrm{n}$
$\Gamma$
$\Lambda_{n}$
Proposition 3.(cf. Hilden, Lemno and Montesinoe Amilibia [3]) Let
by the squares of the elements of An.. Then it follows that
$\Lambda_{n}^{2}$
be the subgroup of
$\Lambda_{n}$
generated
.
$\mathrm{t}\mathrm{r}(\mathrm{A}_{n}^{2})\subset \mathrm{t}\mathrm{r}(\Gamma)\subset \mathrm{t}\mathrm{r}(\Lambda_{n})$
Proposition 4.(cf. Hilden, Lemma and Montesinoe Amilbia [3])
$\mathrm{t}\mathrm{r}(\mathrm{k})$
$=$
$\mathrm{t}\mathrm{r}(\mathrm{A}^{2})=\mathrm{Q}(\mathrm{m}$
Q
$\frac{\pi}{n}$
$(\mathrm{c}\mathrm{o}\mathrm{e}$
,,cog
coe
$\frac{\pi}{2n}$
, coe
$\frac{(n-5)\pi}{2n}$
$\frac{(n-5)\pi}{n},\mathrm{m}$
, os
$\frac{\pi}{2n}\mathrm{m}$
$\frac{\pi}{2})=$
$\frac{(n-3)\pi}{2n}$
Q
$(\mathrm{c}\mathrm{o}\mathrm{e}$
coe
$\frac{\pi}{2n})$
$\frac{\pi}{2})=$
Q
,
$(\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{e}$
$\frac{\pi}{n})$
.
We denote by
the -th side of the regular 2n-g0n. Suppose that the polygon is centered at the
origin such that the middle points of
and
we real.
$k$
$C_{k}$
$\mathrm{o}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\alpha 1$
$8\mathrm{u}\mathrm{c}\mathrm{h}$
$C_{n}$
.
$C_{2n}$
Lemm
Lemma
Let
be ahyperbolic
a hyperbolic translation of the regular 2n-g0n
2-gon identifying apair
a pair of opposite sides
oe
and
. Then the diagonal
goffi entries of
are equal to $1+4\mathrm{m}^{2}(\pi/n)$ .
A proof of this lemm
analogous to that of Lemm
lemma is malrus
Lemma 2.1 in [5].
$\mathrm{g}$
$C_{\mathfrak{n}}$
$F_{n}\mathrm{k}$
$F_{n}$
$C\mathrm{a}_{b}$
$\mathrm{s}.\mathrm{d}$
$F_{n}$
$\mathrm{d}\dot{\mathrm{n}}$
$\mathrm{i}\epsilon$
$\mathrm{p}\mathrm{r}\infty \mathrm{f}$
.
Dffinit
Definition 6. A side pairing
$\dot{\mathrm{n}}\mathrm{n}\epsilon$
$\mathrm{r}.\mathrm{d}\triangleright \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$
$T$
of the regular
Suppmtd by oesLin-AOesLim-Aid1 for &mf.fi Research
and $\mathrm{T}\mathrm{r}$, Japan.
$\cdot$
$\mathrm{k}$
-gon
-g0n is the composite
$2r[]-$
$\mathrm{r}\mathrm{a}\mathrm{e}\mathrm{u}\mathrm{l}\pi 20$
$\cdot$
$\mathrm{f}\alpha.$
$\mathrm{e}$
$\mathrm{R}\infty \mathrm{r}\$
$\dot{\mathrm{r}}$
$\infty \mathrm{m}\mathrm{p}\mathrm{o}\mathrm{o}\mathrm{e}\mathrm{i}\mathrm{t}\mathrm{e}$
$T=R_{n}^{\mathrm{k}}F_{1b}\mathrm{R}_{*}^{-\mathrm{I}}$
of
, The MiniR of Education,, Oultwe,,
$(\mathrm{N}\mathrm{o}.157\mathrm{r}\mathrm{o}\mathfrak{g}5)$
$(\mathrm{N}\mathrm{o}.157\mathrm{r}0\mathfrak{g}5)$
$\mathrm{E}\mathrm{d}\mathrm{u}\varpi \mathrm{t}\dot{\mathrm{n}}$
$\mathrm{n}$
$\mathrm{C}\mathrm{u}\mathrm{h}\mathrm{u}\cdot\vee$
$F_{n}$
and the
$8\mathrm{p}\alpha \mathrm{u}$
\mathrm{U}$
,
$\mathrm{S}\infty.-$
171
rotation
&
arround the origin by angle
$\pi/n$
respectively.
.
Then
$T$
is said to be odd or even if $k-l$ is odd or even,
Theorem 7. Let be a Fuchsian group of signature (3; 0) with a regular 2-gon as a fundamental
if some
polygon. Then
(cooe(\pi /n)) if all side pairings are even, and
side pairing is odd.
See Theorem 2.2 in [5] for a proof.
$\Gamma$
$\mathrm{t}\mathrm{r}(\Gamma)=\mathrm{Q}(\mathrm{c}\mathrm{o}\mathrm{e}(\pi/(2n)))$
$\mathrm{t}\mathrm{r}(\Gamma)=\mathrm{Q}$
By considering the side pairings for each polygons we have the following:
Theorem 8. The polygons only with even
airings are listed as
$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$
$\mathrm{s}\mathrm{i}\mathrm{d}\neq$
$2n$
Sid
. in
$\mathrm{p}$
,
,
$P1$ , $P2$ ,
, $P5$ ,
13,
12 ,
$P_{14}$
$P_{1}$
$P$
$P_{8}$
14
12
Here,
$P_{\mathrm{j}}$
Sid
Sid
,
,
$\mathrm{e}$
$P_{15}$
$P$
$P$
$P$
.
..
$\mathrm{i}$
$\mathrm{p}$
$\mathrm{p}$
$\mathrm{S}’$
$\mathrm{p}$
..
,
,
$P_{17}$
$P$
$P$
14,
1,
’
$P$
,
,
, $P7$ ,
,
,
$P_{1}$
$P$
$P$
.
in
in
,
,
$\tau$
$P_{87}$
,
,
4,
$P_{4}$
$P_{7}$
$P_{8}$
$P_{8}$
,
,
,
,
,
,
,
$P_{18}$
$P$
,
,
$P7$ ,
$P7$ ,
$P$
$\mathrm{V}1$
,
$P$
$P0$ ,
’
10,
11,
6,
7,
,
$P$
’
’
$P$
,
,
,
$P_{3}$
,
,
$P$
$P$
,
,
,
,
$P$
$P$
,
41 ,
’
$\mathrm{p}\mathrm{g}$
$1$
,
$P$
$P_{82}$
,
,
field
$P$
$\mathrm{Q}($
$P_{83}$
$(\pi 15))$
,
$P$
$P$
$\pi 10$
$\mathrm{F}^{\cdot}$
$\mathrm{e}1$
$\mathrm{Q}$
$\mathrm{F}^{\cdot}$
$\mathrm{e}2$
$\mathrm{Q}$
$\pi 7$
$\mathrm{F}^{\cdot}$
$\mathrm{e}3$
$\mathrm{Q}$
$\pi 6$
denotes the 30-gon endowed with -th side-pairing pattern in [6].
$j$
Figure 1:
$\mathfrak{N}\Re \mathrm{n}\mathrm{s}$
only with even side pairings
)
172
Fi ure 2: 14-gons only with even side pairings
Figure 3: 12- oos only with even
-pairings
$\mathrm{s}\mathrm{i}\mathrm{d}\triangleright$
An extremal surface of genus in the sense of C. Bavard has the regular $(12g-6)$-gon as a fundamental
polygon. We see that every extremal surface of genus 3 admitting two extremal disks has the
field
(see Figure 9).
$g$
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$
$\mathrm{Q}(\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{e}(\pi/30))$
References
[1] C. Bavard, Disques extrimaux et
n0.2, 191-202.
surfaces
modulaires, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996),
[2] A. F. Beardon, The geometry of discrete ffoup, Graduate Texts in Mathematics, 91. Springer-Verlag,
New York, 1983.
m
[3] JL Hilden,
Lozano and J. Montaeinoe- milibia, A characterization of Arithmetic subgroups of
, Math. Nachr. 159 (1992), 245-270.
and
$\mathrm{S}\mathrm{L}(2,\mathrm{R})$
$\mathrm{S}\mathrm{L}(2,\mathrm{C})$
[4] T. JOrgensen and M. Naatanen, Sur&cae of genus 2: generic fundamental polygons, Quart. J. Math.
Oxford Ser. (2) 33 (1982), no. 132,
$41\ovalbox{\tt\small REJECT} 1$
[5] M.
.
and T. Kuusalo, On arithmetic genus 28ubaoup of triangle
201 (1997), 21-28.
$\mathrm{N}-\infty-\overline{\mathrm{h}}\mathrm{n}\mathrm{e}\mathrm{n}$
$\mathrm{f}\mathrm{f}\mathrm{e}’ \mathrm{u}\mu$
, Contemp. Math.
[6] G. Nakamura, Generic fundamental polygons for surfaces of genus three, Kodai Math. J. 27
88-1OA
[7] G. Ringel, Map color theorem, springer Verlag, Berlin, 1974.
$(\mathfrak{M}4)$
,
173
Figure 4: 1-goos with odd
$\mathrm{s}\mathrm{i}\mathrm{d}\triangleright \mathrm{p}\mathrm{a}\ddot{\mathrm{m}}\mathrm{n}\mathfrak{B}$
174
0
$\ovalbox{\tt\small REJECT} 1$
33
s
1
$\ovalbox{\tt\small REJECT} 1$
$\otimes 12$
37
34
65
78
89
92
$\ovalbox{\tt\small REJECT} 91$
$0$
Figure 5: 14-gons with odd side pairings
$\mathrm{s}$
175
25
26
53
$9$
$\ovalbox{\tt\small REJECT} 4$
Figure 6:
-gons with odd side-pairings (1)
178
$8$
$\ovalbox{\tt\small REJECT} 97$
$\ovalbox{\tt\small REJECT} 9$
$9$
$\ovalbox{\tt\small REJECT} 9$
$00$
101
$\ovalbox{\tt\small REJECT} 1$
$02$
$\ovalbox{\tt\small REJECT} 1$
$03$
$\ovalbox{\tt\small REJECT} 1$
$4$
$\ovalbox{\tt\small REJECT} 10$
106
$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 113114115116117118119120$
$2$
123
124
125
126
127
128
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT} 12$
129
130
131
132
133
134
135
136
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\mathrm{S}$
$\ovalbox{\tt\small REJECT}$
$\mathfrak{S}$
$137$
138
139
140
141
142
143
144
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\oplus$
$\Theta$
$\mathrm{S}$
$\mathfrak{S}$
$\oplus$
$\oplus$
145
146
147
148
149
150
151
152
$\ovalbox{\tt\small REJECT}$
$\otimes$
$\Phi$
$\otimes$
$\otimes$
$\ovalbox{\tt\small REJECT}$
${ }$
$\ovalbox{\tt\small REJECT}$
$153$
154
155
156
157
158
159
160
$\otimes$
$\mathfrak{S}$
$\ovalbox{\tt\small REJECT}$
$\mathfrak{G}$
$\Theta$
$\ovalbox{\tt\small REJECT}$
$\mathfrak{G}$
$\otimes$
162
163
164
165
166
167
168
$\Phi$
$\mathrm{S}$
$\ovalbox{\tt\small REJECT}$
$\Phi$
$\ovalbox{\tt\small REJECT}$
$\otimes$
$\ovalbox{\tt\small REJECT}$
$\copyright$
$169$
170
174
176
$\ovalbox{\tt\small REJECT}$
$@$
$1$
61
$1$
$\ovalbox{\tt\small REJECT} 17$
172
173
$\ovalbox{\tt\small REJECT} 174$
$\Phi$
$\ovalbox{\tt\small REJECT}$
179
180
181
182
183
184
$\copyright$
$\copyright$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\copyright$
$@$
$\ovalbox{\tt\small REJECT}$
185
186
187
188
189
190
191
192
$\otimes$
$\Phi$
$\otimes$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\copyright$
$\mathfrak{H}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
177
178
$\otimes$
Fi ure 7:
-gons with odd
$\epsilon\dot{u}$
$1\triangleright \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{i}\cdot \mathrm{f}\mathrm{f}$
$(2)$
177
193
194
195
196
197
198
199
200
$\ovalbox{\tt\small REJECT}$
$\Phi$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$201$
202
203
204
205
206
207
208
$\ovalbox{\tt\small REJECT}$
$\copyright$
$\Phi$
$\copyright$
$\copyright$
$\otimes$
$\otimes$
$\ovalbox{\tt\small REJECT}$
$209$
$10$
210
212
213
214
215
216
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
220
221
223
224
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
226
228
229
230
231
232
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT} 2$
$\ovalbox{\tt\small REJECT}$
$8$
219
$\ovalbox{\tt\small REJECT} 21$
$\ovalbox{\tt\small REJECT}$
$25$
$\ovalbox{\tt\small REJECT} 2$
233
234
235
236
237
238
239
240
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\copyright$
$\Phi$
$\ovalbox{\tt\small REJECT}$
$\ovalbox{\tt\small REJECT}$
$\copyright$
$\ovalbox{\tt\small REJECT}$
Figure 8:
-gons with odd
-pairings (3)
$\mathrm{s}.\mathrm{d}\sim$
17\mbox{\boldmath $\theta$}
202
Figure 9: Side pairing patterns which induce extremal surfaces admitting two extremal disks