数理解析研究所講究録1022 巻 1997 年 100-112

100

The elementary Kleinian groups andthree-manifolds
Hiroki SATO佐藤 宏樹
Department of Mathematics
Shizuoka University\S 0. Introduction.
This paper has the following two aims: (1) To describe generators, fimdamental regions
and three-manifolds for the elementary kleinian groups based on the lecture due to Oikawa[2] and Ford [1]; (2) To give
number for the elementary Kleinian groups (Sato$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{e}\mathrm{n}’ \mathrm{s}$[3] .
by the number
In \S 1 we will
the elementary Kleinian
into seventeen
of limit points of the groups. In \S 2 we will consider the finite groups, that is, the polyhedralgroups. In \S 3 we will consider the groups with one limit point and in \S 4 the groups with two
hmit points. In \S 5 we will give
numbers for the elementary Kleinian groups.
In Appendix A we will draw the pictures of fundamental regions and the three-manifolds
for the elementary Kleinilan groups, and in Appendix we will make a table of the groups.
$)$$\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{S}\mathrm{s}\mathrm{i}\Psi$$\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}8$$\mathrm{p}\mathrm{o}\mathrm{u}_{\mathrm{P}^{\mathrm{S}}}$$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}’ \mathrm{S}$$\mathrm{B}$\S 1. Classiflcation of the elementary Kleinian groups.
In this section we give the definition of an elementary group and classify the groups
according to the number of limit points. We denote by M\"ob the set of all M\"obius transformations.

DEFINITION 1.1. A subgroup
points of is finite.

$G$of M\"ob is$G$

ia an elementary group if and only if the number of$\mathrm{s}\dot{u}\mathrm{d}$to be elementary if the number of hmmit$G$

REMARK. We easily see that
hmit points of is $0,1$ or 2.
$G$

101
THEOREM 1.2. The elementary Kleinian groups consist of the follouring seventeen
groups and their conjugate grvups in M\"ob, that is, six finite groups, seven groups with one
limit point and four groups urith two limit points.
(i)$\mathrm{O}$: The finite groups (the polyhedml gmups), that , the groups containing only dliptic$\dot{u}$transformations.

: The tnvial group $G=\{1\}$ .
: The elliptic cydic group of order $n(n\geq 2)$ .
: The dihedral group of order $2n(n\geq 2)$ .
: The tetmhedrd group.
: The octahedrvnl group.
: the icosahedral group.
(ii) I: The groups
urith one limit point, that is, the elementary Kleinian grvups
taining parvnbolic tmnsfomations.
(1) : A parabolic cydic group (A simply periodic group).
(2)
A doublygroup.
(3) I3: The infinite dihedral gmup.
(1)
(2)
(3)
(4)
(5)
(6)

$\mathrm{O}_{1}$$\mathrm{O}_{2}$$\mathrm{O}_{3}$$\mathrm{O}_{4}$$\mathrm{O}_{5}$$\mathrm{O}_{6}$$G$$\omega n-$$\mathrm{I}_{1}$$\mathrm{I}_{2}:$$\mu\dot{n}odic$(4) $\mathrm{I}_{4}:G=(z\mapsto z+\omega, z\mapsto z+\omega’, z\mapsto-z)$ .
(5) I5: $G=(z\mapsto z+\omega,$ $z\mapsto z+\omega’,$.
(6) $\mathrm{I}_{0}:G=(z\mapsto z+\omega, z\mapsto z+\epsilon\omega, z\mapsto\epsilon z)(\epsilon=e^{2\pi}/3)i$.
(7) I7: $G=\langle zarrow>Z+\omega,$.
$zrightarrow iz\rangle$$z\mapsto z+\epsilon\omega,$ $z\mapsto\epsilon z)(\epsilon=e^{\pi i/})3$(iii) II: The groups
with two limit points, that is, the dementary Kleinian groups
containing loxodromic (hyperbolic) transformations.
(1)
A loxodromic (hyperbolic) cydic group.
(2).
(3) $\mathrm{I}\mathrm{h}:G=(z\mapsto Kz, z\mapsto 1/z\rangle(|K|\neq 1)$ .
(4)$z\mapsto 1/z\rangle(|K|\neq 1,n\geq 2)$ .
$G$$\mathrm{I}\mathrm{I}_{1}:$$\mathrm{I}\mathrm{I}_{2}:G=(z\mapsto Kz,$ $z\mapsto e^{2\pi i/}\hslash z\rangle(|K|\neq 1,n\geq 2)$$\mathrm{I}\mathrm{I}_{4}:G=(z\mapsto Kz,$ $z\mapsto e^{2\pi*/n}.Z,$\S 2. Generators, fundamental$\mathrm{r}\mathrm{e}*P\mathrm{i}_{0}\mathrm{n}\mathrm{s}\Phi$and three-manifolds.

In this section we describe generators, fundamental regions and three-manifolds for theelementary Kleinian groups. Let and be the unit ball and its closure, respectively.
We denote by $F(G)$ and $M(G)=(\mathrm{B}\cup\Omega)/G$ a fundamental region and the three-manifold
for an elementary Kleinian group , respectively, where is the region of discontinuty onof .
$\mathrm{B}$$\overline{\mathrm{B}}$$G$$\Omega$$G$$\partial \mathrm{B}$is the elliptic cyclic group of order .
Set $A(z)=e^{2\pi\cdot/n}Z(n\geq 2)$ . Then $G=(1,A,A^{2},$, that is, $G=(A|A^{n}=1\rangle$ . A. fundamental region and three.manifold are as follows: $p(G)=\{z\in \mathrm{C}|0<\arg z<2\pi/n\}$
and
.
(1)

$\mathrm{O}_{2}:G$$n$$\ldots,An-1\rangle$$M(G)=\overline{\mathrm{B}}$102
of order $2n(n\geq 2)$ .
is the dihedraland the South Pole throughjoiming the North
Let be the rotation about the$2\pi/n$the nis through the point$T_{k}(k=1,2,\ldots,n)$aboutrotationbe the. Letulgle ofand the origin. Then
(2)$\mathrm{O}_{3}:G$$\Psi^{\mathrm{o}\mathrm{u}}\mathrm{P}$$\dot{\mathrm{P}}\mathrm{o}\mathrm{l}\mathrm{e}$$S$$\alpha \mathrm{i}\mathrm{s}$$z.=e^{2\mathrm{n}()/\mathfrak{n}}k-1i$$G=\{1,S, S^{2}, \ldots,s^{n-}1,\tau_{1},T_{2,\ldots,\hslash}T\}$.

If we set $G_{0}=\{1, S, S^{2}, \ldots,S^{n}-1\}$ , then $G=G_{0}+T_{1}G_{0}$ . If we set $T=T_{1}$ , then we have
and $G=(S,T|P=1,T^{2}=1)$ . By the stereographic projection, we canconsider and as the following M\"obius transformations of the complex plane:$G_{0}=(S|S^{\mathfrak{n}}=1)$

$T$

$S$

$S(z)=e^{2\pi\dot{\cdot}/n_{Z}}$and$T(Z)=1/z$ .

A fundamental region $F(G)$ and the three-manifold $M(G)$ of.
$\{z\in \mathrm{C}||z|<1,0<\arg z<2\pi/n\}$ and$G$

are as follows: $F(G)=$$M(G)=\overline{\mathrm{B}}$(3)is the tetrahedral group.
: the rotation about the uis joiming$\mathrm{O}_{4}:G$$S_{j}$$v_{j}$

with the center of the opposite face to.hrough $2\pi/3(j=1,2,3,4)$ .
: the rotation about the axis joining the midpoint of the side
. throughof the side

$v_{j}$$\mathrm{t}$$V_{ij}$$v_{k}v_{\ell}(k,\ell\neq i,j)$$b_{ij}$$v_{i}v_{j}$with the midpoint$\pi(V_{12}=V_{3}4, V13=V_{2}4, V14=V_{2\mathrm{s}})$.

Then we have
$G=\{1,S_{i},S^{2}i(i=1,2,3,4), V_{12}, V_{1}3, V_{14}\}$and$\mathrm{o}\mathrm{r}\mathrm{d}(G)=12$

setting. Noting$V_{13}=V_{14}V_{12},S_{2}=V_{12}s_{1}V^{-}12^{1}’ s_{3}=V_{13}S_{1}V_{1}-1S3’ 4=V_{14}s_{1}V^{-}14^{1}$$S=S_{1},$ $V=V_{12},$ $W=V_{14}$, we have$G=(S,V, W|S^{3}=1, V^{2}=1,W^{2}=1)$

and

103
and
$G=(_{S^{2}}^{1}SS^{2}VSVVS^{2}WSWWs^{2}WVs^{WV}WV)$.

Furthermore, since $W=S^{2}VS$ , we have
$G=\langle S, V|S^{3}=1, V^{2}=1, (SV)^{3}=1\rangle$

$G=$

and

By the stereographic projection we can represent$S(z)=i(Z+1)/(z-1)$ ,

$S$

and

$V$

.

as follows:$V(z)=1/z$ .

A fundamental region $F(G)$ and the three-mnifold $M(G)$ are as folows: $p(G)=\{z\in$$-\pi/3<\arg z<\pi/3\}$ and.
$\mathrm{C}||z|<1,$.. (4)$M(G)=\overline{\mathrm{B}}$is the octahedral group.
: the rotation about the axis
through $2\pi/3$ .
: the rotation about the axis joining the midpoint ofand the center of the
opposite face tothrough.
: the rotation about the midpoint of the sideand the midpoint of the opposite
side through .
$\grave{\mathrm{O}}_{5}:G$$S_{j}$$u_{j}v_{j}$$T_{k\dot{g}}$$v_{k}v_{\mathrm{j}}$$v_{k}v_{j}$$\pi/4$$U_{k\mathrm{j}}$$u_{k}v_{j}$

$\pi$

Then
$G=\{1,$$S_{\mathrm{j}},$$S_{j}^{2}(j=1,2,3,4),\tau_{1}jT_{13}^{j\mathrm{j}}T_{\mathfrak{B}}2$”$(j=1,2,3),$$U_{12},$ $U_{1}3,$$U14,$ $U23,$ $U_{2}4,$ $U_{u\}}$104$\mathrm{u}\mathrm{d}\alpha \mathrm{d}(c)=24$.

lf we set $S=S_{1},V=V_{1l}=P_{1l}$ ud $U-U_{12}$ , then$\mathrm{w}$have$G=(_{\tau P}^{S}TSffiT1Tffi\tau g_{SU}\tau^{y}s_{U}^{U}yUTffiUS\mathrm{r}sPUs\tau Us^{Us}UUssS\tau^{S}p_{U}USUTSpU\mathrm{r}\sigma sUs^{U}UsSUSUUU)$

By the$\mathrm{g}\mathrm{t}\mathrm{m}\mathrm{r}\mathrm{a}_{\mathrm{P}^{\mathrm{h}]}\mathrm{C}}$projoetion we can$\mathrm{r}\ovalbox{\tt\small REJECT} \mathrm{t}S_{1}U_{1}\mathrm{r}\mathrm{m}\mathrm{d}\gamma u$$S(z)=i(Z+1)/(_{Z-}1),$ $U(z)=1/z,$

If$\mathrm{m}$set $R(z)=(z+1)/(z-1)$ , thm$\mathrm{w}$$T(Z)=i_{Z},$have$G\approx(R,T|^{p_{-1}},Parrow 1,$ $(\mathrm{r}R)^{\mathrm{a}_{-1}}\}$

A fimrtd$\mathrm{r}\dot{*}\mathrm{m}F\mathrm{t}G$$1,$ $-\pi/4<\mathrm{r}\mathrm{g}z<\pi/4\}$(5)$\mathrm{O}0:G$

is the

) and

and

.

$\mathrm{t}\mathrm{h}\mathrm{r}\infty \mathrm{R}\mathrm{k}\mathrm{H}M\mathrm{t}G$$M(G)=\mathrm{B}$$\mathrm{i}\mathrm{r}\mathrm{d}\mathrm{f}\mathrm{f}\mathrm{o}\mathrm{u}\mathrm{p}$.

.

) are asfollows:$V(Z)\mathrm{r}-z$.

.

$\mathrm{m}\mathrm{w}:F(G)arrow\langle z\mathrm{e}\mathrm{C}||z|<$: the rotation about the ads joinilnngg the center of a face with the oenter of the oppmite
faoe through $2\pi/5$ for $i=1,2,3,4,5,6$.
: the rotation about the axes joining theof a side with the nidpnint of thenide through $\pi \mathrm{b}j=1,2,\ldots,$ .
: the rotation about the ni
in a vertex with the oppmite vertex through $2\pi/3$$10$for $k=1,2,$.
$S_{i}$$T_{j}$$\mathrm{m}\mathrm{i}\mathrm{d}_{\mathrm{P}}\mathrm{r}$$15$$\mathrm{o}\mathrm{p}\mathrm{P}\dot{\infty}\mathrm{t}\mathrm{C}$$U_{k}$$\mathrm{j}\dot{\mathrm{m}}$$\mathrm{g}$$\ldots,$

Then
$G=$

{

$1,S_{l},$fft’$P.sl|’\iota(i=1,2,$ $\ldots,6),T_{j}0=1,2,$ $\ldots,$ $15),\sigma\iota,\nu_{l}(k=1,2,$ $\ldots,10)$

}

105

and

$\mathrm{o}\mathrm{r}\mathrm{d}(G)=60$

.

..

.

$\cdot$Let be the rotation about the horizontal cross-line through . Since $V=(STS3)\tau(s\tau S^{3})-1$
we have$V$$\pi$$G=(_{\tau}^{1}\tau s_{3}2Ts^{4}TSTS^{\cdot}sS\tau_{S^{4}}s\tau s^{3}s_{T}sT\tau sSs2Ss_{2}^{2}\tau S^{2}s^{2}S\tau^{T}s_{2}s_{2}^{2}\tau\tau_{S}s^{3}S4s\tau sS^{3}S^{3}Ts_{3}^{ST}3s_{3}^{3}TS^{3}\tau S^{2}s4^{\cdot}S^{4}Ts_{4}^{4}s\tau sS^{4}S4Ts_{Ts_{3}}\tau S^{2}4S4V\tau^{T}Vs_{S^{4}}VTVTV\tau Vs^{3}S2Vs^{Ts_{2}}Vs\tau Vs_{Ts}^{S\tau}VS\tau_{S^{4}}VVs3Vs^{2}\tau S^{2}VS^{2}\tau s^{3}VS^{2}TsVS^{2}VS2\tau VsTS24VSVs_{3}^{3}Vs3TVs\tau_{S^{2}}^{T}Vs_{3}Vs_{3}^{3}Ts^{3}\tau Ss4VS4TVs\tau_{S}VS^{4}\tau Vs_{3}Vs^{S}V44\tau_{S^{4}}^{T}4s_{4}^{2}S)$and$G=\langle S,T|s^{5}=1,T^{2}=1, (TS)^{3}=1\rangle$.

By the stereographic projection we have$S(z)=e^{2\pi}:/\mathrm{s}_{z}$and$T(z)= \frac{(\sqrt 5+1)z+2}{2z-(\sqrt 5+1)}$.

A fundamental region $F(G)$ and $M(G)$ are as follows: $p(G)=\{z\in.\mathrm{C}||Z|<1,$ $-\pi/5<$$\mathrm{a}r\mathrm{g}z<\pi/5\}$

and$M(G)=\overline{\mathrm{B}}$.

\S 3. The groups with one limit point.
In this section we consider groups with one limit point,groups containing parabolic transformations.

th.at is, elementary Kleinianis a parabolic cyclic group (a simply periodic group).
, where is the axis), where $A(z)=z+\omega(\omega\neq 0)$ . Then
joiming the North Pole with the South Pole. For a fundamental region $F(G)$ see Appendix
A.
(1)$\mathrm{I}_{1}$:$G$$G=\langle A$(2)$M(G)=\overline{\mathrm{B}}\backslash \{L\}$$\mathrm{I}_{2}:G$$G=(A,$$L$is a doubly periodic group.

$B\rangle,$$\mathrm{w}\mathrm{b}\mathrm{e}\Gamma \mathrm{e}A(z)=z+\omega,$

region $p(G)$$B(z)=z+\omega’({\rm Im}(\omega’/\omega)\neq 0,\omega\neq 0,\omega’\neq 0)$. $M(G)=$, where is the central axis in the solid torus. For a fundamental
see Appendix A.

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{S}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{t}_{\mathrm{o}\mathrm{r}}\mathrm{u}\mathrm{s}-\{L\}$$L$

is the infinite dihedral group.
$G=(A,c|C^{2}=1)$ , where $A(z)=z+\omega(\omega\neq 0),C(z)=-Z$ .
, where$F(G)$
seeis the radius joining the North. Pole with the origin. For a fundamental region ,Appendix A. .
(3) I3:$G$$M(G)=\overline{\mathrm{B}}-\{R\}$$R$$\cdot$(4) $\mathrm{L}:G=(A,B,$ $C|C^{2}=1\rangle$ , where $A(z)=z+\omega(\omega\neq 0),$ $B(z)=z+\omega’(\omega’\neq$$C(z)=-Z$ . Then
, where is the center of the ball B.

$0,{\rm Im}(\omega’/\omega)\neq 0),$

$M(G)=\overline{\mathrm{B}}-\{\mathrm{O}\}$$\mathrm{O}$106

For a fundamental region $p(G)$ see Appendix A. The
with four branched points of order two.

Riemam..surfaoe

is the$\Omega/G$$\mathrm{s}.\mathrm{p}$here(5) I5: $G=\langle A, B,C|C^{4}=1\rangle$ , where $A(z)=z+\omega(\omega\neq 0),$ $B(z)=z+i\omega,$ $C(z)=iz$ .
If we set $S=C$ and $T=AC$, then $G=(S,T|s^{4}=1,T^{4}=1,$ $(TS)^{2}=1\rangle$ . Then
, where is the center of the bffi B. For a fundamental region $F(G)$ see
Appendix A. The Riemann surface
is the sphere with three branched points of orders
2, 4, 4.
$M(G)=\overline{\mathrm{B}}-\{\mathrm{O}\}$$\mathrm{O}$$\Omega/G$, where $A(z)=z+\omega,$ $B(z)=z+\epsilon\omega(\epsilon=e^{2\pi 1/3}),$ $C(z)=$. If we set $S=C$ and $T=AC$ , then $G=\langle S,T|s^{3}=1,T^{3}=1, (TS)^{3}=1\rangle$ . Then
$M(G)=b\overline{f}B-\{\mathrm{O}\}$ ,where is the center of the br B. For a fundamental region $F(G)$see Appendix A. The Riemann surfaces
is the sphere with three branched points oforder three.
(6)$\mathrm{I}_{6}:G=\langle A, B, C|C^{3}=1\rangle$$\epsilon z$$\mathrm{O}$$\Omega/G$(7) I7: $G=(A,$ $B,$ $c|c^{6}=1\rangle$ , where $A(z)=z+\omega(\omega\neq 0),$ $B(z)=z+\epsilon\omega(\epsilon=$$C(z)=\epsilon z$ . If we set $S=C$ and $T=A^{-1}BC^{3}$ , then $G=(S,T|s^{6}=1,T^{2}=$$(TS)^{3}=1\rangle$ . Then
, whereis the center of the ball B. For a$e^{\pi i/3}),$$1,$$M(G)=\overline{\mathrm{B}}-\{\mathrm{O}\}$$\mathrm{O}$fimdamental region $F(G)$ see Appendix A. The Riemann surface
three branched points of orders 2, 3, 6.

$\Omega/G$

is the sphere with

\S 4. The groups with two limit points.
In this section we consider groups with two limit points, that is, elementary Kleinian
the upper
groups containing loxodromictransformations. We denote by.
half space.
$(\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{b}_{\mathrm{o}1}\mathrm{i}\mathrm{c})$$H^{3}$

is a loxodromic (hyperbolic) cyclic group.
$G=(A)$ , where $A(z)=Kz(|K|\neq 1)$ . $M(G)$ is the solid torus and $H^{3}\cup\Omega\mapsto M(G)$ is
an unbranched covering map.
(1)$\mathrm{I}\mathrm{I}_{1}:G$, where $A(z)=Kz(|K|\neq 1)$ and $B(z)=e^{2\pi 1/n}z(n\geq 2)$ . $M(G)$ is(2)
the solid torus and $H^{3}\cup\Omega\mapsto M(G)$ is a branched covering map and the mapping is locallyto 1 on the central axis in the solid torus.
$\mathrm{I}\mathrm{I}_{2}:G=(A, B)$$n$, where $A(z)=Kz(|K|\neq 1)$ and $C(z)=1/z$ .
(3)is the sphereand $H^{3}\cup\Omega[]arrow M(c)$ is a branched covering map. The Riemann surface
with four branched points of order 2.
$M(G)=\overline{\mathrm{B}}$$\mathrm{I}\mathrm{I}_{3}:G=(A,\mathit{0}|C^{2}.=1\rangle$$\Omega/G$(4) IL: $G=(A, B,C|B^{\hslash}=1, C^{2}=1)$ , where $A(z)=Kz(|K|\neq 1),B(Z)=eZ2\pi i/\mathfrak{n}(n\geq$2), $C(z)=1/z$ .
and $H^{3}\cup\Omega\mapsto M(G)$ is a branched covering map. The Riemannsurfaceis the sphere with four branched points of order two.
$M(G)=\overline{\mathrm{B}}$$\Omega/G$

107
\S 5.

$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}’ \mathrm{S}$number.

In this section we consider$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}’ \mathrm{s}$number for the elementary Kleinian groups.

DEFINITION 5.1. Let $G=(A, B)$ be a marked two-generator subgroup of M\"ob. We
call$J((A, B\rangle):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA^{-}1B-1)-2|$

Jprgensen’s number for $G=\langle A,$
M\"ob is defined as follows:$B$). The$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{e}\mathrm{n}’ \mathrm{s}$number $||J(G)||$ for a sugroup

$G$

of

.

$||J(G)||:= \inf\{J(\langle A, B\rangle)|(A, B\rangle\subset G, A^{m}\neq B^{m}(m,n\in \mathrm{Z})\}$

Then we have the following.

THEOREM 5.1. (i)(1)$\mathrm{O}_{3}:G=\langle S,$ $T$$||J(G)||=8\sin^{2}\pi/n$

(2)
(3)
(4)

$\mathrm{O}_{4}:G$$\mathrm{O}_{5}:G$$\mathrm{O}_{6}:G$: the Finite groups.
) is the dihedral group, where $S(z)=e^{2\pi}Z:/\hslash$ and $T(z)=1/z$ . Then$\mathrm{O}$.

is the tetrahedral group. Then $||J(G)||=5$ .
is the octahedral group. Then $||J(G)||=3$ .
is the icosahedral group. Then $||J(G)||=4-\sqrt 5$ .

(ii) I: Elementary Kleinian groups urith one limit point.
For all groups $G$ in this type $||J(G)||=0$ .

(iii) II: $Elementar\mathrm{t}/Kleinian$ groups urith two limit points.
, where $A(z)=Kz(|K|\neq 1)$ and $B(z)=e^{2\pi}:/\mathfrak{n}_{Z}(n\geq 2)$ . Then
(1)$\mathrm{I}\mathrm{I}_{2}:G=\langle A, B\rangle$$||J(G)||= \min\{4\sin^{2}\pi/n,$ $|K\mathrm{p}/2k\pi:/e-nK-p/2-k\pi i/n|^{2}e(1\leq p\leq 2\log 3/\log|K|,p\in$$\mathrm{Z};k=0,1,2,$ $\ldots$(2), $n-1$ ))}.
, where $A(z)=Kz(|K|\neq 1)$ and $C(z)=1/z$ . Then $||J(G)||=$$\mathrm{I}\mathrm{I}_{3}:G=(A,$$C\rangle$.

$\min\{4+-|K\mathrm{p}/2-K-p/2|2,2|K^{\mathrm{p}}/2-K^{-p/}2|^{2}(1\leq p\leq 2\log 3/\log|K|,p\in \mathrm{z})\}$, where $A(z)=Kz(|K|\neq 1),$ $B(z)=e^{2\pi}z:/n(n\geq 2)$ and(3)$C(z)=1/z$ . Then $||J(G)||= \min\{4\sin^{2}\pi/n,$ $|Kp/2e:k\pi/n-- K^{-}p/2\pi:/n|e^{-k}2(1\leq p\leq$$\mathrm{I}\mathrm{I}_{4}:G=\langle A, B, C\rangle$$2\log 3/\log|K|,p\in\dot{\mathrm{Z}};k=0,1,2,$ $\ldots,n-1))\}$.

Our proof of this theorem will appear elsewhere.

References[1] L. R. Ford, Automorphic ffinctions, Chelsea, New York, 1951.
[2] K. Oikawa, The Elementary Groups, Lecture Notes, Shizuoka Univ, 1989.

[3] H. Sato,$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{e}\mathrm{n}’ \mathrm{S}$number for the elementary Kleinian groups, in preparation.

108$\mathrm{A}\mathrm{n}\mathrm{n}\Phi \mathrm{n}\mathrm{d}\mathrm{i}\mathrm{X}$A.

Fundamental regions and three-manifolds.

109

110

111

112

$\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{C}\mathrm{e}8$

B. Signature of the elementary groups.

$2g-2+ \sum_{j=1}(1-\frac{1}{\nu_{j}}\mathfrak{n})\leq 0$

where is the genus of the Riemam surface

on .
$R$

$g$

,

$R=\Omega/G\mathrm{a}\mathrm{n}\dot{\mathrm{d}}\nu_{j}\mathrm{i}8$$|$

the order of a$\mathrm{b}\dot{\mathrm{r}}\mathrm{m}\$point