数理解析研究所講究録 1223 巻 2001 年 37-4237The Picard group, the figure-eight knot group andJorgensen groups
Hiroki Sato
佐藤 宏樹 *

Department of Mathematics, Faculty of ScienceShizuoka University

0. Introduction.
In this paper we will state that the Picard group

group$G_{F}$$G_{P}$and the figure eight knot

are tw0-generator groups and Jorgensen groups. Furthermore we will

describe acomplete set of relations for$G_{P}$

as atw0-generator group. The detail willappear elsewhere.
1. The Picard group.

DEFINITION 1.1. The group$G_{P}:= \{\frac{az+b}{cz+d}|a$, , ,$b$$c$$d\in \mathrm{Z}+i\mathrm{Z}$, $ad-bc=1\}$

is the Picard group.
Partly supported by the Grants in-Aid for Scientific and Co operative Research, the Ministryof Education, Science, Sports, Culture and Technology, Japan

2000 Mathematics Subject Classification. Primary$32\mathrm{G}15$;Secondary$20\mathrm{H}10,30\mathrm{F}40$

38
The Picard group

THEOREM A(Magnus [7])four Mobius transformations$S_{m}=(\begin{array}{l}i00-i\end{array})$

THEOREM$S_{m},T_{m}$,$T_{m}=(\begin{array}{l}1-\mathrm{l}0\mathrm{l}\end{array})$,and$V_{m}$

is gene rated by the following

with corresponding matrices$U_{m}=(\begin{array}{l}0-110\end{array})$

The Picard group(Johnson-Weiss [3])$\mathrm{B}$$U_{m}$$G_{P}$$G_{P}$,$V_{m}=(\begin{array}{l}i-10-i\end{array})$.

is generated by the fol-louying three matrices:$B_{j}=(\begin{array}{ll}1 01 1\end{array})$$C_{j}=(\begin{array}{ll}\mathrm{l} 0i 1\end{array})$,.

$S_{j}=(\begin{array}{ll}1 1-1 0\end{array})$

See Johnson-Kellerhals-Ratcliffe Tschantz [1] and Johnson-Weiss [2] for more informations about the Picard group and Coxeter groups.

2.

$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$THEOREMgroups.

. If$\mathrm{C}(\mathrm{J}0\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}[4])$is a non-elementary discrete subgroup of$\langle A, B\rangle$

M\"ob, then$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA^{-1}B^{-1})-2|\geq 1$.

The lower bound 1is best possible.

DEFINITION 2.1.

Let$A$

and$B$

be Mobius transformations. The JOrgensen

number $J(A, B)$ is$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA^{-1}B^{-1})-2|$

DEFINITION 2.2.

Anon-elementary tw0-generator discrete subgroup $G$ of Mob

is aJOrgensen group if $G$ has generators

THEOREM$\mathrm{D}$(Jorgensen-Kiikka [5]).

$A$

andLet$B$

with $J(A, B)=1$ .

$\langle A, B\rangle$

be a non-elementary discretegroup with $J(A, B)=1$ , that is, a Jorgensen group. Then
least seven or$A$.

is parabolic$A$is elliptic of order at39Here we only consider the case where$A$

is parabolic, that is, Jprgensen groupsof parabolic type. Namely, we consider tw0-generator groups$G_{\dot{|}k,\sigma}=\langle A, B_{k,\sigma}.\cdot\rangle$

generated by$A=$

where$k\in \mathrm{R}$andand$(\begin{array}{ll}\mathrm{l} 10 1\end{array})$$\sigma\in \mathrm{C}\backslash \{0\}$$B_{:k,\sigma}=(\begin{array}{ll}ik\sigma -k^{2}\sigma-1/\sigma\sigma ik\sigma\end{array})$.

Let $C$ be the following cylinder:
THEOREM(Sato [9]).

$\mathrm{E}$,$C=\{(\sigma, ik)||\sigma|=1, k\in \mathrm{R}\}$

Every Jorgensen group of type

$G_{ik,\sigma}$.

lies on the cylinderc.
By Theorem$\mathrm{E}$

we consider two generator groups$G_{\mu,\sigma}=\langle A, B_{\mu,\sigma}\rangle$

and $\sigma=-ie^{i\theta}(0\leq\theta<2\pi)$ . For simplicity we set$ik(k\in \mathrm{R})$$G_{k,\theta}=\langle A, B_{k,\sigma}\rangle$

for.

$\sigma=-ie^{\theta}$with $\mu=$$B_{k,\theta}:=B_{ik,\sigma}$and.

We can see that it suffices to consider the case of $(0\leq\theta\leq\pi/2)$ and $k\geq 0$ .
THEOREM$\mathrm{F}$(JOrgensen-Lascurain-Pignataro [6], Sato [9], SatO-Yamada [12]).

Let$A=\{$11aanndd$B_{k,\theta}=(\begin{array}{ll}ke^{\theta} ie^{-i\theta}(k^{2}e^{2\theta}.-1)-ie^{\psi} ke^{1\theta}\end{array})$01
and let Gkio$\sigma\in \mathrm{C}\backslash \{0\}$

(i)
(ii)

$=\langle A, B_{k,\theta}\rangle$

be the group generated by$A$

and Bkye, where. Then

$G_{1/2,\pi/2}$

is a Jorgensen group.

$G_{\sqrt{3}/2,\pi/6}$

is a Jorgensen group.

See Sato $[9,10]$ for JOrgensen groups of parabolic type.

$k\in \mathrm{R}$and403. Theorems.
In this section we will state main theorems. We can prove the theorems by using
Poincar\’e’ polyhedron theorem (cf. Maskit [8]).
$\mathrm{s}$THEOREM 1(Sato [9,11]) (i) The Picard group$G_{P}$is conjugate to$G_{1/2,\pi/2}$, thatis, $G_{P}=RG_{1/2,\pi/2}R^{-1}$ , where$R=(\begin{array}{ll}1 i/20 1\end{array})$(\"u)

The following relations form a complete set of relations for$G_{P}$:$(B^{-1}ABA^{2}BAB^{-1}A^{2}B^{-1}ABA^{2}BAB^{-1}AB)^{2}=1$$(B^{-1}ABA^{2}BAB^{-1}A^{2}B^{-1}ABA)^{2}=1$$(AB^{-1}ABA^{2}BAB^{-1}A^{2}B^{-1}ABA^{2}BAB^{-1}AB)^{2}=1$$(AB^{-1}ABA^{2}BAB^{-1}A^{2}B^{-1}ABA)^{2}=1$$(B^{-1}ABA)^{3}=1$$(AB^{-1}ABA)^{2}=1$$(AB^{-1}ABA^{2}B^{-1}ABA^{2}BAB^{-1}A^{2}B^{-1}ABA^{2}BAB^{-1}AB)^{2}=1$$(AB^{-1}ABA^{2}B^{-1}ABA^{2}BAB^{-1}A^{2}B^{-1}ABA)^{3}=1$,

where $B=RB_{1/2,\pi/2}R^{-1}$ .

COROLLARY.

The Picard group is a twO-generator group and a JOrgensen group.

THEOREM 2(Sato [9,11]).
$G_{\sqrt{3}/2,\pi/6}$

The figure-eight knot group, that is, $G_{F}=RG_{\sqrt{3}/2,\pi/6}R^{-1}$ , where$G_{F}$isconjugate to41

R $=(_{0}1$(ii)$1/21)$

The following relation forms a complete set of relations for$G_{F}$:$ABA^{-1}B^{-1}A=BA^{-1}B^{-1}ABA$ ,where $B=RB_{\sqrt{3}/2,\pi/6}R^{-1}$ .

COROLLARY.

The figure-eight knot group is a twO-generator group and a Jorgensengroup.

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42[7] W. Magnus, Noneuclidean Tesselations and Their Groups, Academic Press,

New York, London, 1974.
[8] B. Maskit, Kleinian Groups, Springer-Verlag, New York, Berlin, Heiderberg,1987.
[9] H. Sato, One-parameter families of edrerne groups for Jorgensen’s inequality,

Contemporary Math. (The First Ahlfors- Bers Colloquium) edited by I. Kraand B. Maskit, 2000, 271-287.
[10] H. Sato, Jorgensen groups of parabolic type, in preparation.
[11] H. Sato, Jorgensen groups and the Pica rd group, in preparation.
[12] H. Sato and R. Yamada, Some exrreme Kleinian groups for JOrgensenys inequality, Rep. Fac. Sci. Shizuoka Univ. 27 (1993), 1-8.

Department of Mathematics
Faculty of Science

Shizuoka University
Ohya Shizuoka 422-8529
Japane-mail smhsato@ipc.shizuoka.ac.jp