数理解析研究所講究録 1270 巻 2002 年 77-83

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The Jorgensen number of the Whitehead link
Hiroki Sato佐藤 宏樹(静岡大学理学部) *ABSTRACT. In this paper we will sketch out the result obtained recently: the Jorgensen
number of the Whitehead link is two. Furthermore we will represent points corresponding

to the Whitehead link by using the cordinates introduced in Sato [7]. The details willappear in Sato [9].
1.

In 1976 JOrgensen obtained the following important theorem called JOrgensen’sinequality, which gives anecessary condition for anon-elementary M\"obius transform-tion group$G=\langle A, B\rangle$

THEOREM A

to be discrete.

. Suppose that the Mobius transformations$(\mathrm{J}0\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}[1])$$A$

and$B$

generate a non-elementary discrete group. 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 1. Let$A$

and $B$ be Mobius transformations. The Jorgensen number

Partly supported by the Grants-in-Aid for Co operative Research as well as Scientific Research,
the Ministry of Education, Science, Sports and Culture, Japan.

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

Key Words and Phrases. Jorgensen’s inequality, Jergensen number, Jorgensen groups, theWhitehead link

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$J(A, B)$ of the ordered pair $(A, B)$ is defined as$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA^{-1}B^{-1})-2|$.

We denote by M\"ob the set of all M\"obius transformations. Throughout this paper

we will always write elements of M\"ob as matrices with determinant 1. We recall thatas the group of conformalofM\"ob $(=PSL(2, \mathrm{C}))$ acts on the upper half space$H^{3}$

isometries of hyperbolic 3-space. Asubgroup
there exists afinite$\mathrm{G}$-orbit inDEFINITION 2. Let$G${ $J(A,$ $B)|$of M\"ob is said to be elementary if.

$\mathrm{R}^{3}$be anon-elementary tw0-generator subgroup of M\"ob. TheJorgensen number $J(G)$ for$\mathrm{J}(\mathrm{G}):=\inf$$G$$\mathrm{R}^{3}$$G$

is defined asand$A$$B$

generate$G$}.

DEFINITION 3. Anon-elementary twoegenerator subgroup

group if

$G$of M\"ob is a Jorgensen

is adiscrete group with $J(G)=1$ .

THEOREM$\mathrm{B}$(-K\"ukka [2]).

$\mathrm{J}0\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$group with $J(A, B)=1$ . Then
If

$G$$\langle A, B\rangle$

$A$

Let

is elliptic

is aJorgensen group such that$\langle A, B\rangle$

be a non-elementary discreteof order at least seven or$A$$A$

is parabolic.

is parabolic, then we call it a Jorgensengroup of parabolic type. Here we only consider Jorgensen groups of parabolic type.
2. Let ( ,$A$

$B\rangle$

we can normalize
$A=$

where
$A$

and

group

be amarked two generator group such that

$B_{\sigma,\mu}$$G_{\sigma,\mu}$:is parabolic. Thenand $B$ as follows:$A$$(\begin{array}{ll}1 10 1\end{array})$$\sigma\in \mathrm{C}\backslash \{0\}$$A$

and$\mu\in \mathrm{C}$$G_{\sigma,\mu}=\langle A,B_{\sigma,\mu}\rangle$

and that

and$G_{\sigma,\mu}$$B:=B_{\sigma_{d^{l}}}=(\begin{array}{lll}\mu\sigma \mu^{2}\sigma -1/\sigma\sigma \mu\sigma\end{array})$. We denote by. We say that

$G_{\sigma,\mu}$$(\sigma, \mu)$,the marked group generated byis the point representing amarked

is the marked group corresponding to apoint$(\sigma,\mu)$.

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In particular, we consider the case of $\mu=ik(k\in \mathrm{R})$ . Namely, we consider markedtw0-generator group$A=$

where$C$

THEOREMand$k$$\in \mathrm{R}$(Sato [7]).

$\mathrm{C}$By Theorem$\mathrm{C}$$(\sigma,ik)$for$C=\{(\sigma, ik)||\sigma|=1, k\in \mathrm{R}\}$If a marked twO-generator grouprepresenting$G_{\sigma},:k$$G_{\sigma},:k$

is a Jorgensen$C$$G_{\sigma\mu}=\langle A, B_{\mu,\sigma}\rangle$

and $\mu=ik(k\in \mathrm{R})$ . For simplicity we set$\sigma=-ie^{:\theta}$.

lies on the cylinder .

$B_{\theta,k}:=\mathrm{G}\mathrm{a},\mathrm{i}\mathrm{k}$

with
and.

There are infinite number of Jprgensen groups (see4.

,.

we consider marked tw0-generator groups$\sigma=-ie^{\dot{*}\theta}(0\leq\theta<2\pi)$$=\langle A, B_{\sigma},:k\rangle$

generated by$B:=\mathrm{G}\mathrm{a},\mathrm{i}\mathrm{k}=(\begin{array}{lll}ik\sigma -k^{2}\sigma -1/\sigma\sigma ik\sigma \end{array})$be the following cylinder:group, then the point

G$,kand$(\begin{array}{ll}\mathrm{l} 10 \mathrm{l}\end{array})$$\sigma\in \mathrm{C}\backslash \{0\}$3. Let$\mathrm{G}\mathrm{a},\mathrm{i}\mathrm{k}=\langle A, B_{\sigma,ik}\rangle$-Lascurain-$\mathrm{J}0\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$Pignataro [3], Sato [7] . The following familier groups are all Jorgensen groups:$)$The modular group, the Picared group (Jprgensen-Lascurain-Pignataro [3], Sato [8,9], SatO-Yamada [10] , the figure-eight knot group (Sato [7]), “the Gehring-Maskit$)$

group” (Sato [7]), where “the Gehring-Maskit group” is the group studied in Maskit[5]. Namely, we have the following theorem:

THEOREM$\mathrm{D}$(Jorgensen-Lascurain-Pignataro [3], Sato [7, 8], SatO-Yamada [10]).

Let

$A=$

and letand$(\begin{array}{ll}1 10 \mathrm{l}\end{array})$$G_{\theta,k}=\langle A, B_{\theta,k}\rangle$$k\in \mathrm{R}$. thenand$B_{\theta,k}=(\begin{array}{ll}ke^{\dot{l}\theta} ie^{-\cdot\theta}.(k^{2}e^{2\theta}.-|1)-ie^{\dot{l}\theta} ke^{1\theta}\end{array})$be the group generated by

$A$

and

$B_{\theta,k}$, where $0\leq\theta<2\pi$

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(i)

$G_{\pi/2,0}$(ii)

is a Jorgensen group.

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

(iii)
(iv)

is a Jorgensen group.

$G_{\pi/6.\sqrt{3}/2}$$G_{0,\sqrt{3}/2}$

is a $J\rho\eta ensen$ group.

is a JOrgensen group.

REMARK (1) The groups$G_{\pi/2,0}G_{\pi/2,1/2}G_{\pi/6,\sqrt{3}/2}$

and$G_{0,\sqrt{3}/2}$

are conjugate to

the modular group, the Picard group, the figure-eight knot group and “the Gehring-Maskit group”, respectively.
(2) See Sato [7] for other Jorgensen groups of parabolic type.

5. Now it gives rise to the following problem.
PROBLEM. Is the Whitehead link aJorgensen group ?
Here we can give the answer to the problem, that is, we have the following theorems.
THEOREM 1(Sato [9]).

The Jorgensen number of the Whitehead link is two.

COROLLARY (Sato [9]). The Whitehead link is not aJOrgensen group.
THEOREN 2(Sato [9]).

generator group$G_{\sigma,\mu}$whereThe Whitehead link is conjugate to the marked tw0$\sigma=\sqrt{2}e^{3\pi\dot{|}/4}$

and $\mu=-1/2$ .

6. The proofs of the theorems will appear elsewhere. Here we only give sketchesof the proofs.

THEOREM$\mathrm{E}$Whitehead link(cf. Wielenberg [11], KrushkaP, Apanasov and$G_{W}$$\mathrm{G}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{s}\mathrm{k}\mathrm{i}_{\dot{1}},[4]$The).

has the following presentation:$G_{W}=\langle A, B|(A^{-1}BAB^{-1})(ABA^{-1}B^{-1})(AB^{-1}A^{-1}B)(A^{-1}B^{-1}AB)=1\rangle$,

8111),

1
A $=(_{0}$$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{P}\mathrm{O}\mathrm{S}\mathrm{I}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}}1$

element $X$ of01).

B $=($ $1-i1$

Let $Gw$ be the Whitehead link defined in Theorem E. Then an

has the following form:$G_{W}$$X=(\begin{array}{ll}1+(1-i)a b_{1}+(1-i)b_{2}(1-i)c 1+(1-i)d\end{array})$

where ,$a$$b_{1}$,$b_{2}$, ,$c$$d\in \mathrm{Z}+i\mathrm{Z}$PROPOSITION 2.
$\langle X, \mathrm{Y}\rangle$

Let, $a+d-b_{1}c+(1-i)(ad-b_{2}c)=0$ .

be the Whitehead link defined in Theorem$G_{W}$

be a non-elementary subgroup generated by $X$ andThen the Jorgensen number of. Let ,$A$$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{P}\mathrm{O}\mathrm{S}\mathrm{I}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}}3$.

$B$$(X, \mathrm{Y})$, where $X$ ,$\mathrm{Y}$$E$

and let$\mathrm{Y}\in G_{W}$is greater than or equal to two: $J(X, \mathrm{Y})\geq 2$ .

be the matrices in Theorem E. Set $C=AB$ . Then

and $C$ generate the Whitehead link.

and$G_{W}$$\mathrm{J}(\mathrm{X}, C)=2$$A$.

Theorem 1follows from Propositons 2and 3.

6. Next we will give asketch of the proof of Theorem 2.
Let $P$ be the regular ideal octahedorn in Ratcliffe [6, p.454].
$S_{A}$,$S_{B}$, $Sc$ ,$S_{D}$,$S_{A’}$,$S_{B’}$, $Sc$’and$S_{D’}$

the side pairing transformations ofto$S_{D’}$$S_{A}$

be the sides of $P$. Let
to$S_{A’}$, of$S_{B}$

to$S_{B’}$, of,$f_{B}$$S_{C}$

to$f_{A}$

Let the sides,$f_{C}$

and$f_{D}$

be$S_{C’}$, and of$S_{D}$, respectively.

PROPOSITION 4.

Let

fined in the above. Thensense of Ratcliffe$f_{A}$$f_{A}$,,$f_{B}$$f_{B}$,,$f_{C}$$f_{C}$

andand$f_{D}$$f_{D}$

be the side pairing transformations degenerate the Whitehead link $G_{W,R}$ in the82$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{P}\mathrm{O}\mathrm{S}\mathrm{I}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}}$

$5$.

Let

$G_{W,R}^{*}=\langle A^{*},B^{*}|A^{*}(B^{*})^{-2}A^{*}B^{*}(A^{*})^{-1}(B^{*})^{-1}$$(A^{*})^{-1}(B^{*})^{2}(A^{*})^{-1}(B^{\cdot})^{-1}A^{*}B^{*}=1\rangle$,

where$A^{*}=(\begin{array}{ll}1 10 \mathrm{l}\end{array})$Then(ii)$G_{W,R}^{*}$,$B^{*}=(\begin{array}{ll}1/2+i/2 3/4+i/4-1+i 1/2+i/2\end{array})$

is conjugate to the Whitehead link $Gw,r$ in the$J(A^{*}, B^{*})=2$

PROpOSITION 6.

.

sense of Ratcliffe.

.
The marked group$G_{W,R}^{*}=\langle A^{*}, B^{*}\rangle$

in Proposition 5corre-sponds to the point $(-1+i, -1/2)$ .

Theorem 2follows from Propositions 5and 6,References[1] T. Jorgensen, On discrete groups of Mobius transformations, Amer. J. Math.

98 (1976) 739-749.
[2] T. Jorgensen and M. Kiikka, Some extreme discrete groups, Ann. Acad. Sci.

Fenn. 1(1975), 245-248.

[3] T. Jorgensen, A. Lascurain and T. Pignataro, Translation extentions of theclassical modular group, Complex Variable 19 (1992), 205-209.
[4] S. L. KrashkaP, B. N. Apanasov and N. A. Gusevskii, Kleinian groups anduniformization in examples and problems, Trams. math. Monographs 62, Amer.
Math. Soc, Providence, Rhode Island, 1986

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[5] B. Maskit, Some special 2-generator Kleinian groups, Proc. Amer. Math. Soc,106 (1989), 175-186.
[6] J. R. Ratcliffe, Foundations of hyperbolic manifolds, GTM 149, Springer-Verlag,

New York, Berlin, Heidelberg, 1994.
[7] H. Sato, One-parameter families of extreme groups for JOrgensen’s inequality,Contemporary Math. 256 (The First Ahlfors -Bers Colloquium) edited by I.
Kra and B. Maskit, 2000, 271-287.
[8] H. Sato, Jorgensen groups and the Picard group, to appear in the Proc. of The

Third ISAAC International Conference, Academic Scientific Publ., 2002.
[9] H. Sato, The Jorgensen number of the Whitehead link, in preparation.

[10] H. Sato and R. Yamada, Some extreme Kleinian groups for Jorgensen’s inequality, Rep. Fac. Sci. Shizuoka Univ. 27 (1993), 1-8.
[11] N. J. Wielenberg, The structure of certain subgroups of the Picard group, Math.

Proc. Cambridge Philos. Soc. 84 (1978), 427-438.

Department of Mathematics

Faculty of Science

Shizuoka University
Ohya Shizuoka 422-8529
Japan$\mathrm{e}$-mail:smhsato@ipc.shizuoka.ac.jp