Li, Changjun

Oichi, Makito

Sato, Hiroki

Kyoto University Research Information Repository

Title Extreme discrete groups of parabolic type for Jorgensen'sinequality (Perspectives of Hyperbolic Spaces II)Author(s)Li, Changjun; Oichi, Makito; Sato, HirokiCitation数理解析研究所講究録 (2004), 1387: 159-169Issue Date2004-07URL http://hdl.handle.net/2433/25797RightType Departmental Bulletin PaperTextversionpublisherKyoto University158Extreme discrete groups of parabolic type forJOrgensen’s inequality(Changjun Li) Shizuoka University(Makito Oichi) Shizuoka University(Hiroki Sato) Shizuoka University$*$AbstractJOrgensen groups of parabolic type parametrized by three real parame-ters are divided into three types: finite type, countably infinite type and un-coutably infinite type. In the previous papers we found all JOrgensen groups offinite type and of countably infinite type. In this paper we announce that wefind all Jgrgensen groups of uncoutably infinite type. Consequently, the problem finding all JOrgensen groups of this parabolic type has been completelysolved. The proofs will appear elthwhere1. JORGENSEN’S INEQUALITY.*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 $30\mathrm{F}40$ ; Secondary $20\mathrm{H}10,32\mathrm{G}15$.Key Words and Phrases. Jorgensen’s inequality, JOrgensen number, JOrgensen group, Kleiniangroup.org $\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$1387 2004 159-1691130It is one of the most important problems in the theory of Kleinian groups todecide whether or not a subgroup $G$ of the Mobius transformation group is discrete.For this problem there are two important and useful theorems. One is Poincare’spolyhedron theorem, which gives a sufficient condition for $G$ to be discrete. Theother is JOrgensen’s inequality, which gives a necessary condition for a twO-generatorMobius transformation group $G=\langle A, B\rangle$ to be discrete.In 1976 JOrgensen obtained the following important theorem called Jprgensen’sinequality theorem, which gives a necessary condition for a non-elementary Mobiustransformtion group $G=$ ’A, $B\rangle$ to be discrete.THEOREM A $(\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}[1])$ . Suppose that the Mobius transformations $A$ and$B$ generate a non-elementary discrete group. Then$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA -1B-1)$ $-2|\geq 1.$The lower bound 1 is best possible.2. JORGENSENS GROUPS.2.1. First we will stste some definitions.DEFINITION 1. Let $A$ and $B$ be Mobius transformations. The Jprgensen num-$berJ(A, B)$ for the ordered pair $(A, B)$ is defined by$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)$ $-4|+|\mathrm{t}\mathrm{r}(ABA -1B-1)$ $-2|$ .That is, this number is the left hand side of Jgrgensen’s inequality.DEFINITION 2. Let $G$ be a non-elementary twO-generator subgroup of Mob.The JOrgensen number $\mathrm{J}\{\mathrm{G}$ ) for $G$ is defined as$J(G):= \inf${ $J(A,$ $B)|$ $A$ and $B$ generate $G$}.$\mathrm{j}[11$DEFINITION 3. A subgroup $G$ of M\"ob is called a Jprgensen group if $G$ satiafiesthe following four conditions:(1) $G$ is a twO-generator group.(2) $G$ is a discrete group.(3) $G$ is a non-elementary group.(4) There exist generators $A$ and $B$ of $G$ such that $J(A, B)=1.$That is, $G$ is a extreme non-elementary discrete group for JOrgensen’s inequality.2.2. JOrgensen and Kiikka showed the following.THEOREM $\mathrm{B}$ (JOrgensen-Kiikka [2]). Let $\langle A, B\rangle$ be a non-elementary discretegroup with $J(A, B)=1.$ Then $A$ is elliptic of order at least seven or $A$ is parabolic.If $\langle A, B\rangle$ is a JOrgensen group such that $A$ is parabolic (resp. elliptic) and$J(A, B)=$ 1, then we call it a Jprgensen group of parabolic type (resp. of el-liptic type). There are infinite number of JOrgensen groups (JOrgensen-Lascurain-Pignataro [3], Sato [8] $)$ .Now it gives rise to the following problem.PROBLEM 1. Find all JOrgensen groups of parabolic type.2.3. Let $\langle A, B\rangle$ be a marked twO-generator group such that $A$ is parabolic.Then we can normalize $A$ and $B$ as follows:$A=(\begin{array}{ll}1 10 1\end{array})$ and $B:=B_{\sigma,\mu}=(\begin{array}{lll}\mu\sigma \mu^{2}\sigma -\mathrm{l}\int\sigma\sigma \mu\sigma\end{array})$where $\sigma\in \mathrm{C}\backslash \{0\}$ and $\mu\in \mathrm{C}$ (see [4] for the detail).182We denote by $G_{\sigma,\mu}$ the marked group generated by $A$ and $B_{\sigma,\mu}:G_{\sigma,\mu}=$ $(A,$ $B_{\sigma,\mu}\rangle$ .We say that $(\sigma, \mu)\in \mathrm{C}\backslash \{0\}\cross \mathrm{C}$ is the point representing a marked group $G_{\sigma,\mu}$ andthat $G_{\sigma,\mu}$ is the marked group coresponding to a point $(\mathrm{a}, \mu)$ .2.4. In [8], Sato considered the case of $\mu=ik(k\in \mathrm{R})$ . Namely, he consideredmarked twO-generator groups $G_{\sigma},:k=\langle A, B_{\sigma},:k\rangle$ generated by$A=$ $(\begin{array}{ll}1 \mathrm{l}0 1\end{array})$ and $B_{\sigma,\dot{\iota}k}=(\begin{array}{lll}ik\sigma -k^{2}\sigma -1/\sigma\sigma ik\sigma \end{array})$where $\sigma\in \mathrm{C}\backslash \{0\}$ and $k\in$ R.Now we have the following conjecture.CONJECTURE. For any JOrgensen group $G$ of parabolic type there exists amarked group $G_{\sigma,ik}$ $(\sigma\in \mathrm{C}_{\mathrm{S}}\{0\}, k\in \mathrm{R})$ such that $G_{\sigma,ik}$ is conjugate to $G$ .If this conjecture is true, then it is suffucient to consider the case of $\mu=ik$ inorder to find all JOrgensen groups of parabolic type. In this paper we only considerthe case of $\mu=ik.$2.5. Let $C$ be the following cylinder:$C=\{(\sigma, ik)||(\mathrm{r}|=1, k\in \mathrm{R}\}$ .THEOREM $\mathrm{C}$ (Sato [8]). If a marked twO-generator group $G_{\sigma,ik}(\sigma\in \mathrm{C}\backslash \{01$ $k\in$R) is a Jprgensen group, then the point $(\sigma, ik)$ representing $G_{\sigma},:k$ lies on the cylinder$C$ .If $(\sigma, ik)$ is a point on the cylinder $C$, then we set $\sigma=-ie$” $(0\leq\theta\leq 2\pi)$ . Forsimplicity we write $B_{\theta,k}$ and $G_{\theta,k}$ for $B_{\sigma,ik}$ and $G_{\sigma,ik}$ with $\sigma=-ie^{:\theta}$ , respectively.If $G_{\theta,k}$ is a JOrgensen group, then we call the group a JOrgensen group of parabolictype $(\theta, k)$ .183Now it gives rise to the following problem.PROBLEM 2. Find all JOrgensen groups of parabolic type $(\theta, k)$ .2.6. We devide JOrgensen groups of this type into three parts as follows:Part 1. $|k|\leq\sqrt{3}/2,0\leq\theta\leq 2\pi$ (finite case).Part 2. $\sqrt{3}/2<|k|\leq 1,0\leq\theta\leq 2\pi$ (countably infinte case).Part 3. $1<|k|$ , $0\leq\theta\leq 2\pi.$ (uncountably infinte case).By some lemmas in [8], it suffices to consider the case of $0\leq\theta\leq\pi/2$ and $k\geq 0$for solvoing Problem 2.In the previous papers $[4, 5]$ we found all JOrgensen groups of finite case and ofcountably infinite case. In this paper we will announce that we found all JOrgensengroups of parabolic type $(\theta, k)(\mathrm{L}\mathrm{i}..$ Oichi - Sato [4,5,6]. That is, Problem 2 iscompletely solved. The proofs will appear elthwhere.3. TEOREMSTHEOREM 1 (Li-Oichi-Sato [4]) (Finite case).Let $D=\{(\theta, \mathrm{c})|0\leq\theta\leq\pi/2,0\leq k\leq\sqrt{3}/2\}$ and $\Omega=\{G_{\theta,k}|(\theta, k)\in D\}$ .Then(1) There are 16 JOrgensen groups $G_{\theta,k}$ in the region $\Omega$ . Nine of them are Kleiniangroups of the first kind and seven groups are of the second kind.(2) The other G$,k in $\Omega$ are not Kleinian groups.The following table describes 16 JOrgensen groups:184165THEOREM 2 (Li-Oichi-Sato [5]) (Countably infinite case).The group $G_{\theta,k}$ with $0\leq\theta\leq\pi/2$ and $\sqrt{3}\prime 2<k\leq 1$ is a JOrgensen group(discrete) if and only if one of the following conditions holds.(a) $\theta=0$ and $k=1.$ In this case, $G_{k}$ is a Kleinian group of the second kind, and$\Omega(G_{\theta,k})/G_{\theta,k}$ is a union of two Riemann surfaces with signature $($0; 2, 3, $\infty)$ .(b) $\theta=0$ and $k=\cos(\pi/n)(n=7,8, \ldots)$ . In this case, $G_{k}$ is a Kleinian groupof the second kind, and $\Omega(G_{k})/G_{k}$ is a union of two Riemann surfaces withsignatures $($0; 2, 3, $n)$ and $($0; 2, 3, $\infty)$ .(c) $\theta=\mathrm{t}\mathrm{t}/4$ and $k=1.$ In this case, $G_{k}$ is a Kleinian group of the first kind, andthe volume $V(G_{\pi/4,1})$ of the 3-orbifold for $G_{\pi/4,1}$ is$V(G_{\pi/4,1})$ $–$ 8(2L(tt/4)-L(tt/12)-L(tt/12).\ he a= pi/4(G_{\pi/4,1})--8(2L(\pi/4)-L \pi/ 2)-L(5\pi/12))$ .(d) (SatO-Yamada [11]) $\theta=\pi/2$ and $k=1.$ In this case, $G_{k}$ is a Kleiniangroup of the second kind, and $\Omega(G_{k})/G_{k}$ is a Riemann surface with signature$($ 0; 3, 3, $\infty)$ .(e) (SatO-Yamada [11]) $\theta=\pi/2$ and $k=\cos(\pi/n)(n=7,8, \ldots)$ . In thiscase, $G_{k}$ is a Kleinian group of the second kind, and $\Omega(G_{k})/G_{k}$ is a Riemannsurface with signature $($ 0; 3, 3, $n)$ .COROLLARY. There are countably infinite JOrgensen groups on the region$\{(\theta, k)|0\leq\theta\leq\pi/2, \sqrt{3}/2<k\leq 1\}$ .THEOREM 3 (Uncountably infinite case).The group $G_{\theta,k}$ with $0\leq\theta\leq\pi/2$ and $1<k$ is a JOrgensen group if and only ifone of the following conditions holds.ee(a) $\theta=0$ and $k>1.$ In this case, $G_{k}$ is a Kleinian group of the second kind, and1 $(G_{\theta,k})/G_{\theta,k}$ is a union of two Riemann surfaces with signatures $($0; 2, 3, $\infty)$and (0; 2, 2, 2, 3).(b) (1) $\theta=r/6$ and $k=\sqrt{3}n/2(n=2,4,6, \ldots)$ . In this case, $G_{k}$ is a Kleiniangroup of the first kind, and the volume $V(G_{\pi/6,\sqrt{3}n/2})$ of the 3-orbifold for$G_{\pi/6,\sqrt{3}n/2}$ is$V(G_{\pi/6,\sqrt{3}n/2})=3L(\pi/3)$ .(2) $\theta=\pi/6$ and $k=\sqrt{3}n/2(n=3,5,7, \ldots)$ . In this case, $G_{k}$ is a Kleiniangroup of the first kind, and the volume $V(G_{\pi/6,\sqrt{3}n/2})$ of the 3-orbifold for$G_{\pi/6,\sqrt{3}n}/2$ is$V(G_{\pi/6,\sqrt{3}n/2})=6L(\pi/3)$ .6,\sqrt{3}n/2})=3L$ ( 13).$kind_{f}$ 3-orbifold/2}(c) (1) $\theta=\pi/4$ and $k=3/2.$ In this case, $G_{k}$ is a Kleinian group of the firstkind, and the volume $V(G_{\pi/4,3/2})$ of the 3-orbifold for $G_{\pi/4,3/2}$ is$V(G_{\pi/4,3/2})$$=3${ $7L(\pi/3)/2-L(\varphi_{0}+\pi/6)+L(\varphi_{0}-$ tt/6)}$=3V(G_{\pi/2,1/2})$ ,where $\varphi_{0}=\sin^{-1}(1/2\sqrt{3})$ .(2) $\theta=r/4$ and $k=1+$ $\mathit{1}/2$ . In this case, $G_{k}$ is a Kleinian group of thei 0=\sin^{-1}(1/2\sqrheta=\pi/4 $k=1+\sqrt{2}/2$first kind, and the volume $V(G_{\pi/4,1+\sqrt{2}/2})$ of $.t$he 3-orbifold for $G\pi/4,1+$ 2is$V(G_{\pi[4,1+\sqrt{2}/2})$$=$ 2(2L(tt/4)-L(5jt/12) $-L(\pi/12)$ )$+4V(G_{\pi/2,1/2})$$=V(G_{\sqrt{2}\pi/2,1/2})+4V(G_{\pi/2,1/2})$ .$\mathfrak{i}67$(3) $\theta=\pi/4$ and $k=(5+\sqrt{5})/4$ . In this case, $G_{k}$ is a Kleinian groupof the first kind, and the volume $V(G_{\pi/4,(5+\sqrt{5})/4})$ of the 3-orbifold for$G_{\pi/4,(5+\sqrt{5})/4}$ is$V(G_{\pi/4,1+(1+\sqrt{5})/4})$$=$ 2L(tt/10) $+2L(2\pi/5)+4V(G_{\pi/2,1/2})$$=V(G_{\pi/4,(1+\sqrt{5})/4})+4V(G_{\pi/2,1/2})$ .(4) $\theta=\pi\oint 4$ and $k=1+\sqrt{3}/2$ . In this case, $G_{k}$ is a Kleinian group of thefirst kind, and the volume $V(G_{\pi/4,1+\sqrt{3}/2})$ of the 3-orbifold for $G_{\pi/4,1+\sqrt{3}/2}$es$V(G_{\pi/4,1+\sqrt{3}/2})$$=5L(\pi/3)+4V(G_{\pi/2,1/2})$$=V(G_{\pi/4,\sqrt{3})/2})+4V(G_{\pi/2,1/2})$ .(5) $\theta=\pi/4$ and $k=1+$ $\cos(\mathrm{w}\mathrm{r}/\mathrm{r}\mathrm{z})$ $(n=7,8, \ldots)$ . In this case, $G_{k}$ areKleinian groups of the second kind, and $l$ $(G_{k})/G_{k}$ is a Riemann surfacewith signature $($ 0; 3, 3, $n)$ .(6) $\theta=\pi/4$ and $k=2.$ In this case, $G_{k}$ is a Kleinian group of the secondkind, and $\Omega(G_{k})/G_{k}$ is a Riemann surface with signature $($ 0; 3, 3, $\infty)$ .(7) $\theta=\pi/4$ and $k>2.$ In this case, $G_{k}$ is a Kleinian group of the secondkind, and $\Omega(G_{k})/G_{k}$ is a Riemann surface with signature (0; 2, 2, 3, 3).(d) $\theta=\pi/3$ and $k=\sqrt{3}n/2(n=2,3, \ldots)$ . In this case, $G_{k}$ is a Kleinian groupof the first kind, and the volume $V(G_{\pi\int 3,\sqrt{3}n/2})$ of the 3-orbifold for $G_{\pi/3,\sqrt{3}n/2}$is$V(G_{\pi/3,\sqrt{3}n/2})=3L$ (\pi 13).198(e) SatO-Yamada [11] $)$ $\theta=\pi/2$ and $k>1.$ In this case, $G_{k}$ is a Kleiniangroup of the second kind, and 1 $(G_{k})/G_{k}$ is a Riemann surface with signature(0; 2, 2, 3, 3).COROLLARY. There are uncountably infinite JOrgensen groups on the region$\{(\theta, \mathrm{C}) |0\leq\theta\leq\pi/2, \mathrm{I}<k\}$ .4. OPEN PROBLEMS1. Find all JOrgensen groups of elliptic type.2. Let $r$ be a real number with $r\geq 1.$ Then is there a discrete group whoseJOrgensen number is equal to $r$ ? (cf. Sato [10])3. The group $G_{\pi/6,\sqrt{3}/2}$ is conjugate to the figure-eight knot group (cf. Sato [8]).The group $G_{\pi/2,0}$ is conjugate to the modular group. The group $G_{\pi/2,1/2}$ is conjugateto the Picard group (cf. Sato [9]). Are JOrgensen groups $G_{\pi/2,\sqrt{2}/2}$ , $G_{\pi/2,(1+\sqrt{5})/4}$ ,and $G_{\pi/2,\sqrt{3}/2}$ conjugate to familiar groups, respectively?References[1] T. JOrgensen, On discrete groups of Mdbius 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. Jgrgensen, A. Lascurain and T. Pignataro, Translation extensions of theclassical modular group, Complex Variables 19 (1992), 205-209.$\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$(197OrgeI63[4] C. Li, M. Oichi and H. Sato, Jprgensen groups of parabolic type $I$ (Finite type),MSRI Koukyuroku, Kyoto Univ. 1293 (2002), 65-77.[5] C. Li, M. Oichi and H. Sato, Jprgensen groups of parabolic type $II$ (Countablyinfinite case), MSRI Koukyuroku, Kyoto Univ. 1329 (2003), 48-57.[6] C. Li, M. Oichi and H. Sato, JOrgensen groups ofparabolic type $III$ (Uncountablyinfinite case), preprint.[7] B. Maskit, Kleinian Groups, Springer-Verlag, New York, Berlin, Heidelberg,1987.opr[8] H. Sato, One-parameter families of extreme groups for Jprgensen’s inequality,Contemporary Math. 256 (The First Ahlfors - Bers Colloquium) edited by I.Kra and B. Maskit, 2000, 271-287.[9] H. Sato, Jprgensen groups and the Picard group, Proc The Third ISAAC In-ternational Congress, Academic Scientific Pub1.(2003), 149-158.[10] H. Sato, Jgtrgensen number of the Whitehead group, to appear in Bol. Soc Math.Mexicana (2004).[11] H. Sato and R. Yamada, Some extreme Kleinian groups for Jprgensen’s inequal-ity, Rep. Fac. Sci. Shizuoka Univ. 27 (1993), 1-8.Or$ity$,C. Lie e-mail:r5144012@ipc.shizuoka.ac.jpM. Oichi e-mail:smohiti@ipc.shizuoka.ac.jpH. Sato $\mathrm{e}$-mail: smhsato@ipc.shizuoka.ac.jp

Extreme discrete groups of parabolic type for Jorgensen's inequality (Perspectives of Hyperbolic Spaces II)

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Extreme discrete groups of parabolic type for Jorgensen's inequality (Perspectives of Hyperbolic Spaces II)

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